Proszę używać tego identyfikatora do cytowań lub wstaw link do tej pozycji:
http://hdl.handle.net/11320/5567
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 (WInf) Formalized Mathematics, 2017, Volume 25, Issue 1 |
Pliki w tej pozycji:
Plik | Opis | Rozmiar | Format | |
---|---|---|---|---|
forma-2017-0003.pdf | 292,86 kB | Adobe PDF | Otwórz |
Pozycja ta dostępna jest na podstawie licencji Licencja Creative Commons CCL