# Liouville Numbers

Manuel Eberl

28 December 2015

### Abstract

Liouville numbers are a class of transcendental numbers that can be approximated particularly well with rational numbers. Historically, they were the first numbers whose transcendence was proven.

In this entry, we define the concept of Liouville numbers as well as the standard construction to obtain Liouville numbers (including Liouville's constant) and we prove their most important properties: irrationality and transcendence.

The proof is very elementary and requires only standard arithmetic, the Mean Value Theorem for polynomials, and the boundedness of polynomials on compact intervals.