Irrationality of e

Statement

The number $e$ 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 $e$ 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 $1$.