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

Theorem: The value of $\pi$ is independent of which circle it is defined for.

Proof: █

Theorem: The real number $\pi$ is irrational.

Proof: █

Theorem

The following formula holds: $$\displaystyle\sum_{k=1}^{\infty} \mathrm{sinc}(k) = \dfrac{\pi-1}{2},$$ where $\mathrm{sinc}$ denotes the $\mathrm{sinc}$ function and $\pi$ denotes pi.

References

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

Proof that $\pi$ exists (video)

Proof that $\pi$ exists

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

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

100 mpmath one-liners for pi