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Tytuł: Introduction to Liouville Numbers
Autorzy: Grabowski, Adam
Korniłowicz, Artur
Słowa kluczowe: Liouville number
Diophantine approximation
transcendental number
Liouville constant
Data wydania: 2017
Data dodania: 5-cze-2017
Wydawca: De Gruyter Open
Źródło: Formalized Mathematics, Volume 25, Issue 1, pp. 39-48
Abstrakt: The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and 0 < x − p q < 1 q n . It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum X∞ k=1 ak b k! for a finite sequence {ak}k∈N and b ∈ N. Based on this definition, we also introduced the so-called Liouville number as L = X∞ k=1 10−k! = 0.110001000000000000000001 . . . , substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https: //en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.
Afiliacja: Grabowski Adam - Institute of Informatics, University of Białystok, Białystok, Poland
Korniłowicz Artur - Institute of Informatics, University of Białystok, Białystok, Poland
URI: http://hdl.handle.net/11320/5567
DOI: 10.1515/forma-2017-0003
ISSN: 1426-2630
1898-9934
Typ Dokumentu: Article
Występuje w kolekcji(ach):Artykuły naukowe (WMiI)
Formalized Mathematics, 2017, Volume 25, Issue 1

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