View source for Liouville number

{{TEX|done}}{{MSC|11J}}

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|Transcendental number]]) follows from the Liouville theorem (cf. [[Liouville theorems|Liouville theorems]]). These numbers were studied by [[Joseph Liouville|J. Liouville]] {{Cite|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====

* {{Ref|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
* {{Ref|2}} A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)
* {{Ref|a1}} O. Perron, "Die Lehre von den Kettenbrüchen", '''1''', Teubner (1977) Sect. 35
* {{Ref|a2}} O. Perron, "Irrationalzahlen", Chelsea, reprint (1948)