# Tag Archives: irrational

## What is a Liouville number?

An irrational number $r$ is called a Liouville number if for every positive integer $n$ there exists integers $p$ and $q \gt 1$ such that $$| r – p/q | \lt 1/q^n.$$ The requirement that $q \gt 1$ is crucial because otherwise, all irrational numbers would satisfy this definition. Liouville numbers are transcendental. In an […]

## The Number e, Part 1: e is Irrational

(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 […]