Fourier’s ingenious proof that e is irrational

Martin McBride
Graphic maths
Published in
7 min readJun 10, 2024

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Joseph Fourier © Guillaume Piolle / CC BY 3.0

Euler’s number, e, is an important number in mathematics that appears in many contexts.

As an example, here are two important properties of $e$. First, the following simple first-order differential equation applies to many situations from population growth to radioactive decay:

The only non-trivial solution to this equation (we will ignore the trivial solution y = 0) is the exponential function e to the power x:

Second, e is involved in Euler’s formula, which is the bedrock of the complex analysis (the study of complex numbers):

If we substitute the value π for θ we get Euler’s identity:

This formula puts e at the heart of mathematics. It links what are arguably the three most important constants in mathematics, pi, e and i, in a single formula.

It turns out that e is an irrational number, and in this article, we will look at Fourier’s ingenious proof of that.

Fourier’s proof

Joseph Fourier’s proof is a proof by contradiction. We start by assuming that e is rational, meaning that, for some positive integers a and b we have:

We will prove that this leads to a contradiction, and therefore cannot be true. This is a similar approach we saw for the proof…

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