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.

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.