The Number e, Part 1: e is Irrational

Posted by Jason Polak on 09. October 2012 · Write a comment · Categories: number-theory · Tags: , ,

(Or would “The Number 1, Part e” be more interesting?!)

Let’s talk about the number $e$, my favourite number. Of course, to talk about it we need a definition, so we define $e$ as

$e = 1/0! + 1/1! + 1/2! + 1/3! + \cdots.$

Exercise: check that this converges! In Part 2, we shall see Hermite’s argument that $e$ is transcendental. In order to warm up, let us prove that $e$ is not an integer first, and then let us prove that $e$ irrational.
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