---
tag: 0G0F7FGCGP
kind: statement
role: theorem
name: Transcendence of pi
aliases: [lindemann-s-theorem]
msc: [11J81]
uses: [00SW8B91E7]
sources:
  - url: https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem
    locator: Transcendence of  e  and π
  - url: https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem
    locator: lead section
---

## Statement

The number $\pi$ is [[transcendental|00SW8B91E7]] over $\mathbb{Q}$.

## Remarks

The statement follows from the Lindemann–Weierstrass theorem: $e^\alpha$ is transcendental for every nonzero algebraic number $\alpha$.
Indeed, if $\pi$ were algebraic, then $i \pi$ would be algebraic and nonzero, but $e^{i \pi} = -1$ is algebraic.
So $\sqrt{\pi}$ is also transcendental, and it is not [[constructible|0FMN2MN7TM]].

## History

Charles Hermite proved the transcendence of $e$ in 1873.
Ferdinand von Lindemann proved in 1882 that $e^\alpha$ is transcendental for every nonzero algebraic $\alpha$, and so that $\pi$ is transcendental.
Karl Weierstrass proved a more general statement in 1885.

## Explanations

### L1 human

The number $\pi$ is not a root of any nonzero polynomial with rational coefficients.
This is much stronger than the irrationality of $\pi$, and the proof is much harder.