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Difference between revisions of "Liouville number"

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====References====
 
====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
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* {{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)
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* {{Ref|2}} A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)
 
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* {{REF|a1}} O. Perron, "Die Lehre von den Kettenbrüchen", '''1''', Teubner (1977) Sect. 35
====Comments====
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* {{REF|a2}} O. Perron, "Irrationalzahlen", Chelsea, reprint (1948)
 
 
====References====
 
<table>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Perron,   "Die Lehre von den Kettenbrüchen" , '''1''' , Teubner (1977) pp. Sect. 35</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Perron,   "Irrationalzahlen" , Chelsea, reprint (1948)</TD></TR>
 
</table>
 

Revision as of 19:07, 24 March 2023

2020 Mathematics Subject Classification: Primary: 11J [MSN][ZBL]

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)
  • Template:REF O. Perron, "Die Lehre von den Kettenbrüchen", 1, Teubner (1977) Sect. 35
  • Template:REF 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=53151
This article was adapted from an original article by S.V. Kotov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article