All Liouville Numbers are Transcendental
| Autor: | Artur Korniłowicz, Adam Grabowski, Adam Naumowicz |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Transcendental equation
Liouville function 0102 computer and information sciences 02 engineering and technology Diophantine approximation 01 natural sciences Liouville number diophantine approximation liouville constant 03b35 QA1-939 0202 electrical engineering electronic engineering information engineering identifier: liouvil1 Liouville field theory transcendental number Mathematical physics Mathematics Liouville's formula 11j81 11k60 Applied Mathematics Mathematical analysis liouville number 020207 software engineering Auxiliary function version: 8.1.05 5.40.1286 Computational Mathematics 010201 computation theory & mathematics Transcendental number |
| Zdroj: | Formalized Mathematics, Vol 25, Iss 1, Pp 49-54 (2017) |
| ISSN: | 1898-9934 |
| Popis: | Summary In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] 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 It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation. |
| Databáze: | OpenAIRE |
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