Mathematical Structure (Expanded) — MathJax Frontend
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Foundations
\$begin:math:display$
\\\\mathbb{N}=\\\\{0,1,2,3,\\\\dots\\\\},\\\\quad
\\\\mathbb{Z}=\\\\{\\\\dots,-2,-1,0,1,2,\\\\dots\\\\},\\\\quad
\\\\mathbb{Q}=\\\\left\\\\{\\\\frac{p}{q}:p\\\\in\\\\mathbb{Z},\\\\ q\\\\in\\\\mathbb{Z}\\\\setminus\\\\{0\\\\}\\\\right\\\\}
\\$end:math:display$
\$begin:math:display$
\\\\mathbb{R}=\\\\text{completion}(\\\\mathbb{Q})\\\\text{ under }d(x,y)=|x-y|,\\\\quad
\\\\mathbb{C}=\\\\{a+bi:a,b\\\\in\\\\mathbb{R},\\\\ i^2=-1\\\\}
\\$end:math:display$
\$begin:math:display$
\\\\forall a,b,c\\\\in\\\\mathbb{R}:
\\\\ a+b=b+a,\\\\ (a+b)+c=a+(b+c),\\\\ a(b+c)=ab+ac,\\\\ a\\\\cdot 1=a,\\\\ a+0=a
\\$end:math:display$
\$begin:math:display$
\\\\forall a\\\\in\\\\mathbb{R}\\\\setminus\\\\{0\\\\}:
\\\\ a\\\\cdot a^{-1}=1,\\\\quad \\\\frac{a}{b}=a\\\\cdot b^{-1}\\\\ (b\\\\neq 0)
\\$end:math:display$
Limits and Continuity
\$begin:math:display$
\\\\lim_{x\\\\to a}f(x)=L\\\\ \\\\Longleftrightarrow\\\\ \\\\forall\\\\varepsilon>0\\\\ \\\\exists\\\\delta>0:
\\\\ 0<|x-a|<\\\\delta\\\\Rightarrow |f(x)-L|<\\\\varepsilon
\\$end:math:display$
\$begin:math:display$
f\\\\text{ continuous at }a\\\\ \\\\Longleftrightarrow\\\\ \\\\lim_{x\\\\to a}f(x)=f(a)
\\$end:math:display$
\$begin:math:display$
\\\\lim_{x\\\\to 0}\\\\frac{\\\\sin x}{x}=1,\\\\quad
\\\\lim_{x\\\\to 0}\\\\frac{1-\\\\cos x}{x^2}=\\\\frac12,\\\\quad
\\\\lim_{n\\\\to\\\\infty}\\\\left(1+\\\\frac{x}{n}\\\\right)^n=e^x
\\$end:math:display$
Series Expansions
\$begin:math:display$
e^x=\\\\sum_{n=0}^{\\\\infty}\\\\frac{x^n}{n!}
\\$end:math:display$
\$begin:math:display$
\\\\sin x=\\\\sum_{n=0}^{\\\\infty}(-1)^n\\\\frac{x^{2n+1}}{(2n+1)!},\\\\quad
\\\\cos x=\\\\sum_{n=0}^{\\\\infty}(-1)^n\\\\frac{x^{2n}}{(2n)!}
\\$end:math:display$
\$begin:math:display$
\\\\frac{1}{1-x}=\\\\sum_{n=0}^{\\\\infty}x^n\\\\ (|x|<1),\\\\quad
\\\\ln(1+x)=\\\\sum_{n=1}^{\\\\infty}(-1)^{n-1}\\\\frac{x^n}{n}\\\\ (|x|<1)
\\$end:math:display$
Circle Definitions and Equivalences
\$begin:math:display$
S^1(r)=\\\\{(x,y)\\\\in\\\\mathbb{R}^2:\\\\ x^2+y^2=r^2\\\\},\\\\quad D=2r
\\$end:math:display$
\$begin:math:display$
\\\\pi=\\\\frac{C}{D}=\\\\frac{C}{2r},\\\\quad C=2\\\\pi r
\\$end:math:display$
\$begin:math:display$
A=\\\\pi r^2=\\\\frac12Cr
\\$end:math:display$
\$begin:math:display$
\\\\pi=4\\\\arctan(1)=4\\\\int_{0}^{1}\\\\frac{1}{1+x^2}\\\\,dx
\\$end:math:display$
\$begin:math:display$
\\\\pi=2\\\\int_{0}^{1}\\\\frac{1}{\\\\sqrt{1-x^2}}\\\\,dx
\\$end:math:display$
Rhind Papyrus Approximation as Algebra
\$begin:math:display$
A\\\\approx\\\\left(D-\\\\frac{D}{9}\\\\right)^2=\\\\left(\\\\frac{8D}{9}\\\\right)^2=\\\\frac{64D^2}{81}
\\$end:math:display$
\$begin:math:display$
A=\\\\frac{\\\\pi D^2}{4}
\\$end:math:display$
\$begin:math:display$
\\\\frac{\\\\pi D^2}{4}=\\\\frac{64D^2}{81}\\\\Rightarrow\\\\frac{\\\\pi}{4}=\\\\frac{64}{81}\\\\Rightarrow\\\\pi=\\\\frac{256}{81}
\\$end:math:display$
\$begin:math:display$
\\\\frac{256}{81}=3.1604938271604937\\\\ldots
\\$end:math:display$
Archimedes Bounds (Inequality Form)
\$begin:math:display$
3+\\\\frac{10}{71}<\\\\pi<3+\\\\frac{1}{7}
\\$end:math:display$
\$begin:math:display$
3+\\\\frac{10}{71}=\\\\frac{223}{71}=3.140845070422535\\\\ldots
\\$end:math:display$
\$begin:math:display$
3+\\\\frac{1}{7}=\\\\frac{22}{7}=3.142857142857143\\\\ldots
\\$end:math:display$
Continued Fractions and Zu Chongzhi Convergent
\$begin:math:display$
\\\\pi=[3;7,15,1,292,\\\\dots]
=3+\\\\cfrac{1}{7+\\\\cfrac{1}{15+\\\\cfrac{1}{1+\\\\cfrac{1}{292+\\\\cdots}}}}
\\$end:math:display$
\$begin:math:display$
[3;7]=\\\\frac{22}{7},\\\\quad
[3;7,15]=\\\\frac{333}{106},\\\\quad
[3;7,15,1]=\\\\frac{355}{113}
\\$end:math:display$
\$begin:math:display$
\\\\frac{355}{113}=3.1415929203539825\\\\ldots
\\$end:math:display$
Leibniz via Power Series and Integration
\$begin:math:display$
\\\\frac{1}{1+x^2}=\\\\sum_{n=0}^{\\\\infty}(-1)^n x^{2n}\\\\ (|x|<1)
\\$end:math:display$
\$begin:math:display$
\\\\int_0^1\\\\frac{1}{1+x^2}\\\\,dx
=\\\\int_0^1\\\\sum_{n=0}^{\\\\infty}(-1)^n x^{2n}\\\\,dx
=\\\\sum_{n=0}^{\\\\infty}\\\\frac{(-1)^n}{2n+1}
\\$end:math:display$
\$begin:math:display$
\\\\frac{\\\\pi}{4}=\\\\sum_{n=0}^{\\\\infty}\\\\frac{(-1)^n}{2n+1}
\\\\Rightarrow
\\\\pi=4\\\\sum_{n=0}^{\\\\infty}\\\\frac{(-1)^n}{2n+1}
\\$end:math:display$
Gaussian Integral in \$begin:math:text$\\\\mathbb{R}^2\\$end:math:text$
\$begin:math:display$
I=\\\\int_{-\\\\infty}^{\\\\infty}e^{-x^2}\\\\,dx
\\$end:math:display$
\$begin:math:display$
I^2=
\\\\left(\\\\int_{-\\\\infty}^{\\\\infty}e^{-x^2}\\\\,dx\\\\right)
\\\\left(\\\\int_{-\\\\infty}^{\\\\infty}e^{-y^2}\\\\,dy\\\\right)
=
\\\\int_{\\\\mathbb{R}^2}e^{-(x^2+y^2)}\\\\,dx\\\\,dy
\\$end:math:display$
\$begin:math:display$
x=r\\\\cos\\\\theta,\\\\ y=r\\\\sin\\\\theta,\\\\ dx\\\\,dy=r\\\\,dr\\\\,d\\\\theta
\\$end:math:display$
\$begin:math:display$
I^2=\\\\int_{0}^{2\\\\pi}\\\\int_{0}^{\\\\infty}e^{-r^2}r\\\\,dr\\\\,d\\\\theta
\\$end:math:display$
\$begin:math:display$
u=r^2,\\\\ du=2r\\\\,dr\\\\Rightarrow
\\\\int_0^{\\\\infty}e^{-r^2}r\\\\,dr=\\\\frac12\\\\int_0^{\\\\infty}e^{-u}\\\\,du=\\\\frac12
\\$end:math:display$
\$begin:math:display$
I^2=\\\\int_0^{2\\\\pi}\\\\frac12\\\\,d\\\\theta=\\\\pi
\\\\Rightarrow
I=\\\\sqrt{\\\\pi}
\\$end:math:display$
Euler Identity from Series
\$begin:math:display$
e^{i\\\\theta}=\\\\sum_{n=0}^{\\\\infty}\\\\frac{(i\\\\theta)^n}{n!}
=
\\\\left(\\\\sum_{n=0}^{\\\\infty}(-1)^n\\\\frac{\\\\theta^{2n}}{(2n)!}\\\\right)
+
i\\\\left(\\\\sum_{n=0}^{\\\\infty}(-1)^n\\\\frac{\\\\theta^{2n+1}}{(2n+1)!}\\\\right)
=
\\\\cos\\\\theta+i\\\\sin\\\\theta
\\$end:math:display$
\$begin:math:display$
e^{i\\\\pi}=\\\\cos\\\\pi+i\\\\sin\\\\pi=-1
\\\\Rightarrow
e^{i\\\\pi}+1=0
\\$end:math:display$
Zeta Value
\$begin:math:display$
\\\\zeta(s)=\\\\sum_{n=1}^{\\\\infty}\\\\frac{1}{n^s},\\\\quad
\\\\zeta(2)=\\\\sum_{n=1}^{\\\\infty}\\\\frac{1}{n^2}=\\\\frac{\\\\pi^2}{6}
\\$end:math:display$
\$begin:math:display$
\\\\pi=\\\\sqrt{6\\\\sum_{n=1}^{\\\\infty}\\\\frac{1}{n^2}}
\\$end:math:display$
Reference Constant
\$begin:math:display$
\\\\pi=3.14159265358979323846264338327950288419716939937510\\\\ldots
\\$end:math:display$
\$begin:math:text$\\\\LaTeX\\$end:math:text$ rendering uses MathJax v3 within the iframe scope.