# Irrationality Measure of Pi

## TL;DR

This paper demonstrates that the irrationality measure of pi is equal to 2, aligning it with almost all irrational numbers, and discusses historical bounds and recent improvements.

## Contribution

It proves that pi's irrationality measure is exactly 2, matching that of almost all irrationals, which was previously only known as an upper bound.

## Key findings

- Irrationality measure of pi is exactly 2.
- Historical bounds on pi's irrationality measure.
- Pi shares the same measure as almost all irrationals.

## Abstract

The first estimate of the upper bound $\mu(\pi)\leq42$ of the irrationality measure of the number $\pi$ was computed by Mahler in 1953, and more recently it was reduced to $\mu(\pi)\leq7.6063$ by Salikhov in 2008. Here, it is shown that $\pi$ has the same irrationality measure $\mu(\pi)=\mu(\alpha)=2$ as almost every irrational number $\alpha>0$.