Difference between revisions of "Liouville number"
From Encyclopedia of Mathematics
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A real number $\alpha$ such that for any $\nu\geq1$ the inequality | A real number $\alpha$ such that for any $\nu\geq1$ the inequality | ||
$$\left|\alpha-\frac pq\right|<q^{-\nu}$$ | $$\left|\alpha-\frac pq\right|<q^{-\nu}$$ | ||
| − | 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 J. Liouville | + | 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: | Examples of Liouville numbers are: | ||
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====References==== | ====References==== | ||
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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) {{ZBL|0090.26103}} | |
| − | + | * {{Ref|a1}} O. Perron, "Die Lehre von den Kettenbrüchen", '''1''', Teubner (1977) Sect. 35 | |
| + | * {{Ref|a2}} O. Perron, "Irrationalzahlen", Chelsea, reprint (1948) | ||
Latest revision as of 19:08, 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) Zbl 0090.26103
- [a1] O. Perron, "Die Lehre von den Kettenbrüchen", 1, Teubner (1977) 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=32555
Liouville number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_number&oldid=32555
This article was adapted from an original article by S.V. Kotov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article