The number $\pi$ is the ratio of a circle's circumference to its diameter. Understanding its various properties and computing its numerical value drove the study of much mathematics throughout history. Questions regarding this special number and its properties fit in here.

$\pi$ is the ratio of a circle's circumference to its diameter. Its definition is modern analysis is (by Karl Weierstrass) $$ \pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}. $$ An alternative definition, popularised by Landau is: Define $\frac{\pi}{2}$ as the smallest positive root of the cosine function.

It can also be given by the Gregory-Leibniz series (exhibits sublinear convergence) $$ \pi = 4 \sum_{j=0}^\infty \frac{(-1)^j}{2j+1}. $$ $\pi$ has the approximate numerical value $3.14159265358979323846\dots$, can be approximated by fractions, for example, $\frac{22}{7}, \frac{333}{106}, \frac{355}{113},\dots$, and is both irrational and transcendental.

It is part of Euler's famous identity:

$$e^{i\pi}+1=0.$$