In a previous post we proved that is irrational. In this post we prove the irrationality of *e.*

A proof of the irrationality of *e* must start by defining *e. *There are some different ways to do that. We’ll take *e* to be the unique positive number *b* such that . (This property itself can be proved from other ways to define *e*. See, for example, Fitzpatrick’s *Advanced Calculus* [1], Section 5.2.) We’ll also make the assumptions that (1) and (2) is an increasing function.

Under this definition of *e*, the Lagrange Remainder Theorem says that, given and , there exists such that

Taking , we obtain, for some ,

.

Now, suppose *e* is rational. Then there exist with . Therefore,

.

Multiplying by , this becomes

.

Since the above inequality is true for all , it is true for . For this value of *n*, is an integer, and .

This means that there is an integer in the interval . However, we know that there is no integer in the interval . Contradiction; thus *e* is irrational.

[1] Patrick M. Fitzpatrick, *Advanced Calculus*, American Mathematical Society, 2006.