Liouville Number

A Liouville number is a transcendental number which has very close rational number approximations. An irrational number beta is called a Liouville number if, for each n, there exist integers p>0 and q>1 such that


Note that the first inequality is true by definition, since it follows immediately from the fact that beta is irrational and hence cannot be equal to p/q for any values of p and q.

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 pi is not a Liouville number.

See also

Exponential Factorial, Irrational Number, Irrationality Measure, Liouville's Constant, Liouville's Approximation Theorem, Roth's Theorem, Transcendental Number

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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 pi." 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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Liouville Number

Cite this as:

Weisstein, Eric W. "Liouville Number." From MathWorld--A Wolfram Web Resource.

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