###
Introduction

There are numerous proofs commonly available online for the fact that the Euler's number

$e$ is irrational.
Then going further we find that $e$ is also a transcendental number
which means that it can not be the root of a polynomial equation with
integral coefficients and thereby transcends the powers of algebra in a
sense. Again the proof that

$e$ is transcendental is also available on various places online.

In this post I am going to present the proof that $e$ is not a quadratic irrationality. This is based on the paper

"Sur l'irrationnalitÃ© du nombre e = 2.718..." by

Joseph Liouville.