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.
." Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes
Math. 15, 30-42, 1953.Wells, D.