 Liouville number

A Liouville number is a real irrational number $\alpha$ such that, for every positive integer $m$, there are integers $p,q$ with $q>1$ and $0<|\alpha -p/q|<q^{-m}$. It admits rational approximations of arbitrarily high polynomial accuracy.

For example, ${\sum }_{n\ge 1}10^{-n!}$ is a Liouville number: truncating after $n!$ leaves a tail smaller than a constant times $10^{-(n+1)!}$, while the denominator is $10^{n!}$. The Liouville approximation theorem bounds how well a fixed algebraic number that is an irrational number can be approximated, proving that every Liouville number is transcendental. The converse fails: transcendence alone does not require such exceptional approximations.

A Liouville number is a type of real number that is particularly significant in the field of number theory, especially in the study of transcendental numbers.