In number theory, a Liouville number is an irrational number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that

A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.