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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 25, Issue 1


All Liouville Numbers are Transcendental

Artur Korniłowicz / Adam Naumowicz / Adam Grabowski
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/forma-2017-0004


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.

MSC: 11J81; 11K60; 03B35

Keywords: Liouville number; Diophantine approximation; transcendental number; Liouville constant

MML: identifier: LIOUVIL1; version: 8.1.05 5.40.1286


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About the article

Received: 2017-02-23

Published Online: 2017-05-11

Published in Print: 2017-03-28

Citation Information: Formalized Mathematics, Volume 25, Issue 1, Pages 49–54, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2017-0004.

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© by Artur Korniłowicz. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License. BY-SA 3.0 LEGALCODE

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