diff --git a/bench/data/groundtruth/math_mathjax_latex_2.jsonl b/bench/data/groundtruth/math_mathjax_latex_2.jsonl index 1efd9c66..175a5a83 100644 --- a/bench/data/groundtruth/math_mathjax_latex_2.jsonl +++ b/bench/data/groundtruth/math_mathjax_latex_2.jsonl @@ -1 +1 @@ -{"url": "https://mathjax.github.io/MathJax-demos-web/equation-numbers.html", "content": "# Equations with automatic AMS numbering\n\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*} Brackets:\n\n$$\nE = mc^2\n$$\n\nBrackets tagged:\n\n$$\nE = mc^2\\tag{x}\n$$\n\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation} Multline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*} Gather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*} Align:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n", "main_html": "
\n\n
\n\n

Equations with automatic AMS numbering

\n\n
\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*}\n\n
\nBrackets:\n\n\\[E = mc^2\\]\n\nBrackets tagged:\n\n\\[E = mc^2\\tag{x}\\]\n\n
\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation}\n\n
\nMultline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*}\n\n
\nGather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*}\n\n
\nAlign:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\n
\n\n\n\n
", "content_list": [[{"type": "title", "raw_content": "

Equations with automatic AMS numbering

", "content": {"title_content": "Equations with automatic AMS numbering", "level": "1"}}, {"type": "paragraph", "raw_content": "
\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*}\n\n
\nBrackets:\n\n
", "content": [{"c": "Equation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*} Brackets:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

\\[E = mc^2\\]

", "content": {"math_content": "E = mc^2", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
\n\nBrackets tagged:\n\n
", "content": [{"c": "Brackets tagged:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

\\[E = mc^2\\tag{x}\\]

", "content": {"math_content": "E = mc^2\\tag{x}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation}\n\n
\nMultline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*}\n\n
\nGather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*}\n\n
\nAlign:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\n
", "content": [{"c": "Split:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation} Multline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*} Gather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*} Align:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}", "t": "text"}]}]], "html": "\n\n\n\n \n \n \n Testing MathJax v3 Equation Numbering\n \n \n \n\n\n\n
\n\n

Equations with automatic AMS numbering

\n\n
\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*}\n\n
\nBrackets:\n\n\\[E = mc^2\\]\n\nBrackets tagged:\n\n\\[E = mc^2\\tag{x}\\]\n\n
\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation}\n\n
\nMultline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*}\n\n
\nGather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*}\n\n
\nAlign:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\n
\n\n\n\n\n", "statics": {"title": 1, "paragraph": 3, "paragraph.text": 3, "equation-interline": 2}} +{"url": "https://mathjax.github.io/MathJax-demos-web/equation-numbers.html", "content": "# Equations with automatic AMS numbering\n\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*} Brackets:\n\n$$\nE = mc^2\n$$\n\nBrackets tagged:\n\n$$\nE = mc^2\\tag{x}\n$$\n\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation} Multline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*} Gather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*} Align:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n", "main_html": "
\n\n
\n\n

Equations with automatic AMS numbering

\n\n
\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*}\n\n
\nBrackets:\n\n\\[E = mc^2\\]\n\nBrackets tagged:\n\n\\[E = mc^2\\tag{x}\\]\n\n
\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation}\n\n
\nMultline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*}\n\n
\nGather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*}\n\n
\nAlign:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\n
\n\n\n\n
", "content_list": [[{"type": "title", "raw_content": "

Equations with automatic AMS numbering

", "content": {"title_content": "Equations with automatic AMS numbering", "level": "1"}}, {"type": "paragraph", "raw_content": "
\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*}\n\n
\nBrackets:\n\n
", "content": [{"c": "Equation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*} Brackets:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

\\[E = mc^2\\]

", "content": {"math_content": "E = mc^2", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
\n\nBrackets tagged:\n\n
", "content": [{"c": "Brackets tagged:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

\\[E = mc^2\\tag{x}\\]

", "content": {"math_content": "E = mc^2\\tag{x}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation}\n\n
\nMultline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*}\n\n
\nGather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*}\n\n
\nAlign:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\n
", "content": [{"c": "Split:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation} Multline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*} Gather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*} Align:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}", "t": "text"}]}]], "html": "\n\n\n\n \n \n \n Testing MathJax v3 Equation Numbering\n \n \n \n\n\n\n
\n\n

Equations with automatic AMS numbering

\n\n
\nEquation:\n\n\\begin{equation}\nE = mc^2\n\\end{equation}\n\nEquation*:\n\n\\begin{equation*}\nE = mc^2\n\\end{equation*}\n\n
\nBrackets:\n\n\\[E = mc^2\\]\n\nBrackets tagged:\n\n\\[E = mc^2\\tag{x}\\]\n\n
\nSplit:\n\n\\begin{equation}\n\\begin{split}\na& =b+c-d\\\\\n& \\quad +e-f\\\\\n& =g+h\\\\\n& =i\n\\end{split}\n\\end{equation}\n\n
\nMultline:\n\n\\begin{multline}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline}\n\nMultline*:\n\n\\begin{multline*}\n a+b+c+d+e+f+g\\\\\n M+N+O+P+Q\\\\\n R+S+T\\\\\n u+v+w+x+y+z\n\\end{multline*}\n\n
\nGather:\n\n\\begin{gather}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather}\n\nGather*:\n\n\\begin{gather*}\na_1=b_1+c_1\\\\\na_2=b_2+c_2-d_2+e_2\n\\end{gather*}\n\n
\nAlign:\n\n\\begin{align}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align}\n\nAlign*:\n\n\\begin{align*}\na_1& =b_1+c_1\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\nAlign:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22}\n\\end{align}\n\nAlign with \\notag and \\tag:\n\n\\begin{align}\na_{11}& =b_{11}& a_{12}& =b_{12}\\notag\\\\\na_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \\tag{y}\n\\end{align}\n\nAlign* with \\tag:\n\n\\begin{align*}\na_1& =b_1+c_1\\tag{z}\\\\\na_2& =b_2+c_2-d_2+e_2\n\\end{align*}\n\n
\n\n\n\n\n", "statics": {"title": 1, "paragraph": 3, "paragraph.text": 3, "equation-interline": 14}} diff --git a/bench/data/groundtruth/math_mathjax_latex_4.jsonl b/bench/data/groundtruth/math_mathjax_latex_4.jsonl index 4edaf8af..ca96f3cf 100644 --- a/bench/data/groundtruth/math_mathjax_latex_4.jsonl +++ b/bench/data/groundtruth/math_mathjax_latex_4.jsonl @@ -1 +1 @@ -{"url": "https://math.stackexchange.com/questions/4082284/solving-for-vector-contained-in-a-diagonal-matrix", "content": "Consider the following system of equations:\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{x}$ , $\\mathbf{y}$ , $\\mathbf{z}$ , and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$ , $D_{\\mathbf{y}}$ , and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$ , $\\mathbf{y}$ , and $\\mathbf{z}$ , respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$ ).\n\nMy question is whether it is possible to solve for $\\mathbf{y}$ here – both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$ . The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$ . The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write\n\n$$\n(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}\n$$\n\nwhere $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$ . So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$ , not $D_{\\mathbf{y}}$ :\n\n$$\n\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}\n$$\n\nEdit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nbecause rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$ . Then we can write\n\n$$\n\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}\n$$\n\nBut then how would solve for $\\mathbf{y}$ in the following?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nI think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:\n\n$$\n\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ – that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$ .\n\nNext, what if we had an equation as follows?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$ , we have some linear combination.\n\nI think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ( $n$ , I believe). Then, we would have a set of vectors ( $n-1$ , I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$ ) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$ .\n\nSo, I have two questions here:\n\n1. Is my general intuition about what this equation would look like correct?\n2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?\n\n## 1 Answer\n\n1\n\nI am not sure what the notation $D_\\mathbf{x}$ , $D_\\mathbf{y}$ , and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}}\n$$\n\n$$\n\\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1}\n$$\n\nNote: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.\n\nEdit 1:\n\nThank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$ , $D_{\\mathbf{y}}$ , and $D_{\\mathbf{z}}$ . Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.\n\nSince $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have\n\n$$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nand multiplying through we have\n\n$$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$\n\nTherefore, this can be written in matrix form as\n\n$$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nor more concisely as\n\n$$\n\\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}}\n$$\n\nwhich is exactly what is given in your edit:\n\n$$\n\\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}\n$$\n\nEdit 2:\n\nAs for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$ , you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):\n\n$$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.\n\n- My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$ . Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$ . Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$ . My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$ . Commented Mar 31, 2021 at 5:45\n- 1 @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck! – Ralff Commented Mar 31, 2021 at 6:15\n- Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on. Commented Mar 31, 2021 at 17:38\n- @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result. – Ralff Commented Mar 31, 2021 at 18:47\n\n## You must log in to answer this question.\n\n## Not the answer you're looking for? Browse other questions tagged\n\n## .\n", "main_html": "
\n \n\t\t\n
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Consider the following system of equations:

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$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

\n

where $\\mathbf{x}$, $\\mathbf{y}$, $\\mathbf{z}$, and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$, $\\mathbf{y}$, and $\\mathbf{z}$, respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$).

\n

My question is whether it is possible to solve for $\\mathbf{y}$ here – both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$. The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

\n

$$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

\n

where $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$. So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$, not $D_{\\mathbf{y}}$:

\n

$$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

\n

Edit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as

\n

$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

\n

because rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$. Then we can write

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$$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

\n

But then how would solve for $\\mathbf{y}$ in the following?

\n

$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

\n

I think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:

\n

$$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

\n

where $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ – that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$.

\n

Next, what if we had an equation as follows?

\n

$$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

\n

where $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$, we have some linear combination.

\n

I think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ($n$, I believe). Then, we would have a set of vectors ($n-1$, I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$.

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So, I have two questions here:

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  1. Is my general intuition about what this equation would look like correct?
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  2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?
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    \n 1 Answer\n \n

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    I am not sure what the notation $D_\\mathbf{x}$, $D_\\mathbf{y}$, and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:

    \n

    $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$\n$$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$\n$$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

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    Note: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.

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    Edit 1:

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    Thank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

    \n

    Since $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have

    \n

    $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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    and multiplying through we have

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    $$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$

    \n

    Therefore, this can be written in matrix form as

    \n

    $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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    or more concisely as

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    $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

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    which is exactly what is given in your edit:

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    $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

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    Edit 2:

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    As for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

    \n

    $$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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    where we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.

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      \n $\\begingroup$\n My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.\n $\\endgroup$\n \n \n Commented\n Mar 31, 2021 at 5:45\n \n \n \n \n
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      \n $\\begingroup$\n @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!\n $\\endgroup$\n
      \n– Ralff\n
      \n \n Commented\n Mar 31, 2021 at 6:15\n \n \n \n \n
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      \n $\\begingroup$\n Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.\n $\\endgroup$\n \n \n Commented\n Mar 31, 2021 at 17:38\n \n
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      \n $\\begingroup$\n @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.\n $\\endgroup$\n
      \n– Ralff\n
      \n \n Commented\n Mar 31, 2021 at 18:47\n \n
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      \n ", "content_list": [[{"type": "paragraph", "raw_content": "

      Consider the following system of equations:

      ", "content": [{"c": "Consider the following system of equations:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

      ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      where \\mathbf{x}, \\mathbf{y}, \\mathbf{z}, and \\mathbf{u} are 1\\times n vectors and D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}} are diagonal n\\times n matrices with \\mathbf{x}, \\mathbf{y}, and \\mathbf{z}, respectively, along their diagonals (i.e., D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{u}", "t": "equation-inline"}, {"c": "are", "t": "text"}, {"c": "1\\times n", "t": "equation-inline"}, {"c": "vectors and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "are diagonal", "t": "text"}, {"c": "n\\times n", "t": "equation-inline"}, {"c": "matrices with", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", respectively, along their diagonals (i.e.,", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": ").", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      My question is whether it is possible to solve for \\mathbf{y} here – both the \\mathbf{y} as a vector and the \\mathbf{y} along the diagonal of D_{\\mathbf{y}}. The problem is that I do not know of operations to pull \\mathbf{y} out of D_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y} out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

      ", "content": [{"c": "My question is whether it is possible to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "here – both the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as a vector and the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The problem is that I do not know of operations to pull", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

      ", "content": {"math_content": "(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      where \\odot is the Hadamard product and e^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out \\mathbf{y} and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to \\mathbf{y}, not D_{\\mathbf{y}}:

      ", "content": [{"c": "where", "t": "text"}, {"c": "\\odot", "t": "equation-inline"}, {"c": "is the Hadamard product and", "t": "text"}, {"c": "e^T = (1,1,\\ldots)\\in\\mathbb R^n", "t": "equation-inline"}, {"c": ". So, would it be possible to distribute out", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", not", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

      ", "content": {"math_content": "\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      Edit: Oh, it seems that extracting \\mathbf{y} in this case would be a simple as rewriting the equation above as

      ", "content": [{"c": "Edit: Oh, it seems that extracting", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in this case would be a simple as rewriting the equation above as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

      ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

      ", "content": [{"c": "because rewriting the equation this way would not change the terms along the diagonal of", "t": "text"}, {"c": "\\mathbf{z}D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". Then we can write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

      ", "content": {"math_content": "\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      But then how would solve for \\mathbf{y} in the following?

      ", "content": [{"c": "But then how would solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in the following?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

      ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      I think that in this case, it would not be possible to solve for a single vector \\mathbf{y} as in the previous case. Instead, we would have a system of polynomials:

      ", "content": [{"c": "I think that in this case, it would not be possible to solve for a single vector", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as in the previous case. Instead, we would have a system of polynomials:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

      ", "content": {"math_content": "\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      where \\mathbf{y^2} is a vector where the elements are the squares of the corresponding elements of \\mathbf{y} – that is, \\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}.

      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "is a vector where the elements are the squares of the corresponding elements of", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "– that is,", "t": "text"}, {"c": "\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Next, what if we had an equation as follows?

      ", "content": [{"c": "Next, what if we had an equation as follows?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

      ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      where \\mathbf{M} is an n \\times n matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of D_{\\mathbf{yM}}, we have some linear combination.

      ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "is an", "t": "text"}, {"c": "n \\times n", "t": "equation-inline"}, {"c": "matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{yM}}", "t": "equation-inline"}, {"c": ", we have some linear combination.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2} again, but this time multiplied by some diagonal matrix based on the elements of \\mathbf{M} and D_{\\mathbf{x}} that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in \\mathbf{y} (e.g., y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in \\mathbf{M} and D_{\\mathbf{x}}.

      ", "content": [{"c": "I think in this case, the polynomial system of equations would be rather complicated. We would have the vector", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "again, but this time multiplied by some diagonal matrix based on the elements of", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "that is multiplied by a scalar (", "t": "text"}, {"c": "n", "t": "equation-inline"}, {"c": ", I believe). Then, we would have a set of vectors (", "t": "text"}, {"c": "n-1", "t": "equation-inline"}, {"c": ", I believe) that each have as elements different products of the elements in", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "(e.g.,", "t": "text"}, {"c": "y_1 y_3", "t": "equation-inline"}, {"c": ") and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      So, I have two questions here:

      ", "content": [{"c": "So, I have two questions here:", "t": "text"}]}, {"type": "list", "raw_content": "
      1. Is my general intuition about what this equation would look like correct?
      2. Are there techniques to solve for the elements of \\mathbf{y} in this system?
      ", "content": {"items": [[[{"c": "Is my general intuition about what this equation would look like correct?", "t": "text"}]], [[{"c": "Are there techniques to solve for the elements of ", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": " in this system?", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "title", "raw_content": "

      \n 1 Answer\n \n

      ", "content": {"title_content": "1 Answer", "level": "2"}}, {"type": "paragraph", "raw_content": "
      \n1
      ", "content": [{"c": "1", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      I am not sure what the notation D_\\mathbf{x}, D_\\mathbf{y}, and D_\\mathbf{z} represent and if the importance is significant, but assuming D_{\\mathbf{x}} is invertible, you may solve for \\mathbf{y} as follows given the original equation:

      ", "content": [{"c": "I am not sure what the notation", "t": "text"}, {"c": "D_\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_\\mathbf{z}", "t": "equation-inline"}, {"c": "represent and if the importance is significant, but assuming", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "is invertible, you may solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as follows given the original equation:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$

      ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

      $$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$

      ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

      $$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

      ", "content": {"math_content": "\\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      Note: I can't verify that your edit is correct because I don't understand what D_{\\mathbf{x} + \\mathbf{z}} represents.

      ", "content": [{"c": "Note: I can't verify that your edit is correct because I don't understand what", "t": "text"}, {"c": "D_{\\mathbf{x} + \\mathbf{z}}", "t": "equation-inline"}, {"c": "represents.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Edit 1:

      ", "content": [{"c": "Edit 1:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Thank you for the response as I didn't realize the meaning of the matrices D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}}. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

      ", "content": [{"c": "Thank you for the response as I didn't realize the meaning of the matrices", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": ". Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Since D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x}) and similarly for the other matrices, we have

      ", "content": [{"c": "Since", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": "and similarly for the other matrices, we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

      ", "content": {"math_content": "\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      and multiplying through we have

      ", "content": [{"c": "and multiplying through we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$

      ", "content": {"math_content": "\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      Therefore, this can be written in matrix form as

      ", "content": [{"c": "Therefore, this can be written in matrix form as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

      ", "content": {"math_content": "\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      or more concisely as

      ", "content": [{"c": "or more concisely as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

      ", "content": {"math_content": "\\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      which is exactly what is given in your edit:

      ", "content": [{"c": "which is exactly what is given in your edit:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

      ", "content": {"math_content": "\\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      Edit 2:

      ", "content": [{"c": "Edit 2:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      As for the case where you have \\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}} where \\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}, you would not be able to solve for \\mathbf{y} as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

      ", "content": [{"c": "As for the case where you have", "t": "text"}, {"c": "\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "where", "t": "text"}, {"c": "\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}", "t": "equation-inline"}, {"c": ", you would not be able to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

      ", "content": {"math_content": "\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

      where we cannot solve for the matrix containing the y variables because we cannot eliminate \\mathbf{x} from the left side of the equation.

      ", "content": [{"c": "where we cannot solve for the matrix containing the", "t": "text"}, {"c": "y", "t": "equation-inline"}, {"c": "variables because we cannot eliminate", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": "from the left side of the equation.", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "My notation for, for example, ", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": " was intended to refer to ", "t": "text"}, {"c": "\\mathrm{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": ". Thus, ", "t": "text"}, {"c": "D_{\\mathbf{x+y}}", "t": "equation-inline"}, {"c": " means ", "t": "text"}, {"c": "\\mathrm{diag}(\\mathbf{x+y})", "t": "equation-inline"}, {"c": ". Sorry if that was unclear. Furthermore, in the solution you provided, ", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": " is still along the diagonal of ", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". My aim is to remove it so we can have an non-implicit expression for ", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ".", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 5:45", "t": "text"}]], [[{"c": "1", "t": "text"}, {"c": "@RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!", "t": "text"}, {"c": "\n– ", "t": "text"}, {"c": "Ralff", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 6:15", "t": "text"}]], [[{"c": "Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 17:38", "t": "text"}]], [[{"c": "@RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the ", "t": "text"}, {"c": "\\mathbf{y}^2", "t": "equation-inline"}, {"c": " case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.", "t": "text"}, {"c": "\n– ", "t": "text"}, {"c": "Ralff", "t": "text"}, {"c": "Commented", "t": "text"}, {"c": "Mar 31, 2021 at 18:47", "t": "text"}]]], "ordered": false, "list_nest_level": "1"}}, {"type": "title", "raw_content": "

      \n You must log in to answer this question.\n

      ", "content": {"title_content": "You must log in to answer this question.", "level": "2"}}, {"type": "title", "raw_content": "

      \nNot the answer you're looking for? Browse other questions tagged

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      .

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      Consider the following system of equations:

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      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

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      where $\\mathbf{x}$, $\\mathbf{y}$, $\\mathbf{z}$, and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$, $\\mathbf{y}$, and $\\mathbf{z}$, respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$).

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      My question is whether it is possible to solve for $\\mathbf{y}$ here – both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$. The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

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      $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

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      where $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$. So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$, not $D_{\\mathbf{y}}$:

      \n

      $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

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      Edit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as

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      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

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      because rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$. Then we can write

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      $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

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      But then how would solve for $\\mathbf{y}$ in the following?

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      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

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      I think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:

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      $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

      \n

      where $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ – that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$.

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      Next, what if we had an equation as follows?

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      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

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      where $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$, we have some linear combination.

      \n

      I think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ($n$, I believe). Then, we would have a set of vectors ($n-1$, I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$.

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      So, I have two questions here:

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      1. Is my general intuition about what this equation would look like correct?
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      2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?
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      \n edited Mar 31, 2021 at 17:37\n
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      \n Ryan da Silva\n
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      \n asked Mar 29, 2021 at 23:19\n
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      I am not sure what the notation $D_\\mathbf{x}$, $D_\\mathbf{y}$, and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:

      \n

      $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$\n$$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$\n$$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

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      Note: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.

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      Edit 1:

      \n

      Thank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

      \n

      Since $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have

      \n

      $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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      and multiplying through we have

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      $$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$

      \n

      Therefore, this can be written in matrix form as

      \n

      $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

      \n

      or more concisely as

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      $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

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      which is exactly what is given in your edit:

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      $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

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      Edit 2:

      \n

      As for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

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      $$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

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      where we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.

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      \n edited Mar 31, 2021 at 18:44\n
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      \n answered Mar 30, 2021 at 8:19\n
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      \"Ralff's
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      \n RalffRalff\n
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        \n $\\begingroup$\n My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.\n $\\endgroup$\n
        \n– Ryan da Silva\n
        \n \n Commented\n Mar 31, 2021 at 5:45\n \n \n \n \n
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        \n $\\begingroup$\n @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!\n $\\endgroup$\n
        \n– Ralff\n
        \n \n Commented\n Mar 31, 2021 at 6:15\n \n \n \n \n
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        \n $\\begingroup$\n Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.\n $\\endgroup$\n
        \n– Ryan da Silva\n
        \n \n Commented\n Mar 31, 2021 at 17:38\n \n
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        \n $\\begingroup$\n @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.\n $\\endgroup$\n
        \n– Ralff\n
        \n \n Commented\n Mar 31, 2021 at 18:47\n \n
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The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$ . The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write\n\n$$\n(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}\n$$\n\nwhere $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$ . So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$ , not $D_{\\mathbf{y}}$ :\n\n$$\n\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}\n$$\n\nEdit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nbecause rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$ . Then we can write\n\n$$\n\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}\n$$\n\nBut then how would solve for $\\mathbf{y}$ in the following?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nI think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:\n\n$$\n\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ – that is,\\$\\mathbf{y^2} =\n\n$$\n\\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}\n$$\n\n\\$.\n\nNext, what if we had an equation as follows?\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$ , we have some linear combination.\n\nI think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ( $n$ , I believe). Then, we would have a set of vectors ( $n-1$ , I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$ ) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$ .\n\nSo, I have two questions here:\n\n1. Is my general intuition about what this equation would look like correct?\n2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?\n\n## 1 Answer\n\n1\n\nI am not sure what the notation $D_\\mathbf{x}$ , $D_\\mathbf{y}$ , and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}\n$$\n\n$$\n\\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}}\n$$\n\n$$\n\\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1}\n$$\n\nNote: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.\n\nEdit 1:\n\nThank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$ , $D_{\\mathbf{y}}$ , and $D_{\\mathbf{z}}$ . Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.\n\nSince $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have\n\n$$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nand multiplying through we have\n\n$$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$\n\nTherefore, this can be written in matrix form as\n\n$$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nor more concisely as\n\n$$\n\\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}}\n$$\n\nwhich is exactly what is given in your edit:\n\n$$\n\\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}\n$$\n\nEdit 2:\n\nAs for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where\\$\\mathbf{y}^2 =\n\n$$\n\\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}\n$$\n\n\\$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):\n\n$$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$\n\nwhere we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.\n\n- My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$ . Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$ . Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$ . My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$ . Commented Mar 31, 2021 at 5:45\n- 1 @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck! – Ralff Commented Mar 31, 2021 at 6:15\n- Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on. Commented Mar 31, 2021 at 17:38\n- @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result. – Ralff Commented Mar 31, 2021 at 18:47\n\n## You must log in to answer this question.\n\n## Not the answer you're looking for? Browse other questions tagged\n\n## .\n", "main_html": "
      \r\n \r\n\t\t\r\n
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      Consider the following system of equations:

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      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

      \r\n

      where $\\mathbf{x}$, $\\mathbf{y}$, $\\mathbf{z}$, and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$, $\\mathbf{y}$, and $\\mathbf{z}$, respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$).

      \r\n

      My question is whether it is possible to solve for $\\mathbf{y}$ here – both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$. The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

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      $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

      \r\n

      where $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$. So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$, not $D_{\\mathbf{y}}$:

      \r\n

      $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

      \r\n

      Edit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as

      \r\n

      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

      \r\n

      because rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$. Then we can write

      \r\n

      $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

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      But then how would solve for $\\mathbf{y}$ in the following?

      \r\n

      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

      \r\n

      I think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:

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      $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

      \r\n

      where $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ – that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$.

      \r\n

      Next, what if we had an equation as follows?

      \r\n

      $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

      \r\n

      where $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$, we have some linear combination.

      \r\n

      I think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ($n$, I believe). Then, we would have a set of vectors ($n-1$, I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$.

      \r\n

      So, I have two questions here:

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      1. Is my general intuition about what this equation would look like correct?
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        I am not sure what the notation $D_\\mathbf{x}$, $D_\\mathbf{y}$, and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:

        \r\n

        $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$\r\n$$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$\r\n$$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

        \r\n

        Note: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.

        \r\n

        Edit 1:

        \r\n

        Thank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

        \r\n

        Since $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have

        \r\n

        $$\r\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

        \r\n

        and multiplying through we have

        \r\n

        $$\r\n\\begin{align}\r\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\r\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\r\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\r\n\\end{align}\r\n$$

        \r\n

        Therefore, this can be written in matrix form as

        \r\n

        $$\r\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

        \r\n

        or more concisely as

        \r\n

        $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

        \r\n

        which is exactly what is given in your edit:

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        $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

        \r\n

        Edit 2:

        \r\n

        As for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

        \r\n

        $$\r\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

        \r\n

        where we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.

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          \r\n $\\begingroup$\r\n My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.\r\n $\\endgroup$\r\n \r\n \r\n Commented\r\n Mar 31, 2021 at 5:45\r\n \r\n \r\n \r\n \r\n
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          \r\n $\\begingroup$\r\n @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!\r\n $\\endgroup$\r\n
          \r\n– Ralff\r\n
          \r\n \r\n Commented\r\n Mar 31, 2021 at 6:15\r\n \r\n \r\n \r\n \r\n
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          \r\n $\\begingroup$\r\n Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.\r\n $\\endgroup$\r\n \r\n \r\n Commented\r\n Mar 31, 2021 at 17:38\r\n \r\n
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          \r\n $\\begingroup$\r\n @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.\r\n $\\endgroup$\r\n
          \r\n– Ralff\r\n
          \r\n \r\n Commented\r\n Mar 31, 2021 at 18:47\r\n \r\n
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        \r\n You must log in to answer this question.\r\n

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          .
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          \r\n ", "content_list": [[{"type": "paragraph", "raw_content": "

          Consider the following system of equations:

          ", "content": [{"c": "Consider the following system of equations:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

          ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          where \\mathbf{x}, \\mathbf{y}, \\mathbf{z}, and \\mathbf{u} are 1\\times n vectors and D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}} are diagonal n\\times n matrices with \\mathbf{x}, \\mathbf{y}, and \\mathbf{z}, respectively, along their diagonals (i.e., D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})).

          ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{u}", "t": "equation-inline"}, {"c": "are", "t": "text"}, {"c": "1\\times n", "t": "equation-inline"}, {"c": "vectors and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "are diagonal", "t": "text"}, {"c": "n\\times n", "t": "equation-inline"}, {"c": "matrices with", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "\\mathbf{z}", "t": "equation-inline"}, {"c": ", respectively, along their diagonals (i.e.,", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": ").", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          My question is whether it is possible to solve for \\mathbf{y} here – both the \\mathbf{y} as a vector and the \\mathbf{y} along the diagonal of D_{\\mathbf{y}}. The problem is that I do not know of operations to pull \\mathbf{y} out of D_{\\mathbf{y}}. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute \\mathbf{y} out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

          ", "content": [{"c": "My question is whether it is possible to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "here – both the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as a vector and the", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The problem is that I do not know of operations to pull", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

          ", "content": {"math_content": "(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          where \\odot is the Hadamard product and e^T = (1,1,\\ldots)\\in\\mathbb R^n. So, would it be possible to distribute out \\mathbf{y} and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to \\mathbf{y}, not D_{\\mathbf{y}}:

          ", "content": [{"c": "where", "t": "text"}, {"c": "\\odot", "t": "equation-inline"}, {"c": "is the Hadamard product and", "t": "text"}, {"c": "e^T = (1,1,\\ldots)\\in\\mathbb R^n", "t": "equation-inline"}, {"c": ". So, would it be possible to distribute out", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": ", not", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ":", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

          ", "content": {"math_content": "\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Edit: Oh, it seems that extracting \\mathbf{y} in this case would be a simple as rewriting the equation above as

          ", "content": [{"c": "Edit: Oh, it seems that extracting", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in this case would be a simple as rewriting the equation above as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

          ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          because rewriting the equation this way would not change the terms along the diagonal of \\mathbf{z}D_{\\mathbf{y}}. Then we can write

          ", "content": [{"c": "because rewriting the equation this way would not change the terms along the diagonal of", "t": "text"}, {"c": "\\mathbf{z}D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ". Then we can write", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

          ", "content": {"math_content": "\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          But then how would solve for \\mathbf{y} in the following?

          ", "content": [{"c": "But then how would solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "in the following?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

          ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          I think that in this case, it would not be possible to solve for a single vector \\mathbf{y} as in the previous case. Instead, we would have a system of polynomials:

          ", "content": [{"c": "I think that in this case, it would not be possible to solve for a single vector", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as in the previous case. Instead, we would have a system of polynomials:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

          ", "content": {"math_content": "\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          where \\mathbf{y^2} is a vector where the elements are the squares of the corresponding elements of \\mathbf{y} – that is, $\\mathbf{y^2} =

          ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "is a vector where the elements are the squares of the corresponding elements of", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "– that is,$\\mathbf{y^2} =", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}

          ", "content": {"math_content": "\\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          $.

          ", "content": [{"c": "$.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Next, what if we had an equation as follows?

          ", "content": [{"c": "Next, what if we had an equation as follows?", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

          ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          where \\mathbf{M} is an n \\times n matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of D_{\\mathbf{yM}}, we have some linear combination.

          ", "content": [{"c": "where", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "is an", "t": "text"}, {"c": "n \\times n", "t": "equation-inline"}, {"c": "matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of", "t": "text"}, {"c": "D_{\\mathbf{yM}}", "t": "equation-inline"}, {"c": ", we have some linear combination.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          I think in this case, the polynomial system of equations would be rather complicated. We would have the vector \\mathbf{y^2} again, but this time multiplied by some diagonal matrix based on the elements of \\mathbf{M} and D_{\\mathbf{x}} that is multiplied by a scalar (n, I believe). Then, we would have a set of vectors (n-1, I believe) that each have as elements different products of the elements in \\mathbf{y} (e.g., y_1 y_3) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in \\mathbf{M} and D_{\\mathbf{x}}.

          ", "content": [{"c": "I think in this case, the polynomial system of equations would be rather complicated. We would have the vector", "t": "text"}, {"c": "\\mathbf{y^2}", "t": "equation-inline"}, {"c": "again, but this time multiplied by some diagonal matrix based on the elements of", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "that is multiplied by a scalar (", "t": "text"}, {"c": "n", "t": "equation-inline"}, {"c": ", I believe). Then, we would have a set of vectors (", "t": "text"}, {"c": "n-1", "t": "equation-inline"}, {"c": ", I believe) that each have as elements different products of the elements in", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "(e.g.,", "t": "text"}, {"c": "y_1 y_3", "t": "equation-inline"}, {"c": ") and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in", "t": "text"}, {"c": "\\mathbf{M}", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          So, I have two questions here:

          ", "content": [{"c": "So, I have two questions here:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. Is my general intuition about what this equation would look like correct?
          2. Are there techniques to solve for the elements of \\mathbf{y} in this system?
          ", "content": {"items": [{"c": "Is my general intuition about what this equation would look like correct?"}, {"c": "Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          \n 1 Answer\n \n

          ", "content": {"title_content": "1 Answer", "level": "2"}}, {"type": "paragraph", "raw_content": "
          \n1
          ", "content": [{"c": "1", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          I am not sure what the notation D_\\mathbf{x}, D_\\mathbf{y}, and D_\\mathbf{z} represent and if the importance is significant, but assuming D_{\\mathbf{x}} is invertible, you may solve for \\mathbf{y} as follows given the original equation:

          ", "content": [{"c": "I am not sure what the notation", "t": "text"}, {"c": "D_\\mathbf{x}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_\\mathbf{y}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_\\mathbf{z}", "t": "equation-inline"}, {"c": "represent and if the importance is significant, but assuming", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": "is invertible, you may solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as follows given the original equation:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$

          ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          $$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$

          ", "content": {"math_content": "\\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          $$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

          ", "content": {"math_content": "\\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Note: I can't verify that your edit is correct because I don't understand what D_{\\mathbf{x} + \\mathbf{z}} represents.

          ", "content": [{"c": "Note: I can't verify that your edit is correct because I don't understand what", "t": "text"}, {"c": "D_{\\mathbf{x} + \\mathbf{z}}", "t": "equation-inline"}, {"c": "represents.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Edit 1:

          ", "content": [{"c": "Edit 1:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Thank you for the response as I didn't realize the meaning of the matrices D_{\\mathbf{x}}, D_{\\mathbf{y}}, and D_{\\mathbf{z}}. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

          ", "content": [{"c": "Thank you for the response as I didn't realize the meaning of the matrices", "t": "text"}, {"c": "D_{\\mathbf{x}}", "t": "equation-inline"}, {"c": ",", "t": "text"}, {"c": "D_{\\mathbf{y}}", "t": "equation-inline"}, {"c": ", and", "t": "text"}, {"c": "D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": ". Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Since D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x}) and similarly for the other matrices, we have

          ", "content": [{"c": "Since", "t": "text"}, {"c": "D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})", "t": "equation-inline"}, {"c": "and similarly for the other matrices, we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

          ", "content": {"math_content": "\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          and multiplying through we have

          ", "content": [{"c": "and multiplying through we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\n\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}\n$$

          ", "content": {"math_content": "\\begin{align}\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\n\\end{align}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Therefore, this can be written in matrix form as

          ", "content": [{"c": "Therefore, this can be written in matrix form as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

          ", "content": {"math_content": "\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          or more concisely as

          ", "content": [{"c": "or more concisely as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

          ", "content": {"math_content": "\\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          which is exactly what is given in your edit:

          ", "content": [{"c": "which is exactly what is given in your edit:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

          ", "content": {"math_content": "\\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Edit 2:

          ", "content": [{"c": "Edit 2:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          As for the case where you have \\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}} where $\\mathbf{y}^2 =

          ", "content": [{"c": "As for the case where you have", "t": "text"}, {"c": "\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}", "t": "equation-inline"}, {"c": "where$\\mathbf{y}^2 =", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}

          ", "content": {"math_content": "\\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          $, you would not be able to solve for \\mathbf{y} as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

          ", "content": [{"c": "$, you would not be able to solve for", "t": "text"}, {"c": "\\mathbf{y}", "t": "equation-inline"}, {"c": "as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          $$\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\n$$

          ", "content": {"math_content": "\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          where we cannot solve for the matrix containing the y variables because we cannot eliminate \\mathbf{x} from the left side of the equation.

          ", "content": [{"c": "where we cannot solve for the matrix containing the", "t": "text"}, {"c": "y", "t": "equation-inline"}, {"c": "variables because we cannot eliminate", "t": "text"}, {"c": "\\mathbf{x}", "t": "equation-inline"}, {"c": "from the left side of the equation.", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [{"c": "My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$ . Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$ . Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$ . My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$ . Commented Mar 31, 2021 at 5:45"}, {"c": "1 @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck! – Ralff Commented Mar 31, 2021 at 6:15"}, {"c": "Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on. Commented Mar 31, 2021 at 17:38"}, {"c": "@RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result. – Ralff Commented Mar 31, 2021 at 18:47"}], "list_attribute": "unordered", "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          \n You must log in to answer this question.\n

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          \nNot the answer you're looking for? Browse other questions tagged

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          Consider the following system of equations:

          \r\n

          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

          \r\n

          where $\\mathbf{x}$, $\\mathbf{y}$, $\\mathbf{z}$, and $\\mathbf{u}$ are $1\\times n$ vectors and $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$ are diagonal $n\\times n$ matrices with $\\mathbf{x}$, $\\mathbf{y}$, and $\\mathbf{z}$, respectively, along their diagonals (i.e., $D_{\\mathbf{x}} = \\mathrm{diag}(\\mathbf{x})$).

          \r\n

          My question is whether it is possible to solve for $\\mathbf{y}$ here – both the $\\mathbf{y}$ as a vector and the $\\mathbf{y}$ along the diagonal of $D_{\\mathbf{y}}$. The problem is that I do not know of operations to pull $\\mathbf{y}$ out of $D_{\\mathbf{y}}$. The one possibility that I have considered is to use the Hadamard product since this can be used to convert a vector into a diagonal matrix and vice versa, as discussed here. But I am not sure how this would work in this case, as it would be necessary to distribute $\\mathbf{y}$ out of the resulting expression, and I don't know if this would be possible given the properties of the Hadamard product. That is, we could write

          \r\n

          $$(\\mathbf{y}e^T) \\odot I_n = D_{\\mathbf{y}}$$

          \r\n

          where $\\odot$ is the Hadamard product and $e^T = (1,1,\\ldots)\\in\\mathbb R^n$. So, would it be possible to distribute out $\\mathbf{y}$ and then move the remaining terms to the other side of the first equation above? For example, we couldn't do something like left-hand side since that would just be equal to $\\mathbf{y}$, not $D_{\\mathbf{y}}$:

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          $$\\mathbf{y}(e^T \\odot I_n) \\neq D_{\\mathbf{y}}$$

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          Edit: Oh, it seems that extracting $\\mathbf{y}$ in this case would be a simple as rewriting the equation above as

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          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{z}}=\\mathbf{u}D_{\\mathbf{z}}$$

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          because rewriting the equation this way would not change the terms along the diagonal of $\\mathbf{z}D_{\\mathbf{y}}$. Then we can write

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          $$\\mathbf{y}=\\mathbf{u}D_{\\mathbf{z}}D_{\\mathbf{x+z}}^{-1}$$

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          But then how would solve for $\\mathbf{y}$ in the following?

          \r\n

          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}}$$

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          I think that in this case, it would not be possible to solve for a single vector $\\mathbf{y}$ as in the previous case. Instead, we would have a system of polynomials:

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          $$\\mathbf{y^2} + \\mathbf{y}D_{\\mathbf{x}} = \\mathbf{u}D_{\\mathbf{z}}$$

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          where $\\mathbf{y^2}$ is a vector where the elements are the squares of the corresponding elements of $\\mathbf{y}$ – that is, $\\mathbf{y^2} = \\begin{pmatrix} y_{1}^2 & y_{2}^2 & \\cdots & y_{n}^2 \\end{pmatrix}$.

          \r\n

          Next, what if we had an equation as follows?

          \r\n

          $$\\mathbf{y}D_{\\mathbf{x}}+\\mathbf{y}D_{\\mathbf{yM}}=\\mathbf{u}D_{\\mathbf{z}}$$

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          where $\\mathbf{M}$ is an $n \\times n$ matrix. Unlike the other matrices, it is not a diagonal matrix. Thus, in each element along the diagonal of $D_{\\mathbf{yM}}$, we have some linear combination.

          \r\n

          I think in this case, the polynomial system of equations would be rather complicated. We would have the vector $\\mathbf{y^2}$ again, but this time multiplied by some diagonal matrix based on the elements of $\\mathbf{M}$ and $D_{\\mathbf{x}}$ that is multiplied by a scalar ($n$, I believe). Then, we would have a set of vectors ($n-1$, I believe) that each have as elements different products of the elements in $\\mathbf{y}$ (e.g., $y_1 y_3$) and are each multiplied by a scalar and a diagonal matrix composed of permutations of the elements in $\\mathbf{M}$ and $D_{\\mathbf{x}}$.

          \r\n

          So, I have two questions here:

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          1. Is my general intuition about what this equation would look like correct?
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          2. Are there techniques to solve for the elements of $\\mathbf{y}$ in this system?
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          \r\n edited Mar 31, 2021 at 17:37\r\n
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          \r\n Ryan da Silva\r\n
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          \r\n asked Mar 29, 2021 at 23:19\r\n
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          I am not sure what the notation $D_\\mathbf{x}$, $D_\\mathbf{y}$, and $D_\\mathbf{z}$ represent and if the importance is significant, but assuming $D_{\\mathbf{x}}$ is invertible, you may solve for $\\mathbf{y}$ as follows given the original equation:

          \r\n

          $$ \\mathbf{y}D_{\\mathbf{x}}+\\mathbf{z}D_{\\mathbf{y}}=\\mathbf{u}D_{\\mathbf{z}} $$\r\n$$ \\mathbf{y}D_{\\mathbf{x}}=\\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} $$\r\n$$ \\mathbf{y}=\\bigr( \\mathbf{u}D_{\\mathbf{z}}-\\mathbf{z}D_{\\mathbf{y}} \\bigr) D_{\\mathbf{x}}^{-1} $$

          \r\n

          Note: I can't verify that your edit is correct because I don't understand what $D_{\\mathbf{x} + \\mathbf{z}}$ represents.

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          Edit 1:

          \r\n

          Thank you for the response as I didn't realize the meaning of the matrices $D_{\\mathbf{x}}$, $D_{\\mathbf{y}}$, and $D_{\\mathbf{z}}$. Yes, your edit is correct, but perhaps, I can provide how I would work the problem if that is of use to you.

          \r\n

          Since $D_{\\mathbf{x}} = \\text{diag}(\\mathbf{x})$ and similarly for the other matrices, we have

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          $$\r\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 & & \\\\ & \\ddots & \\\\ & & x_n\\end{bmatrix} + \\begin{bmatrix} z_1 & \\cdots & z_n \\end{bmatrix} \\begin{bmatrix} y_1 & & \\\\ & \\ddots & \\\\ & & y_n\\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

          \r\n

          and multiplying through we have

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          $$\r\n\\begin{align}\r\n\\begin{bmatrix} y_1 x_1 & \\cdots & y_n x_n \\end{bmatrix} + \\begin{bmatrix} y_1 z_1 & \\cdots & y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}}\\\\\r\n\\begin{bmatrix} y_1 x_1 + y_1 z_1 & \\cdots & y_n x_n + y_n z_n \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\r\n\\begin{bmatrix} y_1 (x_1 + z_1) & \\cdots & y_n(x_n + z_n) \\end{bmatrix} &= \\mathbf{u}D_{\\mathbf{z}} \\\\\r\n\\end{align}\r\n$$

          \r\n

          Therefore, this can be written in matrix form as

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          $$\r\n\\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\begin{bmatrix} x_1 + z_1 & & \\\\ & \\ddots & \\\\ & & x_n + z_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

          \r\n

          or more concisely as

          \r\n

          $$ \\mathbf{y} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) = \\mathbf{u} D_{\\mathbf{z}} $$

          \r\n

          which is exactly what is given in your edit:

          \r\n

          $$ \\mathbf{y} = \\mathbf{u} D_{\\mathbf{z}} (D_{\\mathbf{x}} + D_{\\mathbf{z}} ) ^{-1}$$

          \r\n

          Edit 2:

          \r\n

          As for the case where you have $\\mathbf{y}^2 + \\mathbf{y} D_{\\mathbf{x}} = \\mathbf{u} D_{\\mathbf{z}}$ where $\\mathbf{y}^2 = \\begin{bmatrix} y_1^2 & \\cdots & y_n^2 \\end{bmatrix}$, you would not be able to solve for $\\mathbf{y}$ as far as I can tell... To see this, multiply together the matrices, which would give the following result (skipping intermediate steps):

          \r\n

          $$\r\n\\begin{bmatrix} x_1 & \\cdots & x_n \\end{bmatrix} \\begin{bmatrix} y_1^2 + y_1 & & \\\\ & \\ddots & \\\\ & & y_n^2 + y_n \\end{bmatrix} = \\mathbf{u}D_{\\mathbf{z}}\r\n$$

          \r\n

          where we cannot solve for the matrix containing the $y$ variables because we cannot eliminate $\\mathbf{x}$ from the left side of the equation.

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          \r\n edited Mar 31, 2021 at 18:44\r\n
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          \r\n answered Mar 30, 2021 at 8:19\r\n
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          \"Ralff's
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            \r\n $\\begingroup$\r\n My notation for, for example, $D_{\\mathbf{x}}$ was intended to refer to $\\mathrm{diag}(\\mathbf{x})$. Thus, $D_{\\mathbf{x+y}}$ means $\\mathrm{diag}(\\mathbf{x+y})$. Sorry if that was unclear. Furthermore, in the solution you provided, $\\mathbf{y}$ is still along the diagonal of $D_{\\mathbf{y}}$. My aim is to remove it so we can have an non-implicit expression for $\\mathbf{y}$.\r\n $\\endgroup$\r\n
            \r\n– Ryan da Silva\r\n
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            \r\n $\\begingroup$\r\n @RyandaSilva Ah. Thanks. I apologize I misunderstood the notation. I edited my question of how I would work the problem if that may be of any use to you. Your edit is indeed correct. Best of luck!\r\n $\\endgroup$\r\n
            \r\n– Ralff\r\n
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            \r\n $\\begingroup$\r\n Thank you very much for the help and the kind wishes. If you have time, I have added a bit more to my original post that I would love to get your feedback on.\r\n $\\endgroup$\r\n
            \r\n– Ryan da Silva\r\n
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            \r\n $\\begingroup$\r\n @RyandaSilva You're welcome! I made a mistake, so I updated my answer. Please, see the changes. I provided a hint for your additional edit but only for the $\\mathbf{y}^2$ case. If you have a matrix with off diagonal terms, then the matrix multiplication will be slightly more complicated, but I suggest working out the multiplication by hand, so you can see the result.\r\n $\\endgroup$\r\n
            \r\n– Ralff\r\n
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E}$\n\n$\\newcommand{\\coords}[2]{\\left\\{#1\\right\\}_{#2}}$\n\n$\\newcommand{\\gray}[1]{\\color{gray}{#1}}$\n\n$\\newcommand{\\lgray}[1]{\\color{lightgray}{#1}}$\n\n$\\newcommand{\\rank}{\\operatorname{rank}}$\n\n$\\newcommand{\\row}{\\text{Row}}$\n\n$\\newcommand{\\col}{\\text{Col}}$\n\n$\\renewcommand{\\row}{\\text{Row}}$\n\n$\\newcommand{\\nul}{\\text{Nul}}$\n\n$\\newcommand{\\var}{\\text{Var}}$\n\n$\\newcommand{\\corr}{\\text{corr}}$\n\n$\\newcommand{\\len}[1]{\\left|#1\\right|}$\n\n$\\newcommand{\\bbar}{\\overline{\\bvec}}$\n\n$\\newcommand{\\bhat}{\\widehat{\\bvec}}$\n\n$\\newcommand{\\bperp}{\\bvec^\\perp}$\n\n$\\newcommand{\\xhat}{\\widehat{\\xvec}}$\n\n$\\newcommand{\\vhat}{\\widehat{\\vvec}}$\n\n$\\newcommand{\\uhat}{\\widehat{\\uvec}}$\n\n$\\newcommand{\\what}{\\widehat{\\wvec}}$\n\n$\\newcommand{\\Sighat}{\\widehat{\\Sigma}}$\n\n$\\newcommand{\\lt}{<}$\n\n$\\newcommand{\\gt}{>}$\n\n$\\newcommand{\\amp}{&}$\n\n$\\definecolor{fillinmathshade}{gray}{0.9}$\n\nBy the end of this section, you will be able to:\n\n- Multiply integers\n- Divide integers\n- Simplify expressions with integers\n- Evaluate variable expressions with integers\n- Translate English phrases to algebraic expressions\n- Use integers in applications\n\nA more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.\n\n## Multiply Integers\n\nSince multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.\n\nWe remember that $a\\cdot b$ means add $a,\\, b$ times. Here, we are using the model just to help us discover the pattern.\n\nThe next two examples are more interesting.\n\nWhat does it mean to multiply $5$ by $−3$ ? It means subtract $5, 3$ times. Looking at subtraction as “taking away,” it means to take away $5, 3$ times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away $5$ three times.\n\nIn summary:\n\n$$\n\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\n$$\n\nNotice that for multiplication of two signed numbers, when the:\n\n- signs are the same , the product is positive .\n- signs are different , the product is negative .\n\nWe’ll put this all together in the chart below.\n\nFor multiplication of two signed numbers:\n\n| Same signs | Product | Example |\n|---|---|---|\n| Two positives | Positive | \\(7\\cdot 4 = 28\\) |\n| Two negatives | Positive | \\(-8(-6) = 48\\) |\n\n| Different signs | Product | Example |\n|---|---|---|\n| Positives \\(\\cdot\\) negative | Negative | \\(7(-9) = -63\\) |\n| Negative \\(\\cdot\\) positives | Negative | \\(-5\\cdot 10= -50\\) |\n\nMultiply:\n\n1. $-9\\cdot 3$\n2. $-2(-5)$\n3. $4(-8)$\n4. $7\\cdot 6$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\n$$\n\nMultiply:\n\n1. $-6\\cdot 8$\n2. $-4(-7)$\n3. $9(-7)$\n4. $5\\cdot 12$\n\n1. Answer\n2. $-48$ $28$ $-63$ $60$\n\nMultiply:\n\n1. $-8\\cdot 7$\n2. $-6(-9)$\n3. $7(-4)$\n4. $3\\cdot 13$\n\n1. Answer\n2. $-56$ $54$ $-28$ $39$\n\nWhen we multiply a number by $1$ , the result is the same number. What happens when we multiply a number by $−1$ ? Let’s multiply a positive number and then a negative number by $−1$ to see what we get.\n\n$$\n\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\n$$\n\nEach time we multiply a number by $−1$ , we get its opposite!\n\nMULTIPLICATION BY −1\n\n$$\n−1a=−a\n$$\n\nMultiplying a number by $−1$ gives its opposite.\n\nMultiply:\n\n1. $-1 \\cdot 7$\n2. $-1(-11)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\n$$\n\nMultiply:\n\n1. $-1\\cdot 9$\n2. $-1\\cdot(-17)$\n\n1. Answer\n2. $-9$ $17$\n\nMultiply:\n\n1. $-1\\cdot 8$\n2. $-1\\cdot(-16)$\n\n1. Answer\n2. $-8$ $16$\n\n## Divide Integers\n\nWhat about division? Division is the inverse operation of multiplication. So, $15\\div 3=5$ because $5 \\cdot 3 = 15$ . In words, this expression says that $15$ can be divided into three groups of five each because adding five three times gives $15$ . Look at some examples of multiplying integers, to figure out the rules for dividing integers.\n\n$$\n\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\n$$\n\nDivision follows the same rules as multiplication!\n\nFor division of two signed numbers, when the:\n\n- signs are the same , the quotient is positive .\n- signs are different , the quotient is negative .\n\nAnd remember that we can always check the answer of a division problem by multiplying.\n\nFor multiplication and division of two signed numbers:\n\n- If the signs are the same, the result is positive.\n- If the signs are different, the result is negative.\n\n
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          \n\n1. $-27\\div 3$\n2. $-100\\div (-4)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\n$$\n\nDivide:\n\n1. $-42\\div 6$\n2. $-117\\div (-3)$\n\n1. Answer\n2. $-7$ $39$\n\nDivide:\n\n1. $-63\\div 7$\n2. $-115\\div (-5)$\n\n1. Answer\n2. $-9$ $23$\n\n## Simplify Expressions with Integers\n\nWhat happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?\n\nLet’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.\n\nSimplify:\n\n$7(-2)+4(-7)-6$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\n$$\n\nSimplify:\n\n$8(-3)+5(-7)-4$\n\n1. Answer\n2. $-63$\n\nSimplify:\n\n$9(-3)+7(-8)-1$\n\n1. Answer\n2. $-84$\n\nSimplify:\n\n1. $(-2)^{4}$\n2. $-2^{4}$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\n$$\n\nNotice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the $(−2)$ to the $4^{th}$ power. In part (2), the exponent means to raise just the $2$ to the $4^{th}$ power and then take the opposite.\n\nSimplify:\n\n1. $(-3)^{4}$\n2. $-3^{4}$\n\n1. Answer\n2. $81$ $-81$\n\nSimplify:\n\n1. $(-7)^{2}$\n2. $-7^{2}$\n\n1. Answer\n2. $49$ $-49$\n\nThe next example reminds us to simplify inside parentheses first.\n\nSimplify:\n\n$12-3(9 - 12)$\n\nSolution\n\n$$\n\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\n$$\n\nSimplify:\n\n$17 - 4(8 - 11)$\n\n1. Answer\n2. $29$\n\nSimplify:\n\n$16 - 6(7 - 13)$\n\n1. Answer\n2. $52$\n\nSimplify:\n\n$8(-9)\\div (-2)^{3}$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\n$$\n\nSimplify:\n\n$12(-9)\\div (-3)^{3}$\n\n1. Answer\n2. $4$\n\nSimplify:\n\n$18(-4)\\div (-2)^{3}$\n\n1. Answer\n2. $9$\n\nSimplify:\n\n$-30\\div 2 + (-3)(-7)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\n$$\n\nSimplify:\n\n$-27\\div 3 + (-5)(-6)$\n\n1. Answer\n2. $21$\n\nSimplify:\n\n$-32\\div 4 + (-2)(-7)$\n\n1. Answer\n2. $6$\n\n## Evaluate Variable Expressions with Integers\n\nRemember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.\n\nWhen $n=−5$ , evaluate:\n\n1. $n+1$\n2. $−n+1$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\n$$\n\nWhen $n=−8$ , evaluate:\n\n1. $n+2$\n2. $−n+2$ .\n\n1. Answer\n2. $-6$ $10$\n\nWhen $y=−9$ , evaluate:\n\n1. $y+8$\n2. $−y+8$ .\n\n1. Answer\n2. $-1$ $17$\n\nEvaluate $(x+y)^{2}$ when $x = -18$ and $y = 24$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\n$$\n\nEvaluate $(x+y)^{2}$ when $x = -15$ and $y = 29$ .\n\n1. Answer\n2. $196$\n\nEvaluate $(x+y)^{3}$ when $x = -8$ and $y = 10$ .\n\n1. Answer\n2. $8$\n\nEvaluate $20 -z$ when\n\n1. $z = 12$\n2. $z = -12$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\n$$\n\nEvaluate $17 - k$ when\n\n1. $k = 19$\n2. $k = -19$\n\n1. Answer\n2. $-2$ $36$\n\nEvaluate $-5 - b$ when\n\n1. $b = 14$\n2. $b = -14$\n\n1. Answer\n2. $-19$ $9$\n\nEvaluate:\n\n$2x^{2} + 3x + 8$ when $x = 4$ .\n\nSolution\n\nSubstitute $4$ for $x$ . Use parentheses to show multiplication.\n\n$$\n\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\n$$\n\nEvaluate:\n\n$3x^{2} - 2x + 6$ when $x =-3$ .\n\n1. Answer\n2. $39$\n\nEvaluate:\n\n$4x^{2} - x - 5$ when $x = -2$ .\n\n1. Answer\n2. $13$\n\n## Translate Phrases to Expressions with Integers\n\nOur earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.\n\nTranslate and simplify: the sum of $8$ and $−12$ , increased by $3$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\n$$\n\nTranslate and simplify: the sum of $9$ and $−16$ , increased by $4$ .\n\n1. Answer\n2. $(9 + (-16)) + 4 - 3$\n\nTranslate and simplify: the sum of $-8$ and $−12$ , increased by $7$ .\n\n1. Answer\n2. $(-8 + (-12)) + 7 - 13$\n\nWhen we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.\n\n| \\(a−b\\) |\n|---|\n| \\(a\\) minus \\(b\\) the difference of \\(a\\) and \\(b\\) \\(b\\) subtracted from \\(a\\) \\(b\\) less than \\(a\\) |\n\nBe careful to get a and b in the right order!\n\nTranslate and then simplify\n\n1. the difference of $13$ and $−21$\n2. subtract $24$ from $−19$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\n$$\n\nTranslate and simplify\n\n1. the difference of $14$ and $−23$\n2. subtract $21$ from $−17$ .\n\n1. Answer\n2. $14 - (-23); 37$ $-17 - 21; -38$\n\nTranslate and simplify\n\n1. the difference of $11$ and $−19$\n2. subtract $18$ from $−11$ .\n\n1. Answer\n2. $11 - (-19); 30$ $-11 - 18; -29$\n\nOnce again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “ product ” and for division is “ quotient.”\n\nTranslate to an algebraic expression and simplify if possible: the product of $−2$ and $14$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the product of $−5$ and $12$ .\n\n1. Answer\n2. $-5(12); -60$\n\nTranslate to an algebraic expression and simplify if possible: the product of $8$ and $-13$ .\n\n1. Answer\n2. $-8(13); -104$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $−56$ and $−7$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $−63$ and $−9$ .\n\n1. Answer\n2. $-63\\div (-9); 7$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $−72$ and $−9$ .\n\n1. Answer\n2. $-72\\div (-9); 8$\n\n## Use Integers in Applications\n\nWe’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.\n\nHow to Apply a Strategy to Solve Applications with Integers\n\nThe temperature in Urbana, Illinois one morning was $11$ degrees. By mid-afternoon, the temperature had dropped to $−9$ degrees. What was the difference of the morning and afternoon temperatures?\n\nSolution\n\n| Step 1 . Read the problem. Make sure all the words and ideas are understood. | |\n|---|---|\n| Step 2 . Identify what we are asked to find. | the difference of the morning and afternoon temperatures |\n| Step 3 . Write a phrase that gives the information to find it. | the difference of \\(11\\) and \\(-9\\) |\n| Step 4. Translate the phrase to an expression. | \\(11 - (-9)\\) |\n| Step 5 . Simplify the expression. | \\(20\\) |\n| Step 6 . Write a complete sentence that answers the question. | The difference in temperatures was 20 degrees. |\n\nThe temperature in Anchorage, Alaska one morning was $15$ degrees. By mid-afternoon the temperature had dropped to $30$ degrees below zero. What was the difference in the morning and afternoon temperatures?\n\n1. Answer\n2. The difference in temperatures was $45$ degrees.\n\nThe temperature in Denver was $−6$ degrees at lunchtime. By sunset the temperature had dropped to $−15$ degrees. What was the difference in the lunchtime and sunset temperatures?\n\n1. Answer\n2. The difference in temperatures was $9$ degrees.\n\n1. Read the problem. Make sure all the words and ideas are understood\n2. Identify what we are asked to find.\n3. Write a phrase that gives the information to find it.\n4. Translate the phrase to an expression.\n5. Simplify the expression.\n6. Answer the question with a complete sentence.\n\nThe Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?\n\nSolution\n\n| Step 1 . Read the problem. Make sure all the words and ideas are understood. | |\n|---|---|\n| Step 2 . Identify what we are asked to find. | the number of yards lost |\n| Step 3 . Write a phrase that gives the information to find it. | three times a \\(15\\)-yard penalty |\n| Step 4. Translate the phrase to an expression. | \\(3(-15)\\) |\n| Step 5 . Simplify the expression. | \\(-45\\) |\n| Step 6 . Write a complete sentence that answers the question. | The team lost \\(45\\) yards. |\n\nThe Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of $15$ yards. What is the number of yards lost due to penalties?\n\n1. Answer\n2. The Bears lost $105$ yards.\n\nBill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a \\$2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?\n\n1. Answer\n2. A \\$16 fee was deducted from his checking account.\n\n## Key Concepts\n\n- Multiplication and Division of Two Signed Numbers Same signs—Product is positive Different signs—Product is negative\n- Strategy for Applications Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Answer the question with a complete sentence.\n", "main_html": "
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          \n\\(\\newcommand{\\avec}{\\mathbf a}\\) \\(\\newcommand{\\bvec}{\\mathbf b}\\) \\(\\newcommand{\\cvec}{\\mathbf c}\\) \\(\\newcommand{\\dvec}{\\mathbf d}\\) \\(\\newcommand{\\dtil}{\\widetilde{\\mathbf d}}\\) \\(\\newcommand{\\evec}{\\mathbf e}\\) \\(\\newcommand{\\fvec}{\\mathbf f}\\) \\(\\newcommand{\\nvec}{\\mathbf n}\\) \\(\\newcommand{\\pvec}{\\mathbf p}\\) \\(\\newcommand{\\qvec}{\\mathbf q}\\) \\(\\newcommand{\\svec}{\\mathbf s}\\) \\(\\newcommand{\\tvec}{\\mathbf t}\\) \\(\\newcommand{\\uvec}{\\mathbf u}\\) \\(\\newcommand{\\vvec}{\\mathbf v}\\) \\(\\newcommand{\\wvec}{\\mathbf w}\\) \\(\\newcommand{\\xvec}{\\mathbf x}\\) \\(\\newcommand{\\yvec}{\\mathbf y}\\) \\(\\newcommand{\\zvec}{\\mathbf z}\\) \\(\\newcommand{\\rvec}{\\mathbf r}\\) \\(\\newcommand{\\mvec}{\\mathbf m}\\) \\(\\newcommand{\\zerovec}{\\mathbf 0}\\) \\(\\newcommand{\\onevec}{\\mathbf 1}\\) \\(\\newcommand{\\real}{\\mathbb R}\\) \\(\\newcommand{\\twovec}[2]{\\left[\\begin{array}{r}#1 \\\\ #2 \\end{array}\\right]}\\) \\(\\newcommand{\\ctwovec}[2]{\\left[\\begin{array}{c}#1 \\\\ #2 \\end{array}\\right]}\\) \\(\\newcommand{\\threevec}[3]{\\left[\\begin{array}{r}#1 \\\\ #2 \\\\ #3 \\end{array}\\right]}\\) \\(\\newcommand{\\cthreevec}[3]{\\left[\\begin{array}{c}#1 \\\\ #2 \\\\ #3 \\end{array}\\right]}\\) \\(\\newcommand{\\fourvec}[4]{\\left[\\begin{array}{r}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\end{array}\\right]}\\) \\(\\newcommand{\\cfourvec}[4]{\\left[\\begin{array}{c}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\end{array}\\right]}\\) \\(\\newcommand{\\fivevec}[5]{\\left[\\begin{array}{r}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\\\ #5 \\\\ \\end{array}\\right]}\\) \\(\\newcommand{\\cfivevec}[5]{\\left[\\begin{array}{c}#1 \\\\ #2 \\\\ #3 \\\\ #4 \\\\ #5 \\\\ \\end{array}\\right]}\\) \\(\\newcommand{\\mattwo}[4]{\\left[\\begin{array}{rr}#1 \\amp #2 \\\\ #3 \\amp #4 \\\\ \\end{array}\\right]}\\) \\(\\newcommand{\\laspan}[1]{\\text{Span}\\{#1\\}}\\) \\(\\newcommand{\\bcal}{\\cal B}\\) \\(\\newcommand{\\ccal}{\\cal C}\\) \\(\\newcommand{\\scal}{\\cal S}\\) \\(\\newcommand{\\wcal}{\\cal W}\\) \\(\\newcommand{\\ecal}{\\cal E}\\) \\(\\newcommand{\\coords}[2]{\\left\\{#1\\right\\}_{#2}}\\) \\(\\newcommand{\\gray}[1]{\\color{gray}{#1}}\\) \\(\\newcommand{\\lgray}[1]{\\color{lightgray}{#1}}\\) \\(\\newcommand{\\rank}{\\operatorname{rank}}\\) \\(\\newcommand{\\row}{\\text{Row}}\\) \\(\\newcommand{\\col}{\\text{Col}}\\) \\(\\renewcommand{\\row}{\\text{Row}}\\) \\(\\newcommand{\\nul}{\\text{Nul}}\\) \\(\\newcommand{\\var}{\\text{Var}}\\) \\(\\newcommand{\\corr}{\\text{corr}}\\) \\(\\newcommand{\\len}[1]{\\left|#1\\right|}\\) \\(\\newcommand{\\bbar}{\\overline{\\bvec}}\\) \\(\\newcommand{\\bhat}{\\widehat{\\bvec}}\\) \\(\\newcommand{\\bperp}{\\bvec^\\perp}\\) \\(\\newcommand{\\xhat}{\\widehat{\\xvec}}\\) \\(\\newcommand{\\vhat}{\\widehat{\\vvec}}\\) \\(\\newcommand{\\uhat}{\\widehat{\\uvec}}\\) \\(\\newcommand{\\what}{\\widehat{\\wvec}}\\) \\(\\newcommand{\\Sighat}{\\widehat{\\Sigma}}\\) \\(\\newcommand{\\lt}{<}\\) \\(\\newcommand{\\gt}{>}\\) \\(\\newcommand{\\amp}{&}\\) \\(\\definecolor{fillinmathshade}{gray}{0.9}\\)
          \n
          \n \n \n
          \n
          \n\t\t\t\t
          \n
          \n\n

          By the end of this section, you will be able to:

          \n\n
            \n
          • Multiply integers
            • \n
          • Divide integers
            • \n
          • Simplify expressions with integers
            • \n
          • Evaluate variable expressions with integers
            • \n
          • Translate English phrases to algebraic expressions
            • \n
          • Use integers in applications
            • \n
          \n
          \n\n

          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

          \n\n

          Multiply Integers

          \n\n

          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

          \n\n

          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

          \n\n
          \"Two\n
          Figure \\(\\PageIndex{1}\\)
          \n
          \n\n

          The next two examples are more interesting.

          \n\n

          What does it mean to multiply \\(5\\) by \\(−3\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as “taking away,” it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

          \n\n
          \"This\n
          Figure \\(\\PageIndex{2}\\)
          \n
          \n\n

          In summary:

          \n\n

          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

          \n\n

          Notice that for multiplication of two signed numbers, when the:

          \n\n
            \n
          • signs are the same, the product is positive.
            • \n
          • signs are different, the product is negative.
            • \n
          \n\n

          We’ll put this all together in the chart below.

          \n\n
          \n
          \n\n

          For multiplication of two signed numbers:

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Same signsProductExample
          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
          Table \\(\\PageIndex{1}\\)
          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
          Table \\(\\PageIndex{2}\\)
          \n
          \n\n
          \n
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-9\\cdot 3\\)
            2. \n
          2. \\(-2(-5)\\)
            3. \n
          3. \\(4(-8)\\)
            4. \n
          4. \\(7\\cdot 6\\)
            5. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
            3. \n
          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
            4. \n
          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
            5. \n
          \n
          \n\n
          \n
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-6\\cdot 8\\)
            2. \n
          2. \\(-4(-7)\\)
            3. \n
          3. \\(9(-7)\\)
            4. \n
          4. \\(5\\cdot 12\\)
            5. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-48\\)
            2. \n
          2. \\(28\\)
            3. \n
          3. \\(-63\\)
            4. \n
          4. \\(60\\)
            5. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-8\\cdot 7\\)
            2. \n
          2. \\(-6(-9)\\)
            3. \n
          3. \\(7(-4)\\)
            4. \n
          4. \\(3\\cdot 13\\)
            5. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-56\\)
            2. \n
          2. \\(54\\)
            3. \n
          3. \\(-28\\)
            4. \n
          4. \\(39\\)
            5. \n
          \n
          \n
          \n
          \n\n

          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(−1\\)? Let’s multiply a positive number and then a negative number by \\(−1\\) to see what we get.

          \n\n

          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
          \nEach time we multiply a number by \\(−1\\), we get its opposite!

          \n\n
          \n
           
          \n\n

          MULTIPLICATION BY −1

          \n\n

          \\[−1a=−a\\]

          \n\n

          Multiplying a number by \\(−1\\) gives its opposite.

          \n
          \n\n
          \n
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-1 \\cdot 7\\)
            2. \n
          2. \\(-1(-11)\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-1\\cdot 9\\)
            2. \n
          2. \\(-1\\cdot(-17)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-9\\)
            2. \n
          2. \\(17\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-1\\cdot 8\\)
            2. \n
          2. \\(-1\\cdot(-16)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-8\\)
            2. \n
          2. \\(16\\)
            3. \n
          \n
          \n
          \n
          \n\n

          Divide Integers

          \n\n

          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

          \n\n

          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

          \n\n

          Division follows the same rules as multiplication!

          \n\n

          For division of two signed numbers, when the:

          \n\n
            \n
          • signs are the same, the quotient is positive.
            • \n
          • signs are different, the quotient is negative.
            • \n
          \n\n

          And remember that we can always check the answer of a division problem by multiplying.

          \n\n
          \n
          \n\n

          For multiplication and division of two signed numbers:

          \n\n
            \n
          • If the signs are the same, the result is positive.
            • \n
          • If the signs are different, the result is negative.
            • \n
          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          \n
          \n\n
          \n
          \n\n
            \n
          1. \\(-27\\div 3\\)
            2. \n
          2. \\(-100\\div (-4)\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          \n\n

          Divide:

          \n\n
            \n
          1. \\(-42\\div 6\\)
            2. \n
          2. \\(-117\\div (-3)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-7\\)
            2. \n
          2. \\(39\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Divide:

          \n\n
            \n
          1. \\(-63\\div 7\\)
            2. \n
          2. \\(-115\\div (-5)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-9\\)
            2. \n
          2. \\(23\\)
            3. \n
          \n
          \n
          \n
          \n\n

          Simplify Expressions with Integers

          \n\n

          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

          \n\n

          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(7(-2)+4(-7)-6\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(8(-3)+5(-7)-4\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(-63\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(9(-3)+7(-8)-1\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(-84\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n
            \n
          1. \\((-2)^{4}\\)
            2. \n
          2. \\(-2^{4}\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
            3. \n
          \n\n

          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((−2)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n
            \n
          1. \\((-3)^{4}\\)
            2. \n
          2. \\(-3^{4}\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(81\\)
            2. \n
          2. \\(-81\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n
            \n
          1. \\((-7)^{2}\\)
            2. \n
          2. \\(-7^{2}\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(49\\)
            2. \n
          2. \\(-49\\)
            3. \n
          \n
          \n
          \n
          \n\n

          The next example reminds us to simplify inside parentheses first.

          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(12-3(9 - 12)\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(17 - 4(8 - 11)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(29\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(16 - 6(7 - 13)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(52\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(8(-9)\\div (-2)^{3}\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(12(-9)\\div (-3)^{3}\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(4\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(18(-4)\\div (-2)^{3}\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(9\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(-30\\div 2 + (-3)(-7)\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(-27\\div 3 + (-5)(-6)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(21\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Simplify:

          \n\n

          \\(-32\\div 4 + (-2)(-7)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(6\\)

          \n
          \n
          \n
          \n\n

          Evaluate Variable Expressions with Integers

          \n\n

          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

          \n\n
          \n
          \n\n

          When \\(n=−5\\), evaluate:

          \n\n
            \n
          1. \\(n+1\\)
            2. \n
          2. \\(−n+1\\).
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          \n\n

          When \\(n=−8\\), evaluate:

          \n\n
            \n
          1. \\(n+2\\)
            2. \n
          2. \\(−n+2\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-6\\)
            2. \n
          2. \\(10\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          When \\(y=−9\\), evaluate:

          \n\n
            \n
          1. \\(y+8\\)
            2. \n
          2. \\(−y+8\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-1\\)
            2. \n
          2. \\(17\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(196\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(8\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Evaluate \\(20 -z \\) when

          \n\n
            \n
          1. \\(z = 12\\)
            2. \n
          2. \\(z = -12\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          \n\n

          Evaluate \\(17 - k\\) when

          \n\n
            \n
          1. \\(k = 19\\)
            2. \n
          2. \\(k = -19\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-2\\)
            2. \n
          2. \\(36\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Evaluate \\(-5 - b\\) when

          \n\n
            \n
          1. \\(b = 14\\)
            2. \n
          2. \\(b = -14\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-19\\)
            2. \n
          2. \\(9\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Evaluate:

          \n\n

          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

          \n\n

          Solution

          \n\n

          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

          \n\n

          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Evaluate:

          \n\n

          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(39\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Evaluate:

          \n\n

          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(13\\)

          \n
          \n
          \n
          \n\n

          Translate Phrases to Expressions with Integers

          \n\n

          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

          \n\n
          \n
          \n\n

          Translate and simplify: the sum of \\(8\\) and \\(−12\\), increased by \\(3\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Translate and simplify: the sum of \\(9\\) and \\(−16\\), increased by \\(4\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\((9 + (-16)) + 4 - 3\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Translate and simplify: the sum of \\(-8\\) and \\(−12\\), increased by \\(7\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\((-8 + (-12)) + 7 - 13\\)

          \n
          \n
          \n
          \n\n

          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n
          \\(a−b\\)
          \\(a\\) minus \\(b\\)
          \n the difference of \\(a\\) and \\(b\\)
          \n \\(b\\) subtracted from \\(a\\)
          \n \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          \n\n

          Be careful to get a and b in the right order!

          \n\n
          \n
          \n\n

          Translate and then simplify

          \n\n
            \n
          1. the difference of \\(13\\) and \\(−21\\)
            2. \n
          2. subtract \\(24\\) from \\(−19\\).
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          \n\n

          Translate and simplify

          \n\n
            \n
          1. the difference of \\(14\\) and \\(−23\\)
            2. \n
          2. subtract \\(21\\) from \\(−17\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(14 - (-23); 37\\)
            2. \n
          2. \\(-17 - 21; -38\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Translate and simplify

          \n\n
            \n
          1. the difference of \\(11\\) and \\(−19\\)
            2. \n
          2. subtract \\(18\\) from \\(−11\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(11 - (-19); 30\\)
            2. \n
          2. \\(-11 - 18; -29\\)
            3. \n
          \n
          \n
          \n
          \n\n

          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

          \n\n
          \n
          \n\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−2\\) and \\(14\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−5\\) and \\(12\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-5(12); -60\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-8(13); -104\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−56\\) and \\(−7\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

          \n
          \n\n
          \n
          \n\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−63\\) and \\(−9\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-63\\div (-9); 7\\)

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−72\\) and \\(−9\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-72\\div (-9); 8\\)

          \n
          \n
          \n
          \n\n

          Use Integers in Applications

          \n\n

          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

          \n\n

          How to Apply a Strategy to Solve Applications with Integers

          \n\n
          \n
          \n\n

          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(−9\\) degrees. What was the difference of the morning and afternoon temperatures?

          \n\n

          Solution

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5. Simplify the expression.\\(20\\)
          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          \n
          \n\n
          \n
          \n\n

          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

          \n\n
          \n
          Answer
          \n
          \n

          The difference in temperatures was \\(45\\) degrees.

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          The temperature in Denver was \\(−6\\) degrees at lunchtime. By sunset the temperature had dropped to \\(−15\\) degrees. What was the difference in the lunchtime and sunset temperatures?

          \n\n
          \n
          Answer
          \n
          \n

          The difference in temperatures was \\(9\\) degrees.

          \n
          \n
          \n
          \n\n
          \n
          \n\n
            \n
          1. Read the problem. Make sure all the words and ideas are understood
            2. \n
          2. Identify what we are asked to find.
            3. \n
          3. Write a phrase that gives the information to find it.
            4. \n
          4. Translate the phrase to an expression.
            5. \n
          5. Simplify the expression.
            6. \n
          6. Answer the question with a complete sentence.
            7. \n
          \n
          \n\n
          \n
          \n\n

          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

          \n\n

          Solution

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the number of yards lost
          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5. Simplify the expression.\\(-45\\)
          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          \n
          \n\n
          \n
          \n\n

          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

          \n\n
          \n
          Answer
          \n
          \n

          The Bears lost \\(105\\) yards.

          \n
          \n
          \n
          \n\n
          \n
          \n\n

          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

          \n\n
          \n
          Answer
          \n
          \n

          A $16 fee was deducted from his checking account.

          \n
          \n
          \n
          \n\n

          Key Concepts

          \n\n
            \n
          • Multiplication and Division of Two Signed Numbers\n\n
              \n
            • Same signs—Product is positive
              • \n
            • Different signs—Product is negative
              • \n
            \n
            • \n
          • Strategy for Applications\n
              \n
            1. Identify what you are asked to find.
              2. \n
            2. Write a phrase that gives the information to find it.
              3. \n
            3. Translate the phrase to an expression.
              4. \n
            4. Simplify the expression.
              5. \n
            5. Answer the question with a complete sentence.
              6. \n
            \n
            • \n
          \n
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          \\definecolor{fillinmathshade}{gray}{0.9}
          ", "content": [{"c": "\\definecolor{fillinmathshade}{gray}{0.9}", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          By the end of this section, you will be able to:

          ", "content": [{"c": "By the end of this section, you will be able to:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "Multiply integers", "t": "text"}]], [[{"c": "Divide integers", "t": "text"}]], [[{"c": "Simplify expressions with integers", "t": "text"}]], [[{"c": "Evaluate variable expressions with integers", "t": "text"}]], [[{"c": "Translate English phrases to algebraic expressions", "t": "text"}]], [[{"c": "Use integers in applications", "t": "text"}]]], "ordered": false, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

          ", "content": [{"c": "A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.", "t": "text"}]}, {"type": "title", "raw_content": "

          Multiply Integers

          ", "content": {"title_content": "Multiply Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

          ", "content": [{"c": "Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          We remember that a\\cdot b means add a,\\, b times. Here, we are using the model just to help us discover the pattern.

          ", "content": [{"c": "We remember that", "t": "text"}, {"c": "a\\cdot b", "t": "equation-inline"}, {"c": "means add", "t": "text"}, {"c": "a,\\, b", "t": "equation-inline"}, {"c": "times. Here, we are using the model just to help us discover the pattern.", "t": "text"}]}, {"type": "image", "raw_content": "
          \"Two
          Figure \\PageIndex{1}
          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17395/CNX_ElemAlg_Figure_01_04_001_img_new.jpg?revision=1", "data": null, "alt": "Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

          The next two examples are more interesting.

          ", "content": [{"c": "The next two examples are more interesting.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          What does it mean to multiply 5 by −3? It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

          ", "content": [{"c": "What does it mean to multiply", "t": "text"}, {"c": "5", "t": "equation-inline"}, {"c": "by", "t": "text"}, {"c": "−3", "t": "equation-inline"}, {"c": "? It means subtract", "t": "text"}, {"c": "5, 3", "t": "equation-inline"}, {"c": "times. Looking at subtraction as “taking away,” it means to take away", "t": "text"}, {"c": "5, 3", "t": "equation-inline"}, {"c": "times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away", "t": "text"}, {"c": "5", "t": "equation-inline"}, {"c": "three times.", "t": "text"}]}, {"type": "image", "raw_content": "
          \"This
          Figure \\PageIndex{2}
          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17306/CNX_ElemAlg_Figure_01_04_002_img_new.jpg?revision=1", "data": null, "alt": "This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

          In summary:

          ", "content": [{"c": "In summary:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Notice that for multiplication of two signed numbers, when the:

          ", "content": [{"c": "Notice that for multiplication of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "signs are the ", "t": "text"}, {"c": "same", "t": "text"}, {"c": ", the product is ", "t": "text"}, {"c": "positive", "t": "text"}, {"c": ".", "t": "text"}]], [[{"c": "signs are ", "t": "text"}, {"c": "different", "t": "text"}, {"c": ", the product is ", "t": "text"}, {"c": "negative", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": false, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          We’ll put this all together in the chart below.

          ", "content": [{"c": "We’ll put this all together in the chart below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For multiplication of two signed numbers:

          ", "content": [{"c": "For multiplication of two signed numbers:", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          Same signsProductExample
          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
          Table \\(\\PageIndex{1}\\)
          ", "content": {"html": "
          Same signsProductExample
          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
          Table \\(\\PageIndex{1}\\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "simple_table", "raw_content": "
          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
          Table \\(\\PageIndex{2}\\)
          ", "content": {"html": "
          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
          Table \\(\\PageIndex{2}\\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -9\\cdot 3
          2. -2(-5)
          3. 4(-8)
          4. 7\\cdot 6
          ", "content": {"items": [[[{"c": "-9\\cdot 3", "t": "equation-inline"}]], [[{"c": "-2(-5)", "t": "equation-inline"}]], [[{"c": "4(-8)", "t": "equation-inline"}]], [[{"c": "7\\cdot 6", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -6\\cdot 8
          2. -4(-7)
          3. 9(-7)
          4. 5\\cdot 12
          ", "content": {"items": [[[{"c": "-6\\cdot 8", "t": "equation-inline"}]], [[{"c": "-4(-7)", "t": "equation-inline"}]], [[{"c": "9(-7)", "t": "equation-inline"}]], [[{"c": "5\\cdot 12", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -48
          2. 28
          3. -63
          4. 60
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-48", "t": "equation-inline"}, {"c": "28", "t": "equation-inline"}, {"c": "-63", "t": "equation-inline"}, {"c": "60", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -8\\cdot 7
          2. -6(-9)
          3. 7(-4)
          4. 3\\cdot 13
          ", "content": {"items": [[[{"c": "-8\\cdot 7", "t": "equation-inline"}]], [[{"c": "-6(-9)", "t": "equation-inline"}]], [[{"c": "7(-4)", "t": "equation-inline"}]], [[{"c": "3\\cdot 13", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -56
          2. 54
          3. -28
          4. 39
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-56", "t": "equation-inline"}, {"c": "54", "t": "equation-inline"}, {"c": "-28", "t": "equation-inline"}, {"c": "39", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          When we multiply a number by 1, the result is the same number. What happens when we multiply a number by −1? Let’s multiply a positive number and then a negative number by −1 to see what we get.

          ", "content": [{"c": "When we multiply a number by", "t": "text"}, {"c": "1", "t": "equation-inline"}, {"c": ", the result is the same number. What happens when we multiply a number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": "? Let’s multiply a positive number and then a negative number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": "to see what we get.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "


          \nEach time we multiply a number by −1, we get its opposite!

          ", "content": [{"c": "Each time we multiply a number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": ", we get its opposite!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          MULTIPLICATION BY −1

          ", "content": [{"c": "MULTIPLICATION BY −1", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[−1a=−a\\]

          ", "content": {"math_content": "−1a=−a", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Multiplying a number by −1 gives its opposite.

          ", "content": [{"c": "Multiplying a number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": "gives its opposite.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -1 \\cdot 7
          2. -1(-11)
          ", "content": {"items": [[[{"c": "-1 \\cdot 7", "t": "equation-inline"}]], [[{"c": "-1(-11)", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -1\\cdot 9
          2. -1\\cdot(-17)
          ", "content": {"items": [[[{"c": "-1\\cdot 9", "t": "equation-inline"}]], [[{"c": "-1\\cdot(-17)", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -9
          2. 17
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-9", "t": "equation-inline"}, {"c": "17", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -1\\cdot 8
          2. -1\\cdot(-16)
          ", "content": {"items": [[[{"c": "-1\\cdot 8", "t": "equation-inline"}]], [[{"c": "-1\\cdot(-16)", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -8
          2. 16
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-8", "t": "equation-inline"}, {"c": "16", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "title", "raw_content": "

          Divide Integers

          ", "content": {"title_content": "Divide Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          What about division? Division is the inverse operation of multiplication. So, 15\\div 3=5 because 5 \\cdot 3 = 15. In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

          ", "content": [{"c": "What about division? Division is the inverse operation of multiplication. So,", "t": "text"}, {"c": "15\\div 3=5", "t": "equation-inline"}, {"c": "because", "t": "text"}, {"c": "5 \\cdot 3 = 15", "t": "equation-inline"}, {"c": ". In words, this expression says that", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "can be divided into three groups of five each because adding five three times gives", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": ". Look at some examples of multiplying integers, to figure out the rules for dividing integers.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Division follows the same rules as multiplication!

          ", "content": [{"c": "Division follows the same rules as multiplication!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For division of two signed numbers, when the:

          ", "content": [{"c": "For division of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "signs are the ", "t": "text"}, {"c": "same", "t": "text"}, {"c": ", the quotient is ", "t": "text"}, {"c": "positive", "t": "text"}, {"c": ".", "t": "text"}]], [[{"c": "signs are ", "t": "text"}, {"c": "different", "t": "text"}, {"c": ", the quotient is ", "t": "text"}, {"c": "negative", "t": "text"}, {"c": ".", "t": "text"}]]], "ordered": false, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          And remember that we can always check the answer of a division problem by multiplying.

          ", "content": [{"c": "And remember that we can always check the answer of a division problem by multiplying.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For multiplication and division of two signed numbers:

          ", "content": [{"c": "For multiplication and division of two signed numbers:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "If the signs are the same, the result is positive.", "t": "text"}]], [[{"c": "If the signs are different, the result is negative.", "t": "text"}]]], "ordered": false, "list_nest_level": "1"}}, {"type": "complex_table", "raw_content": "
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          ", "content": {"html": "
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "complex_table", "raw_content": "
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          ", "content": {"html": "
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "list", "raw_content": "
          1. -27\\div 3
          2. -100\\div (-4)
          ", "content": {"items": [[[{"c": "-27\\div 3", "t": "equation-inline"}]], [[{"c": "-100\\div (-4)", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Divide:

          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -42\\div 6
          2. -117\\div (-3)
          ", "content": {"items": [[[{"c": "-42\\div 6", "t": "equation-inline"}]], [[{"c": "-117\\div (-3)", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -7
          2. 39
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-7", "t": "equation-inline"}, {"c": "39", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Divide:

          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -63\\div 7
          2. -115\\div (-5)
          ", "content": {"items": [[[{"c": "-63\\div 7", "t": "equation-inline"}]], [[{"c": "-115\\div (-5)", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -9
          2. 23
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-9", "t": "equation-inline"}, {"c": "23", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "title", "raw_content": "

          Simplify Expressions with Integers

          ", "content": {"title_content": "Simplify Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

          ", "content": [{"c": "What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

          ", "content": [{"c": "Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          7(-2)+4(-7)-6

          ", "content": [{"c": "7(-2)+4(-7)-6", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          8(-3)+5(-7)-4

          ", "content": [{"c": "8(-3)+5(-7)-4", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          -63

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-63", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          9(-3)+7(-8)-1

          ", "content": [{"c": "9(-3)+7(-8)-1", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          -84

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-84", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. (-2)^{4}
          2. -2^{4}
          ", "content": {"items": [[[{"c": "(-2)^{4}", "t": "equation-inline"}]], [[{"c": "-2^{4}", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the (−2) to the 4^{th} power. In part (2), the exponent means to raise just the 2 to the 4^{th} power and then take the opposite.

          ", "content": [{"c": "Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the", "t": "text"}, {"c": "(−2)", "t": "equation-inline"}, {"c": "to the", "t": "text"}, {"c": "4^{th}", "t": "equation-inline"}, {"c": "power. In part (2), the exponent means to raise just the", "t": "text"}, {"c": "2", "t": "equation-inline"}, {"c": "to the", "t": "text"}, {"c": "4^{th}", "t": "equation-inline"}, {"c": "power and then take the opposite.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. (-3)^{4}
          2. -3^{4}
          ", "content": {"items": [[[{"c": "(-3)^{4}", "t": "equation-inline"}]], [[{"c": "-3^{4}", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 81
          2. -81
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "81", "t": "equation-inline"}, {"c": "-81", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. (-7)^{2}
          2. -7^{2}
          ", "content": {"items": [[[{"c": "(-7)^{2}", "t": "equation-inline"}]], [[{"c": "-7^{2}", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 49
          2. -49
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "49", "t": "equation-inline"}, {"c": "-49", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          The next example reminds us to simplify inside parentheses first.

          ", "content": [{"c": "The next example reminds us to simplify inside parentheses first.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          12-3(9 - 12)

          ", "content": [{"c": "12-3(9 - 12)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          17 - 4(8 - 11)

          ", "content": [{"c": "17 - 4(8 - 11)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          29

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "29", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          16 - 6(7 - 13)

          ", "content": [{"c": "16 - 6(7 - 13)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          52

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "52", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          8(-9)\\div (-2)^{3}

          ", "content": [{"c": "8(-9)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          12(-9)\\div (-3)^{3}

          ", "content": [{"c": "12(-9)\\div (-3)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          4

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "4", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          18(-4)\\div (-2)^{3}

          ", "content": [{"c": "18(-4)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          9

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "9", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          -30\\div 2 + (-3)(-7)

          ", "content": [{"c": "-30\\div 2 + (-3)(-7)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          -27\\div 3 + (-5)(-6)

          ", "content": [{"c": "-27\\div 3 + (-5)(-6)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          21

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "21", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          -32\\div 4 + (-2)(-7)

          ", "content": [{"c": "-32\\div 4 + (-2)(-7)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          6

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "6", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Evaluate Variable Expressions with Integers

          ", "content": {"title_content": "Evaluate Variable Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

          ", "content": [{"c": "Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          When n=−5, evaluate:

          ", "content": [{"c": "When", "t": "text"}, {"c": "n=−5", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. n+1
          2. −n+1.
          ", "content": {"items": [[[{"c": "n+1", "t": "equation-inline"}]], [[{"c": "−n+1", "t": "equation-inline"}, {"c": ".", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          When n=−8, evaluate:

          ", "content": [{"c": "When", "t": "text"}, {"c": "n=−8", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. n+2
          2. −n+2.
          ", "content": {"items": [[[{"c": "n+2", "t": "equation-inline"}]], [[{"c": "−n+2", "t": "equation-inline"}, {"c": ".", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -6
          2. 10
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-6", "t": "equation-inline"}, {"c": "10", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          When y=−9, evaluate:

          ", "content": [{"c": "When", "t": "text"}, {"c": "y=−9", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. y+8
          2. −y+8.
          ", "content": {"items": [[[{"c": "y+8", "t": "equation-inline"}]], [[{"c": "−y+8", "t": "equation-inline"}, {"c": ".", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -1
          2. 17
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-1", "t": "equation-inline"}, {"c": "17", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Evaluate (x+y)^{2} when x = -18 and y = 24.

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{2}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -18", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 24", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Evaluate (x+y)^{2} when x = -15 and y = 29.

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{2}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -15", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 29", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          196

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "196", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Evaluate (x+y)^{3} when x = -8 and y = 10.

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{3}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 10", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          8

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "8", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Evaluate 20 -z when

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "20 -z ", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
          1. z = 12
          2. z = -12
          ", "content": {"items": [[[{"c": "z = 12", "t": "equation-inline"}]], [[{"c": "z = -12", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Evaluate 17 - k when

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "17 - k", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
          1. k = 19
          2. k = -19
          ", "content": {"items": [[[{"c": "k = 19", "t": "equation-inline"}]], [[{"c": "k = -19", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -2
          2. 36
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-2", "t": "equation-inline"}, {"c": "36", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Evaluate -5 - b when

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "-5 - b", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
          1. b = 14
          2. b = -14
          ", "content": {"items": [[[{"c": "b = 14", "t": "equation-inline"}]], [[{"c": "b = -14", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -19
          2. 9
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-19", "t": "equation-inline"}, {"c": "9", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Evaluate:

          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          2x^{2} + 3x + 8 when x = 4.

          ", "content": [{"c": "2x^{2} + 3x + 8", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = 4", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Substitute 4 for x. Use parentheses to show multiplication.

          ", "content": [{"c": "Substitute", "t": "text"}, {"c": "4", "t": "equation-inline"}, {"c": "for", "t": "text"}, {"c": "x", "t": "equation-inline"}, {"c": ". Use parentheses to show multiplication.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Evaluate:

          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          3x^{2} - 2x + 6 when x =-3.

          ", "content": [{"c": "3x^{2} - 2x + 6", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x =-3", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          39

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "39", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Evaluate:

          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          4x^{2} - x - 5 when x = -2.

          ", "content": [{"c": "4x^{2} - x - 5", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -2", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          13

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "13", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Translate Phrases to Expressions with Integers

          ", "content": {"title_content": "Translate Phrases to Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

          ", "content": [{"c": "Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Translate and simplify: the sum of 8 and −12, increased by 3.

          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−12", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "3", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify: the sum of 9 and −16, increased by 4.

          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "9", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−16", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "4", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          (9 + (-16)) + 4 - 3

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "(9 + (-16)) + 4 - 3", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify: the sum of -8 and −12, increased by 7.

          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "-8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−12", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "7", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          (-8 + (-12)) + 7 - 13

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "(-8 + (-12)) + 7 - 13", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

          ", "content": [{"c": "When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          \\(a−b\\)
          \\(a\\) minus \\(b\\)
          \n the difference of \\(a\\) and \\(b\\)
          \n \\(b\\) subtracted from \\(a\\)
          \n \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          ", "content": {"html": "
          \\(a−b\\)
          \\(a\\) minus \\(b\\) the difference of \\(a\\) and \\(b\\) \\(b\\) subtracted from \\(a\\) \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Be careful to get a and b in the right order!

          ", "content": [{"c": "Be careful to get a and b in the right order!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Translate and then simplify

          ", "content": [{"c": "Translate and then simplify", "t": "text"}]}, {"type": "list", "raw_content": "
          1. the difference of 13 and −21
          2. subtract 24 from −19.
          ", "content": {"items": [[[{"c": "the difference of ", "t": "text"}, {"c": "13", "t": "equation-inline"}, {"c": " and ", "t": "text"}, {"c": "−21", "t": "equation-inline"}]], [[{"c": "subtract ", "t": "text"}, {"c": "24", "t": "equation-inline"}, {"c": " from ", "t": "text"}, {"c": "−19", "t": "equation-inline"}, {"c": ".", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify

          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
          1. the difference of 14 and −23
          2. subtract 21 from −17.
          ", "content": {"items": [[[{"c": "the difference of ", "t": "text"}, {"c": "14", "t": "equation-inline"}, {"c": " and ", "t": "text"}, {"c": "−23", "t": "equation-inline"}]], [[{"c": "subtract ", "t": "text"}, {"c": "21", "t": "equation-inline"}, {"c": " from ", "t": "text"}, {"c": "−17", "t": "equation-inline"}, {"c": ".", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 14 - (-23); 37
          2. -17 - 21; -38
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "14 - (-23); 37", "t": "equation-inline"}, {"c": "-17 - 21; -38", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify

          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
          1. the difference of 11 and −19
          2. subtract 18 from −11.
          ", "content": {"items": [[[{"c": "the difference of ", "t": "text"}, {"c": "11", "t": "equation-inline"}, {"c": " and ", "t": "text"}, {"c": "−19", "t": "equation-inline"}]], [[{"c": "subtract ", "t": "text"}, {"c": "18", "t": "equation-inline"}, {"c": " from ", "t": "text"}, {"c": "−11", "t": "equation-inline"}, {"c": ".", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 11 - (-19); 30
          2. -11 - 18; -29
          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "11 - (-19); 30", "t": "equation-inline"}, {"c": "-11 - 18; -29", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

          ", "content": [{"c": "Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “ product ” and for division is “ quotient.”", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the product of −2 and 14.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "−2", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "14", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the product of −5 and 12.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "−5", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "12", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -5(12); -60

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-5(12); -60", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the product of 8 and -13.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "-13", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -8(13); -104

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-8(13); -104", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the quotient of −56 and −7.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "−56", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−7", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the quotient of −63 and −9.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "−63", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−9", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -63\\div (-9); 7

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-63\\div (-9); 7", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the quotient of −72 and −9.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "−72", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−9", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -72\\div (-9); 8

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "-72\\div (-9); 8", "t": "equation-inline"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Use Integers in Applications

          ", "content": {"title_content": "Use Integers in Applications", "level": "2"}}, {"type": "paragraph", "raw_content": "

          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

          ", "content": [{"c": "We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          How to Apply a Strategy to Solve Applications with Integers

          ", "content": [{"c": "How to Apply a Strategy to Solve Applications with Integers", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          The temperature in Urbana, Illinois one morning was 11 degrees. By mid-afternoon, the temperature had dropped to −9 degrees. What was the difference of the morning and afternoon temperatures?

          ", "content": [{"c": "The temperature in Urbana, Illinois one morning was", "t": "text"}, {"c": "11", "t": "equation-inline"}, {"c": "degrees. By mid-afternoon, the temperature had dropped to", "t": "text"}, {"c": "−9", "t": "equation-inline"}, {"c": "degrees. What was the difference of the morning and afternoon temperatures?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          Step 1. Read the problem. Make sure all the words and ideas are understood.
          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5. Simplify the expression.\\(20\\)
          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          ", "content": {"html": "
          Step 1 . Read the problem. Make sure all the words and ideas are understood.
          Step 2 . Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3 . Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5 . Simplify the expression.\\(20\\)
          Step 6 . Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

          ", "content": [{"c": "The temperature in Anchorage, Alaska one morning was", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "degrees. By mid-afternoon the temperature had dropped to", "t": "text"}, {"c": "30", "t": "equation-inline"}, {"c": "degrees below zero. What was the difference in the morning and afternoon temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          The difference in temperatures was 45 degrees.

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The difference in temperatures was ", "t": "text"}, {"c": "45", "t": "equation-inline"}, {"c": " degrees.", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The temperature in Denver was −6 degrees at lunchtime. By sunset the temperature had dropped to −15 degrees. What was the difference in the lunchtime and sunset temperatures?

          ", "content": [{"c": "The temperature in Denver was", "t": "text"}, {"c": "−6", "t": "equation-inline"}, {"c": "degrees at lunchtime. By sunset the temperature had dropped to", "t": "text"}, {"c": "−15", "t": "equation-inline"}, {"c": "degrees. What was the difference in the lunchtime and sunset temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          The difference in temperatures was 9 degrees.

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The difference in temperatures was ", "t": "text"}, {"c": "9", "t": "equation-inline"}, {"c": " degrees.", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          1. Read the problem. Make sure all the words and ideas are understood
          2. Identify what we are asked to find.
          3. Write a phrase that gives the information to find it.
          4. Translate the phrase to an expression.
          5. Simplify the expression.
          6. Answer the question with a complete sentence.
          ", "content": {"items": [[[{"c": "Read the problem. Make sure all the words and ideas are understood", "t": "text"}]], [[{"c": "Identify what we are asked to find.", "t": "text"}]], [[{"c": "Write a phrase that gives the information to find it.", "t": "text"}]], [[{"c": "Translate the phrase to an expression.", "t": "text"}]], [[{"c": "Simplify the expression.", "t": "text"}]], [[{"c": "Answer the question with a complete sentence.", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

          ", "content": [{"c": "The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          Step 1. Read the problem. Make sure all the words and ideas are understood.
          Step 2. Identify what we are asked to find.the number of yards lost
          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5. Simplify the expression.\\(-45\\)
          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          ", "content": {"html": "
          Step 1 . Read the problem. Make sure all the words and ideas are understood.
          Step 2 . Identify what we are asked to find.the number of yards lost
          Step 3 . Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5 . Simplify the expression.\\(-45\\)
          Step 6 . Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

          ", "content": [{"c": "The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "yards. What is the number of yards lost due to penalties?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          The Bears lost 105 yards.

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "The Bears lost ", "t": "text"}, {"c": "105", "t": "equation-inline"}, {"c": " yards.", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

          ", "content": [{"c": "Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          A $16 fee was deducted from his checking account.

          ", "content": {"items": [[[{"c": "Answer", "t": "text"}]], [[{"c": "A $16 fee was deducted from his checking account.", "t": "text"}]]], "ordered": true, "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Key Concepts

          ", "content": {"title_content": "Key Concepts", "level": "2"}}, {"type": "list", "raw_content": "", "content": {"items": [[[{"c": "Multiplication and Division of Two Signed Numbers", "t": "text"}, {"c": "Same signs—Product is positive", "t": "text"}, {"c": "Different signs—Product is negative", "t": "text"}]], [[{"c": "Strategy for Applications", "t": "text"}, {"c": "Identify what you are asked to find.", "t": "text"}, {"c": "Write a phrase that gives the information to find it.", "t": "text"}, {"c": "Translate the phrase to an expression.", "t": "text"}, {"c": "Simplify the expression.", "t": "text"}, {"c": "Answer the question with a complete sentence.", "t": "text"}]]], "ordered": false, "list_nest_level": "2"}}]], "html": "\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n \n \n \n\n 1.5: Multiply and Divide Integers - Mathematics LibreTexts\n\n\n \n\n \n\n\n \n\n\n \n \n\n\n \n\n\n\n\n\n\nSkip to main content
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          1: Foundations
          MTH 098 Elementary Algebra
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          Mon, 06 Jan 2020 03:19:01 GMT
          1.5: Multiply and Divide Integers
          30345
          30345
          Paul Seeburger
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          \n \"Mathematics\n
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          \n\n\n

          \n 1.5: Multiply and Divide Integers\n

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            Last updated
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          2. \n Save as PDF\n
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        • Page ID
          30345
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          Learning Objectives
          \n\n

          By the end of this section, you will be able to:

          \n\n
            \n
          • Multiply integers
            • \n
          • Divide integers
            • \n
          • Simplify expressions with integers
            • \n
          • Evaluate variable expressions with integers
            • \n
          • Translate English phrases to algebraic expressions
            • \n
          • Use integers in applications
            • \n
          \n
          \n\n

          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

          \n\n

          Multiply Integers

          \n\n

          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

          \n\n

          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

          \n\n
          \"Two\n
          Figure \\(\\PageIndex{1}\\)
          \n
          \n\n

          The next two examples are more interesting.

          \n\n

          What does it mean to multiply \\(5\\) by \\(−3\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as “taking away,” it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

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          \"This\n
          Figure \\(\\PageIndex{2}\\)
          \n
          \n\n

          In summary:

          \n\n

          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

          \n\n

          Notice that for multiplication of two signed numbers, when the:

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          • signs are the same, the product is positive.
            • \n
          • signs are different, the product is negative.
            • \n
          \n\n

          We’ll put this all together in the chart below.

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          MULTIPLICATION OF SIGNED NUMBERS
          \n\n

          For multiplication of two signed numbers:

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Same signsProductExample
          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
          Table \\(\\PageIndex{1}\\)
          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
          Table \\(\\PageIndex{2}\\)
          \n
          \n\n
          \n
          Example \\(\\PageIndex{1}\\)
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-9\\cdot 3\\)
            2. \n
          2. \\(-2(-5)\\)
            3. \n
          3. \\(4(-8)\\)
            4. \n
          4. \\(7\\cdot 6\\)
            5. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
            3. \n
          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
            4. \n
          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
            5. \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{2}\\)
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-6\\cdot 8\\)
            2. \n
          2. \\(-4(-7)\\)
            3. \n
          3. \\(9(-7)\\)
            4. \n
          4. \\(5\\cdot 12\\)
            5. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-48\\)
            2. \n
          2. \\(28\\)
            3. \n
          3. \\(-63\\)
            4. \n
          4. \\(60\\)
            5. \n
          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{3}\\)
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-8\\cdot 7\\)
            2. \n
          2. \\(-6(-9)\\)
            3. \n
          3. \\(7(-4)\\)
            4. \n
          4. \\(3\\cdot 13\\)
            5. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-56\\)
            2. \n
          2. \\(54\\)
            3. \n
          3. \\(-28\\)
            4. \n
          4. \\(39\\)
            5. \n
          \n
          \n
          \n
          \n\n

          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(−1\\)? Let’s multiply a positive number and then a negative number by \\(−1\\) to see what we get.

          \n\n

          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
          \nEach time we multiply a number by \\(−1\\), we get its opposite!

          \n\n
          \n
           
          \n\n

          MULTIPLICATION BY −1

          \n\n

          \\[−1a=−a\\]

          \n\n

          Multiplying a number by \\(−1\\) gives its opposite.

          \n
          \n\n
          \n
          Example \\(\\PageIndex{4}\\)
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-1 \\cdot 7\\)
            2. \n
          2. \\(-1(-11)\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{5}\\)
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-1\\cdot 9\\)
            2. \n
          2. \\(-1\\cdot(-17)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-9\\)
            2. \n
          2. \\(17\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{6}\\)
          \n\n

          Multiply:

          \n\n
            \n
          1. \\(-1\\cdot 8\\)
            2. \n
          2. \\(-1\\cdot(-16)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-8\\)
            2. \n
          2. \\(16\\)
            3. \n
          \n
          \n
          \n
          \n\n

          Divide Integers

          \n\n

          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

          \n\n

          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

          \n\n

          Division follows the same rules as multiplication!

          \n\n

          For division of two signed numbers, when the:

          \n\n
            \n
          • signs are the same, the quotient is positive.
            • \n
          • signs are different, the quotient is negative.
            • \n
          \n\n

          And remember that we can always check the answer of a division problem by multiplying.

          \n\n
          \n
          MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
          \n\n

          For multiplication and division of two signed numbers:

          \n\n
            \n
          • If the signs are the same, the result is positive.
            • \n
          • If the signs are different, the result is negative.
            • \n
          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          \n
          \n\n
          \n
          Example \\(\\PageIndex{7}\\)
          \n\n
            \n
          1. \\(-27\\div 3\\)
            2. \n
          2. \\(-100\\div (-4)\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{8}\\)
          \n\n

          Divide:

          \n\n
            \n
          1. \\(-42\\div 6\\)
            2. \n
          2. \\(-117\\div (-3)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-7\\)
            2. \n
          2. \\(39\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{9}\\)
          \n\n

          Divide:

          \n\n
            \n
          1. \\(-63\\div 7\\)
            2. \n
          2. \\(-115\\div (-5)\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-9\\)
            2. \n
          2. \\(23\\)
            3. \n
          \n
          \n
          \n
          \n\n

          Simplify Expressions with Integers

          \n\n

          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

          \n\n

          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

          \n\n
          \n
          Example \\(\\PageIndex{10}\\)
          \n\n

          Simplify:

          \n\n

          \\(7(-2)+4(-7)-6\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{11}\\)
          \n\n

          Simplify:

          \n\n

          \\(8(-3)+5(-7)-4\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(-63\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{12}\\)
          \n\n

          Simplify:

          \n\n

          \\(9(-3)+7(-8)-1\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(-84\\)

          \n
          \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{13}\\)
          \n\n

          Simplify:

          \n\n
            \n
          1. \\((-2)^{4}\\)
            2. \n
          2. \\(-2^{4}\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
            3. \n
          \n\n

          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((−2)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{14}\\)
          \n\n

          Simplify:

          \n\n
            \n
          1. \\((-3)^{4}\\)
            2. \n
          2. \\(-3^{4}\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(81\\)
            2. \n
          2. \\(-81\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{15}\\)
          \n\n

          Simplify:

          \n\n
            \n
          1. \\((-7)^{2}\\)
            2. \n
          2. \\(-7^{2}\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(49\\)
            2. \n
          2. \\(-49\\)
            3. \n
          \n
          \n
          \n
          \n\n

          The next example reminds us to simplify inside parentheses first.

          \n\n
          \n
          Example \\(\\PageIndex{16}\\)
          \n\n

          Simplify:

          \n\n

          \\(12-3(9 - 12)\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{17}\\)
          \n\n

          Simplify:

          \n\n

          \\(17 - 4(8 - 11)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(29\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{18}\\)
          \n\n

          Simplify:

          \n\n

          \\(16 - 6(7 - 13)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(52\\)

          \n
          \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{19}\\)
          \n\n

          Simplify:

          \n\n

          \\(8(-9)\\div (-2)^{3}\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{20}\\)
          \n\n

          Simplify:

          \n\n

          \\(12(-9)\\div (-3)^{3}\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(4\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{21}\\)
          \n\n

          Simplify:

          \n\n

          \\(18(-4)\\div (-2)^{3}\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(9\\)

          \n
          \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{22}\\)
          \n\n

          Simplify:

          \n\n

          \\(-30\\div 2 + (-3)(-7)\\)

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{23}\\)
          \n\n

          Simplify:

          \n\n

          \\(-27\\div 3 + (-5)(-6)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(21\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{24}\\)
          \n\n

          Simplify:

          \n\n

          \\(-32\\div 4 + (-2)(-7)\\)

          \n\n
          \n
          Answer
          \n
          \n

          \\(6\\)

          \n
          \n
          \n
          \n\n

          Evaluate Variable Expressions with Integers

          \n\n

          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

          \n\n
          \n
          Example \\(\\PageIndex{25}\\)
          \n\n

          When \\(n=−5\\), evaluate:

          \n\n
            \n
          1. \\(n+1\\)
            2. \n
          2. \\(−n+1\\).
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{26}\\)
          \n\n

          When \\(n=−8\\), evaluate:

          \n\n
            \n
          1. \\(n+2\\)
            2. \n
          2. \\(−n+2\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-6\\)
            2. \n
          2. \\(10\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{27}\\)
          \n\n

          When \\(y=−9\\), evaluate:

          \n\n
            \n
          1. \\(y+8\\)
            2. \n
          2. \\(−y+8\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-1\\)
            2. \n
          2. \\(17\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{28}\\)
          \n\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{29}\\)
          \n\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(196\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{30}\\)
          \n\n

          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(8\\)

          \n
          \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{31}\\)
          \n\n

          Evaluate \\(20 -z \\) when

          \n\n
            \n
          1. \\(z = 12\\)
            2. \n
          2. \\(z = -12\\)
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{32}\\)
          \n\n

          Evaluate \\(17 - k\\) when

          \n\n
            \n
          1. \\(k = 19\\)
            2. \n
          2. \\(k = -19\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-2\\)
            2. \n
          2. \\(36\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{33}\\)
          \n\n

          Evaluate \\(-5 - b\\) when

          \n\n
            \n
          1. \\(b = 14\\)
            2. \n
          2. \\(b = -14\\)
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(-19\\)
            2. \n
          2. \\(9\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{34}\\)
          \n\n

          Evaluate:

          \n\n

          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

          \n\n

          Solution

          \n\n

          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

          \n\n

          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{35}\\)
          \n\n

          Evaluate:

          \n\n

          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(39\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{36}\\)
          \n\n

          Evaluate:

          \n\n

          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(13\\)

          \n
          \n
          \n
          \n\n

          Translate Phrases to Expressions with Integers

          \n\n

          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

          \n\n
          \n
          Example \\(\\PageIndex{37}\\)
          \n\n

          Translate and simplify: the sum of \\(8\\) and \\(−12\\), increased by \\(3\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{38}\\)
          \n\n

          Translate and simplify: the sum of \\(9\\) and \\(−16\\), increased by \\(4\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\((9 + (-16)) + 4 - 3\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{39}\\)
          \n\n

          Translate and simplify: the sum of \\(-8\\) and \\(−12\\), increased by \\(7\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\((-8 + (-12)) + 7 - 13\\)

          \n
          \n
          \n
          \n\n

          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n
          \\(a−b\\)
          \\(a\\) minus \\(b\\)
          \n the difference of \\(a\\) and \\(b\\)
          \n \\(b\\) subtracted from \\(a\\)
          \n \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          \n\n

          Be careful to get a and b in the right order!

          \n\n
          \n
          Example \\(\\PageIndex{40}\\)
          \n\n

          Translate and then simplify

          \n\n
            \n
          1. the difference of \\(13\\) and \\(−21\\)
            2. \n
          2. subtract \\(24\\) from \\(−19\\).
            3. \n
          \n\n

          Solution

          \n\n
            \n
          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
            2. \n
          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
            3. \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{41}\\)
          \n\n

          Translate and simplify

          \n\n
            \n
          1. the difference of \\(14\\) and \\(−23\\)
            2. \n
          2. subtract \\(21\\) from \\(−17\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(14 - (-23); 37\\)
            2. \n
          2. \\(-17 - 21; -38\\)
            3. \n
          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{42}\\)
          \n\n

          Translate and simplify

          \n\n
            \n
          1. the difference of \\(11\\) and \\(−19\\)
            2. \n
          2. subtract \\(18\\) from \\(−11\\).
            3. \n
          \n\n
          \n
          Answer
          \n
          \n
            \n
          1. \\(11 - (-19); 30\\)
            2. \n
          2. \\(-11 - 18; -29\\)
            3. \n
          \n
          \n
          \n
          \n\n

          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

          \n\n
          \n
          Example \\(\\PageIndex{43}\\)
          \n\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−2\\) and \\(14\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{44}\\)
          \n\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−5\\) and \\(12\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-5(12); -60\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{45}\\)
          \n\n

          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-8(13); -104\\)

          \n
          \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{46}\\)
          \n\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−56\\) and \\(−7\\).

          \n\n

          Solution

          \n\n

          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

          \n
          \n\n
          \n
          Try It \\(\\PageIndex{47}\\)
          \n\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−63\\) and \\(−9\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-63\\div (-9); 7\\)

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{48}\\)
          \n\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−72\\) and \\(−9\\).

          \n\n
          \n
          Answer
          \n
          \n

          \\(-72\\div (-9); 8\\)

          \n
          \n
          \n
          \n\n

          Use Integers in Applications

          \n\n

          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

          \n\n

          How to Apply a Strategy to Solve Applications with Integers

          \n\n
          \n
          Example \\(\\PageIndex{49}\\)
          \n\n

          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(−9\\) degrees. What was the difference of the morning and afternoon temperatures?

          \n\n

          Solution

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5. Simplify the expression.\\(20\\)
          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{50}\\)
          \n\n

          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

          \n\n
          \n
          Answer
          \n
          \n

          The difference in temperatures was \\(45\\) degrees.

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{51}\\)
          \n\n

          The temperature in Denver was \\(−6\\) degrees at lunchtime. By sunset the temperature had dropped to \\(−15\\) degrees. What was the difference in the lunchtime and sunset temperatures?

          \n\n
          \n
          Answer
          \n
          \n

          The difference in temperatures was \\(9\\) degrees.

          \n
          \n
          \n
          \n\n
          \n
          APPLY A STRATEGY TO SOLVE APPLICATIONS WITH INTEGERS.
          \n\n
            \n
          1. Read the problem. Make sure all the words and ideas are understood
            2. \n
          2. Identify what we are asked to find.
            3. \n
          3. Write a phrase that gives the information to find it.
            4. \n
          4. Translate the phrase to an expression.
            5. \n
          5. Simplify the expression.
            6. \n
          6. Answer the question with a complete sentence.
            7. \n
          \n
          \n\n
          \n
          Example \\(\\PageIndex{52}\\)
          \n\n

          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

          \n\n

          Solution

          \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the number of yards lost
          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5. Simplify the expression.\\(-45\\)
          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{53}\\)
          \n\n

          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

          \n\n
          \n
          Answer
          \n
          \n

          The Bears lost \\(105\\) yards.

          \n
          \n
          \n
          \n\n
          \n
          Try It \\(\\PageIndex{54}\\)
          \n\n

          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

          \n\n
          \n
          Answer
          \n
          \n

          A $16 fee was deducted from his checking account.

          \n
          \n
          \n
          \n\n

          Key Concepts

          \n\n
            \n
          • Multiplication and Division of Two Signed Numbers\n\n
              \n
            • Same signs—Product is positive
              • \n
            • Different signs—Product is negative
              • \n
            \n
            • \n
          • Strategy for Applications\n
              \n
            1. Identify what you are asked to find.
              2. \n
            2. Write a phrase that gives the information to find it.
              3. \n
            3. Translate the phrase to an expression.
              4. \n
            4. Simplify the expression.
              5. \n
            5. Answer the question with a complete sentence.
              6. \n
            \n
            • \n
          \n
          \n\t\t\t\t
          \n\n\n\n\n
          \n\n\n\n\n\n\n\n\n

          1.5: Multiply and Divide Integers is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.

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          \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "statics": {"title": 8, "list": 65, "list.text": 128, "paragraph": 205, "paragraph.equation-inline": 175, "paragraph.text": 174, "image": 2, "equation-interline": 29, "simple_table": 5, "complex_table": 2, "complex_table.complex": 2}} +{"url": "https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_pseeburger/MTH_098_Elementary_Algebra/1%3A_Foundations/1.5%3A_Multiply_and_Divide_Integers", "content": "# 1.5: Multiply and Divide Integers\n\n1. \nLast updated\n\nPage ID\n30345\n\nBy the end of this section, you will be able to:\n\n- Multiply integers\n- Divide integers\n- Simplify expressions with integers\n- Evaluate variable expressions with integers\n- Translate English phrases to algebraic expressions\n- Use integers in applications\n\nA more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.\n\n## Multiply Integers\n\nSince multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.\n\nWe remember that $a\\cdot b$ means add $a,\\, b$ times. Here, we are using the model just to help us discover the pattern.\n\nThe next two examples are more interesting.\n\nWhat does it mean to multiply $5$ by $−3$ ? It means subtract $5, 3$ times. Looking at subtraction as “taking away,” it means to take away $5, 3$ times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away $5$ three times.\n\nIn summary:\n\n$$\n\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\n$$\n\nNotice that for multiplication of two signed numbers, when the:\n\n- signs are the same , the product is positive .\n- signs are different , the product is negative .\n\nWe’ll put this all together in the chart below.\n\nFor multiplication of two signed numbers:\n\n| Same signs | Product | Example |\n|---|---|---|\n| Two positives | Positive | \\(7\\cdot 4 = 28\\) |\n| Two negatives | Positive | \\(-8(-6) = 48\\) |\n\n| Different signs | Product | Example |\n|---|---|---|\n| Positives \\(\\cdot\\) negative | Negative | \\(7(-9) = -63\\) |\n| Negative \\(\\cdot\\) positives | Negative | \\(-5\\cdot 10= -50\\) |\n\nMultiply:\n\n1. $-9\\cdot 3$\n2. $-2(-5)$\n3. $4(-8)$\n4. $7\\cdot 6$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\n$$\n\nMultiply:\n\n1. $-6\\cdot 8$\n2. $-4(-7)$\n3. $9(-7)$\n4. $5\\cdot 12$\n\nAnswer\n\n1. $-48$\n2. $28$\n3. $-63$\n4. $60$\n\nMultiply:\n\n1. $-8\\cdot 7$\n2. $-6(-9)$\n3. $7(-4)$\n4. $3\\cdot 13$\n\nAnswer\n\n1. $-56$\n2. $54$\n3. $-28$\n4. $39$\n\nWhen we multiply a number by $1$ , the result is the same number. What happens when we multiply a number by $−1$ ? Let’s multiply a positive number and then a negative number by $−1$ to see what we get.\n\n$$\n\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\n$$\n\nEach time we multiply a number by $−1$ , we get its opposite!\n\nMULTIPLICATION BY −1\n\n$$\n−1a=−a\n$$\n\nMultiplying a number by $−1$ gives its opposite.\n\nMultiply:\n\n1. $-1 \\cdot 7$\n2. $-1(-11)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\n$$\n\nMultiply:\n\n1. $-1\\cdot 9$\n2. $-1\\cdot(-17)$\n\nAnswer\n\n1. $-9$\n2. $17$\n\nMultiply:\n\n1. $-1\\cdot 8$\n2. $-1\\cdot(-16)$\n\nAnswer\n\n1. $-8$\n2. $16$\n\n## Divide Integers\n\nWhat about division? Division is the inverse operation of multiplication. So, $15\\div 3=5$ because $5 \\cdot 3 = 15$ . In words, this expression says that $15$ can be divided into three groups of five each because adding five three times gives $15$ . Look at some examples of multiplying integers, to figure out the rules for dividing integers.\n\n$$\n\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\n$$\n\nDivision follows the same rules as multiplication!\n\nFor division of two signed numbers, when the:\n\n- signs are the same , the quotient is positive .\n- signs are different , the quotient is negative .\n\nAnd remember that we can always check the answer of a division problem by multiplying.\n\nFor multiplication and division of two signed numbers:\n\n- If the signs are the same, the result is positive.\n- If the signs are different, the result is negative.\n\n
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          \n\n
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          \n\n1. $-27\\div 3$\n2. $-100\\div (-4)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\n$$\n\nDivide:\n\n1. $-42\\div 6$\n2. $-117\\div (-3)$\n\nAnswer\n\n1. $-7$\n2. $39$\n\nDivide:\n\n1. $-63\\div 7$\n2. $-115\\div (-5)$\n\nAnswer\n\n1. $-9$\n2. $23$\n\n## Simplify Expressions with Integers\n\nWhat happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?\n\nLet’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.\n\nSimplify:\n\n$7(-2)+4(-7)-6$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\n$$\n\nSimplify:\n\n$8(-3)+5(-7)-4$\n\nAnswer\n$-63$\n\nSimplify:\n\n$9(-3)+7(-8)-1$\n\nAnswer\n$-84$\n\nSimplify:\n\n1. $(-2)^{4}$\n2. $-2^{4}$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\n$$\n\nNotice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the $(−2)$ to the $4^{th}$ power. In part (2), the exponent means to raise just the $2$ to the $4^{th}$ power and then take the opposite.\n\nSimplify:\n\n1. $(-3)^{4}$\n2. $-3^{4}$\n\nAnswer\n\n1. $81$\n2. $-81$\n\nSimplify:\n\n1. $(-7)^{2}$\n2. $-7^{2}$\n\nAnswer\n\n1. $49$\n2. $-49$\n\nThe next example reminds us to simplify inside parentheses first.\n\nSimplify:\n\n$12-3(9 - 12)$\n\nSolution\n\n$$\n\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\n$$\n\nSimplify:\n\n$17 - 4(8 - 11)$\n\nAnswer\n$29$\n\nSimplify:\n\n$16 - 6(7 - 13)$\n\nAnswer\n$52$\n\nSimplify:\n\n$8(-9)\\div (-2)^{3}$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\n$$\n\nSimplify:\n\n$12(-9)\\div (-3)^{3}$\n\nAnswer\n$4$\n\nSimplify:\n\n$18(-4)\\div (-2)^{3}$\n\nAnswer\n$9$\n\nSimplify:\n\n$-30\\div 2 + (-3)(-7)$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\n$$\n\nSimplify:\n\n$-27\\div 3 + (-5)(-6)$\n\nAnswer\n$21$\n\nSimplify:\n\n$-32\\div 4 + (-2)(-7)$\n\nAnswer\n$6$\n\n## Evaluate Variable Expressions with Integers\n\nRemember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.\n\nWhen $n=−5$ , evaluate:\n\n1. $n+1$\n2. $−n+1$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\n$$\n\nWhen $n=−8$ , evaluate:\n\n1. $n+2$\n2. $−n+2$ .\n\nAnswer\n\n1. $-6$\n2. $10$\n\nWhen $y=−9$ , evaluate:\n\n1. $y+8$\n2. $−y+8$ .\n\nAnswer\n\n1. $-1$\n2. $17$\n\nEvaluate $(x+y)^{2}$ when $x = -18$ and $y = 24$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\n$$\n\nEvaluate $(x+y)^{2}$ when $x = -15$ and $y = 29$ .\n\nAnswer\n$196$\n\nEvaluate $(x+y)^{3}$ when $x = -8$ and $y = 10$ .\n\nAnswer\n$8$\n\nEvaluate $20 -z$ when\n\n1. $z = 12$\n2. $z = -12$\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\n$$\n\nEvaluate $17 - k$ when\n\n1. $k = 19$\n2. $k = -19$\n\nAnswer\n\n1. $-2$\n2. $36$\n\nEvaluate $-5 - b$ when\n\n1. $b = 14$\n2. $b = -14$\n\nAnswer\n\n1. $-19$\n2. $9$\n\nEvaluate:\n\n$2x^{2} + 3x + 8$ when $x = 4$ .\n\nSolution\n\nSubstitute $4$ for $x$ . Use parentheses to show multiplication.\n\n$$\n\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\n$$\n\nEvaluate:\n\n$3x^{2} - 2x + 6$ when $x =-3$ .\n\nAnswer\n$39$\n\nEvaluate:\n\n$4x^{2} - x - 5$ when $x = -2$ .\n\nAnswer\n$13$\n\n## Translate Phrases to Expressions with Integers\n\nOur earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.\n\nTranslate and simplify: the sum of $8$ and $−12$ , increased by $3$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\n$$\n\nTranslate and simplify: the sum of $9$ and $−16$ , increased by $4$ .\n\nAnswer\n$(9 + (-16)) + 4 - 3$\n\nTranslate and simplify: the sum of $-8$ and $−12$ , increased by $7$ .\n\nAnswer\n$(-8 + (-12)) + 7 - 13$\n\nWhen we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.\n\n| \\(a−b\\) |\n|---|\n| \\(a\\) minus \\(b\\) the difference of \\(a\\) and \\(b\\) \\(b\\) subtracted from \\(a\\) \\(b\\) less than \\(a\\) |\n\nBe careful to get a and b in the right order!\n\nTranslate and then simplify\n\n1. the difference of $13$ and $−21$\n2. subtract $24$ from $−19$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\n$$\n\n$$\n\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\n$$\n\nTranslate and simplify\n\n1. the difference of $14$ and $−23$\n2. subtract $21$ from $−17$ .\n\nAnswer\n\n1. $14 - (-23); 37$\n2. $-17 - 21; -38$\n\nTranslate and simplify\n\n1. the difference of $11$ and $−19$\n2. subtract $18$ from $−11$ .\n\nAnswer\n\n1. $11 - (-19); 30$\n2. $-11 - 18; -29$\n\nOnce again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”\n\nTranslate to an algebraic expression and simplify if possible: the product of $−2$ and $14$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the product of $−5$ and $12$ .\n\nAnswer\n$-5(12); -60$\n\nTranslate to an algebraic expression and simplify if possible: the product of $8$ and $-13$ .\n\nAnswer\n$-8(13); -104$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $−56$ and $−7$ .\n\nSolution\n\n$$\n\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\n$$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $−63$ and $−9$ .\n\nAnswer\n$-63\\div (-9); 7$\n\nTranslate to an algebraic expression and simplify if possible: the quotient of $−72$ and $−9$ .\n\nAnswer\n$-72\\div (-9); 8$\n\n## Use Integers in Applications\n\nWe’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.\n\nHow to Apply a Strategy to Solve Applications with Integers\n\nThe temperature in Urbana, Illinois one morning was $11$ degrees. By mid-afternoon, the temperature had dropped to $−9$ degrees. What was the difference of the morning and afternoon temperatures?\n\nSolution\n\n| Step 1 . Read the problem. Make sure all the words and ideas are understood. | |\n|---|---|\n| Step 2 . Identify what we are asked to find. | the difference of the morning and afternoon temperatures |\n| Step 3 . Write a phrase that gives the information to find it. | the difference of \\(11\\) and \\(-9\\) |\n| Step 4. Translate the phrase to an expression. | \\(11 - (-9)\\) |\n| Step 5 . Simplify the expression. | \\(20\\) |\n| Step 6 . Write a complete sentence that answers the question. | The difference in temperatures was 20 degrees. |\n\nThe temperature in Anchorage, Alaska one morning was $15$ degrees. By mid-afternoon the temperature had dropped to $30$ degrees below zero. What was the difference in the morning and afternoon temperatures?\n\nAnswer\nThe difference in temperatures was $45$ degrees.\n\nThe temperature in Denver was $−6$ degrees at lunchtime. By sunset the temperature had dropped to $−15$ degrees. What was the difference in the lunchtime and sunset temperatures?\n\nAnswer\nThe difference in temperatures was $9$ degrees.\n\n1. Read the problem. Make sure all the words and ideas are understood\n2. Identify what we are asked to find.\n3. Write a phrase that gives the information to find it.\n4. Translate the phrase to an expression.\n5. Simplify the expression.\n6. Answer the question with a complete sentence.\n\nThe Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?\n\nSolution\n\n| Step 1 . Read the problem. Make sure all the words and ideas are understood. | |\n|---|---|\n| Step 2 . Identify what we are asked to find. | the number of yards lost |\n| Step 3 . Write a phrase that gives the information to find it. | three times a \\(15\\)-yard penalty |\n| Step 4. Translate the phrase to an expression. | \\(3(-15)\\) |\n| Step 5 . Simplify the expression. | \\(-45\\) |\n| Step 6 . Write a complete sentence that answers the question. | The team lost \\(45\\) yards. |\n\nThe Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of $15$ yards. What is the number of yards lost due to penalties?\n\nAnswer\nThe Bears lost $105$ yards.\n\nBill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a \\$2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?\n\nAnswer\nA $16 fee was deducted from his checking account.\n\n## Key Concepts\n\n- Multiplication and Division of Two Signed Numbers\n - Same signs—Product is positive\n - Different signs—Product is negative\n- Strategy for Applications\n 1. Identify what you are asked to find.\n 2. Write a phrase that gives the information to find it.\n 3. Translate the phrase to an expression.\n 4. Simplify the expression.\n 5. Answer the question with a complete sentence.\n", "main_html": "
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          By the end of this section, you will be able to:

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          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

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          Multiply Integers

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          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

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          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

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          The next two examples are more interesting.

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          What does it mean to multiply \\(5\\) by \\(−3\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as “taking away,” it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

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          In summary:

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          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

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          Notice that for multiplication of two signed numbers, when the:

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          We’ll put this all together in the chart below.

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          For multiplication of two signed numbers:

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          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
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          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
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          Multiply:

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          Solution

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          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
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          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
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          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
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          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
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          Multiply:

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          1. \\(-6\\cdot 8\\)
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          2. \\(-4(-7)\\)
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          3. \\(9(-7)\\)
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          1. \\(-48\\)
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          2. \\(28\\)
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          3. \\(-63\\)
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          Multiply:

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          1. \\(-8\\cdot 7\\)
            2. \r\n
          2. \\(-6(-9)\\)
            3. \r\n
          3. \\(7(-4)\\)
            4. \r\n
          4. \\(3\\cdot 13\\)
            5. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-56\\)
            2. \r\n
          2. \\(54\\)
            3. \r\n
          3. \\(-28\\)
            4. \r\n
          4. \\(39\\)
            5. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(−1\\)? Let’s multiply a positive number and then a negative number by \\(−1\\) to see what we get.

          \r\n\r\n

          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
          \r\nEach time we multiply a number by \\(−1\\), we get its opposite!

          \r\n\r\n
          \r\n
           
          \r\n\r\n

          MULTIPLICATION BY −1

          \r\n\r\n

          \\[−1a=−a\\]

          \r\n\r\n

          Multiplying a number by \\(−1\\) gives its opposite.

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-1 \\cdot 7\\)
            2. \r\n
          2. \\(-1(-11)\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-1\\cdot 9\\)
            2. \r\n
          2. \\(-1\\cdot(-17)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-9\\)
            2. \r\n
          2. \\(17\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-1\\cdot 8\\)
            2. \r\n
          2. \\(-1\\cdot(-16)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-8\\)
            2. \r\n
          2. \\(16\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Divide Integers

          \r\n\r\n

          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

          \r\n\r\n

          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

          \r\n\r\n

          Division follows the same rules as multiplication!

          \r\n\r\n

          For division of two signed numbers, when the:

          \r\n\r\n
            \r\n
          • signs are the same, the quotient is positive.
            • \r\n
          • signs are different, the quotient is negative.
            • \r\n
          \r\n\r\n

          And remember that we can always check the answer of a division problem by multiplying.

          \r\n\r\n
          \r\n
          \r\n\r\n

          For multiplication and division of two signed numbers:

          \r\n\r\n
            \r\n
          • If the signs are the same, the result is positive.
            • \r\n
          • If the signs are different, the result is negative.
            • \r\n
          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n
            \r\n
          1. \\(-27\\div 3\\)
            2. \r\n
          2. \\(-100\\div (-4)\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Divide:

          \r\n\r\n
            \r\n
          1. \\(-42\\div 6\\)
            2. \r\n
          2. \\(-117\\div (-3)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-7\\)
            2. \r\n
          2. \\(39\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Divide:

          \r\n\r\n
            \r\n
          1. \\(-63\\div 7\\)
            2. \r\n
          2. \\(-115\\div (-5)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-9\\)
            2. \r\n
          2. \\(23\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Simplify Expressions with Integers

          \r\n\r\n

          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

          \r\n\r\n

          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(7(-2)+4(-7)-6\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(8(-3)+5(-7)-4\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-63\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(9(-3)+7(-8)-1\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-84\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n
            \r\n
          1. \\((-2)^{4}\\)
            2. \r\n
          2. \\(-2^{4}\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
            3. \r\n
          \r\n\r\n

          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((−2)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n
            \r\n
          1. \\((-3)^{4}\\)
            2. \r\n
          2. \\(-3^{4}\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(81\\)
            2. \r\n
          2. \\(-81\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n
            \r\n
          1. \\((-7)^{2}\\)
            2. \r\n
          2. \\(-7^{2}\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(49\\)
            2. \r\n
          2. \\(-49\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          The next example reminds us to simplify inside parentheses first.

          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(12-3(9 - 12)\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(17 - 4(8 - 11)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(29\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(16 - 6(7 - 13)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(52\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(8(-9)\\div (-2)^{3}\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(12(-9)\\div (-3)^{3}\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(4\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(18(-4)\\div (-2)^{3}\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(9\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(-30\\div 2 + (-3)(-7)\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(-27\\div 3 + (-5)(-6)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(21\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(-32\\div 4 + (-2)(-7)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(6\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Evaluate Variable Expressions with Integers

          \r\n\r\n

          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

          \r\n\r\n
          \r\n
          \r\n\r\n

          When \\(n=−5\\), evaluate:

          \r\n\r\n
            \r\n
          1. \\(n+1\\)
            2. \r\n
          2. \\(−n+1\\).
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          When \\(n=−8\\), evaluate:

          \r\n\r\n
            \r\n
          1. \\(n+2\\)
            2. \r\n
          2. \\(−n+2\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-6\\)
            2. \r\n
          2. \\(10\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          When \\(y=−9\\), evaluate:

          \r\n\r\n
            \r\n
          1. \\(y+8\\)
            2. \r\n
          2. \\(−y+8\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-1\\)
            2. \r\n
          2. \\(17\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(196\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(8\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate \\(20 -z \\) when

          \r\n\r\n
            \r\n
          1. \\(z = 12\\)
            2. \r\n
          2. \\(z = -12\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate \\(17 - k\\) when

          \r\n\r\n
            \r\n
          1. \\(k = 19\\)
            2. \r\n
          2. \\(k = -19\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-2\\)
            2. \r\n
          2. \\(36\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate \\(-5 - b\\) when

          \r\n\r\n
            \r\n
          1. \\(b = 14\\)
            2. \r\n
          2. \\(b = -14\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-19\\)
            2. \r\n
          2. \\(9\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate:

          \r\n\r\n

          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate:

          \r\n\r\n

          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(39\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Evaluate:

          \r\n\r\n

          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(13\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Translate Phrases to Expressions with Integers

          \r\n\r\n

          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate and simplify: the sum of \\(8\\) and \\(−12\\), increased by \\(3\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate and simplify: the sum of \\(9\\) and \\(−16\\), increased by \\(4\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\((9 + (-16)) + 4 - 3\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate and simplify: the sum of \\(-8\\) and \\(−12\\), increased by \\(7\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\((-8 + (-12)) + 7 - 13\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          \\(a−b\\)
          \\(a\\) minus \\(b\\)
          \r\n the difference of \\(a\\) and \\(b\\)
          \r\n \\(b\\) subtracted from \\(a\\)
          \r\n \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          \r\n\r\n

          Be careful to get a and b in the right order!

          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate and then simplify

          \r\n\r\n
            \r\n
          1. the difference of \\(13\\) and \\(−21\\)
            2. \r\n
          2. subtract \\(24\\) from \\(−19\\).
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate and simplify

          \r\n\r\n
            \r\n
          1. the difference of \\(14\\) and \\(−23\\)
            2. \r\n
          2. subtract \\(21\\) from \\(−17\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(14 - (-23); 37\\)
            2. \r\n
          2. \\(-17 - 21; -38\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate and simplify

          \r\n\r\n
            \r\n
          1. the difference of \\(11\\) and \\(−19\\)
            2. \r\n
          2. subtract \\(18\\) from \\(−11\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(11 - (-19); 30\\)
            2. \r\n
          2. \\(-11 - 18; -29\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−2\\) and \\(14\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−5\\) and \\(12\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-5(12); -60\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-8(13); -104\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−56\\) and \\(−7\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−63\\) and \\(−9\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-63\\div (-9); 7\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−72\\) and \\(−9\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-72\\div (-9); 8\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Use Integers in Applications

          \r\n\r\n

          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

          \r\n\r\n

          How to Apply a Strategy to Solve Applications with Integers

          \r\n\r\n
          \r\n
          \r\n\r\n

          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(−9\\) degrees. What was the difference of the morning and afternoon temperatures?

          \r\n\r\n

          Solution

          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5. Simplify the expression.\\(20\\)
          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          The difference in temperatures was \\(45\\) degrees.

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          The temperature in Denver was \\(−6\\) degrees at lunchtime. By sunset the temperature had dropped to \\(−15\\) degrees. What was the difference in the lunchtime and sunset temperatures?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          The difference in temperatures was \\(9\\) degrees.

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n
            \r\n
          1. Read the problem. Make sure all the words and ideas are understood
            2. \r\n
          2. Identify what we are asked to find.
            3. \r\n
          3. Write a phrase that gives the information to find it.
            4. \r\n
          4. Translate the phrase to an expression.
            5. \r\n
          5. Simplify the expression.
            6. \r\n
          6. Answer the question with a complete sentence.
            7. \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

          \r\n\r\n

          Solution

          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the number of yards lost
          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5. Simplify the expression.\\(-45\\)
          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          The Bears lost \\(105\\) yards.

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          \r\n\r\n

          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          A $16 fee was deducted from his checking account.

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Key Concepts

          \r\n\r\n
            \r\n
          • Multiplication and Division of Two Signed Numbers\r\n\r\n
              \r\n
            • Same signs—Product is positive
              • \r\n
            • Different signs—Product is negative
              • \r\n
            \r\n
            • \r\n
          • Strategy for Applications\r\n
              \r\n
            1. Identify what you are asked to find.
              2. \r\n
            2. Write a phrase that gives the information to find it.
              3. \r\n
            3. Translate the phrase to an expression.
              4. \r\n
            4. Simplify the expression.
              5. \r\n
            5. Answer the question with a complete sentence.
              6. \r\n
            \r\n
            • \r\n
          \r\n
          \r\n\t\t\t\t
          \r\n\r\n\r\n\r\n\r\n \r\n
          \r\n\r\n\r\n \r\n\r\n
          \r\n \r\n", "content_list": [[{"type": "title", "raw_content": "

          \n 1.5: Multiply and Divide Integers\n

          ", "content": {"title_content": "1.5: Multiply and Divide Integers", "level": "1"}}, {"type": "list", "raw_content": "
          1. Last updated
          ", "content": {"items": [{"child_list": {"list_attribute": "definition", "items": [{"c": "Last updated"}]}}], "list_attribute": "ordered", "list_nest_level": "2"}}, {"type": "list", "raw_content": "
          Page ID
          30345
          ", "content": {"items": [{"c": "Page ID"}, {"c": "30345"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          By the end of this section, you will be able to:

          ", "content": [{"c": "By the end of this section, you will be able to:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [{"c": "Multiply integers"}, {"c": "Divide integers"}, {"c": "Simplify expressions with integers"}, {"c": "Evaluate variable expressions with integers"}, {"c": "Translate English phrases to algebraic expressions"}, {"c": "Use integers in applications"}], "list_attribute": "unordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

          ", "content": [{"c": "A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.", "t": "text"}]}, {"type": "title", "raw_content": "

          Multiply Integers

          ", "content": {"title_content": "Multiply Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

          ", "content": [{"c": "Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          We remember that a\\cdot b means add a,\\, b times. Here, we are using the model just to help us discover the pattern.

          ", "content": [{"c": "We remember that", "t": "text"}, {"c": "a\\cdot b", "t": "equation-inline"}, {"c": "means add", "t": "text"}, {"c": "a,\\, b", "t": "equation-inline"}, {"c": "times. Here, we are using the model just to help us discover the pattern.", "t": "text"}]}, {"type": "image", "raw_content": "
          \"Two
          Figure \\PageIndex{1}
          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17395/CNX_ElemAlg_Figure_01_04_001_img_new.jpg?revision=1", "data": null, "alt": "Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

          The next two examples are more interesting.

          ", "content": [{"c": "The next two examples are more interesting.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          What does it mean to multiply 5 by −3? It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

          ", "content": [{"c": "What does it mean to multiply", "t": "text"}, {"c": "5", "t": "equation-inline"}, {"c": "by", "t": "text"}, {"c": "−3", "t": "equation-inline"}, {"c": "? It means subtract", "t": "text"}, {"c": "5, 3", "t": "equation-inline"}, {"c": "times. Looking at subtraction as “taking away,” it means to take away", "t": "text"}, {"c": "5, 3", "t": "equation-inline"}, {"c": "times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away", "t": "text"}, {"c": "5", "t": "equation-inline"}, {"c": "three times.", "t": "text"}]}, {"type": "image", "raw_content": "
          \"This
          Figure \\PageIndex{2}
          ", "content": {"url": "https://math.libretexts.org/@api/deki/files/17306/CNX_ElemAlg_Figure_01_04_002_img_new.jpg?revision=1", "data": null, "alt": "This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.", "title": null, "caption": null}}, {"type": "paragraph", "raw_content": "

          In summary:

          ", "content": [{"c": "In summary:", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Notice that for multiplication of two signed numbers, when the:

          ", "content": [{"c": "Notice that for multiplication of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [{"c": "signs are the same , the product is positive ."}, {"c": "signs are different , the product is negative ."}], "list_attribute": "unordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          We’ll put this all together in the chart below.

          ", "content": [{"c": "We’ll put this all together in the chart below.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For multiplication of two signed numbers:

          ", "content": [{"c": "For multiplication of two signed numbers:", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          Same signsProductExample
          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
          Table \\(\\PageIndex{1}\\)
          ", "content": {"html": "
          Same signsProductExample
          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
          Table \\(\\PageIndex{1}\\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "simple_table", "raw_content": "
          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
          Table \\(\\PageIndex{2}\\)
          ", "content": {"html": "
          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
          Table \\(\\PageIndex{2}\\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -9\\cdot 3
          2. -2(-5)
          3. 4(-8)
          4. 7\\cdot 6
          ", "content": {"items": [{"c": "$-9\\cdot 3$"}, {"c": "$-2(-5)$"}, {"c": "$4(-8)$"}, {"c": "$7\\cdot 6$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -6\\cdot 8
          2. -4(-7)
          3. 9(-7)
          4. 5\\cdot 12
          ", "content": {"items": [{"c": "$-6\\cdot 8$"}, {"c": "$-4(-7)$"}, {"c": "$9(-7)$"}, {"c": "$5\\cdot 12$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -48
          2. 28
          3. -63
          4. 60
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-48$"}, {"c": "$28$"}, {"c": "$-63$"}, {"c": "$60$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -8\\cdot 7
          2. -6(-9)
          3. 7(-4)
          4. 3\\cdot 13
          ", "content": {"items": [{"c": "$-8\\cdot 7$"}, {"c": "$-6(-9)$"}, {"c": "$7(-4)$"}, {"c": "$3\\cdot 13$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -56
          2. 54
          3. -28
          4. 39
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-56$"}, {"c": "$54$"}, {"c": "$-28$"}, {"c": "$39$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          When we multiply a number by 1, the result is the same number. What happens when we multiply a number by −1? Let’s multiply a positive number and then a negative number by −1 to see what we get.

          ", "content": [{"c": "When we multiply a number by", "t": "text"}, {"c": "1", "t": "equation-inline"}, {"c": ", the result is the same number. What happens when we multiply a number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": "? Let’s multiply a positive number and then a negative number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": "to see what we get.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "


          \nEach time we multiply a number by −1, we get its opposite!

          ", "content": [{"c": "Each time we multiply a number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": ", we get its opposite!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          MULTIPLICATION BY −1

          ", "content": [{"c": "MULTIPLICATION BY −1", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[−1a=−a\\]

          ", "content": {"math_content": "−1a=−a", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Multiplying a number by −1 gives its opposite.

          ", "content": [{"c": "Multiplying a number by", "t": "text"}, {"c": "−1", "t": "equation-inline"}, {"c": "gives its opposite.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -1 \\cdot 7
          2. -1(-11)
          ", "content": {"items": [{"c": "$-1 \\cdot 7$"}, {"c": "$-1(-11)$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -1\\cdot 9
          2. -1\\cdot(-17)
          ", "content": {"items": [{"c": "$-1\\cdot 9$"}, {"c": "$-1\\cdot(-17)$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -9
          2. 17
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-9$"}, {"c": "$17$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Multiply:

          ", "content": [{"c": "Multiply:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -1\\cdot 8
          2. -1\\cdot(-16)
          ", "content": {"items": [{"c": "$-1\\cdot 8$"}, {"c": "$-1\\cdot(-16)$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -8
          2. 16
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-8$"}, {"c": "$16$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "title", "raw_content": "

          Divide Integers

          ", "content": {"title_content": "Divide Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          What about division? Division is the inverse operation of multiplication. So, 15\\div 3=5 because 5 \\cdot 3 = 15. In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

          ", "content": [{"c": "What about division? Division is the inverse operation of multiplication. So,", "t": "text"}, {"c": "15\\div 3=5", "t": "equation-inline"}, {"c": "because", "t": "text"}, {"c": "5 \\cdot 3 = 15", "t": "equation-inline"}, {"c": ". In words, this expression says that", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "can be divided into three groups of five each because adding five three times gives", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": ". Look at some examples of multiplying integers, to figure out the rules for dividing integers.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Division follows the same rules as multiplication!

          ", "content": [{"c": "Division follows the same rules as multiplication!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For division of two signed numbers, when the:

          ", "content": [{"c": "For division of two signed numbers, when the:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [{"c": "signs are the same , the quotient is positive ."}, {"c": "signs are different , the quotient is negative ."}], "list_attribute": "unordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          And remember that we can always check the answer of a division problem by multiplying.

          ", "content": [{"c": "And remember that we can always check the answer of a division problem by multiplying.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          For multiplication and division of two signed numbers:

          ", "content": [{"c": "For multiplication and division of two signed numbers:", "t": "text"}]}, {"type": "list", "raw_content": "", "content": {"items": [{"c": "If the signs are the same, the result is positive."}, {"c": "If the signs are different, the result is negative."}], "list_attribute": "unordered", "list_nest_level": "1"}}, {"type": "complex_table", "raw_content": "
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          ", "content": {"html": "
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "complex_table", "raw_content": "
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          ", "content": {"html": "
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          ", "is_complex": true, "table_nest_level": "1"}}, {"type": "list", "raw_content": "
          1. -27\\div 3
          2. -100\\div (-4)
          ", "content": {"items": [{"c": "$-27\\div 3$"}, {"c": "$-100\\div (-4)$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Divide:

          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -42\\div 6
          2. -117\\div (-3)
          ", "content": {"items": [{"c": "$-42\\div 6$"}, {"c": "$-117\\div (-3)$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -7
          2. 39
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-7$"}, {"c": "$39$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Divide:

          ", "content": [{"c": "Divide:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. -63\\div 7
          2. -115\\div (-5)
          ", "content": {"items": [{"c": "$-63\\div 7$"}, {"c": "$-115\\div (-5)$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -9
          2. 23
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-9$"}, {"c": "$23$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "title", "raw_content": "

          Simplify Expressions with Integers

          ", "content": {"title_content": "Simplify Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

          ", "content": [{"c": "What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

          ", "content": [{"c": "Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          7(-2)+4(-7)-6

          ", "content": [{"c": "7(-2)+4(-7)-6", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          8(-3)+5(-7)-4

          ", "content": [{"c": "8(-3)+5(-7)-4", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          -63

          ", "content": {"items": [{"c": "Answer"}, {"c": "$-63$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          9(-3)+7(-8)-1

          ", "content": [{"c": "9(-3)+7(-8)-1", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          -84

          ", "content": {"items": [{"c": "Answer"}, {"c": "$-84$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. (-2)^{4}
          2. -2^{4}
          ", "content": {"items": [{"c": "$(-2)^{4}$"}, {"c": "$-2^{4}$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the (−2) to the 4^{th} power. In part (2), the exponent means to raise just the 2 to the 4^{th} power and then take the opposite.

          ", "content": [{"c": "Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the", "t": "text"}, {"c": "(−2)", "t": "equation-inline"}, {"c": "to the", "t": "text"}, {"c": "4^{th}", "t": "equation-inline"}, {"c": "power. In part (2), the exponent means to raise just the", "t": "text"}, {"c": "2", "t": "equation-inline"}, {"c": "to the", "t": "text"}, {"c": "4^{th}", "t": "equation-inline"}, {"c": "power and then take the opposite.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. (-3)^{4}
          2. -3^{4}
          ", "content": {"items": [{"c": "$(-3)^{4}$"}, {"c": "$-3^{4}$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 81
          2. -81
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$81$"}, {"c": "$-81$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. (-7)^{2}
          2. -7^{2}
          ", "content": {"items": [{"c": "$(-7)^{2}$"}, {"c": "$-7^{2}$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 49
          2. -49
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$49$"}, {"c": "$-49$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          The next example reminds us to simplify inside parentheses first.

          ", "content": [{"c": "The next example reminds us to simplify inside parentheses first.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          12-3(9 - 12)

          ", "content": [{"c": "12-3(9 - 12)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          17 - 4(8 - 11)

          ", "content": [{"c": "17 - 4(8 - 11)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          29

          ", "content": {"items": [{"c": "Answer"}, {"c": "$29$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          16 - 6(7 - 13)

          ", "content": [{"c": "16 - 6(7 - 13)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          52

          ", "content": {"items": [{"c": "Answer"}, {"c": "$52$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          8(-9)\\div (-2)^{3}

          ", "content": [{"c": "8(-9)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          12(-9)\\div (-3)^{3}

          ", "content": [{"c": "12(-9)\\div (-3)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          4

          ", "content": {"items": [{"c": "Answer"}, {"c": "$4$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          18(-4)\\div (-2)^{3}

          ", "content": [{"c": "18(-4)\\div (-2)^{3}", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          9

          ", "content": {"items": [{"c": "Answer"}, {"c": "$9$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          -30\\div 2 + (-3)(-7)

          ", "content": [{"c": "-30\\div 2 + (-3)(-7)", "t": "equation-inline"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          -27\\div 3 + (-5)(-6)

          ", "content": [{"c": "-27\\div 3 + (-5)(-6)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          21

          ", "content": {"items": [{"c": "Answer"}, {"c": "$21$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Simplify:

          ", "content": [{"c": "Simplify:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          -32\\div 4 + (-2)(-7)

          ", "content": [{"c": "-32\\div 4 + (-2)(-7)", "t": "equation-inline"}]}, {"type": "list", "raw_content": "
          Answer

          6

          ", "content": {"items": [{"c": "Answer"}, {"c": "$6$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Evaluate Variable Expressions with Integers

          ", "content": {"title_content": "Evaluate Variable Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

          ", "content": [{"c": "Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          When n=−5, evaluate:

          ", "content": [{"c": "When", "t": "text"}, {"c": "n=−5", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. n+1
          2. −n+1.
          ", "content": {"items": [{"c": "$n+1$"}, {"c": "$−n+1$ ."}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          When n=−8, evaluate:

          ", "content": [{"c": "When", "t": "text"}, {"c": "n=−8", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. n+2
          2. −n+2.
          ", "content": {"items": [{"c": "$n+2$"}, {"c": "$−n+2$ ."}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -6
          2. 10
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-6$"}, {"c": "$10$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          When y=−9, evaluate:

          ", "content": [{"c": "When", "t": "text"}, {"c": "y=−9", "t": "equation-inline"}, {"c": ", evaluate:", "t": "text"}]}, {"type": "list", "raw_content": "
          1. y+8
          2. −y+8.
          ", "content": {"items": [{"c": "$y+8$"}, {"c": "$−y+8$ ."}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -1
          2. 17
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-1$"}, {"c": "$17$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Evaluate (x+y)^{2} when x = -18 and y = 24.

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{2}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -18", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 24", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Evaluate (x+y)^{2} when x = -15 and y = 29.

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{2}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -15", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 29", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          196

          ", "content": {"items": [{"c": "Answer"}, {"c": "$196$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Evaluate (x+y)^{3} when x = -8 and y = 10.

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "(x+y)^{3}", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "y = 10", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          8

          ", "content": {"items": [{"c": "Answer"}, {"c": "$8$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Evaluate 20 -z when

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "20 -z ", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
          1. z = 12
          2. z = -12
          ", "content": {"items": [{"c": "$z = 12$"}, {"c": "$z = -12$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Evaluate 17 - k when

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "17 - k", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
          1. k = 19
          2. k = -19
          ", "content": {"items": [{"c": "$k = 19$"}, {"c": "$k = -19$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -2
          2. 36
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-2$"}, {"c": "$36$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Evaluate -5 - b when

          ", "content": [{"c": "Evaluate", "t": "text"}, {"c": "-5 - b", "t": "equation-inline"}, {"c": "when", "t": "text"}]}, {"type": "list", "raw_content": "
          1. b = 14
          2. b = -14
          ", "content": {"items": [{"c": "$b = 14$"}, {"c": "$b = -14$"}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. -19
          2. 9
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$-19$"}, {"c": "$9$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Evaluate:

          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          2x^{2} + 3x + 8 when x = 4.

          ", "content": [{"c": "2x^{2} + 3x + 8", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = 4", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Substitute 4 for x. Use parentheses to show multiplication.

          ", "content": [{"c": "Substitute", "t": "text"}, {"c": "4", "t": "equation-inline"}, {"c": "for", "t": "text"}, {"c": "x", "t": "equation-inline"}, {"c": ". Use parentheses to show multiplication.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Evaluate:

          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          3x^{2} - 2x + 6 when x =-3.

          ", "content": [{"c": "3x^{2} - 2x + 6", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x =-3", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          39

          ", "content": {"items": [{"c": "Answer"}, {"c": "$39$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Evaluate:

          ", "content": [{"c": "Evaluate:", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          4x^{2} - x - 5 when x = -2.

          ", "content": [{"c": "4x^{2} - x - 5", "t": "equation-inline"}, {"c": "when", "t": "text"}, {"c": "x = -2", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          13

          ", "content": {"items": [{"c": "Answer"}, {"c": "$13$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Translate Phrases to Expressions with Integers

          ", "content": {"title_content": "Translate Phrases to Expressions with Integers", "level": "2"}}, {"type": "paragraph", "raw_content": "

          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

          ", "content": [{"c": "Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Translate and simplify: the sum of 8 and −12, increased by 3.

          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−12", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "3", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify: the sum of 9 and −16, increased by 4.

          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "9", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−16", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "4", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          (9 + (-16)) + 4 - 3

          ", "content": {"items": [{"c": "Answer"}, {"c": "$(9 + (-16)) + 4 - 3$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify: the sum of -8 and −12, increased by 7.

          ", "content": [{"c": "Translate and simplify: the sum of", "t": "text"}, {"c": "-8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−12", "t": "equation-inline"}, {"c": ", increased by", "t": "text"}, {"c": "7", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          (-8 + (-12)) + 7 - 13

          ", "content": {"items": [{"c": "Answer"}, {"c": "$(-8 + (-12)) + 7 - 13$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

          ", "content": [{"c": "When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          \\(a−b\\)
          \\(a\\) minus \\(b\\)
          \n the difference of \\(a\\) and \\(b\\)
          \n \\(b\\) subtracted from \\(a\\)
          \n \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          ", "content": {"html": "
          \\(a−b\\)
          \\(a\\) minus \\(b\\) the difference of \\(a\\) and \\(b\\) \\(b\\) subtracted from \\(a\\) \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Be careful to get a and b in the right order!

          ", "content": [{"c": "Be careful to get a and b in the right order!", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Translate and then simplify

          ", "content": [{"c": "Translate and then simplify", "t": "text"}]}, {"type": "list", "raw_content": "
          1. the difference of 13 and −21
          2. subtract 24 from −19.
          ", "content": {"items": [{"c": "the difference of $13$ and $−21$"}, {"c": "subtract $24$ from $−19$ ."}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify

          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
          1. the difference of 14 and −23
          2. subtract 21 from −17.
          ", "content": {"items": [{"c": "the difference of $14$ and $−23$"}, {"c": "subtract $21$ from $−17$ ."}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 14 - (-23); 37
          2. -17 - 21; -38
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$14 - (-23); 37$"}, {"c": "$-17 - 21; -38$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Translate and simplify

          ", "content": [{"c": "Translate and simplify", "t": "text"}]}, {"type": "list", "raw_content": "
          1. the difference of 11 and −19
          2. subtract 18 from −11.
          ", "content": {"items": [{"c": "the difference of $11$ and $−19$"}, {"c": "subtract $18$ from $−11$ ."}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          Answer
          1. 11 - (-19); 30
          2. -11 - 18; -29
          ", "content": {"items": [{"c": "Answer"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "$11 - (-19); 30$"}, {"c": "$-11 - 18; -29$"}]}}], "list_attribute": "definition", "list_nest_level": "2"}}, {"type": "paragraph", "raw_content": "

          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

          ", "content": [{"c": "Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the product of −2 and 14.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "−2", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "14", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the product of −5 and 12.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "−5", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "12", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -5(12); -60

          ", "content": {"items": [{"c": "Answer"}, {"c": "$-5(12); -60$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the product of 8 and -13.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the product of", "t": "text"}, {"c": "8", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "-13", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -8(13); -104

          ", "content": {"items": [{"c": "Answer"}, {"c": "$-8(13); -104$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the quotient of −56 and −7.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "−56", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−7", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

          ", "content": {"math_content": "\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the quotient of −63 and −9.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "−63", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−9", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -63\\div (-9); 7

          ", "content": {"items": [{"c": "Answer"}, {"c": "$-63\\div (-9); 7$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Translate to an algebraic expression and simplify if possible: the quotient of −72 and −9.

          ", "content": [{"c": "Translate to an algebraic expression and simplify if possible: the quotient of", "t": "text"}, {"c": "−72", "t": "equation-inline"}, {"c": "and", "t": "text"}, {"c": "−9", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          -72\\div (-9); 8

          ", "content": {"items": [{"c": "Answer"}, {"c": "$-72\\div (-9); 8$"}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Use Integers in Applications

          ", "content": {"title_content": "Use Integers in Applications", "level": "2"}}, {"type": "paragraph", "raw_content": "

          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

          ", "content": [{"c": "We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          How to Apply a Strategy to Solve Applications with Integers

          ", "content": [{"c": "How to Apply a Strategy to Solve Applications with Integers", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          The temperature in Urbana, Illinois one morning was 11 degrees. By mid-afternoon, the temperature had dropped to −9 degrees. What was the difference of the morning and afternoon temperatures?

          ", "content": [{"c": "The temperature in Urbana, Illinois one morning was", "t": "text"}, {"c": "11", "t": "equation-inline"}, {"c": "degrees. By mid-afternoon, the temperature had dropped to", "t": "text"}, {"c": "−9", "t": "equation-inline"}, {"c": "degrees. What was the difference of the morning and afternoon temperatures?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          Step 1. Read the problem. Make sure all the words and ideas are understood.
          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5. Simplify the expression.\\(20\\)
          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          ", "content": {"html": "
          Step 1 . Read the problem. Make sure all the words and ideas are understood.
          Step 2 . Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3 . Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5 . Simplify the expression.\\(20\\)
          Step 6 . Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

          ", "content": [{"c": "The temperature in Anchorage, Alaska one morning was", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "degrees. By mid-afternoon the temperature had dropped to", "t": "text"}, {"c": "30", "t": "equation-inline"}, {"c": "degrees below zero. What was the difference in the morning and afternoon temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          The difference in temperatures was 45 degrees.

          ", "content": {"items": [{"c": "Answer"}, {"c": "The difference in temperatures was $45$ degrees."}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The temperature in Denver was −6 degrees at lunchtime. By sunset the temperature had dropped to −15 degrees. What was the difference in the lunchtime and sunset temperatures?

          ", "content": [{"c": "The temperature in Denver was", "t": "text"}, {"c": "−6", "t": "equation-inline"}, {"c": "degrees at lunchtime. By sunset the temperature had dropped to", "t": "text"}, {"c": "−15", "t": "equation-inline"}, {"c": "degrees. What was the difference in the lunchtime and sunset temperatures?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          The difference in temperatures was 9 degrees.

          ", "content": {"items": [{"c": "Answer"}, {"c": "The difference in temperatures was $9$ degrees."}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "list", "raw_content": "
          1. Read the problem. Make sure all the words and ideas are understood
          2. Identify what we are asked to find.
          3. Write a phrase that gives the information to find it.
          4. Translate the phrase to an expression.
          5. Simplify the expression.
          6. Answer the question with a complete sentence.
          ", "content": {"items": [{"c": "Read the problem. Make sure all the words and ideas are understood"}, {"c": "Identify what we are asked to find."}, {"c": "Write a phrase that gives the information to find it."}, {"c": "Translate the phrase to an expression."}, {"c": "Simplify the expression."}, {"c": "Answer the question with a complete sentence."}], "list_attribute": "ordered", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

          ", "content": [{"c": "The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?", "t": "text"}]}, {"type": "paragraph", "raw_content": "

          Solution

          ", "content": [{"c": "Solution", "t": "text"}]}, {"type": "simple_table", "raw_content": "
          Step 1. Read the problem. Make sure all the words and ideas are understood.
          Step 2. Identify what we are asked to find.the number of yards lost
          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5. Simplify the expression.\\(-45\\)
          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          ", "content": {"html": "
          Step 1 . Read the problem. Make sure all the words and ideas are understood.
          Step 2 . Identify what we are asked to find.the number of yards lost
          Step 3 . Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5 . Simplify the expression.\\(-45\\)
          Step 6 . Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

          ", "content": [{"c": "The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of", "t": "text"}, {"c": "15", "t": "equation-inline"}, {"c": "yards. What is the number of yards lost due to penalties?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          The Bears lost 105 yards.

          ", "content": {"items": [{"c": "Answer"}, {"c": "The Bears lost $105$ yards."}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "paragraph", "raw_content": "

          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

          ", "content": [{"c": "Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?", "t": "text"}]}, {"type": "list", "raw_content": "
          Answer

          A $16 fee was deducted from his checking account.

          ", "content": {"items": [{"c": "Answer"}, {"c": "A $16 fee was deducted from his checking account."}], "list_attribute": "definition", "list_nest_level": "1"}}, {"type": "title", "raw_content": "

          Key Concepts

          ", "content": {"title_content": "Key Concepts", "level": "2"}}, {"type": "list", "raw_content": "", "content": {"items": [{"child_list": {"list_attribute": "unordered", "items": [{"c": "Same signs—Product is positive"}, {"c": "Different signs—Product is negative"}]}, "c": "Multiplication and Division of Two Signed Numbers"}, {"child_list": {"list_attribute": "ordered", "items": [{"c": "Identify what you are asked to find."}, {"c": "Write a phrase that gives the information to find it."}, {"c": "Translate the phrase to an expression."}, {"c": "Simplify the expression."}, {"c": "Answer the question with a complete sentence."}]}, "c": "Strategy for Applications"}], "list_attribute": "unordered", "list_nest_level": "2"}}]], "html": "\r\n\r\n\r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n\r\n \r\n \r\n \r\n\r\n 1.5: Multiply and Divide Integers - Mathematics LibreTexts\r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n \r\n\r\n\r\n \r\n \r\n\r\n\r\n \r\n\r\n\r\n\r\n\r\n\r\n\r\nSkip to main content
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          1: Foundations
          MTH 098 Elementary Algebra
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          Mon, 06 Jan 2020 03:19:01 GMT
          1.5: Multiply and Divide Integers
          30345
          30345
          Paul Seeburger
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          https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FUnder_Construction%2FPurgatory%2FRemixer_University%2FUsername%253A_pseeburger%2FMTH_098_Elementary_Algebra%2F1%253A_Foundations%2F1.5%253A_Multiply_and_Divide_Integers
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          \r\n 1.5: Multiply and Divide Integers\r\n

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        • Page ID
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          Learning Objectives
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          By the end of this section, you will be able to:

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          • Multiply integers
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          • Divide integers
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          • Simplify expressions with integers
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          • Evaluate variable expressions with integers
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          • Translate English phrases to algebraic expressions
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          • Use integers in applications
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          A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers.

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          Multiply Integers

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          Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

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          We remember that \\(a\\cdot b\\) means add \\(a,\\, b\\) times. Here, we are using the model just to help us discover the pattern.

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          Figure \\(\\PageIndex{1}\\)
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          The next two examples are more interesting.

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          What does it mean to multiply \\(5\\) by \\(−3\\)? It means subtract \\(5, 3\\) times. Looking at subtraction as “taking away,” it means to take away \\(5, 3\\) times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away \\(5\\) three times.

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          In summary:

          \r\n\r\n

          \\[\\begin{array} {ll} {5 \\cdot 3 = 15} &{-5(3) = -15} \\\\ {5(-3) = -15} &{(-5)(-3) = 15} \\end{array}\\]

          \r\n\r\n

          Notice that for multiplication of two signed numbers, when the:

          \r\n\r\n
            \r\n
          • signs are the same, the product is positive.
            • \r\n
          • signs are different, the product is negative.
            • \r\n
          \r\n\r\n

          We’ll put this all together in the chart below.

          \r\n\r\n
          \r\n
          MULTIPLICATION OF SIGNED NUMBERS
          \r\n\r\n

          For multiplication of two signed numbers:

          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Same signsProductExample
          Two positivesPositive\\(7\\cdot 4 = 28\\)
          Two negativesPositive\\(-8(-6) = 48\\)
          Table \\(\\PageIndex{1}\\)
          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Different signsProductExample
          Positives \\(\\cdot\\) negativeNegative\\(7(-9) = -63\\)
          Negative \\(\\cdot\\) positivesNegative\\(-5\\cdot 10= -50\\)
          Table \\(\\PageIndex{2}\\)
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{1}\\)
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-9\\cdot 3\\)
            2. \r\n
          2. \\(-2(-5)\\)
            3. \r\n
          3. \\(4(-8)\\)
            4. \r\n
          4. \\(7\\cdot 6\\)
            5. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{-9\\cdot 3} \\\\ {\\text{Multiply, noting that the signs are different, so the product is negative.}} &{-27} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{-2(-5)} \\\\ {\\text{Multiply, noting that the signs are same, so the product is positive.}} &{10} \\end{array}\\]
            3. \r\n
          3. \\[\\begin{array} {ll} {} &{4(-8)} \\\\ {\\text{Multiply, with different signs.}} &{-32} \\end{array}\\]
            4. \r\n
          4. \\[\\begin{array} {ll} {} &{7\\cdot 6} \\\\ {\\text{Multiply, with different signs.}} &{42} \\end{array}\\]
            5. \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{2}\\)
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-6\\cdot 8\\)
            2. \r\n
          2. \\(-4(-7)\\)
            3. \r\n
          3. \\(9(-7)\\)
            4. \r\n
          4. \\(5\\cdot 12\\)
            5. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-48\\)
            2. \r\n
          2. \\(28\\)
            3. \r\n
          3. \\(-63\\)
            4. \r\n
          4. \\(60\\)
            5. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{3}\\)
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-8\\cdot 7\\)
            2. \r\n
          2. \\(-6(-9)\\)
            3. \r\n
          3. \\(7(-4)\\)
            4. \r\n
          4. \\(3\\cdot 13\\)
            5. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-56\\)
            2. \r\n
          2. \\(54\\)
            3. \r\n
          3. \\(-28\\)
            4. \r\n
          4. \\(39\\)
            5. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          When we multiply a number by \\(1\\), the result is the same number. What happens when we multiply a number by \\(−1\\)? Let’s multiply a positive number and then a negative number by \\(−1\\) to see what we get.

          \r\n\r\n

          \\[\\begin{array} {lll} {} &{-1\\cdot 4} &{-1(-3)}\\\\ {\\text{Multiply.}} &{-4} &{3} \\\\ {} &{-4\\text{ is the opposite of 4.}} &{3\\text{ is the opposite of } -3} \\end{array}\\]
          \r\nEach time we multiply a number by \\(−1\\), we get its opposite!

          \r\n\r\n
          \r\n
           
          \r\n\r\n

          MULTIPLICATION BY −1

          \r\n\r\n

          \\[−1a=−a\\]

          \r\n\r\n

          Multiplying a number by \\(−1\\) gives its opposite.

          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{4}\\)
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-1 \\cdot 7\\)
            2. \r\n
          2. \\(-1(-11)\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{-1\\cdot 7} \\\\ {\\text{Multiply, noting that the signs are different}} &{-7} \\\\ {\\text{so the product is negative.}} &{-7\\text{ is the opposite of 7.}} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{-1(-11)} \\\\ {\\text{Multiply, noting that the signs are different}} &{11} \\\\ {\\text{so the product is positive.}} &{11\\text{ is the opposite of -11.}} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{5}\\)
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-1\\cdot 9\\)
            2. \r\n
          2. \\(-1\\cdot(-17)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-9\\)
            2. \r\n
          2. \\(17\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{6}\\)
          \r\n\r\n

          Multiply:

          \r\n\r\n
            \r\n
          1. \\(-1\\cdot 8\\)
            2. \r\n
          2. \\(-1\\cdot(-16)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-8\\)
            2. \r\n
          2. \\(16\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Divide Integers

          \r\n\r\n

          What about division? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5 \\cdot 3 = 15\\). In words, this expression says that \\(15\\) can be divided into three groups of five each because adding five three times gives \\(15\\). Look at some examples of multiplying integers, to figure out the rules for dividing integers.

          \r\n\r\n

          \\[\\begin{array} {ll} {5\\cdot 3 = 15\\text{ so }15\\div 3 = 5} &{-5(3) = -15\\text{ so }-15\\div 3 = -5} \\\\ {(-5)(-3) = 15\\text{ so }15\\div (-3) = -5} &{5(-3) = -15\\text{ so }-15\\div (-3) = 5} \\end{array}\\]

          \r\n\r\n

          Division follows the same rules as multiplication!

          \r\n\r\n

          For division of two signed numbers, when the:

          \r\n\r\n
            \r\n
          • signs are the same, the quotient is positive.
            • \r\n
          • signs are different, the quotient is negative.
            • \r\n
          \r\n\r\n

          And remember that we can always check the answer of a division problem by multiplying.

          \r\n\r\n
          \r\n
          MULTIPLICATION AND DIVISION OF SIGNED NUMBERS
          \r\n\r\n

          For multiplication and division of two signed numbers:

          \r\n\r\n
            \r\n
          • If the signs are the same, the result is positive.
            • \r\n
          • If the signs are different, the result is negative.
            • \r\n
          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Same signsResult
          Two positivesPositive
          Two negativesPositive
          If the signs are the same, the result is positive.
          Table \\(\\PageIndex{3}\\)
          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Different signsResult
          Positive and negativeNegative
          Negative and positiveNegative
          If the signs are different, the result is negative.
          Table \\(\\PageIndex{4}\\)
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{7}\\)
          \r\n\r\n
            \r\n
          1. \\(-27\\div 3\\)
            2. \r\n
          2. \\(-100\\div (-4)\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{-27 \\div 3} \\\\ {\\text{Divide, with different signs, the quotient is}} &{-9} \\\\ {\\text{negative.}} &{} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{-100 \\div (-4)} \\\\ {\\text{Divide, with signs that are the same the}} &{25} \\\\ {\\text{ quotient is negative.}} &{} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{8}\\)
          \r\n\r\n

          Divide:

          \r\n\r\n
            \r\n
          1. \\(-42\\div 6\\)
            2. \r\n
          2. \\(-117\\div (-3)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-7\\)
            2. \r\n
          2. \\(39\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{9}\\)
          \r\n\r\n

          Divide:

          \r\n\r\n
            \r\n
          1. \\(-63\\div 7\\)
            2. \r\n
          2. \\(-115\\div (-5)\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-9\\)
            2. \r\n
          2. \\(23\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Simplify Expressions with Integers

          \r\n\r\n

          What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

          \r\n\r\n

          Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{10}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(7(-2)+4(-7)-6\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{7(-2)+4(-7)-6} \\\\ {\\text{Multiply first.}} &{-14+(-28)-6} \\\\ {\\text{Add.}} &{-42-6} \\\\{\\text{Subtract}} &{-48} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{11}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(8(-3)+5(-7)-4\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-63\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{12}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(9(-3)+7(-8)-1\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-84\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{13}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n
            \r\n
          1. \\((-2)^{4}\\)
            2. \r\n
          2. \\(-2^{4}\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{(-2)^{4}} \\\\ {\\text{Write in expanded form.}} &{(-2)(-2)(-2)(-2)} \\\\ {\\text{Multiply}} &{4(-2)(-2)} \\\\{\\text{Multiply}} &{-8(-2)} \\\\{\\text{Multiply}} &{16} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{-2^{4}} \\\\ {\\text{Write in expanded form. We are asked to find the opposite of }2^{4}.} &{-(2\\cdot 2\\cdot 2 \\cdot 2)} \\\\ {\\text{Multiply}} &{-(4\\cdot 2\\cdot 2)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Multiply}} &{-16} \\end{array}\\]
            3. \r\n
          \r\n\r\n

          Notice the difference in parts (1) and (2). In part (1), the exponent means to raise what is in the parentheses, the \\((−2)\\) to the \\(4^{th}\\) power. In part (2), the exponent means to raise just the \\(2\\) to the \\(4^{th}\\) power and then take the opposite.

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{14}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n
            \r\n
          1. \\((-3)^{4}\\)
            2. \r\n
          2. \\(-3^{4}\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(81\\)
            2. \r\n
          2. \\(-81\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{15}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n
            \r\n
          1. \\((-7)^{2}\\)
            2. \r\n
          2. \\(-7^{2}\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(49\\)
            2. \r\n
          2. \\(-49\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          The next example reminds us to simplify inside parentheses first.

          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{16}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(12-3(9 - 12)\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {llll} {} &{12-3(9 - 12)} \\\\ {\\text{Subtract parentheses first}} &{12-3(-3)} \\\\ {\\text{Multiply.}} &{12-(-9)} \\\\{\\text{Multiply}} &{-(8\\cdot 2)} \\\\{\\text{Subtract}} &{21} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{17}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(17 - 4(8 - 11)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(29\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{18}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(16 - 6(7 - 13)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(52\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{19}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(8(-9)\\div (-2)^{3}\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{8(-9)\\div(-2)^{3}} \\\\ {\\text{Exponents first}} &{8(-9)\\div(-8)} \\\\ {\\text{Multiply.}} &{-72\\div (-8)} \\\\{\\text{Divide}} &{9} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{20}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(12(-9)\\div (-3)^{3}\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(4\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{21}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(18(-4)\\div (-2)^{3}\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(9\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{22}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(-30\\div 2 + (-3)(-7)\\)

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{-30\\div 2 + (-3)(-7)} \\\\ {\\text{Multiply and divide left to right, so divide first.}} &{-15+(-3)(-7)} \\\\ {\\text{Multiply.}} &{-15+ 21} \\\\{\\text{Add}} &{6} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{23}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(-27\\div 3 + (-5)(-6)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(21\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{24}\\)
          \r\n\r\n

          Simplify:

          \r\n\r\n

          \\(-32\\div 4 + (-2)(-7)\\)

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(6\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Evaluate Variable Expressions with Integers

          \r\n\r\n

          Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{25}\\)
          \r\n\r\n

          When \\(n=−5\\), evaluate:

          \r\n\r\n
            \r\n
          1. \\(n+1\\)
            2. \r\n
          2. \\(−n+1\\).
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{\\color{red}{-5}}+1 \\\\ {\\text{Simplify.}} &{-4} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{n+ 1} \\\\ {\\text{Substitute }{ \\color{red}{-5}}\\text{ for } n} &{- {\\color{red}{(-5)}} +1} \\\\ {\\text{Simplify.}} &{5+1} \\\\{\\text{Add.}} &{6} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{26}\\)
          \r\n\r\n

          When \\(n=−8\\), evaluate:

          \r\n\r\n
            \r\n
          1. \\(n+2\\)
            2. \r\n
          2. \\(−n+2\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-6\\)
            2. \r\n
          2. \\(10\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{27}\\)
          \r\n\r\n

          When \\(y=−9\\), evaluate:

          \r\n\r\n
            \r\n
          1. \\(y+8\\)
            2. \r\n
          2. \\(−y+8\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-1\\)
            2. \r\n
          2. \\(17\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{28}\\)
          \r\n\r\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -18\\) and \\(y = 24\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{(x+y)^{2}} \\\\ {\\text{Substitute }-18\\text{ for }x \\text{ and } 24 \\text{ for } y} &{(-18 + 24)^{2}} \\\\ {\\text{Add inside parentheses}} &{(6)^{2}} \\\\{\\text{Simplify.}} &{36} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{29}\\)
          \r\n\r\n

          Evaluate \\((x+y)^{2}\\) when \\(x = -15\\) and \\(y = 29\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(196\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{30}\\)
          \r\n\r\n

          Evaluate \\((x+y)^{3}\\) when \\(x = -8\\) and \\(y = 10\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(8\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{31}\\)
          \r\n\r\n

          Evaluate \\(20 -z \\) when

          \r\n\r\n
            \r\n
          1. \\(z = 12\\)
            2. \r\n
          2. \\(z = -12\\)
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }12\\text{ for }z.} &{20 - 12} \\\\ {\\text{Subtract}} &{8} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &{20 - z} \\\\ {\\text{Substitute }-12\\text{ for }z.} &{20 - (-12)} \\\\ {\\text{Subtract}} &{32} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{32}\\)
          \r\n\r\n

          Evaluate \\(17 - k\\) when

          \r\n\r\n
            \r\n
          1. \\(k = 19\\)
            2. \r\n
          2. \\(k = -19\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-2\\)
            2. \r\n
          2. \\(36\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{33}\\)
          \r\n\r\n

          Evaluate \\(-5 - b\\) when

          \r\n\r\n
            \r\n
          1. \\(b = 14\\)
            2. \r\n
          2. \\(b = -14\\)
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(-19\\)
            2. \r\n
          2. \\(9\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{34}\\)
          \r\n\r\n

          Evaluate:

          \r\n\r\n

          \\(2x^{2} + 3x + 8\\) when \\(x = 4\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          Substitute \\(4\\) for \\(x\\). Use parentheses to show multiplication.

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{2x^{2} + 3x + 8} \\\\ {\\text{Substitute }} &{2(4)^{2} + 3(4) + 8} \\\\ {\\text{Evaluate exponents.}} &{2(16) + 3(4) + 8} \\\\ {\\text{Multiply.}} &{32 + 12 + 8} \\\\{\\text{Add.}} &{52} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{35}\\)
          \r\n\r\n

          Evaluate:

          \r\n\r\n

          \\(3x^{2} - 2x + 6\\) when \\(x =-3\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(39\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{36}\\)
          \r\n\r\n

          Evaluate:

          \r\n\r\n

          \\(4x^{2} - x - 5\\) when \\(x = -2\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(13\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Translate Phrases to Expressions with Integers

          \r\n\r\n

          Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{37}\\)
          \r\n\r\n

          Translate and simplify: the sum of \\(8\\) and \\(−12\\), increased by \\(3\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{sum} \\text{of 8 and -12, increased by 3}} \\\\ {\\text{Translate.}} &{[8 + (-12)] + 3} \\\\ {\\text{Simplify. Be careful not to confuse the}} &{(-4) + 3} \\\\{\\text{brackets with an absolute value sign.}} \\\\{\\text{Add.}} &{-1} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{38}\\)
          \r\n\r\n

          Translate and simplify: the sum of \\(9\\) and \\(−16\\), increased by \\(4\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\((9 + (-16)) + 4 - 3\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{39}\\)
          \r\n\r\n

          Translate and simplify: the sum of \\(-8\\) and \\(−12\\), increased by \\(7\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\((-8 + (-12)) + 7 - 13\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          \\(a−b\\)
          \\(a\\) minus \\(b\\)
          \r\n the difference of \\(a\\) and \\(b\\)
          \r\n \\(b\\) subtracted from \\(a\\)
          \r\n \\(b\\) less than \\(a\\)
          Table \\(\\PageIndex{5}\\)
          \r\n\r\n

          Be careful to get a and b in the right order!

          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{40}\\)
          \r\n\r\n

          Translate and then simplify

          \r\n\r\n
            \r\n
          1. the difference of \\(13\\) and \\(−21\\)
            2. \r\n
          2. subtract \\(24\\) from \\(−19\\).
            3. \r\n
          \r\n\r\n

          Solution

          \r\n\r\n
            \r\n
          1. \\[\\begin{array} {ll} {} &{\\text{the } \\textbf{difference } \\text{of 13 and -21}} \\\\ {\\text{Translate.}} &{13 - (-21)} \\\\ {\\text{Simplify.}} &{34} \\end{array}\\]
            2. \r\n
          2. \\[\\begin{array} {ll} {} &\\textbf{subtract }24 \\textbf{ from }-19 \\\\ {\\text{Translate.}} &{-19 - 24} \\\\ {\\text{Remember, subtract b from a means }a - b} &{} \\\\{\\text{Simplify.}} &{-43} \\end{array}\\]
            3. \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{41}\\)
          \r\n\r\n

          Translate and simplify

          \r\n\r\n
            \r\n
          1. the difference of \\(14\\) and \\(−23\\)
            2. \r\n
          2. subtract \\(21\\) from \\(−17\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(14 - (-23); 37\\)
            2. \r\n
          2. \\(-17 - 21; -38\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{42}\\)
          \r\n\r\n

          Translate and simplify

          \r\n\r\n
            \r\n
          1. the difference of \\(11\\) and \\(−19\\)
            2. \r\n
          2. subtract \\(18\\) from \\(−11\\).
            3. \r\n
          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n
            \r\n
          1. \\(11 - (-19); 30\\)
            2. \r\n
          2. \\(-11 - 18; -29\\)
            3. \r\n
          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{43}\\)
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−2\\) and \\(14\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{\\text{the product of }-2 \\text{ and } 14} \\\\ {\\text{Translate.}} &{(-2)(14)} \\\\{\\text{Simplify.}} &{-28} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{44}\\)
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the product of \\(−5\\) and \\(12\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-5(12); -60\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{45}\\)
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the product of \\(8\\) and \\(-13\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-8(13); -104\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{46}\\)
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−56\\) and \\(−7\\).

          \r\n\r\n

          Solution

          \r\n\r\n

          \\[\\begin{array} {ll} {} &{\\text{the quotient of }-56 \\text{ and } -7} \\\\ {\\text{Translate.}} &{-56\\div(-7)} \\\\{\\text{Simplify.}} &{8} \\end{array}\\]

          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{47}\\)
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−63\\) and \\(−9\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-63\\div (-9); 7\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{48}\\)
          \r\n\r\n

          Translate to an algebraic expression and simplify if possible: the quotient of \\(−72\\) and \\(−9\\).

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          \\(-72\\div (-9); 8\\)

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Use Integers in Applications

          \r\n\r\n

          We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

          \r\n\r\n

          How to Apply a Strategy to Solve Applications with Integers

          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{49}\\)
          \r\n\r\n

          The temperature in Urbana, Illinois one morning was \\(11\\) degrees. By mid-afternoon, the temperature had dropped to \\(−9\\) degrees. What was the difference of the morning and afternoon temperatures?

          \r\n\r\n

          Solution

          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the difference of the morning and afternoon temperatures
          Step 3. Write a phrase that gives the information to find it.the difference of \\(11\\) and \\(-9\\)
          Step 4. Translate the phrase to an expression.\\(11 - (-9)\\)
          Step 5. Simplify the expression.\\(20\\)
          Step 6. Write a complete sentence that answers the question.The difference in temperatures was 20 degrees.
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{50}\\)
          \r\n\r\n

          The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to \\(30\\) degrees below zero. What was the difference in the morning and afternoon temperatures?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          The difference in temperatures was \\(45\\) degrees.

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{51}\\)
          \r\n\r\n

          The temperature in Denver was \\(−6\\) degrees at lunchtime. By sunset the temperature had dropped to \\(−15\\) degrees. What was the difference in the lunchtime and sunset temperatures?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          The difference in temperatures was \\(9\\) degrees.

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          APPLY A STRATEGY TO SOLVE APPLICATIONS WITH INTEGERS.
          \r\n\r\n
            \r\n
          1. Read the problem. Make sure all the words and ideas are understood
            2. \r\n
          2. Identify what we are asked to find.
            3. \r\n
          3. Write a phrase that gives the information to find it.
            4. \r\n
          4. Translate the phrase to an expression.
            5. \r\n
          5. Simplify the expression.
            6. \r\n
          6. Answer the question with a complete sentence.
            7. \r\n
          \r\n
          \r\n\r\n
          \r\n
          Example \\(\\PageIndex{52}\\)
          \r\n\r\n

          The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

          \r\n\r\n

          Solution

          \r\n\r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n
          Step 1. Read the problem. Make sure all the words and ideas are understood. 
          Step 2. Identify what we are asked to find.the number of yards lost
          Step 3. Write a phrase that gives the information to find it.three times a \\(15\\)-yard penalty
          Step 4. Translate the phrase to an expression.\\(3(-15)\\)
          Step 5. Simplify the expression.\\(-45\\)
          Step 6. Write a complete sentence that answers the question.The team lost \\(45\\) yards.
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{53}\\)
          \r\n\r\n

          The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of \\(15\\) yards. What is the number of yards lost due to penalties?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          The Bears lost \\(105\\) yards.

          \r\n
          \r\n
          \r\n
          \r\n\r\n
          \r\n
          Try It \\(\\PageIndex{54}\\)
          \r\n\r\n

          Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

          \r\n\r\n
          \r\n
          Answer
          \r\n
          \r\n

          A $16 fee was deducted from his checking account.

          \r\n
          \r\n
          \r\n
          \r\n\r\n

          Key Concepts

          \r\n\r\n
            \r\n
          • Multiplication and Division of Two Signed Numbers\r\n\r\n
              \r\n
            • Same signs—Product is positive
              • \r\n
            • Different signs—Product is negative
              • \r\n
            \r\n
            • \r\n
          • Strategy for Applications\r\n
              \r\n
            1. Identify what you are asked to find.
              2. \r\n
            2. Write a phrase that gives the information to find it.
              3. \r\n
            3. Translate the phrase to an expression.
              4. \r\n
            4. Simplify the expression.
              5. \r\n
            5. Answer the question with a complete sentence.
              6. \r\n
            \r\n
            • \r\n
          \r\n
          \r\n\t\t\t\t
          \r\n\r\n\r\n\r\n\r\n
          \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n

          1.5: Multiply and Divide Integers is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.

          \r\n\r\n\r\n\r\n\r\n
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          The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.

          ", "content": [{"c": "The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.\n\n", "t": "text"}]}, {"type": "title", "raw_content": "

          Insert Math Editor\u200b

          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!", "content": {"title_content": "Insert Math Editor\u200b", "level": "3"}}, {"type": "paragraph", "raw_content": "
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!

          \nAnd then you can insert the math as either a Block or Inline.

          \nScreenshots below:

          ", "content": [{"c": "In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!\n\n And then you can insert the math as either a Block or Inline.\n\n Screenshots below:\n\n", "t": "text"}]}, {"type": "image", "raw_content": "\"LaTeX", "content": {"url": "https://physicshelpforum.com/data/attachments/4/4740-7fe09219b74ce4ca36a512e8383c412c.jpg", "data": null, "alt": "LaTeX Math Editor", "title": "LaTeX Math Editor", "caption": null}}, {"type": "image", "raw_content": "\"LaTeX", "content": {"url": "https://physicshelpforum.com/data/attachments/4/4739-460ec05f3727868e315d20ce55ce9340.jpg", "data": null, "alt": "LaTeX Math Equations for Physics", "title": "LaTeX Math Equations for Physics", "caption": null}}, {"type": "paragraph", "raw_content": "
          Reactions:
          ", "content": [{"c": "Reactions:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          2 users
          ", "content": [{"c": "2 users", "t": "text"}]}, {"type": "title", "raw_content": "

          Writing Math Equations\u200b

          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.", "content": {"title_content": "Writing Math Equations\u200b", "level": "3"}}, {"type": "paragraph", "raw_content": "
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.

          \nSame as before, you can wrap your Latex equations with any of the following tags:

          ", "content": [{"c": "And when you want to write math equations directly in your post, you are not required to use the Insert Math button.\n\n Same as before, you can wrap your Latex equations with any of the following tags:\n\n", "t": "text"}]}, {"type": "title", "raw_content": "

          Block Math\u200b

          ", "content": {"title_content": "Block Math\u200b", "level": "4"}}, {"type": "paragraph", "raw_content": "
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          Inline Math\u200b

          ", "content": {"title_content": "Inline Math\u200b", "level": "4"}}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[IMATH] [/IMATH]\n[ILATEX] [/ILATEX]\n[ITEX] [/ITEX]\n## ##", "inline": false, "content": {"code_content": "[IMATH] [/IMATH]\n[ILATEX] [/ILATEX]\n[ITEX] [/ITEX]\n## ##", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "
          Reactions:
          ", "content": [{"c": "Reactions:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          1 users
          ", "content": [{"c": "1 users", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\n\n Last edited: \n\n\n
          ", "content": [{"c": "Last edited:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          And some example Latex Equations, with the math expressions used for writing, to see it working...

          Newton's Second Law of Motion
          [math]\\vec{F} = m\\vec{a}[/math]

          ", "content": [{"c": "And some example Latex Equations, with the math expressions used for writing, to see it working...\n\n Newton's Second Law of Motion\n", "t": "text"}, {"c": "[math]\\vec{F} = m\\vec{a}[/math]", "t": "code-inline"}, {"c": "\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\vec{F} = m\\vec{a}", "content": {"math_content": "\\vec{F} = m\\vec{a}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Gauss's Law for Electrcity
          ", "content": [{"c": "Gauss's Law for Electrcity", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          [math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]
          ", "content": [{"c": "[math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]", "t": "code-inline"}]}, {"type": "equation-interline", "raw_content": "\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}", "content": {"math_content": "\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Einstein's Field Equations of General Relativity
          ", "content": [{"c": "Einstein's Field Equations of General Relativity", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          [math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]
          ", "content": [{"c": "[math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]", "t": "code-inline"}]}, {"type": "equation-interline", "raw_content": "G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}", "content": {"math_content": "G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          The Dirac Equation
          ", "content": [{"c": "The Dirac Equation", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          [math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]
          ", "content": [{"c": "[math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]", "t": "code-inline"}]}, {"type": "equation-interline", "raw_content": "(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0", "content": {"math_content": "(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          The Schr\u00f6dinger Equation in Quantum Mechanics (Time-Dependent Form)
          ", "content": [{"c": "The Schr\u00f6dinger Equation in Quantum Mechanics (Time-Dependent Form)", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          [math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]
          ", "content": [{"c": "[math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]", "t": "code-inline"}]}, {"type": "equation-interline", "raw_content": "i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)", "content": {"math_content": "i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Reactions:
          ", "content": [{"c": "Reactions:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          2 users
          ", "content": [{"c": "2 users", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          Long Math Equations
          \nFor very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...

          \nFor example, here is a very long equation without the align tags used in MathJax v3:

          \n\t\tCode:\n\t
          ", "content": [{"c": "Long Math Equations\n For very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...\n\n For example, here is a very long equation without the", "t": "text"}, {"c": "align", "t": "code-inline"}, {"c": "tags used in MathJax v3:\n\n Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]", "inline": false, "content": {"code_content": "[math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

          \nAnd this is how the equation looks with the horizontal scrollbar:

          ", "content": [{"c": "And this is how the equation looks with the horizontal scrollbar:\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n", "content": {"math_content": "P(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "


          \nThe better way to write the exact same equation would be to add the align tags used in MathJax v3, to recognize the \\\\ as line breaks:

          \n\t\tCode:\n\t
          ", "content": [{"c": "The better way to write the exact same equation would be to add the", "t": "text"}, {"c": "align", "t": "code-inline"}, {"c": "tags used in MathJax v3, to recognize the", "t": "text"}, {"c": "\\\\", "t": "code-inline"}, {"c": "as line breaks:\n\n Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]", "inline": false, "content": {"code_content": "[math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

          \nAnd then it display in a more readable format, as seen here:

          ", "content": [{"c": "And then it display in a more readable format, as seen here:\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n", "content": {"math_content": "\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Reactions:
          ", "content": [{"c": "Reactions:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          2 users
          ", "content": [{"c": "2 users", "t": "text"}]}, {"type": "title", "raw_content": "
          Writing Physics Equations\u200b
          We have also installed a physics science package into MathJax, for physics notation and equation support.", "content": {"title_content": "Writing Physics Equations\u200b", "level": "3"}}, {"type": "paragraph", "raw_content": "
          We have also installed a physics science package into MathJax, for physics notation and equation support.

          \nHere are a few physics-related examples:

          Derivatives
          \n\t\tCode:\n\t
          ", "content": [{"c": "We have also installed a physics science package into MathJax, for physics notation and equation support.\n\n Here are a few physics-related examples:\n\n Derivatives\n Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]", "inline": false, "content": {"code_content": "[math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}", "content": {"math_content": "\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Partial Derivatives
          ", "content": [{"c": "Partial Derivatives", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]", "inline": false, "content": {"code_content": "[math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}", "content": {"math_content": "\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Vectors
          ", "content": [{"c": "Vectors", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]", "inline": false, "content": {"code_content": "[math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}", "content": {"math_content": "\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Gradient, Divergence, Curl
          ", "content": [{"c": "Gradient, Divergence, Curl", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]", "inline": false, "content": {"code_content": "[math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}", "content": {"math_content": "\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}", "content": {"math_content": "\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}", "content": {"math_content": "\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Bra-Ket Notation
          ", "content": [{"c": "Bra-Ket Notation", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]", "inline": false, "content": {"code_content": "[math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}", "content": {"math_content": "\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Commutators and Anticommutators
          ", "content": [{"c": "Commutators and Anticommutators", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]", "inline": false, "content": {"code_content": "[math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}", "content": {"math_content": "[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}", "content": {"math_content": "\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Matrices
          ", "content": [{"c": "Matrices", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]", "inline": false, "content": {"code_content": "[math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n", "content": {"math_content": "\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Dirac Notation (Ket, Bra, Operators)
          ", "content": [{"c": "Dirac Notation (Ket, Bra, Operators)", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]", "inline": false, "content": {"code_content": "[math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}", "content": {"math_content": "|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}", "content": {"math_content": "\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}", "math_type": "latex", "by": "mathjax"}}, {"type": "equation-interline", "raw_content": "\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}", "content": {"math_content": "\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Quick Quadratic Equation
          ", "content": [{"c": "Quick Quadratic Equation", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]", "inline": false, "content": {"code_content": "[math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)", "content": {"math_content": "ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)", "math_type": "latex", "by": "mathjax"}}, {"type": "paragraph", "raw_content": "
          Reactions:
          ", "content": [{"c": "Reactions:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          1 users
          ", "content": [{"c": "1 users", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          This is awesome!
          ", "content": [{"c": "This is awesome!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          Reactions:
          ", "content": [{"c": "Reactions:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          2 users
          ", "content": [{"c": "2 users", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          Joined Nov 2013
          ", "content": [{"c": "Joined Nov 2013", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          852 Posts | 104+
          ", "content": [{"c": "852 Posts | 104+", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          New Zealand
          ", "content": [{"c": "New Zealand", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          That looks nice!
          ", "content": [{"c": "That looks nice!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          Reactions:
          ", "content": [{"c": "Reactions:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          2 users
          ", "content": [{"c": "2 users", "t": "text"}]}]], "main_html": "
          The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.

          Insert Math Editor\u200b
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!

          \nAnd then you can insert the math as either a Block or Inline.

          \nScreenshots below:

          \"LaTeX\"LaTeX
          Reactions:
          2 users
          Writing Math Equations\u200b
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.

          \nSame as before, you can wrap your Latex equations with any of the following tags:

          Block Math\u200b
          \n\t\tCode:\n\t
          [MATH]      [/MATH]\n[LATEX]     [/LATEX]\n[TEX]       [/TEX]\n$$          $$
          Inline Math\u200b
          \n\t\tCode:\n\t
          [IMATH]     [/IMATH]\n[ILATEX]    [/ILATEX]\n[ITEX]      [/ITEX]\n##          ##
          Reactions:
          1 users
          \n\n\n            Last edited: \n\n\n
          And some example Latex Equations, with the math expressions used for writing, to see it working...

          Newton's Second Law of Motion
          [math]\\vec{F} = m\\vec{a}[/math]

          \\vec{F} = m\\vec{a}
          Gauss's Law for Electrcity
          [math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]
          \\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}
          Einstein's Field Equations of General Relativity
          [math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]
          G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}
          The Dirac Equation
          [math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]
          (i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0
          The Schr\u00f6dinger Equation in Quantum Mechanics (Time-Dependent Form)
          [math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]
          i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)
          Reactions:
          2 users
          Long Math Equations
          \nFor very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...

          \nFor example, here is a very long equation without the align tags used in MathJax v3:

          \n\t\tCode:\n\t
          [math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]

          \nAnd this is how the equation looks with the horizontal scrollbar:

          \nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n


          \nThe better way to write the exact same equation would be to add the align tags used in MathJax v3, to recognize the \\\\ as line breaks:

          \n\t\tCode:\n\t
          [math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]

          \nAnd then it display in a more readable format, as seen here:

          \n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n
          Reactions:
          2 users
          Writing Physics Equations\u200b
          We have also installed a physics science package into MathJax, for physics notation and equation support.
          We have also installed a physics science package into MathJax, for physics notation and equation support.

          \nHere are a few physics-related examples:

          Derivatives
          \n\t\tCode:\n\t
          [math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}
          Partial Derivatives
          \n\t\tCode:\n\t
          [math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}
          Vectors
          \n\t\tCode:\n\t
          [math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}
          Gradient, Divergence, Curl
          \n\t\tCode:\n\t
          [math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}
          Bra-Ket Notation
          \n\t\tCode:\n\t
          [math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}
          Commutators and Anticommutators
          \n\t\tCode:\n\t
          [math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}
          Matrices
          \n\t\tCode:\n\t
          [math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n
          Dirac Notation (Ket, Bra, Operators)
          \n\t\tCode:\n\t
          [math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}
          Quick Quadratic Equation
          \n\t\tCode:\n\t
          [math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)
          Reactions:
          1 users
          This is awesome!
          Reactions:
          2 users
          Joined Nov 2013
          852 Posts | 104+
          New Zealand
          That looks nice!
          Reactions:
          2 users
          ", "statics": {"paragraph": 55, "paragraph.text": 55, "title": 5, "image": 2, "code": 13, "paragraph.code-inline": 8, "equation-interline": 21}, "url": "https://physicshelpforum.com/t/latex-upgrade-physics-forum-powered-by-mathjax-v3.17489/", "content": "The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.\n\n### Insert Math Editor\u200b\n\nIn the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!\n\n And then you can insert the math as either a Block or Inline.\n\n Screenshots below:\n\nReactions:\n\n2 users\n\n### Writing Math Equations\u200b\n\nAnd when you want to write math equations directly in your post, you are not required to use the Insert Math button.\n\n Same as before, you can wrap your Latex equations with any of the following tags:\n\n#### Block Math\u200b\n\nCode:\n\n```\n[MATH]      [/MATH]\n[LATEX]     [/LATEX]\n[TEX]       [/TEX]\n$$          $$\n```\n\n#### Inline Math\u200b\n\nCode:\n\n```\n[IMATH]     [/IMATH]\n[ILATEX]    [/ILATEX]\n[ITEX]      [/ITEX]\n##          ##\n```\n\nReactions:\n\n1 users\n\nLast edited:\n\nAnd some example Latex Equations, with the math expressions used for writing, to see it working...\n\n Newton's Second Law of Motion `[math]\\vec{F} = m\\vec{a}[/math]`\n\n$$\n\\vec{F} = m\\vec{a}\n$$\n\nGauss's Law for Electrcity\n\n`[math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]`\n\n$$\n\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}\n$$\n\nEinstein's Field Equations of General Relativity\n\n`[math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]`\n\n$$\nG_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}\n$$\n\nThe Dirac Equation\n\n`[math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]`\n\n$$\n(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0\n$$\n\nThe Schr\u00f6dinger Equation in Quantum Mechanics (Time-Dependent Form)\n\n`[math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]`\n\n$$\ni\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)\n$$\n\nReactions:\n\n2 users\n\nLong Math Equations\n For very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...\n\n For example, here is a very long equation without the `align` tags used in MathJax v3:\n\n Code:\n\n```\n[math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]\n```\n\nAnd this is how the equation looks with the horizontal scrollbar:\n\n$$\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n$$\n\nThe better way to write the exact same equation would be to add the `align` tags used in MathJax v3, to recognize the `\\\\` as line breaks:\n\n Code:\n\n```\n[math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]\n```\n\nAnd then it display in a more readable format, as seen here:\n\n$$\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n$$\n\nReactions:\n\n2 users\n\n### Writing Physics Equations\u200b\n\nWe have also installed a physics science package into MathJax, for physics notation and equation support.\n\n Here are a few physics-related examples:\n\n Derivatives\n Code:\n\n```\n[math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]\n```\n\n$$\n\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}\n$$\n\nPartial Derivatives\n\nCode:\n\n```\n[math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]\n```\n\n$$\n\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}\n$$\n\nVectors\n\nCode:\n\n```\n[math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]\n```\n\n$$\n\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}\n$$\n\nGradient, Divergence, Curl\n\nCode:\n\n```\n[math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]\n```\n\n$$\n\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}\n$$\n\n$$\n\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}\n$$\n\n$$\n\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}\n$$\n\nBra-Ket Notation\n\nCode:\n\n```\n[math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]\n```\n\n$$\n\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}\n$$\n\nCommutators and Anticommutators\n\nCode:\n\n```\n[math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]\n```\n\n$$\n[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}\n$$\n\n$$\n\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}\n$$\n\nMatrices\n\nCode:\n\n```\n[math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]\n```\n\n$$\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n$$\n\nDirac Notation (Ket, Bra, Operators)\n\nCode:\n\n```\n[math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]\n```\n\n$$\n|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}\n$$\n\n$$\n\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}\n$$\n\n$$\n\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}\n$$\n\nQuick Quadratic Equation\n\nCode:\n\n```\n[math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]\n```\n\n$$\nax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)\n$$\n\nReactions:\n\n1 users\n\nThis is awesome!\n\nReactions:\n\n2 users\n\nJoined Nov 2013\n\n852 Posts | 104+\n\nNew Zealand\n\nThat looks nice!\n\nReactions:\n\n2 users\n", "html": "\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\t\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\t\n\t\t\n\t\t\n\n\n\n\t\tInfo - Latex Upgrade - Physics Forum Powered by MathJax v3 | Physics 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          You are using an out of date browser. It  may not display this or other websites correctly.
          You should upgrade or use an alternative browser.
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          Info Latex Upgrade - Physics Forum Powered by MathJax v3
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          The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.
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          Insert Math Editor​
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!
          \n
          \nAnd then you can insert the math as either a Block or Inline.
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          \nScreenshots below:
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          Writing Math Equations​
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.
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          \nSame as before, you can wrap your Latex equations with any of the following tags:
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          Block Math​
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          [MATH]      [/MATH]\n[LATEX]     [/LATEX]\n[TEX]       [/TEX]\n$$          $$
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          Inline Math​
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          [IMATH]     [/IMATH]\n[ILATEX]    [/ILATEX]\n[ITEX]      [/ITEX]\n##          ##
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          And some example Latex Equations, with the math expressions used for writing, to see it working...
          \n
          \nNewton's Second Law of Motion
          \n[math]\\vec{F} = m\\vec{a}[/math]
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          \n\n\t\\vec{F} = m\\vec{a}\n\n\n
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          \nGauss's Law for Electrcity
          \n[math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]
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          \n\n\t\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}\n\n\n
          \n
          \nEinstein's Field Equations of General Relativity
          \n[math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]
          \n
          \n\n\tG_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}\n\n\n
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          \nThe Dirac Equation
          \n[math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]
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          \n\n\t(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0\n\n\n
          \n
          \nThe Schr\u00f6dinger Equation in Quantum Mechanics (Time-Dependent Form)
          \n[math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]
          \n
          \n\n\ti\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)\n\n\n
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          Long Math Equations
          \nFor very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...
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          \nFor example, here is a very long equation without the align tags used in MathJax v3:
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          [math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]
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          \nAnd this is how the equation looks with the horizontal scrollbar:
          \n
          \n\n\t\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\n\n\n
          \n
          \nThe better way to write the exact same equation would be to add the align tags used in MathJax v3, to recognize the \\\\ as line breaks:
          \n
          \n\n\n\n\n\n
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          \n\t\tCode:\n\t
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          [math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]
          \n\t
          \n

          \nAnd then it display in a more readable format, as seen here:
          \n
          \n\n\t\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n\n\n\n
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          chip
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          Writing Physics Equations​
          We have also installed a physics science package into MathJax, for physics notation and equation support.
          \n
          \nHere are a few physics-related examples:
          \n
          \nDerivatives
          \n\n\n\n\n\n
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          \n\t\tCode:\n\t
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          [math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]
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          \n\n\t\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}\n\n\n
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          \nPartial Derivatives
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          [math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]
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          \n\n\t\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}\n\n\n
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          \nVectors
          \n\n\n\n\n\n
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          [math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]
          \n\t
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          \n\n\t\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}\n\n\n
          \n
          \nGradient, Divergence, Curl
          \n\n\n\n\n\n
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          [math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]
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          \n

          \n\n\t\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}\n\n\n
          \n\n\t\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}\n\n\n
          \n\n\t\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}\n\n\n
          \n
          \nBra-Ket Notation
          \n\n\n\n\n\n
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          \n\t\tCode:\n\t
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          [math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]
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          \n

          \n\n\t\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}\n\n\n
          \n
          \nCommutators and Anticommutators
          \n\n\n\n\n\n
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          \n\t\tCode:\n\t
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          [math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]
          \n\t
          \n

          \n\n\t[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}\n\n\n
          \n\n\t\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}\n\n\n
          \n
          \nMatrices
          \n\n\n\n\n\n
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          \n\t\tCode:\n\t
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          [math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]
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          \n\n\t\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n\n\n\n
          \n
          \nDirac Notation (Ket, Bra, Operators)
          \n\n\n\n\n\n
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          \n\t\tCode:\n\t
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          [math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]
          \n\t
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          \n\n\t|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}\n\n\n
          \n\n\t\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}\n\n\n
          \n\n\t\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}\n\n\n
          \n
          \nQuick Quadratic Equation
          \n\n\n\n\n\n
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          [math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]
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          \n\n\tax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)\n\n\n
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          benit13
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          Glasgow
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          This is awesome!
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          kiwiheretic
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          New Zealand
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          That looks nice!
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          \n\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tLogin or Register / Reply\n\t\t\t\t\t\t\t\n\n\t\t\t\t\t
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+{"url": "https://physicshelpforum.com/t/latex-upgrade-physics-forum-powered-by-mathjax-v3.17489/", "content": "The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.\n\n### Insert Math Editor​\n\nIn the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!\n\n\n\n And then you can insert the math as either a Block or Inline.\n\n\n\n Screenshots below:\n\nReactions: 2 users\n\n### Writing Math Equations​\n\nAnd when you want to write math equations directly in your post, you are not required to use the Insert Math button.\n\n\n\n Same as before, you can wrap your Latex equations with any of the following tags:\n\n#### Block Math​\n\nCode:\n\n```\n[MATH]      [/MATH]\n[LATEX]     [/LATEX]\n[TEX]       [/TEX]\n$$          $$\n```\n\n#### Inline Math​\n\nCode:\n\n```\n[IMATH]     [/IMATH]\n[ILATEX]    [/ILATEX]\n[ITEX]      [/ITEX]\n##          ##\n```\n\nReactions: 1 users\n\nLast edited:\n\nAnd some example Latex Equations, with the math expressions used for writing, to see it working...\n\n\n\n Newton's Second Law of Motion `[math]\\vec{F} = m\\vec{a}[/math]`\n\n$$\n\\vec{F} = m\\vec{a}\n$$\n\nGauss's Law for Electrcity `[math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]`\n\n$$\n\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}\n$$\n\nEinstein's Field Equations of General Relativity `[math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]`\n\n$$\nG_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}\n$$\n\nThe Dirac Equation `[math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]`\n\n$$\n(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0\n$$\n\nThe Schrödinger Equation in Quantum Mechanics (Time-Dependent Form) `[math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]`\n\n$$\ni\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)\n$$\n\nReactions: 2 users\n\nLong Math Equations\n\n For very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...\n\n\n\n For example, here is a very long equation without the `align` tags used in MathJax v3:\n\nCode:\n\n```\n[math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]\n```\n\nAnd this is how the equation looks with the horizontal scrollbar:\n\n$$\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n$$\n\nThe better way to write the exact same equation would be to add the `align` tags used in MathJax v3, to recognize the `\\\\` as line breaks:\n\nCode:\n\n```\n[math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]\n```\n\nAnd then it display in a more readable format, as seen here:\n\n$$\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n$$\n\nReactions: 2 users\n\n### Writing Physics Equations​\n\nWe have also installed a physics science package into MathJax, for physics notation and equation support.\n\n\n\n Here are a few physics-related examples:\n\n\n\n Derivatives\n\nCode:\n\n```\n[math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]\n```\n\n$$\n\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}\n$$\n\nPartial Derivatives\n\nCode:\n\n```\n[math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]\n```\n\n$$\n\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}\n$$\n\nVectors\n\nCode:\n\n```\n[math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]\n```\n\n$$\n\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}\n$$\n\nGradient, Divergence, Curl\n\nCode:\n\n```\n[math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]\n```\n\n$$\n\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}\n$$\n\n$$\n\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}\n$$\n\n$$\n\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}\n$$\n\nBra-Ket Notation\n\nCode:\n\n```\n[math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]\n```\n\n$$\n\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}\n$$\n\nCommutators and Anticommutators\n\nCode:\n\n```\n[math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]\n```\n\n$$\n[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}\n$$\n\n$$\n\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}\n$$\n\nMatrices\n\nCode:\n\n```\n[math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]\n```\n\n$$\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n$$\n\nDirac Notation (Ket, Bra, Operators)\n\nCode:\n\n```\n[math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]\n```\n\n$$\n|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}\n$$\n\n$$\n\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}\n$$\n\n$$\n\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}\n$$\n\nQuick Quadratic Equation\n\nCode:\n\n```\n[math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]\n```\n\n$$\nax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)\n$$\n\nReactions: 1 users\n\nThis is awesome!\n\nReactions: 2 users\n\nJoined Nov 2013\n\n852 Posts | 104+\n\nNew Zealand\n\nThat looks nice!\n\nReactions: 2 users\n", "main_html": "
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          The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.
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          Insert Math Editor​
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!
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          Writing Math Equations​
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.
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          And some example Latex Equations, with the math expressions used for writing, to see it working...
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          \r\nNewton's Second Law of Motion
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          \r\nThe Schrödinger Equation in Quantum Mechanics (Time-Dependent Form)
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          \r\n\r\n\r\n\r\n\t\t\r\n\r\n\r\n\r\nReactions:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t2 users\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t
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          Long Math Equations
          \r\nFor very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...
          \r\n
          \r\nFor example, here is a very long equation without the align tags used in MathJax v3:
          \r\n
          \r\n\r\n\r\n\r\n\r\n\r\n
          \r\n\t
          \r\n\t\tCode:\r\n\t
          \r\n\t
          \r\n\t\t
          [math]\r\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n[/math]
          \r\n\t
          \r\n

          \r\nAnd this is how the equation looks with the horizontal scrollbar:
          \r\n
          \r\n\r\n\t\r\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n\r\n\r\n\r\n
          \r\n
          \r\nThe better way to write the exact same equation would be to add the align tags used in MathJax v3, to recognize the \\\\ as line breaks:
          \r\n
          \r\n\r\n\r\n\r\n\r\n\r\n
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          \r\n\t\tCode:\r\n\t
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          [math]\r\n\\begin{align*}\r\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n\\end{align*}\r\n[/math]
          \r\n\t
          \r\n

          \r\nAnd then it display in a more readable format, as seen here:
          \r\n
          \r\n\r\n\t\r\n\\begin{align*}\r\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n\\end{align*}\r\n\r\n\r\n\r\n
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          Writing Physics Equations​
          We have also installed a physics science package into MathJax, for physics notation and equation support.
          \r\n
          \r\nHere are a few physics-related examples:
          \r\n
          \r\nDerivatives
          \r\n\r\n\r\n\r\n\r\n\r\n
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          \r\n\t\tCode:\r\n\t
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          [math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}\r\n\r\n\r\n
          \r\n
          \r\nPartial Derivatives
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          [math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}\r\n\r\n\r\n
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          \r\nVectors
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          [math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}\r\n\r\n\r\n
          \r\n
          \r\nGradient, Divergence, Curl
          \r\n\r\n\r\n\r\n\r\n\r\n
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          [math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\r\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\r\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}\r\n\r\n\r\n
          \r\n\r\n\t\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}\r\n\r\n\r\n
          \r\n\r\n\t\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}\r\n\r\n\r\n
          \r\n
          \r\nBra-Ket Notation
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          \r\n\t\tCode:\r\n\t
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          [math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}\r\n\r\n\r\n
          \r\n
          \r\nCommutators and Anticommutators
          \r\n\r\n\r\n\r\n\r\n\r\n
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          \r\n\t\tCode:\r\n\t
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          [math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\r\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}\r\n\r\n\r\n
          \r\n\r\n\t\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}\r\n\r\n\r\n
          \r\n
          \r\nMatrices
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          [math]\r\n\\begin{pmatrix}\r\na & b \\\\\r\nc & d\r\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\r\n[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t\r\n\\begin{pmatrix}\r\na & b \\\\\r\nc & d\r\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\r\n\r\n\r\n\r\n
          \r\n
          \r\nDirac Notation (Ket, Bra, Operators)
          \r\n\r\n\r\n\r\n\r\n\r\n
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          \r\n\t\tCode:\r\n\t
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          [math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\r\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\r\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]
          \r\n\t
          \r\n

          \r\n\r\n\t|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}\r\n\r\n\r\n
          \r\n\r\n\t\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}\r\n\r\n\r\n
          \r\n\r\n\t\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}\r\n\r\n\r\n
          \r\n
          \r\nQuick Quadratic Equation
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          \r\n\t\tCode:\r\n\t
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          [math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]
          \r\n\t
          \r\n

          \r\n\r\n\tax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)\r\n\r\n\r\n
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          \r\n\r\n\r\n\r\n\t\t\r\n\r\n\r\n\r\nReactions:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t1 users\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t
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          This is awesome!
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          \r\n\r\n\r\n\r\n\t\t\r\n\r\n\r\n\r\nReactions:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t2 users\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t
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          \r\n\r\n\r\n\t\t\t\t\r\n\r\n\r\n\t\t\r\n\r\n\r\n\t\t\t
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          Joined Nov 2013
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          852 Posts | 104+
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          New Zealand
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          \r\n\r\n
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          \r\n\r\n\r\n\r\n\r\n\t\t\t
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          That looks nice!
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          \r\n\r\n\r\n\r\n\t\t\r\n\r\n\r\n\r\nReactions:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t2 users\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t
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          \r\n\t
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          \r\n\r\n\r\n\r\n\r\n\t\r\n\t\r\n\r\n\r\n\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          \r\n\r\n\r\n\r\n\r\n
          \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t\t
          \r\n\t\t\t\t\t\t\t\t\t\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t
          \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t", "content_list": [[{"type": "paragraph", "raw_content": "
          The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.

          ", "content": [{"c": "The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.\n\n\n\n", "t": "text"}]}, {"type": "title", "raw_content": "
          Insert Math Editor​
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!
          ", "content": {"title_content": "Insert Math Editor​", "level": "3"}}, {"type": "paragraph", "raw_content": "
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!

          \nAnd then you can insert the math as either a Block or Inline.

          \nScreenshots below:

          ", "content": [{"c": "In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!\n\n\n\n And then you can insert the math as either a Block or Inline.\n\n\n\n Screenshots below:\n\n\n\n", "t": "text"}]}, {"type": "image", "raw_content": "\"LaTeX", "content": {"url": "https://physicshelpforum.com/data/attachments/4/4740-7fe09219b74ce4ca36a512e8383c412c.jpg", "data": null, "alt": "LaTeX Math Editor", "title": "LaTeX Math Editor", "caption": null}}, {"type": "image", "raw_content": "\"LaTeX", "content": {"url": "https://physicshelpforum.com/data/attachments/4/4739-460ec05f3727868e315d20ce55ce9340.jpg", "data": null, "alt": "LaTeX Math Equations for Physics", "title": "LaTeX Math Equations for Physics", "caption": null}}, {"type": "paragraph", "raw_content": "
          Reactions:2 users
          ", "content": [{"c": "Reactions: 2 users", "t": "text"}]}, {"type": "title", "raw_content": "
          Writing Math Equations​
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.
          ", "content": {"title_content": "Writing Math Equations​", "level": "3"}}, {"type": "paragraph", "raw_content": "
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.

          \nSame as before, you can wrap your Latex equations with any of the following tags:

          ", "content": [{"c": "And when you want to write math equations directly in your post, you are not required to use the Insert Math button.\n\n\n\n Same as before, you can wrap your Latex equations with any of the following tags:\n\n\n\n", "t": "text"}]}, {"type": "title", "raw_content": "
          Block Math​
          ", "content": {"title_content": "Block Math​", "level": "4"}}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[MATH]      [/MATH]\n[LATEX]     [/LATEX]\n[TEX]       [/TEX]\n$$          $$", "inline": false, "content": {"code_content": "[MATH]      [/MATH]\n[LATEX]     [/LATEX]\n[TEX]       [/TEX]\n$$          $$", "by": "tag_pre_code"}}, {"type": "title", "raw_content": "
          Inline Math​
          ", "content": {"title_content": "Inline Math​", "level": "4"}}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[IMATH]     [/IMATH]\n[ILATEX]    [/ILATEX]\n[ITEX]      [/ITEX]\n##          ##", "inline": false, "content": {"code_content": "[IMATH]     [/IMATH]\n[ILATEX]    [/ILATEX]\n[ITEX]      [/ITEX]\n##          ##", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "
          Reactions:1 users
          ", "content": [{"c": "Reactions: 1 users", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\n\n            Last edited: \n\n\n
          ", "content": [{"c": "Last edited:", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          And some example Latex Equations, with the math expressions used for writing, to see it working...

          Newton's Second Law of Motion
          [math]\\vec{F} = m\\vec{a}[/math]

          ", "content": [{"c": "And some example Latex Equations, with the math expressions used for writing, to see it working...\n\n\n\n Newton's Second Law of Motion\n\n", "t": "text"}, {"c": "[math]\\vec{F} = m\\vec{a}[/math]", "t": "code-inline"}, {"c": "\n\n\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\vec{F} = m\\vec{a}", "content": {"math_content": "\\vec{F} = m\\vec{a}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Gauss's Law for Electrcity
          [math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]

          ", "content": [{"c": "Gauss's Law for Electrcity\n\n", "t": "text"}, {"c": "[math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]", "t": "code-inline"}, {"c": "\n\n\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}", "content": {"math_content": "\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Einstein's Field Equations of General Relativity
          [math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]

          ", "content": [{"c": "Einstein's Field Equations of General Relativity\n\n", "t": "text"}, {"c": "[math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]", "t": "code-inline"}, {"c": "\n\n\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}", "content": {"math_content": "G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          The Dirac Equation
          [math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]

          ", "content": [{"c": "The Dirac Equation\n\n", "t": "text"}, {"c": "[math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]", "t": "code-inline"}, {"c": "\n\n\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0", "content": {"math_content": "(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          The Schrödinger Equation in Quantum Mechanics (Time-Dependent Form)
          [math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]

          ", "content": [{"c": "The Schrödinger Equation in Quantum Mechanics (Time-Dependent Form)\n\n", "t": "text"}, {"c": "[math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]", "t": "code-inline"}, {"c": "\n\n\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)", "content": {"math_content": "i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "
          Reactions:2 users
          ", "content": [{"c": "Reactions: 2 users", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          Long Math Equations
          \nFor very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...

          \nFor example, here is a very long equation without the align tags used in MathJax v3:

          ", "content": [{"c": "Long Math Equations\n\n For very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...\n\n\n\n For example, here is a very long equation without the", "t": "text"}, {"c": "align", "t": "code-inline"}, {"c": "tags used in MathJax v3:\n\n\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]", "inline": false, "content": {"code_content": "[math]\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n[/math]", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

          \nAnd this is how the equation looks with the horizontal scrollbar:

          ", "content": [{"c": "And this is how the equation looks with the horizontal scrollbar:\n\n\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n", "content": {"math_content": "P(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          \nThe better way to write the exact same equation would be to add the align tags used in MathJax v3, to recognize the \\\\ as line breaks:

          ", "content": [{"c": "The better way to write the exact same equation would be to add the", "t": "text"}, {"c": "align", "t": "code-inline"}, {"c": "tags used in MathJax v3, to recognize the", "t": "text"}, {"c": "\\\\", "t": "code-inline"}, {"c": "as line breaks:\n\n\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]", "inline": false, "content": {"code_content": "[math]\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n[/math]", "by": "tag_pre_code"}}, {"type": "paragraph", "raw_content": "

          \nAnd then it display in a more readable format, as seen here:

          ", "content": [{"c": "And then it display in a more readable format, as seen here:\n\n\n\n", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\n\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}\n", "content": {"math_content": "\\begin{align*}\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\n\\end{align*}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "
          Reactions:2 users
          ", "content": [{"c": "Reactions: 2 users", "t": "text"}]}, {"type": "title", "raw_content": "
          Writing Physics Equations​
          We have also installed a physics science package into MathJax, for physics notation and equation support.
          ", "content": {"title_content": "Writing Physics Equations​", "level": "3"}}, {"type": "paragraph", "raw_content": "
          We have also installed a physics science package into MathJax, for physics notation and equation support.

          \nHere are a few physics-related examples:

          Derivatives
          ", "content": [{"c": "We have also installed a physics science package into MathJax, for physics notation and equation support.\n\n\n\n Here are a few physics-related examples:\n\n\n\n Derivatives\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]", "inline": false, "content": {"code_content": "[math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}", "content": {"math_content": "\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Partial Derivatives
          ", "content": [{"c": "Partial Derivatives\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]", "inline": false, "content": {"code_content": "[math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}", "content": {"math_content": "\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Vectors
          ", "content": [{"c": "Vectors\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]", "inline": false, "content": {"code_content": "[math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}", "content": {"math_content": "\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Gradient, Divergence, Curl
          ", "content": [{"c": "Gradient, Divergence, Curl\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]", "inline": false, "content": {"code_content": "[math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}", "content": {"math_content": "\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}", "math_type": "latex", "by": ""}}, {"type": "equation-interline", "raw_content": "\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}", "content": {"math_content": "\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}", "math_type": "latex", "by": ""}}, {"type": "equation-interline", "raw_content": "\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}", "content": {"math_content": "\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Bra-Ket Notation
          ", "content": [{"c": "Bra-Ket Notation\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]", "inline": false, "content": {"code_content": "[math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}", "content": {"math_content": "\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Commutators and Anticommutators
          ", "content": [{"c": "Commutators and Anticommutators\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]", "inline": false, "content": {"code_content": "[math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}", "content": {"math_content": "[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}", "math_type": "latex", "by": ""}}, {"type": "equation-interline", "raw_content": "\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}", "content": {"math_content": "\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Matrices
          ", "content": [{"c": "Matrices\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]", "inline": false, "content": {"code_content": "[math]\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\n", "content": {"math_content": "\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Dirac Notation (Ket, Bra, Operators)
          ", "content": [{"c": "Dirac Notation (Ket, Bra, Operators)\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]", "inline": false, "content": {"code_content": "[math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}", "content": {"math_content": "|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}", "math_type": "latex", "by": ""}}, {"type": "equation-interline", "raw_content": "\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}", "content": {"math_content": "\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}", "math_type": "latex", "by": ""}}, {"type": "equation-interline", "raw_content": "\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}", "content": {"math_content": "\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "


          Quick Quadratic Equation
          ", "content": [{"c": "Quick Quadratic Equation\n\n", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          \n\t\tCode:\n\t
          ", "content": [{"c": "Code:", "t": "text"}]}, {"type": "code", "raw_content": "[math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]", "inline": false, "content": {"code_content": "[math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]", "by": "tag_pre_code"}}, {"type": "equation-interline", "raw_content": "ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)", "content": {"math_content": "ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)", "math_type": "latex", "by": ""}}, {"type": "paragraph", "raw_content": "
          Reactions:1 users
          ", "content": [{"c": "Reactions: 1 users", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          This is awesome!
          ", "content": [{"c": "This is awesome!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          Reactions:2 users
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          Joined Nov 2013
          ", "content": [{"c": "Joined Nov 2013", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          852 Posts | 104+
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          New Zealand
          ", "content": [{"c": "New Zealand", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          That looks nice!
          ", "content": [{"c": "That looks nice!", "t": "text"}]}, {"type": "paragraph", "raw_content": "
          Reactions:2 users
          ", "content": [{"c": "Reactions: 2 users", "t": "text"}]}]], "html": "\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\t\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\t\t\r\n\t\t\r\n\t\t\r\n\r\n\r\n\r\n\t\tInfo - Latex Upgrade - Physics Forum Powered by MathJax v3 | Physics Forum\r\n\t\t\r\n\r\n\t\t\t\r\n\t\t\t\r\n\r\n\t\t\r\n\r\n\t\t\t\r\n\r\n\r\n\r\n\r\n\t\t\r\n\t\t\r\n\t\t\r\n\r\n\r\n\t\t\t\r\n\r\n\t\t\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t\r\n\t\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t\r\n\t\t\r\n\t\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\r\n\t\t\r\n\t\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t\r\n\r\n\r\n\t\t\r\n\r\n\r\n\t\r\n\r\n\r\n\r\n\t\t\t\r\n\r\n\r\n\t\t\t\r\n\r\n\r\n\r\n\t\t\r\n\r\n\t\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\r\n\t\r\n\r\n\t\r\n\t\t\r\n\r\n\r\n\t\t
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          The entire LaTeX rendering system has been upgraded to MathJax v3, and we have created a tool to see a preview render of your equations before posting them.
          \r\n
          \r\n
          Insert Math Editor​
          In the toolbar, there is an Math Editor button, to make it easy for writing your equations in LaTeX and seeing the rendering in real time!
          \r\n
          \r\nAnd then you can insert the math as either a Block or Inline.
          \r\n
          \r\nScreenshots below:
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          Writing Math Equations​
          And when you want to write math equations directly in your post, you are not required to use the Insert Math button.
          \r\n
          \r\nSame as before, you can wrap your Latex equations with any of the following tags:
          \r\n
          \r\n
          Block Math​
          \r\n\r\n\r\n\r\n\r\n
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          [MATH]      [/MATH]\r\n[LATEX]     [/LATEX]\r\n[TEX]       [/TEX]\r\n$$          $$
          \r\n\t
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          Inline Math​
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          [IMATH]     [/IMATH]\r\n[ILATEX]    [/ILATEX]\r\n[ITEX]      [/ITEX]\r\n##          ##
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          34 Posts | 39+
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          And some example Latex Equations, with the math expressions used for writing, to see it working...
          \r\n
          \r\nNewton's Second Law of Motion
          \r\n[math]\\vec{F} = m\\vec{a}[/math]
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          \r\n\r\n\t\\vec{F} = m\\vec{a}\r\n\r\n\r\n
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          \r\nGauss's Law for Electrcity
          \r\n[math]\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}[/math]
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          \r\n\r\n\t\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\varepsilon_0}\r\n\r\n\r\n
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          \r\nEinstein's Field Equations of General Relativity
          \r\n[math]G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}[/math]
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          \r\n\r\n\tG_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}\r\n\r\n\r\n
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          \r\nThe Dirac Equation
          \r\n[math](i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0[/math]
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          \r\n\r\n\t(i\\gamma^\\mu \\partial_\\mu - m)\\psi = 0\r\n\r\n\r\n
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          \r\nThe Schrödinger Equation in Quantum Mechanics (Time-Dependent Form)
          \r\n[math]i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)[/math]
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          \r\n\r\n\ti\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}, t) = \\hat{H}\\Psi(\\mathbf{r}, t)\r\n\r\n\r\n
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          34 Posts | 39+
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          Long Math Equations
          \r\nFor very long equations, it's recommended to use line breaks appropriately. But if you don't add line breaks, and the equation is extremely long (or wider than the display you're using), a horizontal scrollbar will appear below the equation, to show that you can hover your pointer over the equation and scroll horizontally...
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          \r\nFor example, here is a very long equation without the align tags used in MathJax v3:
          \r\n
          \r\n\r\n\r\n\r\n\r\n\r\n
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          \r\n\t\tCode:\r\n\t
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          [math]\r\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n[/math]
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          \r\n

          \r\nAnd this is how the equation looks with the horizontal scrollbar:
          \r\n
          \r\n\r\n\t\r\nP(\\text{16 out of 20}) = \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) = \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) = \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) = \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) = \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n\r\n\r\n\r\n
          \r\n
          \r\nThe better way to write the exact same equation would be to add the align tags used in MathJax v3, to recognize the \\\\ as line breaks:
          \r\n
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          [math]\r\n\\begin{align*}\r\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n\\end{align*}\r\n[/math]
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          \r\n

          \r\nAnd then it display in a more readable format, as seen here:
          \r\n
          \r\n\r\n\t\r\n\\begin{align*}\r\nP(\\text{16 out of 20}) &= \\frac{20!}{16! \\times 4!} \\times 0.85^{16} \\times 0.15^{4} = 0.182122 \\\\\r\nP(\\text{17 out of 20}) &= \\frac{20!}{17! \\times 3!} \\times 0.85^{17} \\times 0.15^{3} = 0.242829 \\\\\r\nP(\\text{18 out of 20}) &= \\frac{20!}{18! \\times 2!} \\times 0.85^{18} \\times 0.15^{2} = 0.229338 \\\\\r\nP(\\text{19 out of 20}) &= \\frac{20!}{19! \\times 1!} \\times 0.85^{19} \\times 0.15^{1} = 0.136798 \\\\\r\nP(\\text{20 out of 20}) &= \\frac{20!}{20! \\times 0!} \\times 0.85^{20} \\times 0.15^{0} = 0.0387595\r\n\\end{align*}\r\n\r\n\r\n\r\n
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          • \"Like\"
            • \r\n\r\n\t\t
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          chip
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          Joined Oct 2019
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          34 Posts | 39+
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          Washington
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          Writing Physics Equations​
          We have also installed a physics science package into MathJax, for physics notation and equation support.
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          \r\nHere are a few physics-related examples:
          \r\n
          \r\nDerivatives
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          [math]\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}[/math]
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          \r\n\r\n\t\\frac{d}{dx} f(x) \\quad \\text{versus} \\quad \\dv{f}{x}\r\n\r\n\r\n
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          \r\nPartial Derivatives
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          [math]\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}[/math]
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          \r\n\r\n\t\\frac{\\partial}{\\partial x} f(x,y) \\quad \\text{versus} \\quad \\pdv{f}{x}\r\n\r\n\r\n
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          \r\nVectors
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          [math]\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}[/math]
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          \r\n\r\n\t\\vec{v} \\quad \\text{versus} \\quad \\vb{v} \\quad \\text{and} \\quad \\vec{v} \\cdot \\vec{w} \\quad \\text{versus} \\quad \\vb{v} \\vdot \\vb{w}\r\n\r\n\r\n
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          \r\nGradient, Divergence, Curl
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          [math]\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}[/math]\r\n[math]\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}[/math]\r\n[math]\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}[/math]
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          \r\n

          \r\n\r\n\t\\vec{\\nabla} \\phi \\quad \\text{versus} \\quad \\grad{\\phi}\r\n\r\n\r\n
          \r\n\r\n\t\\vec{\\nabla} \\cdot \\vec{v} \\quad \\text{versus} \\quad \\div{\\vb{v}}\r\n\r\n\r\n
          \r\n\r\n\t\\vec{\\nabla} \\times \\vec{v} \\quad \\text{versus} \\quad \\curl{\\vb{v}}\r\n\r\n\r\n
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          \r\nBra-Ket Notation
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          [math]\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}[/math]
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          \r\n\r\n\t\\langle \\psi | \\phi \\rangle \\quad \\text{versus} \\quad \\braket{\\psi}{\\phi}\r\n\r\n\r\n
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          \r\nCommutators and Anticommutators
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          [math][\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}[/math]\r\n[math]\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}[/math]
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          \r\n\r\n\t[\\hat{A}, \\hat{B}] \\quad \\text{versus} \\quad \\comm{\\hat{A}}{\\hat{B}}\r\n\r\n\r\n
          \r\n\r\n\t\\{\\hat{A}, \\hat{B}\\} \\quad \\text{versus} \\quad \\acomm{\\hat{A}}{\\hat{B}}\r\n\r\n\r\n
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          \r\nMatrices
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          [math]\r\n\\begin{pmatrix}\r\na & b \\\\\r\nc & d\r\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\r\n[/math]
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          \r\n\r\n\t\r\n\\begin{pmatrix}\r\na & b \\\\\r\nc & d\r\n\\end{pmatrix} \\quad \\text{versus} \\quad \\mqty(a & b \\\\ c & d)\r\n\r\n\r\n\r\n
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          \r\nDirac Notation (Ket, Bra, Operators)
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          [math]|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}[/math]\r\n[math]\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}[/math]\r\n[math]\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}[/math]
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          \r\n\r\n\t|\\psi\\rangle \\quad \\text{versus} \\quad \\ket{\\psi}\r\n\r\n\r\n
          \r\n\r\n\t\\langle\\psi| \\quad \\text{versus} \\quad \\bra{\\psi}\r\n\r\n\r\n
          \r\n\r\n\t\\hat{H}|\\psi\\rangle \\quad \\text{versus} \\quad \\op{H}{\\psi}\r\n\r\n\r\n
          \r\n
          \r\nQuick Quadratic Equation
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          [math]ax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)[/math]
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          \r\n\r\n\tax^2 + bx + c = 0 \\quad \\text{versus} \\quad \\qty(a x^2 + b x + c = 0)\r\n\r\n\r\n
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          • \"Like\"
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          \r\n\r\n\r\n\r\nReactions:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t1 users\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t
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          benit13
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          Joined Oct 2017
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          1K Posts | 768+
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          Glasgow
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          This is awesome!
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          • \"Like\"
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          \r\n\r\n\r\n\r\nReactions:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t2 users\r\n\r\n\r\n\r\n\r\n\t\t\t\t\t\t\t\t
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          kiwiheretic
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          Joined Nov 2013
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          852 Posts | 104+
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          New Zealand
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          That looks nice!
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          • \"Like\"
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          \r\n\r\n\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\tLogin or Register / Reply\r\n\t\t\t\t\t\t\t\r\n\r\n\t\t\t\t\t
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          \r\n        
          Physics Forum
          \r\n        
            \r\n            
          • Physics Forum for Physics Discussions, Physics Homework Help, High School Physics, University Physics, Advanced Physics, Science, Engineering, Academic Studies
            • \r\n        
          \r\n\t\t\"Physics\r\n    
          \r\n
          \r\n\r\n\r\n\t
          \r\n    
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          High School Physics
          \r\n        
          \r\n            Kinematics / Dynamics\r\n\t\t\tThermodynamics / Fluid Mechanics\r\n\t\t\tElectricity / Magnetism\r\n\t\t\tWork, Energy, Power\r\n        
          \r\n    
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          \r\n        
          \r\n\t\t\tRelativity\r\n\t\t\tAdvanced Mechanics\r\n\t\t\tAdvanced Thermodynamics\r\n\t\t\tQuantum Physics\r\n\r\n        
          \r\n    
          \r\n
          \r\n\r\n\r\n\t
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          \r\n        
          Education Forums
          \r\n        
          \r\n            Math Forums\r\n\t\t\tHistory Forum\r\n\t\t\tPsychology Forum\r\n        
          \r\n    
          \r\n
          \r\n\r\n\r\n\r\n\t\t
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          \r\n\t\t\t\t\t\t\t\tCopyright © 2008-2024 Physics Forum. All rights reserved.\r\n\t\t\t\t\t\t
          \r\n\r\n\r\n\t\t\t\t
          \r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\r\n\r\n\r\n\t\t\t
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          \r\n\r\n\t\t\r\n\r\n\r\n\r\n\t\r\n\t\r\n\t\r\n\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\r\n\r\n\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t\t\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\t\t\t\r\n\r\n\r\n\r\n\t\r\n\t\r\n\r\n", "statics": {"paragraph": 47, "paragraph.text": 55, "title": 5, "image": 2, "code": 13, "paragraph.code-inline": 8, "equation-interline": 21}}
diff --git a/docs/llm_web_kit/model/lang_id.md b/docs/llm_web_kit/model/lang_id.md
index a7e987e6..cbe864b8 100644
--- a/docs/llm_web_kit/model/lang_id.md
+++ b/docs/llm_web_kit/model/lang_id.md
@@ -4,7 +4,7 @@
 
 is_218e为True时使用lid218e模型，在多个小语种中有更好的表现，除个别容易使模型混淆的情况外，会返回正常的language_details字段，若该参数为False，则language_details字段为空，默认值为True
 
-is_cn_specific为True时，会对文本中的中文文本进行细分，分为zho-Hans(简体中文)或zho-Hant(繁体中文)，结果在language_details字段中，默认值为False
+is_cn_specific为True时，会对文本中的中文文本进行细分，分为zho-Hans(简体中文)或zho-Hant(繁体中文),结果在language_details字段中，默认值为False,如果需要使用，请先pip install langdetect_zh==1.0.4,该package使用langdetect的方法,并针对中文进行了特调,能有效识别简体中文和繁体中文
 
 ## 配置文件需要改动的部分
 
@@ -120,3 +120,89 @@ print(update_language_by_str(text, is_cn_specific=True))
 总时间: 1.3538 秒
 
 处理速度: 443.91 条/秒
+
+## 性能说明
+
+测试集使用gsarti/flores_101，该数据集包含102种语言的并行句子，每个语种2009条测试集路径：https://huggingface.co/datasets/gsarti/flores_101
+
+下表所示lid176为单模型结果，模型路径为s3://web-parse-huawei/shared_resource/language/lid176.bin
+
+lid218e也为单模型结果，模型路径为s3://web-parse-huawei/shared_resource/language/lid218e.bin
+
+级联方案即为该代码调用方案，使用lid176判断zh, en, ja, ko，使用lid218e判断其他语种，使用langdetect_zh区分简体中文与繁体中文
+
+该表统计了三种模型在102种语言上错误的次数，其中lid176繁体中文全错是考虑到该模型无法区分简体中文和繁体中文
+
+| 级联方案  |          | lid176   |          | lid218e   |          |
+| --------- | -------- | -------- | -------- | --------- | -------- |
+| 真实语言  | 错误次数 | 真实语言 | 错误次数 | 真实语言  | 错误次数 |
+| bos       | 1079     | zho_trad | 2009     | bos       | 1079     |
+| kam       | 767      | ful      | 2009     | kam       | 765      |
+| zho_trad  | 18       | lug      | 2009     | zho_trad  | 623      |
+| hrv       | 197      | hau      | 2009     | zho_simpl | 229      |
+| nya       | 165      | ibo      | 2009     | hrv       | 197      |
+| kea       | 145      | kea      | 2009     | nya       | 161      |
+| msa       | 145      | kam      | 2009     | kea       | 145      |
+| ful       | 67       | lin      | 2009     | msa       | 145      |
+| xho       | 51       | luo      | 2009     | ful       | 56       |
+| umb       | 46       | mri      | 2009     | umb       | 46       |
+| zul       | 38       | nso      | 2009     | jpn       | 46       |
+| fas       | 38       | nya      | 2009     | fas       | 38       |
+| ind       | 37       | orm      | 2009     | ind       | 37       |
+| mri       | 27       | sna      | 2009     | xho       | 32       |
+| wol       | 22       | umb      | 2009     | zul       | 16       |
+| ast       | 16       | wol      | 2009     | ast       | 16       |
+| dan       | 13       | xho      | 2009     | dan       | 13       |
+| nob       | 13       | zul      | 2009     | wol       | 13       |
+| nso       | 12       | bos      | 1879     | nob       | 13       |
+| luo       | 11       | ast      | 1373     | nso       | 11       |
+| lug       | 11       | som      | 1184     | luo       | 8        |
+| jav       | 9        | msa      | 1131     | lug       | 7        |
+| sna       | 9        | yor      | 943      | pus       | 7        |
+| ibo       | 8        | oci      | 753      | glg       | 6        |
+| afr       | 7        | hrv      | 609      | jav       | 6        |
+| pus       | 7        | jav      | 590      | mri       | 5        |
+| glg       | 6        | afr      | 574      | hin       | 4        |
+| som       | 4        | glg      | 294      | swe       | 3        |
+| swh       | 4        | uzb      | 188      | yor       | 3        |
+| hin       | 4        | ltz      | 151      | lin       | 3        |
+| yor       | 4        | ceb      | 144      | lao       | 2        |
+| lin       | 4        | nob      | 137      | oci       | 2        |
+| ceb       | 3        | swh      | 118      | som       | 2        |
+| swe       | 3        | mlt      | 109      | ceb       | 2        |
+| lao       | 2        | dan      | 90       | khm       | 2        |
+| oci       | 2        | slv      | 56       | slv       | 1        |
+| uzb       | 2        | ind      | 48       | uzb       | 1        |
+| orm       | 2        | slk      | 41       | npi       | 1        |
+| nld       | 2        | pus      | 37       | tgl       | 1        |
+| hau       | 2        | gle      | 26       | bul       | 1        |
+| slv       | 1        | npi      | 18       | fra       | 1        |
+| zho_simpl | 1        | azj      | 17       | hau       | 1        |
+| npi       | 1        | asm      | 14       | ita       | 1        |
+| eng       | 1        | tgk      | 13       | ltz       | 1        |
+| tgl       | 1        | isl      | 13       | kaz       | 1        |
+| est       | 1        | est      | 12       | por       | 1        |
+| bul       | 1        | snd      | 12       | afr       | 1        |
+| fra       | 1        | cym      | 11       | spa       | 1        |
+| ita       | 1        | cat      | 11       |           |          |
+| khm       | 1        | srp      | 11       |           |          |
+| ltz       | 1        | kir      | 10       |           |          |
+| kaz       | 1        | nld      | 5        |           |          |
+| por       | 1        | por      | 5        |           |          |
+| spa       | 1        | swe      | 4        |           |          |
+|           |          | mkd      | 4        |           |          |
+|           |          | lav      | 4        |           |          |
+|           |          | urd      | 3        |           |          |
+|           |          | tgl      | 3        |           |          |
+|           |          | kaz      | 2        |           |          |
+|           |          | ron      | 2        |           |          |
+|           |          | ita      | 2        |           |          |
+|           |          | bel      | 2        |           |          |
+|           |          | bul      | 2        |           |          |
+|           |          | lit      | 2        |           |          |
+|           |          | lao      | 1        |           |          |
+|           |          | ckb      | 1        |           |          |
+
+根据统计表格，lid176准确率0.7715，lid218e准确率为0.9817，级联方案准确率为0.9853，准确率公式为：1-sum(错误次数)/(102\*2009)
+
+级联方案相比于lid176提升了多语种的准确率，同时也解决了lid218e针对部分语种（中文简体、中文繁体、日语）的错误
diff --git a/docs/llm_web_kit/model/politics_detector.md b/docs/llm_web_kit/model/politics_detector.md
index ed99f31c..350523c7 100644
--- a/docs/llm_web_kit/model/politics_detector.md
+++ b/docs/llm_web_kit/model/politics_detector.md
@@ -1,8 +1,8 @@
 ## 作用
 
-识别中文或英文文本中的涉政内容，目前包含了新旧两类接口，旧的接口接收单条数据，并返回该数据的涉政分数，分数接近1代表不涉政，分数接近0则代表涉政。目前旧的接口仅支持CPU模型。
+识别中文或英文文本中的涉政内容，目前包含了新旧两类接口，25m3_cpu模型接口接收单条数据，并返回该数据的涉政分数，分数接近1代表不涉政，分数接近0则代表涉政。目前25m3_cpu模型接口仅支持CPU模型。
 
-新的接口检测结果以ModelResponse类返回，该类包含is_remained和details两个字段，其中is_remained代表数据是否需要保留，details则是一个包含涉政分数等详细信息的字典。新的接口支持CPU和GPU两种模型。
+25m3模型接口检测结果以ModelResponse类返回，该类包含is_remained和details两个字段，其中is_remained代表数据是否需要保留，details则是一个包含涉政分数等详细信息的字典。25m3模型接口支持GPU模型。
 
 ## 配置文件需要改动的部分
 
@@ -13,20 +13,20 @@
         "common":{
             "cache_path": "~/.llm_web_kit_cache"
         },
-        "political-24m7":{
-            "download_path": "s3://web-parse-huawei/shared_resource/political/24m7.zip",
-            "md5": "97eabb56268a3af3f68e8a96a50d5f80",
-        },
         "political-25m3":{
             "download_path": "s3://web-parse-huawei/shared_resource/political/25m3.zip",
             "md5": "d0d14a561f987763d654165b536b5858",
         },
+        "political-25m3_cpu":{
+            "download_path": "s3://web-parse-huawei/shared_resource/political/25m3_cpu.zip",
+            "md5": "926359a393de6a36c1b4be403711767f",
+        },
     },
 ```
 
 ## 调用方法
 
-1. 旧的接口调用方法如下：
+1. 25m3_cpu模型接口调用方法如下：
 
 ```python
 from llm_web_kit.model.politics_detector import *
@@ -81,7 +81,7 @@ print(political_filter_cpu(text, "en"))
 # 输出结果为：{'political_prob': 1.0000100135803223}
 ```
 
-2. 新的接口调用方法如下：
+2. 25m3模型接口调用方法如下：
 
 ```python
 from llm_web_kit.model.model_impl import ModelFactory, ModelType, DeviceType
@@ -113,7 +113,7 @@ for i in range(0, len(requests), batch_size):
 
 ## 运行时间
 
-1. 旧的接口（political_filter_cpu）
+1. 25m3_cpu模型接口（political_filter_cpu）
 
    使用型号为`AMD EPYC 7742`的cpu单核进行测试，测试集总共有 77861 条数据（均是中英文的数据），下面只统计了political_filter_cpu接口本身的耗时，排除了数据读取的时间。
 
@@ -127,7 +127,7 @@ for i in range(0, len(requests), batch_size):
 
    每秒可处理: 416.3049条数据
 
-2. 新的接口（predictor.predict_batch）
+2. 25m3模型接口（predictor.predict_batch）
 
    使用单卡NVIDIA A100测试涉政的GPU模型，测试集共有39111条数据，下面统计了不同batch_size下，predictor.predict_batch接口的速度，该接口内部包括tokenize和模型推理操作。
 
@@ -159,3 +159,29 @@ for i in range(0, len(requests), batch_size):
    | 128        | 31.580092769179686 |
    | 256        | 24.26296225431703  |
    | 512        | cuda out of memory |
+
+## 性能说明
+
+25m3_cpu模型（threshold=0.5）：
+
+测试集路径：s3://xyz-process-ylk2/xyz-users/huyucheng1/political_data_202502/test/
+
+| 指标          | 新模型值             | 旧模型值             |
+| ------------- | -------------------- | -------------------- |
+| **F1**        | 0.9089603520041284   | 0.8831507760632497   |
+| **Accuracy**  | 0.8624864742896118   | 0.8013861609546715   |
+| **Precision** | 0.9041776426882809   | 0.7913184992146802   |
+| **Recall**    | 0.9137939273134369   | 0.999095513748191    |
+| **TN**        | 68641                | 19820                |
+| **FP**        | 28373                | 77194                |
+| **FN**        | 25257                | 265                  |
+| **TP**        | 267727               | 292719               |
+| **Prec_Pos**  | 0.9041776426882809   | 0.7913184992146802   |
+| **Recl_Pos**  | 0.9137939273134369   | 0.999095513748191    |
+| **F1_Pos**    | 0.9089603520041284   | 0.8831507760632497   |
+| **Prec_Neg**  | 0.7310166350720995   | 0.986806074184715    |
+| **Recl_Neg**  | 0.7075370565073082   | 0.204300410250067    |
+| **F1_Neg**    | 0.719085232986926    | 0.3385169813576546   |
+| **qps**       | 1493.477337807 条/秒 | 1674.157845704 条/秒 |
+
+注：上述指标均是在集群中得出，单核运行时间请参考运行时间第一小节
diff --git a/docs/llm_web_kit/model/rule_based_safety_module.md b/docs/llm_web_kit/model/rule_based_safety_module.md
index 3c8917e0..8d81e541 100644
--- a/docs/llm_web_kit/model/rule_based_safety_module.md
+++ b/docs/llm_web_kit/model/rule_based_safety_module.md
@@ -10,8 +10,8 @@
             "cache_path": "~/.llm_web_kit_cache"
         },
         "unsafe_words":{
-            "download_path": "s3://web-parse-huawei/shared_resource/political/unsafe_words.jsonl",
-            "md5": "e81dd1050a79f68b9d9b3f66baadde66",
+            "download_path": "s3://web-parse-huawei/shared_resource/unsafe_words/unsafe_words_porn_politics.jsonl",
+            "md5": "ef51faf114353d987ec97b211a8d2b06",
         },
         "xyz_internal_unsafe_words":{
             "download_path": "s3://web-parse-huawei/shared_resource/political/xyz_internal_unsafe_words.jsonl",
@@ -51,6 +51,32 @@ m.process("your content",
  'safety_infos': {'domain_level': '', 'hit_unsafe_words': False}}
 ```
 
+### 敏感词检测模块用法示例
+
+```python
+from llm_web_kit.model.unsafe_words_detector import *
+
+checker = UnsafeWordChecker(language="zh-en")
+
+content = "64式销售QQ"
+unsafe_words = checker.check_unsafe_words(
+    content_str=content,
+)
+print(unsafe_words)
+[{'word': '64式', 'type': '违禁品', 'level': 'L3', 'language': 'zh', 'count': 1.0}, {'word': '64式销售', 'type': '违禁品', 'level': 'L3', 'language': 'zh', 'count': 1.0}, {'word': '销售', 'type': '广告营销', 'level': 'L3', 'language': 'zh', 'count': 1.0}, {'word': '64式销售qq', 'type': '违禁品', 'level': 'L1', 'language': 'zh', 'count': 1.0}]
+
+checker = UnsafeWordsFilter()
+content = "64式销售QQ"
+#from_safe_source:是否来自安全来源。如果是，直接返回安全。
+#from_domestic_source: 是否来自国内来源。如果是，仅检查 L1 级别的不安全词；否则检查 L1 和 L2 级别。
+result = checker.filter(
+    content,
+    'zh',
+    from_safe_source = False,
+    from_domestic_source = True,
+)
+```
+
 ## 速度
 
 ### 整体速度：
diff --git a/llm_web_kit/extractor/config.py b/llm_web_kit/extractor/config.py
index 7afd68e3..c34075b0 100644
--- a/llm_web_kit/extractor/config.py
+++ b/llm_web_kit/extractor/config.py
@@ -6,4 +6,5 @@
     {'url': 'stackexchange.com', 'tag': '//*[contains(@class, "d-none")]'},  # 任意标签，class包含d-none，限制在stackexchange.com网站
     {'url': 'mathoverflow.net', 'tag': '//*[contains(@class, "d-none")]'},  # 任意标签，class包含d-none，限制在mathoverflow.net网站
     {'url': 'blog.csdn.net', 'tag': '//span[contains(@class, "katex-html")]'},  # 仅针对 blog.csdn.net 域名，删除所有 class 包含 katex-html 的  标签及其内容（用于移除数学公式渲染的 HTML 部分）
+    {'url': 'math.libretexts.org', 'tag': '//div[contains(@class, "Headertext")]'},  # 仅针对 bmath.libretexts.org 域名，删除所有 class 包含 Headertext 的 
           标签及其内容
 ]
diff --git a/llm_web_kit/extractor/html/recognizer/cc_math/common.py b/llm_web_kit/extractor/html/recognizer/cc_math/common.py
index 712902bc..3ba0de72 100644
--- a/llm_web_kit/extractor/html/recognizer/cc_math/common.py
+++ b/llm_web_kit/extractor/html/recognizer/cc_math/common.py
@@ -88,6 +88,10 @@ class ZHIHU:
     DOMAIN = 'zhihu.com'
 
 
+class MATHINSIGHT:
+    DOMAIN = 'mathinsight.org'
+
+
 # 行内行间公式，MathJax中一般也可以通过配置来区分行内行间公式
 EQUATION_INLINE = DocElementType.EQUATION_INLINE
 EQUATION_INTERLINE = DocElementType.EQUATION_INTERLINE
diff --git a/llm_web_kit/extractor/html/recognizer/cc_math/config/mathsight/formula_conversion.py b/llm_web_kit/extractor/html/recognizer/cc_math/config/mathsight/formula_conversion.py
new file mode 100644
index 00000000..18d52525
--- /dev/null
+++ b/llm_web_kit/extractor/html/recognizer/cc_math/config/mathsight/formula_conversion.py
@@ -0,0 +1,28 @@
+import re
+
+
+def convert_to_standard_latex(text):
+    # 创建替换规则字典
+    replacements = {
+        # 向量表示
+        r'\\vc{([^}]*)}': r'\\mathbf{\1}',
+        # 雅可比矩阵
+        r'\\jacm{([^}]*)}': r'D\1',
+        # 其他常见宏的替换
+        r'\\diff{([^}]*)}{([^}]*)}': r'\\frac{\\mathrm{d} \1}{\\mathrm{d} \2}',
+        r'\\pdiff{([^}]*)}{([^}]*)}': r'\\frac{\\partial \1}{\\partial \2}',
+        r'\\norm{([^}]*)}': r'\\|\1\\|',
+        # 积分宏替换 (简化版，根据需要可以扩展)
+        r'\\lint{([^}]*)}{([^}]*)}': r'\\int_{\1} \2 \\cdot d\\mathbf{s}',
+        r'\\slint{([^}]*)}{([^}]*)}': r'\\int_{\1} \2 \\,ds',
+        # 默认字母替换
+        r'\\dlvf': r'\\mathbf{F}',
+        r'\\dlc': r'C',
+        r'\\dlsi': r'f',
+        # 实数集合符号
+        r'\\R': r'\\mathbb{R}',
+    }
+    # 应用所有替换规则
+    for pattern, replacement in replacements.items():
+        text = re.sub(pattern, replacement, text)
+    return text
diff --git a/llm_web_kit/extractor/html/recognizer/cc_math/config/mathsight/midefault.js b/llm_web_kit/extractor/html/recognizer/cc_math/config/mathsight/midefault.js
new file mode 100644
index 00000000..310b640a
--- /dev/null
+++ b/llm_web_kit/extractor/html/recognizer/cc_math/config/mathsight/midefault.js
@@ -0,0 +1,239 @@
+MathJax.Hub.Config({
+    tex2jax: {inlineMath: [["$","$"],["\\(","\\)"],["`","`"]]},
+
+    "HTML-CSS": { linebreaks: { automatic: true, width: "65% container" } },
+    SVG: { linebreaks: { automatic: true, width: "65% container" } },
+
+    TeX: {
+	equationNumbers: {  autoNumber: "AMS"  },
+
+    Macros: {
+	goodbreak: '\\mmlToken{mo}[linebreak="goodbreak"]{}',
+	badbreak: ['\\mmlToken{mo}[linebreak="badbreak"]{#1}',1],
+	nobreak: ['\\mmlToken{mo}[linebreak="nobreak"]{#1}',1],
+	invisibletimes: ['\\mmlToken{mo}{\u2062}'],
+	cdotbadbreak: '\\badbreak{\u22C5}',
+	timesbadbreak: '\\badbreak{\u00D7}',
+
+	diff: ['\\frac{\\mathrm{d} #1}{\\mathrm{d} #2}',2],
+	diffn: ['\\frac{\\mathrm{d}^{#3} #1}{\\mathrm{d} #2^{#3}}',3],
+	pdiff: ['\\frac{\\partial #1}{\\partial #2}',2],
+	pdiffn: ['\\frac{\\partial^{#3} #1}{\\partial #2^{#3}}',3],
+	vc: ['\\mathbf{#1}',1],
+	R: '\\mathbf{R}',
+	tr: '\\mathop{\\rm tr}',
+	blue: '\\color{blue}{\\textbf{blue}}',
+	red: '\\color{red}{\\textbf{red}}',
+	green: '\\color{green}{\\textbf{green}}',
+	cyan: '\\color{cyan}{\\textbf{cyan}}',
+	magenta: '\\color{magenta}{\\textbf{magenta}}',
+	curl: '\\mathop{\\text{curl}}',
+	div: '\\mathop{\\text{div}}',
+	norm: ['\\|#1\\|',1],
+
+	//arclength symbol
+	arcLengthSymbol: 's',
+	als: '\\arcLengthSymbol',
+
+	//symbol for differential in vector-valued line integral
+	lineIntegralSymbol: '\\vc{s}',
+	lis: '\\lineIntegralSymbol',
+
+	//surace area symbol
+	surfaceAreaSymbol: 'S',
+	sas: '\\surfaceAreaSymbol',
+
+	//symbol for differential in vector-valued surface integral
+	surfaceIntegralDifferential: '\\vc{S}',
+	sid: '\\surfaceIntegralDifferential',
+
+	///////////////////////////////
+	//formats for various integrals
+	///////////////////////////////
+
+	//vector line integrals
+	lineIntegral: ['\\int_{#1} #2  \\cdot d\\lis', 2],
+	lint: ['\\lineIntegral{#1}{#2}',2],
+
+	closedLineIntegral: ['\\oint_{#1} #2 \\cdot d\\lis',2],
+	clint: ['\\closedLineIntegral{#1}{#2}',2],
+
+    	//scalar line integrals
+	scalarLineIntegral: ['\\int_{#1} #2 \\,d\\als',2],
+	slint: ['\\scalarLineIntegral{#1}{#2}',2],
+
+	closedScalarLineIntegral: ['\\oint_{#1} #2 \\,d\\als',2],
+	cslint: ['\\closedScalarLineIntegral{#1}{#2}',2],
+
+	//vector surface integrals
+	surfaceIntegral: ['\\iint_{#1} #2 \\cdot d\\sid',2],
+	sint: ['\\surfaceIntegral{#1}{#2}',2],
+
+	//scalar surface interals
+	scalarSurfaceIntegral: ['\\iint_{#1} #2 \\,d\\sas',2],
+	ssint: ['\\scalarSurfaceIntegral{#1}{#2}',2],
+
+	//parametrized versions of integrals
+	//vector line integrals
+	parametrizedLineIntegral: ['\\int_{#1}^{#2} #3(#4(t)) \\cdot #4\'(t) dt',4],
+	plint: ['\\parametrizedLineIntegral{#1}{#2}{#3}{#4}',4],
+
+	//scalar line integrals
+	parametrizedScalarLineIntegral: ['\\int_{#1}^{#2} #3(#4(t)) \\norm{#4\'(t)} dt',4],
+	pslint: ['\\parametrizedScalarLineIntegral{#1}{#2}{#3}{#4}',4],
+
+	//vector surface integrals
+	parametrizedSurfaceIntegral: ['\\int_{#1}^{#2}\\int_{#3}^{#4} #5(#6(#7,#8)) \\cdot \\left( \\pdiff{#6}{#7}(#7,#8) \\times \\pdiff{#6}{#8}(#7,#8) \\right) d#7\\,d#8',8],
+	psint: ['\\parametrizedSurfaceIntegral{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}', 8],
+
+	//vector surface integrals, reverse normal vector
+	parametrizedSurfaceIntegralReverseNormal: ['\\int_{#1}^{#2}\\int_{#3}^{#4} #5(#6(#7,#8)) \\cdot \\left( \\pdiff{#6}{#8}(#7,#8) \\times \\pdiff{#6}{#7}(#7,#8) \\right) d#7\\,d#8', 8],
+	psintrn: ['\\parametrizedSurfaceIntegralReverseNormal{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}', 8],
+
+	//vector surface integrals, reverse normal vector, reverse integration order
+	parametrizedSurfaceIntegralReverseNormalReverseOrder: ['\\int_{#1}^{#2}\\int_{#3}^{#4} #5(#6(#7,#8)) \\cdot \\left( \\pdiff{#6}{#8}(#7,#8) \\times \\pdiff{#6}{#7}(#7,#8) \\right) d#8\\,d#7',8],
+	psintrnro: ['\\parametrizedSurfaceIntegralReverseNormalReverseOrder{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}', 8],
+
+	//vector surface integrals, reverse integration order
+	parametrizedSurfaceIntegralReverseOrder: ['\\int_{#1}^{#2}\\int_{#3}^{#4} #5(#6(#7,#8)) \\cdot \\left( \\pdiff{#6}{#7}(#7,#8) \\times \\pdiff{#6}{#8}(#7,#8) \\right) d#8\\,d#7',8],
+	psintro: ['\\parametrizedSurfaceIntegralReverseOrder{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}', 8],
+
+	//vector surface integrals, over region
+	parametrizedSurfaceIntegralOverRegion: ['\\iint_{#1} #2(#3(#4,#5)) \\cdot \\left( \\pdiff{#3}{#4}(#4,#5) \\times \\pdiff{#3}{#5}(#4,#5) \\right) d#4\\,d#5',5],
+	psintor: ['\\parametrizedSurfaceIntegralOverRegion{#1}{#2}{#3}{#4}{#5}',5],
+
+	//scalar surface integrals
+	parametrizedScalarSurfaceIntegral: ['\\int_{#1}^{#2}\\!\\!\\!\\int_{#3}^{#4} #5(#6(#7,#8)) \\invisibletimes \\left\\| \\pdiff{#6}{#7}(#7,#8) \\times \\pdiff{#6}{#8}(#7,#8) \\right\\| d#7\\,d#8',8],
+	pssint: ['\\parametrizedScalarSurfaceIntegral{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}', 8],
+
+	//scalar surface integrals, reverse integration order
+	parametrizedScalarSurfaceIntegralReverseOrder: ['\\int_{#1}^{#2}\\int_{#3}^{#4} #5(#6(#7,#8)) \\left\\| \\pdiff{#6}{#7}(#7,#8) \\times \\pdiff{#6}{#8}(#7,#8) \\right\\| d#8\\,d#7',8],
+	pssintro: ['\\parametrizedScalarSurfaceIntegralReverseOrder{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}', 8],
+
+	//scalar surface integrals, over region
+	parametrizedScalarSurfaceIntegralOverRegion: ['\\iint_{#1} #2(#3(#4,#5)) \\left\\| \\pdiff{#3}{#4}(#4,#5) \\times \\pdiff{#3}{#5}(#4,#5) \\right\\| d#4\\,d#5',5],
+	pssintor: ['\\parametrizedScalarSurfaceIntegralOverRegion{#1}{#2}{#3}{#4}{#5}',5],
+
+
+	//default letters for objects
+	//regions (2D)
+	defaultLetterRegion: 'D',
+	dlr: '\\defaultLetterRegion',
+
+	//volumes (3D)
+	defaultLetterVolume: 'W',
+	dlv: '\\defaultLetterVolume',
+
+	//curves
+	defaultLetterCurve: 'C',
+	dlc: '\\defaultLetterCurve',
+	alternateDefaultLetterCurve: 'B',
+	adlc: '\\alternateDefaultLetterCurve',
+	secondAlternateDefaultLetterCurve: 'E',
+	sadlc: '\\secondAlternateDefaultLetterCurve',
+
+	//surfaces
+	defaultLetterSurface: 'S',
+	dls: '\\defaultLetterSurface',
+
+	//vector fields
+	defaultLetterVectorField: '\\vc{F}',
+	dlvf: '\\defaultLetterVectorField',
+	defaultLetterVectorFieldComponent: 'F',
+	dlvfc: '\\defaultLetterVectorFieldComponent',
+	alternateDefaultLetterVectorField: '\\vc{G}',
+	adlvf: '\\alternateDefaultLetterVectorField',
+	alternateDefaultLetterVectorFieldComponent: 'G',
+	adlvfc: '\\alternateDefaultLetterVectorFieldComponent',
+
+	//scalar integrals
+	defaultLetterScalarIntegrand: 'f',
+	dlsi: '\\defaultLetterScalarIntegrand',
+
+	//line parameterization
+	defaultLetterLineParameterization: '\\vc{c}',
+	dllp: '\\defaultLetterLineParameterization',
+	defaultLetterLineParameterizationComponent: 'c',
+	dllpc: '\\defaultLetterLineParameterizationComponent',
+	alternateDefaultLetterLineParameterization: '\\vc{p}',
+	adllp: '\\alternateDefaultLetterLineParameterization',
+	alternateDefaultLetterLineParameterizationComponent: 'p',
+	adllpc: '\\alternateDefaultLetterLineParameterizationComponent',
+	secondAlternateDefaultLetterLineParameterization: '\\vc{q}',
+	sadllp: '\\secondAlternateDefaultLetterLineParameterization',
+	secondAlternateDefaultLetterLineParameterizationComponent: 'q',
+	sadllpc: '\\secondAlternateDefaultLetterLineParameterizationComponent',
+	thirdAlternateDefaultLetterLineParameterization: '\\vc{d}',
+	tadllp: '\\thirdAlternateDefaultLetterLineParameterization',
+	thirdAlternateDefaultLetterLineParameterizationComponent: 'd',
+	tadllpc: '\\thirdAlternateDefaultLetterLineParameterizationComponent',
+
+	//surface parameterization
+	defaultLetterSurfaceParameterization: '\\vc{\\Phi}',
+	dlsp: '\\defaultLetterSurfaceParameterization',
+	defaultLetterSurfaceParameterizationComponent: '\\Phi',
+	dlspc: '\\defaultLetterSurfaceParameterizationComponent',
+
+	//variables for surface parameters
+	surfaceParameterFirstVariable: 'u',
+	spfv: '\\surfaceParameterFirstVariable',
+	surfaceParameterSecondVariable: 'v',
+	spsv: '\\surfaceParameterSecondVariable',
+
+	//change of variable function
+	changeVariableFunction: '\\vc{T}',
+	cvarf: '\\changeVariableFunction',
+	changeVariableFunctionComponent: 'T',
+	cvarfc: '\\changeVariableFunctionComponent',
+
+	//default variables for change of variables
+	changeVariableFirstVariable: 'u',
+	cvarfv: '\\changeVariableFirstVariable',
+	changeVariableSecondVariable: 'v',
+	cvarsv: '\\changeVariableSecondVariable',
+	changeVariableThirdVariable: 'w',
+	cvartv: '\\changeVariableThirdVariable',
+
+	//potential function
+	defaultLetterPotentialFunction: 'f',
+	dlpf: '\\defaultLetterPotentialFunction',
+
+	//default combo shortcuts (combining the above)
+	//vector line integral
+	defaultLineIntegral: '\\lint{\\dlc}{\\dlvf}',
+	dlint: '\\defaultLineIntegral',
+	defaultClosedLineIntegral: '\\clint{\\dlc}{\\dlvf}',
+	dclint: '\\defaultClosedLineIntegral',
+	defaultParametrizedLineIntegral: '\\plint{a}{b}{\\dlvf}{\\dllp}',
+	dplint: '\\defaultParametrizedLineIntegral',
+
+	//scalar line integral
+	defaultScalarLineIntegral: '\\slint{\\dlc}{\\dlsi}',
+	dslint: '\\defaultScalarLineIntegral',
+	defaultClosedScalarLineIntegral: '\\cslint{\\dlc}{\\dlsi}',
+	dcslint: '\\defaultClosedScalarLineIntegral',
+	defaultParametrizedScalarLineIntegral: '\\pslint{a}{b}{\\dlsi}{\\dllp}',
+	dpslint: '\\defaultParametrizedScalarLineIntegral',
+
+	//vector surface integral
+	defaultSurfaceIntegral: '\\sint{\\dls}{\\dlvf}',
+	dsint: '\\defaultSurfaceIntegral',
+	defaultParametrizedSurfaceIntegral: '\\psintor{\\dlr}{\\dlvf}{\\dlsp}{\\spfv}{\\spsv}',
+	dpsint: '\\defaultParametrizedSurfaceIntegral',
+
+	//scalar surface integral
+	defaultScalarSurfaceIntegral: '\\ssint{\\dls}{\\dlsi}',
+	dssint: '\\defaultScalarSurfaceIntegral',
+	defaultParametrizedScalarSurfaceIntegral: '\\pssintor{\\dlr}{\\dlsi}{\\dlsp}{\\spfv}{\\spsv}',
+	dpssint: '\\defaultParametrizedScalarSurfaceIntegral',
+
+	//jacobian matrix
+	JacobianMatrix: ['D{#1}',1],
+	jacm: ['\\JacobianMatrix{#1}',1],
+
+    }
+  }
+
+});
+
+MathJax.Ajax.loadComplete("/static/mathjaxconfig/midefault.js");
diff --git a/llm_web_kit/extractor/html/recognizer/cc_math/render/mathjax.py b/llm_web_kit/extractor/html/recognizer/cc_math/render/mathjax.py
index c55b7d27..3200b438 100644
--- a/llm_web_kit/extractor/html/recognizer/cc_math/render/mathjax.py
+++ b/llm_web_kit/extractor/html/recognizer/cc_math/render/mathjax.py
@@ -1,6 +1,9 @@
 import re
 from typing import Any, Dict, List
 
+from llm_web_kit.extractor.html.recognizer.cc_math.common import MATHINSIGHT
+from llm_web_kit.extractor.html.recognizer.cc_math.config.mathsight.formula_conversion import \
+    convert_to_standard_latex
 from llm_web_kit.extractor.html.recognizer.cc_math.render.render import (
     BaseMathRender, MathRenderType)
 from llm_web_kit.libs.html_utils import HtmlElement, html_to_element
@@ -34,6 +37,7 @@ def __init__(self):
             'config': '',
             'version': ''
         }
+        self.url = ''  # 添加url属性
 
     def get_render_type(self) -> str:
         """获取渲染器类型."""
@@ -332,8 +336,22 @@ def _process_math_in_text(
         if not text:
             return text
 
-        # 查找所有匹配
+        # 首先查找所有分隔符形式的匹配
         matches = list(pattern.finditer(text))
+
+        # 如果是行间公式且没有分隔符匹配，并且processEnvironments为True，才尝试查找 \begin{xxx}...\end{xxx} 格式
+        if is_display and not matches and MATHJAX_OPTIONS.get('processEnvironments', True):
+            # 检查是否是mjx-container元素，如果是则跳过此特殊处理
+            parent = element.getparent()
+            if parent is not None and parent.tag == 'mjx-container':
+                return text
+
+            # mjx-container标签中\begin{xxx}...\end{xxx}包裹的公式在如下逻辑中无法提取
+            begin_matches = list(re.finditer(r'\\begin\{([^}]+)}(.*?)\\end\{\1}', text, re.DOTALL))
+            if begin_matches:
+                return self._process_begin_end_matches(element, text, begin_matches, is_tail)
+
+        # 如果没有匹配到分隔符形式的公式，直接返回原文本
         if not matches:
             return text
 
@@ -359,6 +377,9 @@ def _process_math_in_text(
             if self._is_escaped_delimiter(text, match.start()):
                 continue
 
+            # 限定MATHINSIGHT域名
+            if MATHINSIGHT.DOMAIN in self.url:
+                formula = convert_to_standard_latex(formula)
             start_pos = match.start()
             end_pos = match.end()
 
@@ -407,6 +428,61 @@ def _process_math_in_text(
         # 返回处理后的文本
         return result
 
+    def _process_begin_end_matches(self, element, text, begin_matches, is_tail):
+        """处理begin{},end{}包裹的数学公式."""
+        from llm_web_kit.extractor.html.recognizer.cc_math.common import \
+            MathType
+        from llm_web_kit.extractor.html.recognizer.recognizer import CCTag
+        from llm_web_kit.libs.html_utils import build_cc_element
+
+        result = text
+        last_position = len(result)
+        parent = element.getparent()
+
+        for match in reversed(begin_matches):
+            env_name = match.group(1)  # 环境名称 (如 align, equation 等)
+            formula = match.group(2)  # 公式内容
+            formula = normalize_ctl_text(formula)
+            if not formula.strip():
+                continue  # 跳过空公式
+
+            # 构建完整的公式文本
+            formula = f'\\begin{{{env_name}}}{formula}\\end{{{env_name}}}'
+            # 将自定义LaTeX转换为标准LaTeX格式
+            if MATHINSIGHT.DOMAIN in self.url:
+                formula = convert_to_standard_latex(formula)
+
+            start_pos = match.start()
+            end_pos = match.end()
+            suffix = result[end_pos:last_position]
+
+            math_node = build_cc_element(
+                html_tag_name=CCTag.CC_MATH_INTERLINE,  # 始终是行间公式
+                text=formula,  # 使用标准化后的公式
+                tail=suffix,
+                type=MathType.LATEX,
+                by=self.render_type,
+                html=match.group(0)
+            )
+
+            result = result[:start_pos]
+
+            if is_tail:
+                element.tail = result
+                if parent is not None:
+                    parent_index = list(parent).index(element)
+                    parent.insert(parent_index + 1, math_node)
+            else:
+                element.text = result
+                if len(element) > 0:
+                    element.insert(0, math_node)
+                else:
+                    element.append(math_node)
+
+            last_position = start_pos
+
+        return result
+
     def _is_escaped_delimiter(self, text: str, pos: int) -> bool:
         """检查分隔符是否被转义.
 
diff --git a/llm_web_kit/extractor/html/recognizer/cc_math/render/render.py b/llm_web_kit/extractor/html/recognizer/cc_math/render/render.py
index 189b0d14..ccfc5bff 100644
--- a/llm_web_kit/extractor/html/recognizer/cc_math/render/render.py
+++ b/llm_web_kit/extractor/html/recognizer/cc_math/render/render.py
@@ -25,6 +25,15 @@ def __init__(self):
         """初始化渲染器基类."""
         self.options = {}
         self.render_type = None
+        self.url = ''  # 添加url属性的正确方式
+
+    def set_url(self, url: str) -> None:
+        """设置URL。
+
+        Args:
+            url: 当前页面的URL
+        """
+        self.url = url
 
     @abstractmethod
     def get_render_type(self) -> str:
diff --git a/llm_web_kit/extractor/html/recognizer/ccmath.py b/llm_web_kit/extractor/html/recognizer/ccmath.py
index ed971da8..90ee2b0d 100644
--- a/llm_web_kit/extractor/html/recognizer/ccmath.py
+++ b/llm_web_kit/extractor/html/recognizer/ccmath.py
@@ -42,6 +42,9 @@ def recognize(self, base_url: str, main_html_lst: List[Tuple[HtmlElement, HtmlEl
         # 获取数学公式渲染器
         base_render = BaseMathRender()
         math_render = base_render.get_math_render(raw_html)
+        # 设置URL
+        if math_render:
+            math_render.url = base_url
         # TODO: 自定义配置目前只支持mathjax
         if math_render and math_render.render_type == MathRenderType.MATHJAX:
             math_render.get_options(raw_html)
diff --git a/llm_web_kit/main_html_parser/simplify_html/simplify_html.py b/llm_web_kit/main_html_parser/simplify_html/simplify_html.py
index 679b0742..bfff8f29 100644
--- a/llm_web_kit/main_html_parser/simplify_html/simplify_html.py
+++ b/llm_web_kit/main_html_parser/simplify_html/simplify_html.py
@@ -47,6 +47,9 @@
     # '-header', '_header',  # 有特例，可能自定义的header中有标题，先注释
 }
 
+# 自定义标签
+tail_block_tag = 'cc-alg-uc-text'
+
 
 def add_data_uids(dom: html.HtmlElement) -> None:
     """为DOM所有节点添加data-uid属性（递归所有子节点）"""
@@ -762,7 +765,7 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html
                                 # trailing_text = last_child.tail
 
                                 # 创建wrapper元素
-                                wrapper = etree.Element('cc-alg-uc-tex')
+                                wrapper = etree.Element(tail_block_tag)
                                 wrapper.set('_item_id', current_id)
 
                                 # 设置前面的文本
@@ -798,7 +801,7 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html
                                 # 检查父节点的text
                                 if original_parent.text and original_parent.text.strip() == root_for_xpath.text.strip():
                                     # 创建wrapper
-                                    wrapper = etree.Element('cc-alg-uc-tex')
+                                    wrapper = etree.Element(tail_block_tag)
                                     wrapper.set('_item_id', current_id)
                                     wrapper.text = original_parent.text
 
@@ -818,7 +821,7 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html
                                     for child in original_parent.iterchildren():
                                         if child.tail and child.tail.strip() == root_for_xpath.text.strip():
                                             # 创建wrapper
-                                            wrapper = etree.Element('cc-alg-uc-tex')
+                                            wrapper = etree.Element(tail_block_tag)
                                             wrapper.set('_item_id', current_id)
                                             wrapper.text = child.tail
 
@@ -830,21 +833,8 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html
                                             index = parent.index(child)
                                             parent.insert(index + 1, wrapper)
 
-                                            found = True
                                             break
 
-                                # 如果没有找到匹配的文本节点，使用父节点作为包裹对象
-                                if not found:
-                                    wrapper = etree.Element('cc-alg-uc-tex')
-                                    wrapper.set('_item_id', current_id)
-                                    wrapper.text = root_for_xpath.text
-                                    # 将父节点的内容移动到wrapper中
-                                    for child in list(original_parent.iterchildren()):
-                                        wrapper.append(child)
-                                        original_parent.remove(child)
-
-                                    # 添加wrapper到父节点
-                                    original_parent.append(wrapper)
             else:
                 # 块级元素直接设置属性
                 original_parent.set('_item_id', current_id)
diff --git a/llm_web_kit/model/model_impl.py b/llm_web_kit/model/model_impl.py
index 7e4c0f3b..6f5c58f6 100644
--- a/llm_web_kit/model/model_impl.py
+++ b/llm_web_kit/model/model_impl.py
@@ -112,7 +112,7 @@ def convert_result_to_response(self, result: dict) -> ModelResponse:
         # raise NotImplementedError
         # TODO convert result to response ensure the threshold
         return PoliticalResponse(
-            is_remained=result['political_prob'] > 0.99, details=result
+            is_remained=result['political_prob'] > 0.89, details=result
         )
 
 
diff --git a/llm_web_kit/model/politics_detector.py b/llm_web_kit/model/politics_detector.py
index 72cfe63a..571101ca 100644
--- a/llm_web_kit/model/politics_detector.py
+++ b/llm_web_kit/model/politics_detector.py
@@ -27,7 +27,7 @@ def __init__(self, model_path: str = None):
         if not model_path:
             model_path = self.auto_download()
         model_bin_path = os.path.join(model_path, 'model.bin')
-        tokenizer_path = os.path.join(model_path, 'internlm2-chat-20b')
+        tokenizer_path = os.path.join(model_path, 'qwen2.5_7b_tokenizer')
 
         self.model = fasttext.load_model(model_bin_path)
         self.tokenizer = transformer.AutoTokenizer.from_pretrained(
@@ -35,12 +35,12 @@ def __init__(self, model_path: str = None):
         )
 
     def auto_download(self):
-        """Default download the 24m7.zip model."""
-        resource_name = 'political-24m7'
+        """Default download the 25m3_cpu.zip model."""
+        resource_name = 'political-25m3_cpu'
         resource_config = load_config()['resources']
-        political_24m7_config: dict = resource_config[resource_name]
-        political_24m7_s3 = political_24m7_config['download_path']
-        political_24m7_md5 = political_24m7_config.get('md5', '')
+        political_25m3_cpu_config: dict = resource_config[resource_name]
+        political_25m3_cpu_s3 = political_25m3_cpu_config['download_path']
+        political_25m3_cpu_md5 = political_25m3_cpu_config.get('md5', '')
         # get the zip path calculated by the s3 path
         zip_path = os.path.join(CACHE_DIR, f'{resource_name}.zip')
         # the unzip path is calculated by the zip path
@@ -52,9 +52,9 @@ def auto_download(self):
             logger.info(f'try to unzip from zip_path: {zip_path}')
             if not os.path.exists(zip_path):
                 logger.info(f'zip_path: {zip_path} does not exist')
-                logger.info(f'downloading {political_24m7_s3}')
+                logger.info(f'downloading {political_25m3_cpu_s3}')
                 zip_path = download_auto_file(
-                    political_24m7_s3, zip_path, political_24m7_md5
+                    political_25m3_cpu_s3, zip_path, political_25m3_cpu_md5
                 )
             logger.info(f'unzipping {zip_path}')
             unzip_path = unzip_local_file(zip_path, unzip_path)
@@ -195,7 +195,7 @@ def get_singleton_political_detect() -> PoliticalDetector:
 def decide_political_by_prob(
     predictions: Tuple[str], probabilities: Tuple[float]
 ) -> float:
-    idx = predictions.index('__label__normal')
+    idx = predictions.index('__label__positive')
     normal_score = probabilities[idx]
     return float(normal_score)
 
@@ -226,8 +226,6 @@ def political_filter_cpu(data_dict: Dict[str, Any], language: str):
 
 if __name__ == '__main__':
     test_cases = []
-    test_cases.append('你好，我很高兴见到你！')
-    test_cases.append('hello, nice to meet you!')
     test_cases.append('你好，唔該幫我一個忙？')
     test_cases.append('Bawo ni? Mo nife Yoruba. ')
     test_cases.append(
diff --git a/llm_web_kit/model/porn_detector.py b/llm_web_kit/model/porn_detector.py
index f06251bf..d4830fb6 100644
--- a/llm_web_kit/model/porn_detector.py
+++ b/llm_web_kit/model/porn_detector.py
@@ -172,7 +172,7 @@ def __init__(self, model_path: str = None) -> None:
             model_config = json.load(reader)
         self.cls_index = int(model_config.get('cls_index', 1))
         self.use_sigmoid = bool(model_config.get('use_sigmoid', False))
-        self.max_tokens = int(model_config.get('max_tokens', 300))
+        self.max_tokens = int(model_config.get('max_tokens', 512))
         self.remain_tail = min(
             self.max_tokens - 1, int(model_config.get('remain_tail', -1))
         )
diff --git a/llm_web_kit/model/unsafe_words_detector.py b/llm_web_kit/model/unsafe_words_detector.py
index 28556fc5..194a618b 100644
--- a/llm_web_kit/model/unsafe_words_detector.py
+++ b/llm_web_kit/model/unsafe_words_detector.py
@@ -68,7 +68,7 @@ def get_ac(language='zh-en'):
     unsafe_words_file_path = auto_download(language)
     t2 = time.time()
     print(
-        f'-----------------auto_download cost time: {t2-t1} , language: {language}------------------'
+        f'-----------------auto_download cost time: {t2 - t1} , language: {language}------------------'
     )
     with open(unsafe_words_file_path, 'r') as f:
         lines = f.readlines()
@@ -85,6 +85,8 @@ def get_ac(language='zh-en'):
     words = {}
     for line in lines:
         w = json_loads(line)
+        if w.get('tag') == 'delete':
+            continue
         word = str(w.get('word') or '').lower()
         if not word:
             continue
@@ -163,7 +165,7 @@ def __init__(self, language='zh-en') -> None:
         self.ac = get_ac(language)
         t2 = time.time()
         print(
-            f'---------------UnsafeWordChecker init time: {t2-t1} , language: {language}-----------------'
+            f'---------------UnsafeWordChecker init time: {t2 - t1} , language: {language}-----------------'
         )
 
     def check_unsafe_words(self, content_str: str) -> list:
diff --git a/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html/math_mathinsight_custom_latex.html b/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html/math_mathinsight_custom_latex.html
new file mode 100644
index 00000000..17af7e85
--- /dev/null
+++ b/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html/math_mathinsight_custom_latex.html
@@ -0,0 +1,1228 @@
+
+
+
+
+
+Subtleties of differentiability in higher dimensions - Math Insight
+	
+	
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+      
          Math Insight
          
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+  
          
+    
          
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+      
          
+	
          Subtleties of differentiability in higher dimensions
          
+      
          
+    
          
+
+    
          
+      
          
+
+	
+
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+      
          
+    
          
+  
          
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+      
          
+        
          
+          
          
+
+
+
          The definition of differentiability
          
+
+
          The definition of differentiability in higher dimensions looks
+fairly intimidating at first glance.
+For this reason, we suggest beginning by reading the page about
+the
+intuition behind this definition.
+We repeat the definition from the end of that page.
          
+
+
+
          Definition: The function $\vc{f}: \R^n \to \R^m$ is
+differentiable
+ at the point $\vc{a}$ if there exists a linear transformation $\vc{T}: \R^n \to \R^m$ that satisfies the condition
+$$\lim_{\vc{x} \to \vc{a}}
+\frac{\| \vc{f}(\vc{x})-\vc{f}(\vc{a}) - \vc{T}(\vc{x}-\vc{a})\|}{\|\vc{x}-\vc{a}\|} = 0.$$
+
          
+
+
          The $m \times n$ 
+matrix associated with the linear transformation
+$\vc{T}$ is the matrix of partial derivatives, which we denote
+by $\jacm{f}(\vc{a})$.
+We can refer to $\jacm{f}(\vc{a})$ as the total derivative
+(or simply the derivative) of $\vc{f}$.
+
          
+
+
+
          The trouble with limits in higher dimensions
          
+
+
          The differentiabililty definition is based on the limit
+$\vc{x} \to \vc{a}$.
+In order for this limit to exist,
+we have to get the same result, no matter the route $\vc{x}$
+takes on its way to $\vc{a}$.
+If we find two routes that give different values for this limit,
+then we can conclude the limit does not exist.
+In many of the subtle cases where a function fails to be differentiable,
+one can find linear transformations $\vc{T}$ where
+$$\frac{\| \vc{f}(\vc{x})-\vc{f}(\vc{a}) - \vc{T}(\vc{x}-\vc{a})\|}{\|\vc{x}-\vc{a}\|}$$
+does go to zero when $\vc{x}$ approaches $\vc{a}$ along some routes,
+but it does not go to zero when $\vc{x}$ approaches $\vc{a}$ along other routes.
+Don't make the mistake concluding a function must be differentiable
+after checking that you get zero along just a few paths of $\vc{x} \to \vc{a}$.
+
          
+
+
          For a scalar-valued function of one variable, $f(x)$,
+checking that this limit is zero isn't too difficult.
+On the real line, there are only two routes for $x$ to get to $a$.
+It can approach $a$ from above or from below.
+If we find a candidate tangent line
+$$L(x)=f(a)+T(x-a)=f(a)+m(x-a)$$
+where the upper and lower limits
+are both zero, then we can conclude that we really found a tangent line and the function is differentiable.
          
+
+
          On the other hand, as pictured below, we may be able to find one
+candidate tangent line (in blue) where the limit is zero from the left
+but not the right.  And we may find a second candidate tangent line (in red)
+where the limit is zero from the right and not the left.
+But in this case, it easy to see what is happening and to identify
+the kink in $f(x)$ that makes it non-differentiable at $x=a$.
+
          
+
+
          Two different tangents from the left and the right
          
+
+
          Increasing the number of dimensions to two or more makes a limit
+become much more complicated.
+Now, one has to consider all the possible paths $\vc{x}$ could take to the point $\vc{a}$, such as the ones illustrated below for the case of two dimensions.
+The challenge is the fact that there are an infinite number of such paths,
+and one needs to get the same limiting value on all these paths in order for the limit to exist.
+
          
+
+
          A limit involves many paths in two or more dimensions
          
+
+
+
+
          Understanding the differentiability condition in two dimensions
          
+
+
          To develop an intuition of the subtle ways in which the differentiability
+condition could be violated, it's enough to stick with
+scalar-valued functions of two variables, $f: \R^2 \to \R$.
+In this case, we can plot the graph of $f$ as a surface,
+and we understand that the linear approximation
+$$L(\vc{x}) = f(\vc{a}) + T(\vc{x}-\vc{a})$$
+is a tangent plane.
          
+
+
          We will also leave our formal definition of differentiability aside,
+and just focus on the geometry of the tangent plane.
+You can gather intuition about the correspondence between this definition
+and tangent planes in the
+page on the definition of differentiability.
+The differentiability definition requires the tangent plane to be tangent to the graph
+along any path of mathin that goes to the point $\vc{a}$.
          
+
+
+
          The important consequence of this fact is that the existence of a
+derivative is much stronger than the existence of partial derivatives.
+Partial derivatives involve the limit of $\vc{f}$ only along
+directions parallel to the coordinate axes (in the above picture, such
+limits would be arrows coming straight in from the left, right, top,
+and bottom). 
          
+
+
          If a function varies smoothly along the paths coming into
+$\vc{a}$ from the positive and negative $x$ directions, the partial
+derivative with respect to $x$ at the point $\vc{a}$ will exist.  If a
+function varies smoothly along the paths coming in from the positive
+and negative $y$ directions, the partial derivative with respect to $y$
+at the point $\vc{a}$ will exist.
          
+
+
          But just because the function behaves “nicely” along those four
+directions, it doesn't mean it behaves nicely along every path coming
+into $\vc{a}$.  
          
+
+
          For example, look at the following surface.  At the origin (i.e.,
+$\vc{a}=(0,0)$), the partial derivatives exist and are zero. (If one
+moves in the positive or negative $x$ or $y$ direction, the function
+is constant.)
          
+
+
          Applet: Non-differentiable function with partial derivatives
          Applet loading
          Non-differentiable function with partial derivatives. The partial derivatives of this function $f(x,y)$ exist at the origin, $\pdiff{f}{x}(0,0)=0$ and $\pdiff{f}{y}(0,0)=0$, as the function is constant along the $x$ and $y$ axis, $f(x,0)=f(0,y)=0$.  However, the slopes coming into the origin from other directions are non-zero.  If there were a tangent plane at the origin, it would have to be the horizontal plane $z=0$, as that is the only plane that would be tangent along the $x$ and $y$ directions.  Clearly, the slopes would not match in other directions, so this plane is not tangent.   Therefore, there is no tangent plane at $\vc{a}=(0,0)$, and the function is not differentiable there.  You can drag the blue point on the slider to remove the folds in the surface, but that does not change the partial derivatives at the origin.  The wrinkle at the origin is enough to make the function non-differentiable.
          More information about applet. 
          
+
+
          However, if you approach the origin from a path coming from any other
+direction, the slope of the path will be nonzero.  You'll be climbing
+uphill as you reach the origin (or possibly going downhill if you changed the surface to remove the folds).
          
+
+
          If the function were differentiable at the origin, it would have a
+tangent plane at the origin.
+If a tangent plane existed, the slopes would have to match the partial
+derivatives.  In this case, since the
+partial derivatives are zero, the only option for the tangent plane is
+the horizontal plane, as shown below.  Clearly this plane is not a
+tangent plane, as it is not tangent to paths approaching the
+origin from all directions.  This surface is a graph of a function
+that has partial derivatives at the origin but is not differentiable
+at the origin.
          
+
+
          Applet: Non-differentiable function with partial derivatives and no tangent plane
          Applet loading
          Non-differentiable function with partial derivatives and no tangent plane. Since the partial derivatives of this function $f(x,y)$ exist and are zero at the origin, the only possible candidate for a tangent plane is the horizontal plane $z=0$ shown here.  Although this plane is tangent along the $x$ and $y$ directions, the slopes of the function clearly do not match in other directions.  Therefore, there is no tangent plane at $\vc{a}=(0,0)$, and the function is not differentiable there.  You can drag the blue point on the slider to remove the folds in the surface, but that does not change the partial derivatives at the origin.  The wrinkle at the origin is enough to make the function non-differentiable.
          More information about applet. 
          
+
+
          For the above figures, if you drag the blue dot on the sliders, you can
+look at other surfaces that are not differentiable. The partial
+derivatives at the origin do not change, so the candidate for the
+tangent plane is still the horizontal.  However, you can see that the
+horizontal plane is not a tangent plane.
          
+
+
          Note that for these functions, you could write down the matrix of
+partial derivatives.  It would  simply  be $[0 \, 0]$.  But that
+matrix would not correspond to the derivative.
          
+
+
+
          Further example
          
+
+
          The following is another example function that has partial derivatives at
+the origin but is not differentiable.  For this example, we have an
+equation for the function.
+\begin{align*}
+  f(x,y) =
+  \begin{cases}
+    \displaystyle
+    \frac{x^2y}{x^2+y^2} & \text{if } (x,y) \ne (0,0)\\
+    0 & \text{if } (x,y) = (0.0)
+  \end{cases}
+\end{align*}
+
          
+
+
          
+The graph of the function is shown below.
          
+
+
          Applet: Non-differentiable function with partial derivatives
          Applet loading
          Non-differentiable function with partial derivatives. The partial derivatives of this function $f(x,y)$ are zero at the origin, $\pdiff{f}{x}(0,0)=\pdiff{f}{y}(0,0)=0$.  Therefore, the only possibility for a tangent plane would be a horizontal plane.  However, since the slopes of this function coming to the origin along different directions are not zero, a horizontal plane cannot be tangent.  We conclude that no tangent plane exists at the origin and this function is not differentiable there.
          More information about applet. 
          
+
+
          We can show partial derivatives exist at (0,0) but that function is not
+differentiable at (0,0).  Since this function is defined in piecewise fashion
+around the origin, there are no simple formulas for the partial derivatives.
+We have to use the limit definition of the partial derivatives.
+For the partial derivative with respect to $x$, this formula is
+\begin{align*}
+  \pdiff{f}{x}(0,0) &=
+  \lim_{h \rightarrow 0} \frac{f(0+h,0)-f(0,0)}{h}.
+\end{align*}
+\begin{align*}
+\frac{\| \vc{f}(\vc{x})-\vc{f}(\vc{a}) - \vc{T}(\vc{x}-\vc{a})\|}{\|\vc{x}-\vc{a}\|}
+\end{align*}
+Since $f(0,0)=0$ and  $f(0+h,0)=f(h,0) = 0$, we calculate
+\begin{align*}
+  \pdiff{f}{x}(0,0) &=
+  \lim_{h \rightarrow 0} \frac{f(0+h,0)-f(0,0)}{h}\\
+  &= \lim_{h \rightarrow 0}
+  \frac{0- 0}{h}
+  = \lim_{h \rightarrow 0}
+  0 =0.
+\end{align*}
+Similarly, you can show that $\displaystyle \pdiff{f}{y}(0,0)=0$.
+
          
+
+
+
          If the function had a tangent plane at (0,0), then it would have to
+be the flat plane with equation $z=f(0,0)=0$.
          
+
+
          Applet: Non-differentiable function with partial derivatives and no tangent plane
          Applet loading
          Non-differentiable function with partial derivatives and no tangent plane. A function that is not differentiable at the origin is shown with the only possibility for a tangent plane at the origin. As the partial derivatives are zero at the origin, this candidate plane is a horizontal plane.  However, this plane is not tangent since the slopes of this function coming to the origin along different directions are not zero.
          More information about applet. 
          
+
+
+
          As you may see from the graph, if one approaches the origin along a diagonal,
+the tangent to the path would not be in the plane $z=0$.
          
+
+
          Overcoming subtleties
          
+
+
          Thankfully, in many cases, there is a simple way to overcome these subtleties and know for sure that a function is differentiable.
+The differentiability theorem tells us that a function with continuous partial derivatives must be differentiable and therefore cannot have any of the crazy behavior of the above examples.
+By the same token, since the above functions are not differentiable, they must have discontinuous partial derivatives.
+For the latter example, we can visualize how it indeed has discontinuous partial derivatives.
+
+          
          
+        
          
+        
          
+          
          
+
+
+
+
          
+
          Thread navigation
          
+
+
+
          Multivariable calculus
          
+
+
+
+
          Math 2374
          
+
+
+
          
+
+
+
+
+
          
+
          Similar pages
          
+
+
          
+
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+          
          
+        
          
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+
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+      
+      
          
+        
          
+          
          
+            
          
+              
          
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+
          See also
          
+
+
          
+
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+              
          
+            
          
+            
          
+              
          
+
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+
+
          
+
          Lighten up
          
+
+
          
+
+
+              
          
+            
          
+          
          
+        
          
+        
          
+          
          
+
+
+
+
          
+
          Go deeper
          
+
+
          
+
+
+          
          
+        
          
+      
          
+
+    
          
+  
          
+
          
+
          
+  
          
+    
          
+
+
+
+
          Cite this as
          
+
+
          Nykamp DQ, “Subtleties of differentiability in higher dimensions.” From Math Insight.
+  http://mathinsight.org/differentiability_multivariable_subtleties
          
+
          
+
+
          Keywords:
+derivative, differentiability, linear transformation, partial derivative
+
          
+
+
          
+
          Send us a message about “Subtleties of differentiability in higher dimensions”
          
+
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+
+
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+
+  
          
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+
          
+
          
+
          
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+
          
+
+
          
+ 
+
          
+
          
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+
          
+
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+
          
+
          Credits
          
+Thanks to David Arnold for correcting errors.
+
          
+
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+    
          
+  
          
+
          
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+
\ No newline at end of file
diff --git a/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html_data_input.jsonl b/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html_data_input.jsonl
index 34b7bf74..9aa54bf7 100644
--- a/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html_data_input.jsonl
+++ b/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html_data_input.jsonl
@@ -96,3 +96,4 @@
 {"track_id": "8ec3baea-f43f-4f57-bde4-b67507bd56c1", "dataset_name": "CC", "url": "http://www.rbej.com/content/10/1/90","data_source_category": "HTML",  "path":"sub_sup_exception.html", "file_bytes": 1000, "page_layout_type":"artical", "meta_info": {"input_datetime": "2020-01-01 00:00:00"}}
 {"track_id": "8061e636-31c3-4cfa-ac1a-ad1a2c38360c", "dataset_name": "CC", "url": "https://blue-reg.com/categorie/p/","data_source_category": "HTML",  "path":"list_item_notext.html", "file_bytes": 1000, "page_layout_type":"artical", "meta_info": {"input_datetime": "2020-01-01 00:00:00"}}
 {"track_id": "test_zhihu_custom_tag", "dataset_name": "test_zhihu_custom_tag", "url": "https://zhuanlan.zhihu.com/p/22925478480","data_source_category": "HTML",  "path":"math_zhihu_custom_tag.html", "file_bytes": 1000, "page_layout_type":"artical", "meta_info": {"input_datetime": "2020-01-01 00:00:00"}}
+{"track_id": "test_math_insight_custom_latex", "dataset_name": "test_math_insight_custom_latex", "url": "https://mathinsight.org/differentiability_multivariable_subtleties","data_source_category": "HTML",  "path":"math_mathinsight_custom_latex.html", "file_bytes": 1000, "page_layout_type":"artical", "meta_info": {"input_datetime": "2020-01-01 00:00:00"}}
diff --git a/tests/llm_web_kit/extractor/html/recognizer/test_math.py b/tests/llm_web_kit/extractor/html/recognizer/test_math.py
index 1ecadfdc..75548377 100644
--- a/tests/llm_web_kit/extractor/html/recognizer/test_math.py
+++ b/tests/llm_web_kit/extractor/html/recognizer/test_math.py
@@ -2,6 +2,8 @@
 from pathlib import Path
 
 from llm_web_kit.exception.exception import HtmlMathRecognizerException
+from llm_web_kit.extractor.html.pre_extractor import \
+    HTMLFileFormatCleanTagsPreExtractor
 from llm_web_kit.extractor.html.recognizer.cc_math.common import (
     CCMATH_INLINE, CSDN, ZHIHU)
 from llm_web_kit.extractor.html.recognizer.cc_math.tag_script import (
@@ -450,7 +452,7 @@ def test_math_recognizer_html(self):
             raw_html_path = base_dir.joinpath(test_case['input'][0])
             # print('raw_html_path::::::::', raw_html_path)
             base_url = test_case['base_url']
-            raw_html = raw_html_path.read_text()
+            raw_html = raw_html_path.read_text(encoding='utf-8')
             parts = self.math_recognizer.recognize(base_url, [(html_to_element(raw_html), html_to_element(raw_html))], raw_html)
             # print(parts)
             # 将parts列表中第一个元素拼接保存到文件，带随机数
@@ -458,12 +460,24 @@ def test_math_recognizer_html(self):
             # with open('parts'+str(random.randint(1, 100))+".html", 'w') as f:
             #     for part in parts:
             #         f.write(str(part[0]))
+            # 创建预处理器并清理隐藏元素
+            pre_extractor = HTMLFileFormatCleanTagsPreExtractor({})
+            data_json = {'html': raw_html, 'url': base_url}
+            data_json = pre_extractor._do_pre_extract(data_json)
+            cleaned_html = data_json['html']
+
+            # 使用清理后的HTML进行公式识别
+            parts = self.math_recognizer.recognize(
+                base_url,
+                [(html_to_element(cleaned_html), html_to_element(cleaned_html))],
+                cleaned_html
+            )
             # 检查行间公式抽取正确性
             new_parts = []
             for part in parts:
                 new_parts.append((element_to_html(part[0]), element_to_html(part[1])))
             parts = [part[0] for part in new_parts if CCTag.CC_MATH_INTERLINE in part[0]]
-            expect_text = base_dir.joinpath(test_case['expected']).read_text().strip()
+            expect_text = base_dir.joinpath(test_case['expected']).read_text(encoding='utf-8').strip()
             expect_formulas = [formula for formula in expect_text.split('\n') if formula]
             self.assertEqual(len(parts), len(expect_formulas))
             # answers = []
diff --git a/tests/llm_web_kit/extractor/test_extractor_chain.py b/tests/llm_web_kit/extractor/test_extractor_chain.py
index a7578fbf..c01284b0 100644
--- a/tests/llm_web_kit/extractor/test_extractor_chain.py
+++ b/tests/llm_web_kit/extractor/test_extractor_chain.py
@@ -64,7 +64,7 @@ def setUp(self):
                     continue
                 self.data_json.append(json.loads(line))
 
-        assert len(self.data_json) == 98
+        assert len(self.data_json) == 99
 
         # Config for HTML extraction
         self.config = load_pipe_tpl('html-test')
@@ -188,13 +188,15 @@ def test_html_pipeline_suit_2(self):
         # Create DataJson from test data
         input_data = DataJson(test_data)
         result = chain.extract(input_data)
-
+        # 打印content_list内容
+        content_list = result.get_content_list()._get_data()
+        print('Content List:', json.dumps(content_list, ensure_ascii=False, indent=2))
         # Verify basic properties
         self.assertEqual(result.get_dataset_name(), 'test_pipeline_suit')
         self.assertEqual(result['track_id'], 'stackoverflow_math')
 
         html_content_list = result.get_content_list()[0]
-        assert len(html_content_list) == 22
+        assert len(html_content_list) == 24
 
     def test_mathlab_html_to_md(self):
         """测试第二个数据：这个数据会丢失一些文本信息."""
@@ -778,6 +780,21 @@ def test_zhihu_custom_tag(self):
         self.assertIn(r'$\mathrm{return}=0+0+0+1=1.$', md_content)
         self.assertIn(r'\begin{aligned} V(s) & =\mathbb{E}[G_t|S_t=s] \\  & =\mathbb{E}[R_t+\gamma R_{t+1}+\gamma^2R_{t+2}+\ldots|S_t=s] \\  & =\mathbb{E}[R_t+\gamma(R_{t+1}+\gamma R_{t+2}+\ldots)|S_t=s] \\  & =\mathbb{E}[R_t+\gamma G_{t+1}|S_t=s] \\  & =\mathbb{E}[R_t+\gamma V(S_{t+1})|S_t=s] \end{aligned}', md_content)
 
+    def test_mathinsight_custom_latex(self):
+        """测试mathinsight自定义latex."""
+        chain = ExtractSimpleFactory.create(self.config)
+        self.assertIsNotNone(chain)
+        test_data = self.data_json[98]
+        # 验证URL中包含mathinsight.org
+        # self.assertIn('mathinsight.org', test_data['url'])
+        input_data = DataJson(test_data)
+        result = chain.extract(input_data)
+        md_content = result.get_content_list().to_nlp_md()
+        with open('output_mathinsight.md', 'w', encoding='utf-8') as f:
+            f.write(md_content)
+        self.assertIn(r'L(\mathbf{x}) = f(\mathbf{a}) + T(\mathbf{x}-\mathbf{a})', md_content)
+        self.assertIn(r'\frac{x^2y}{x^2+y^2} & \text{if } (x,y) \ne (0,0)\\',md_content)
+
     def test_double_ul(self):
         """测试双重ul标签."""
         chain = ExtractSimpleFactory.create(self.config)
diff --git a/tests/llm_web_kit/model/test_model_impl.py b/tests/llm_web_kit/model/test_model_impl.py
index e06ce093..402ec4fc 100644
--- a/tests/llm_web_kit/model/test_model_impl.py
+++ b/tests/llm_web_kit/model/test_model_impl.py
@@ -92,14 +92,14 @@ def test_convert_result_to_response(self, mock_load_model):
         mock_load_model.return_value = MagicMock()
         model = PoliticalCPUModel()
 
-        # Test case where political_prob > 0.99 (should be flagged)
-        result = {'political_prob': 0.995}
+        # Test case where political_prob > 0.89 (should be flagged)
+        result = {'political_prob': 0.9}
         response = model.convert_result_to_response(result)
         assert response.is_remained
         assert response.details == result
 
-        # Test case where political_prob <= 0.99 (should not be flagged)
-        result = {'political_prob': 0.985}
+        # Test case where political_prob <= 0.89 (should not be flagged)
+        result = {'political_prob': 0.88}
         response = model.convert_result_to_response(result)
         assert not response.is_remained
         assert response.details == result
diff --git a/tests/llm_web_kit/model/test_politicis_detector.py b/tests/llm_web_kit/model/test_politicis_detector.py
index d5db5c77..1f824a4c 100644
--- a/tests/llm_web_kit/model/test_politicis_detector.py
+++ b/tests/llm_web_kit/model/test_politicis_detector.py
@@ -23,7 +23,7 @@
 from llm_web_kit.model.resource_utils import CACHE_DIR
 
 
-class TestPoliticalDetector:
+class TestPoliticalDetector(TestCase):
 
     @patch('transformers.AutoTokenizer.from_pretrained')
     @patch('llm_web_kit.model.politics_detector.fasttext.load_model')
@@ -34,7 +34,7 @@ def test_init(self, mock_auto_download, mock_load_model, mock_auto_tokenizer):
         _ = PoliticalDetector()
         mock_load_model.assert_called_once_with('/fake/model/path/model.bin')
         mock_auto_tokenizer.assert_called_once_with(
-            '/fake/model/path/internlm2-chat-20b',
+            '/fake/model/path/qwen2.5_7b_tokenizer',
             use_fast=False,
             trust_remote_code=True,
         )
@@ -45,7 +45,7 @@ def test_init(self, mock_auto_download, mock_load_model, mock_auto_tokenizer):
         _ = PoliticalDetector('custom_model_path')
         mock_load_model.assert_called_once_with(os.path.join('custom_model_path', 'model.bin'))
         mock_auto_tokenizer.assert_called_once_with(
-            os.path.join('custom_model_path', 'internlm2-chat-20b'),
+            os.path.join('custom_model_path', 'qwen2.5_7b_tokenizer'),
             use_fast=False,
             trust_remote_code=True,
         )
@@ -60,6 +60,62 @@ def test_predict(self, mock_auto_download, mock_load_model, mock_auto_tokenizer)
         assert predictions == ['label1', 'label2']
         assert probabilities == [0.9, 0.1]
 
+import logging
+import os
+from unittest import TestCase
+from unittest.mock import MagicMock, patch
+
+from loguru import logger
+
+# 假设你的类和函数在 llm_web_kit.model.politics_detector 模块中
+from llm_web_kit.model.politics_detector import PoliticalDetector
+
+
+class TestPoliticalDetectorWithAutoDownload(TestCase):
+
+    @classmethod
+    def setUpClass(cls):
+        # 禁用所有日志输出，防止 loguru 报错
+        logger.disable('llm_web_kit')
+
+    @patch('llm_web_kit.model.politics_detector.load_config')
+    @patch('llm_web_kit.model.politics_detector.os.path.exists', return_value=False)
+    @patch('llm_web_kit.model.politics_detector.download_auto_file', return_value='/tmp/cache/political-25m3_cpu.zip')
+    @patch('llm_web_kit.model.politics_detector.unzip_local_file', return_value='/tmp/cache/political-25m3_cpu')
+    @patch('llm_web_kit.model.politics_detector.logger.info')
+    def test_auto_download_triggers_config_access_and_logging(
+        self,
+        mock_logger_info,
+        mock_unzip_local_file,
+        mock_download_auto_file,
+        mock_os_path_exists,
+        mock_load_config
+    ):
+        # 构造一个假的配置返回值
+        mock_load_config.return_value = {
+            'resources': {
+                'political-25m3_cpu': {
+                    'download_path': 's3://fake-bucket/political-25m3_cpu.zip',
+                    'md5': 'fake_md5_hash'
+                }
+            }
+        }
+
+        # 创建 detector 实例，这会触发 auto_download()
+        with patch('transformers.AutoTokenizer.from_pretrained'), \
+             patch('llm_web_kit.model.politics_detector.fasttext.load_model'):
+
+            detector = PoliticalDetector()
+
+        # 验证 auto_download 返回的路径是否正确
+        self.assertEqual(detector.auto_download(), '/tmp/cache/political-25m3_cpu')
+
+        # 验证 load_config 是否至少被调用了一次
+        self.assertGreaterEqual(mock_load_config.call_count, 1)
+
+        # 验证 logger.info 是否被调用，并包含 download_path
+        mock_logger_info.assert_any_call('downloading s3://fake-bucket/political-25m3_cpu.zip')
+
 class TestGTEModel(TestCase):
     @patch('llm_web_kit.model.politics_detector.GTEModel.auto_download')
     @patch('llm_web_kit.model.politics_detector.import_transformer')
@@ -228,11 +284,11 @@ def test_predict(self, mock_get_key, mock_torch, mock_pre_process):
 
 
 def test_decide_political_by_prob():
-    predictions = ['__label__normal', '__label__political']
+    predictions = ['__label__positive', '__label__negative']
     probabilities = [0.6, 0.4]
     assert decide_political_by_prob(predictions, probabilities) == 0.6
 
-    predictions = ['__label__political', '__label__normal']
+    predictions = ['__label__negative', '__label__positive']
     probabilities = [0.7, 0.3]
     assert decide_political_by_prob(predictions, probabilities) == 0.3
 
@@ -240,7 +296,7 @@ def test_decide_political_by_prob():
 def test_decide_political_func():
     political_detect = MagicMock()
     political_detect.predict.return_value = (
-        ['__label__normal', '__label__political'],
+        ['__label__positive', '__label__negative'],
         [0.6, 0.4],
     )
     test_str = 'test text'
diff --git a/tests/st/test_st.py b/tests/st/test_st.py
index a89d471b..625e6b83 100644
--- a/tests/st/test_st.py
+++ b/tests/st/test_st.py
@@ -78,7 +78,7 @@ def test_st_bench(self):
             output_path=output_path,
         )
 
-        with open(self.pipeline_data_path, 'r') as f:
+        with open(self.pipeline_data_path, 'r', encoding='utf-8') as f:
             for line in f:
                 data_json = json.loads(line.strip())
                 # files结构是{'filename': {'url': '', 'filepath': ''}}，获取filepath
@@ -89,7 +89,7 @@ def test_st_bench(self):
                 try:
                     output, content_list, statics = eval_ours_extract_html(self.chainConfig, data_json)
                     # 断言statics中的元素数量和groundtruth_filepath中的元素数量一致
-                    with open(groundtruth_filepath, 'r') as f:
+                    with open(groundtruth_filepath, 'r', encoding='utf-8') as f:
                         groundtruth = json.loads(f.readline().strip())
                     # 断言equation-interline, paragraph.equation-inline和list.equation-inline元素数一致
                     self.assertEqual(statics.get('equation-interline'), groundtruth.get('statics', {}).get('equation-interline'), msg=f'{fileName}抽取equation-interline数量和groundtruth:{groundtruth_filepath}不一致')