# Pi

A circle in Euclidean plane geometry is defined to be the set of points equidistant from a center point. The length around a circle is called its circumference and the length a line from the circle through the center is called a diameter of the circle. All diameters have the same length by definition of the circle. Let $A$ be a circle. The number $\pi$ is defined to be the ratio $\dfrac{C}{D}$ where $C$ is the circumference of $A$ and $D$ the diameter of $A$. It requires proof to show that the value obtained from the circle $A$, call this $\pi_A$, is the same number one obtains from another circle $B$, the value $\pi_B$.

# Properties

Pi is irrational

Sum of values of sinc

Wallis product

# Videos

The story of $\pi$ by Tom Apostol (1995)

Proof that $\pi$ exists (2014)

# References

Proof that $\pi$ is constant for all circles without using limits

Proof that $\pi$ exists

A simple proof that $\pi$ is irrational by Ivan Niven

100 mpmath one-liners for pi