# Fourier’s ingenious proof that e is irrational

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…