Liouville number

Noun

Meaning

Liouville number (plural Liouville numbers)

(number theory) An irrational number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that




0
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x
−


p
q



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1

q

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{\displaystyle \textstyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{n}}}.}

Source: en.wiktionary.org

Related terms

Liouville-Arnold theorem Liouville's formula Liouville's equation Liouville's theorem

Examples

A Liouville number is irrational but does not have this property, so it can't be algebraic and must be transcendental. Source: Internet

It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number ); the proof was given by Charles Hermite in 1873. Source: Internet

Kurt Mahler showed in 1953 that π is also not a Liouville number. Source: Internet

Mahler's classification Kurt Mahler in 1932 partitioned the transcendental numbers into 3 classes, called S, T, and U. Bugeaud (2012) p.250 Definition of these classes draws on an extension of the idea of a Liouville number (cited above). Source: Internet

Proof of assertion: As a consequence of this lemma, let x be a Liouville number; as noted in the article text, x is then irrational. Source: Internet

Since every rational number can be represented as such c/d, we will have proven that no Liouville number can be rational. Source: Internet

Close letter words and terms

Liouguei Lioujia Lioujiao Liouville's equation Liouville's formula Liouville's theorem