Namespaces
Variants
Actions

Liouville number

From Encyclopedia of Mathematics
Revision as of 17:14, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A real number such that for any the inequality

has infinitely many integer solutions and satisfying the conditions , . The fact that a Liouville number is transcendental (cf. Transcendental number) follows from the Liouville theorem (cf. Liouville theorems). These numbers were studied by J. Liouville [1].

Examples of Liouville numbers are:

References

[1] J. Liouville, "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques" C.R. Acad. Sci. Paris , 18 (1844) pp. 883–885
[2] A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)


Comments

References

[a1] O. Perron, "Die Lehre von den Kettenbrüchen" , 1 , Teubner (1977) pp. Sect. 35
[a2] O. Perron, "Irrationalzahlen" , Chelsea, reprint (1948)
How to Cite This Entry:
Liouville number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_number&oldid=32555
This article was adapted from an original article by S.V. Kotov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article