A Liouville number is a transcendental number which has very close rational number approximations. An irrational number is called a Liouville number if, for each , there exist integers and such that
Note that the first inequality is true by definition, since it follows immediately from the fact that is irrational and hence cannot be equal to for any values of and .
Liouville's constant is an example of a Liouville number and is sometimes called "the" Liouville number or "Liouville's number" (Wells 1986, p. 26). Mahler (1953) proved that is not a Liouville number.