### Definition Until calculus, π is defined as the ratio of a circle's [[circumference]] to its [[Diameter]]. This is circular reasoning, so in [[Calculus]] we can define it a little more directly as the arc length of half of the unit circle (Weierstrass) $\pi=\int_{-1}^{1} \frac{d x}{\sqrt{1-x^{2}}}$ It can also be defined as half of the distance (absolute value of the difference) between terms in the sequence of solutions to $\exp z=1$ More informally, the magnitude of the derivative of the homomorphism that maps the quotient group $\mathbb{R}/\mathbb{Z}$ (reals modulo integers) onto complex numbers on the unit circle is twice the value defined as π Regardless, all of these definitions point at the same concept. π shows up to explain the differences between "straight" and "circular" distances Some alternative representations, which inspire curiosity: Leibniz formula (2s and Square Odds): $\begin{aligned} \pi&=\frac{4}{1+\frac{1^{2}}{2+\frac{3^{2}}{2+\frac{5^{2}}{2+\ddots}}}}\\&=\sum_{n=0}^{\infty} \frac{4(-1)^{n}}{2 n+1}\\&=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+-\cdots \end{aligned}$ Somayaji formula (6s and Square Odds): $\begin{aligned}\pi&=3+\frac{1^{2}}{6+\frac{3^{2}}{6+\frac{5^{2}}{6+\ddots}}}\\&=3-\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n(n+1)(2 n+1)}\\&=3+\frac{1}{1 \cdot 2 \cdot 3}-\frac{1}{2 \cdot 3 \cdot 5}+\frac{1}{3 \cdot 4 \cdot 7}-+\cdots\end{aligned}$ Linear precision (Odds and Squares) $\begin{aligned}\pi&=\frac{4}{1+\frac{1^{2}}{3+\frac{2^{2}}{5+\frac{3^{2}}{7+\ddots}}}}\\&=4-1+\frac{1}{6}-\frac{1}{34}+\frac{16}{3145}-\frac{4}{4551}+\frac{1}{6601}-\frac{1}{38341}+-\cdots\end{aligned}$ Arcsine Formula $\begin{aligned}\pi&=6 \sin ^{-1}\left(\frac{1}{2}\right)\\&=\sum_{n=0}^{\infty} \frac{3 \cdot\binom{{2n}}{n}}{16^{n}(2 n+1)}\\&=\frac{3}{16^{0} \cdot 1}+\frac{6}{16^{1} \cdot 3}+\frac{18}{16^{2} \cdot 5}+\frac{60}{16^{3} \cdot 7}+\cdots\end{aligned}$ Linear combination of arc tangents (representing sector areas adding to quarters of unit circles: $\begin{aligned} \frac{\pi}{4}&=\arctan \frac{1}{2}+\arctan \frac{1}{3}&\text{Euler}\\ &=2 \arctan \frac{1}{2}-\arctan \frac{1}{7}&\text{Hermann}\\ &=2 \arctan \frac{1}{3}+\arctan \frac{1}{7} & \text{Hutton/Vega}\\ &=4 \arctan \frac{1}{5}-\arctan \frac{1}{239} & \text{Machin} \end{aligned}$