Liouville number

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. 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.

Where this leads

1 page builds on this one.

Liouville's theorem on Diophantine approximation

Tag	`0J1R5NP36S`
URI	`m2://0J1R5NP36S`
Kind	definition
Writing	W1 stated
MSC	11J81
Defines	Liouville number

Liouville number

Definition

Remarks

Where this leads

Further reading

What this page needs for W2

What links here

Used by