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.

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 147, 1997.

Mahler, K. "On the Approximation of ." Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes Math. 15, 30-42, 1953.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

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Weisstein, Eric W. "Liouville Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LiouvilleNumber.html