Liouville number

As Liouville number , named after Joseph Liouville , is known in the theory of numbers a real number that satisfies the condition that for all positive integers integers and with there, so ${\displaystyle x,}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle q>1}$

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

Irrationality and transcendence

All Liouville numbers are irrational : For every rational number with an integer numerator and a positive integer denominator there is a positive integer with If now and integers with and are, then is ${\displaystyle x={\tfrac {c}{d}}}$ ${\displaystyle c}$ ${\displaystyle d}$ ${\displaystyle n}$ ${\displaystyle 2^{n-1}>d.}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle q>1}$ ${\displaystyle {\tfrac {p}{q}}\neq {\tfrac {c}{d}}}$

${\displaystyle \left|x-{\frac {p}{q}}\right|=\left|{\frac {c}{d}}-{\frac {p}{q}}\right|=\left|{\frac {c\,q-p\,d}{d\,q}}\right|\geq {\frac {1}{d\,q}}>{\frac {1}{2^{n-1}q}}\geq {\frac {1}{q^{n}}}\ .}$

In 1844, Liouville showed that numbers with this property are not only irrational but also transcendent . This was the first proof of the transcendence of a number, Liouville's constant:

${\displaystyle c=\sum _{j=1}^{\infty }10^{-(j!)}=0{,}11000{\text{ }}10000{\text{ }}00000{\text{ }}00000{\text{ }}00010{\text{ }}\ldots }$ (Follow A012245 in OEIS )

All Liouville numbers are transcendent, but not all transcendent numbers are Liouville. For example, Euler's number is transcendent, but not Liouvillesch.

literature

• Joseph Liouville: Nouvelle demonstration d'un théoreme sur les irrationalles algébriques, inséré dans le compte rendu de la dernière séance . In: Comptes rendus de l'Académie des sciences . tape 18 , 1844, pp. 910-911 .
• SV Kotov: Liouville number . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Web links

• Eric W. Weisstein : Liouvilles Constant . In: MathWorld (English).