List of mathematical formulae

Math These are the mathematical formulae used by mathematicians for serious proofs.

\(e^{i\pi}=-1\)
Due to Euler's identity, people who support tau, or \(2\pi\), think the better alternative is \(e^{i\tau}=1\), which, they say, is easier and should replace the identity.
 * \(e^{i\pi}=-1\), also known as Euler's identity.

However, there are other types:
 * If n is odd, \(e^{ni\pi}=-1\).
 * If n is even, \(e^{ni\pi}=1\).

This makes this discussion useless.

Quadratic formula
The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) when \(ax^2+bx+c=0\).

Algebraic identities

 * (x-y)(x+y) = x² - y². This is the difference of 2 square numbers formula.
 * \((a+b)^2=a^2+b^2+2ab\). This is a useful identity with the previous identity.
 * \(x^ax^b=x^{a+b}\). This is one rule for exponentiation.

e
\[e=\lim_{n\rightarrow\infty}({1\over n})^n\]

\[e=\sum_{n=0}^{\infty} \frac{1}{n!}\]

\(\pi\)
\[\pi=4(4\arctan{1\over5}-\arctan{1\over239})\]

\[\pi=4\arctan{1}\]

\[\pi=\sqrt{6(\sum_{n=1}^{\infty} \frac{1}{n^2})}\] This uses \({\pi^2\over6}=\zeta({2})\).

\[\pi=({\frac{2\sqrt{2}}{9801}\sum_{k=0}^{\infty} \frac{(4k)!(1103+26390k)}{(k!)^4396^{4k}}})^{-1}\]

\(\phi\)
\[\phi= {1+\sqrt{5}}\over{2}\]

\[\phi=\sqrt{1\frac14}+\frac12\]

\(\delta_S\), Silver ratio
\[\delta_S=1+\sqrt{2}\] \[\delta_S=\frac{2+\sqrt{8}}{2}\]