<abstract xmlns="http://www.w3.org/1999/xhtml"> 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<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/forma_39_1.jpg"></inline-graphic>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.</abstract>

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 andIt 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.

All Liouville Numbers are Transcendental

Formalized Mathematics

This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

{"article-title":"All Liouville Numbers are Transcendental"}

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...

All Liouville Numbers are Transcendental

Published Online: May 11, 2017

Page range: 49 - 54

Received: Feb 23, 2017

DOI: https://doi.org/10.1515/forma-2017-0004

Keywords
Liouville number, Diophantine approximation, transcendental number, Liouville constant

© by Artur Korniłowicz

This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

All Liouville Numbers are Transcendental

Published Online: May 11, 2017

Page range: 49 - 54

Received: Feb 23, 2017

DOI: https://doi.org/10.1515/forma-2017-0004

KeywordsLiouville number, Diophantine approximation, transcendental number, Liouville constant

© by Artur Korniłowicz

This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License.

Keywords
Liouville number, Diophantine approximation, transcendental number, Liouville constant