---
tag: 0N3WFZEFPM
kind: statement
role: theorem
name: Liouville's theorem on Diophantine approximation
msc: [11J68]
uses: [0J1R5NP36S, 0MB4TGMXG6, 079EGSD32Y]
sources:
  - url: https://en.wikipedia.org/wiki/Liouville_number
    locator: Liouville numbers and transcendence
---

## Statement

Let $\alpha$ be a real [[algebraic number|0MB4TGMXG6]] of degree $d \ge 2$.
Then there is a constant $c > 0$ such that $\left| \alpha - \frac{p}{q} \right| > \frac{c}{q^d}$ for all integers $p$ and $q$ with $q > 0$.
As a consequence, each [[Liouville number|0J1R5NP36S]] is [[transcendental|079EGSD32Y]].

## Remarks

The proof uses the minimal polynomial $f$ of $\alpha$: $q^d f(p/q)$ is a nonzero integer, and the mean value theorem bounds $f(p/q)$ by $|\alpha - p/q|$.
For a Liouville number $x$, the approximations are better than $c / q^d$ for every $d$, so $x$ is not algebraic.
Roth's theorem improves the exponent $d$ to $2 + \varepsilon$ for each $\varepsilon > 0$.

## History

Joseph Liouville proved the bound in 1844. With it, he gave the first proof that transcendental numbers exist.

## Explanations

### L1 human

Algebraic numbers cannot be approximated by fractions too well: the error is at least a constant times $1/q^d$.
A number with better approximations of every order cannot be algebraic.
This gave the first explicit transcendental numbers.