---
tag: 0J1R5NP36S
kind: definition
name: Liouville number
msc: [11J81]
defines: 0PYEZM6CN9
sources:
  - url: https://en.wikipedia.org/wiki/Liouville_number
    locator: lead section
---

## Definition

A real number $x$ is a Liouville number if for each positive integer $n$ there are integers $p$ and $q$ with $q > 1$ and
$$0 < \left| x - \frac{p}{q} \right| < \frac{1}{q^n}.$$

## Remarks

The standard example is Liouville's constant $\sum_{k=1}^{\infty} 10^{-k!} = 0.110001000\ldots$.
Each Liouville number is irrational, and in fact [[transcendental|079EGSD32Y]].
The numbers $e$ and $\pi$ are transcendental, but they are not Liouville numbers.
The Liouville numbers are uncountable, but they form a set of Lebesgue measure zero.

## Explanations

### L1 human

A Liouville number has rational approximations that are extremely good, better than any fixed power of $1/q$.
Algebraic numbers cannot be approximated this well, so a Liouville number is not algebraic.