---
tag: 0W4GQSG7B6
kind: statement
role: theorem
name: Irrationality of e
msc: [11J72]
uses: [0MVCFMW7S4, 0G4D3ZKTP8]
sources:
  - url: https://en.wikipedia.org/wiki/Proof_that_e_is_irrational
    locator: Fourier's proof
---

## Statement

The number [[$e$|0MVCFMW7S4]] is irrational.

## Remarks

Assume $e = a/b$ with positive integers $a, b$. Then $x = b! \big( e - \sum_{n=0}^{b} \frac{1}{n!} \big)$ is an integer.
But the [[series for $e$|0G4D3ZKTP8]] gives $0 < x < \sum_{k=1}^{\infty} (b + 1)^{-k} = \frac{1}{b} \le 1$. No integer lies strictly between $0$ and $1$.
This short proof is due to Joseph Fourier.

## History

Leonhard Euler wrote the first proof in 1737. It was published seven years later.

## Explanations

### L1 human

The series for $e$ converges so fast that $e$ cannot be a fraction.
If $e = a/b$, a suitable multiple of the tail of the series would be an integer between $0$ and $1$.