Difference between revisions of "Liouville number"
From Encyclopedia of Mathematics
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| − | A real number | + | {{TEX|done}} |
| + | A real number $\alpha$ such that for any $\nu\geq1$ the inequality | ||
| − | < | + | $$\left|\alpha-\frac pq\right|<q^{-\nu}$$ |
| − | has infinitely many integer solutions | + | has infinitely many integer solutions $p$ and $q$ satisfying the conditions $q>0$, $(p,q)=1$. The fact that a Liouville number is transcendental (cf. [[Transcendental number|Transcendental number]]) follows from the Liouville theorem (cf. [[Liouville theorems|Liouville theorems]]). These numbers were studied by J. Liouville [[#References|[1]]]. |
Examples of Liouville numbers are: | Examples of Liouville numbers are: | ||
| − | + | $$\alpha_1=\sum_{n=1}^\infty2^{-n!},$$ | |
| − | + | $$\alpha_2=\sum_{n=1}^\infty(-1)^n2^{-3^n},$$ | |
| − | + | $$\alpha_3=\sum_{n=1}^\infty(10^{n!})^{-1}.$$ | |
====References==== | ====References==== | ||
Revision as of 08:59, 27 July 2014
A real number $\alpha$ such that for any $\nu\geq1$ the inequality
$$\left|\alpha-\frac pq\right|<q^{-\nu}$$
has infinitely many integer solutions $p$ and $q$ satisfying the conditions $q>0$, $(p,q)=1$. 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:
$$\alpha_1=\sum_{n=1}^\infty2^{-n!},$$
$$\alpha_2=\sum_{n=1}^\infty(-1)^n2^{-3^n},$$
$$\alpha_3=\sum_{n=1}^\infty(10^{n!})^{-1}.$$
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=15912
Liouville number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_number&oldid=15912
This article was adapted from an original article by S.V. Kotov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article