# All Liouville Numbers are Transcendental

Open access

## 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.

[1] Tom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 2nd edition, 1997.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[4] Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and _ as an application of multivariate and symmetric polynomials. In Jeremy Avigad and Adam Chlipala, editors, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pages 76-87. ACM, 2016. doi: 10.1145/2854065.2854072.

[5] Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4:71-84, 2011.

[6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.

[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.

[8] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.

[9] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[10] J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, 1996.

[11] Manuel Eberl. Liouville numbers. Archive of Formal Proofs, December 2015. http://isa-afp.org/entries/Liouville_Numbers.shtml, Formal proof development.

[12] Adam Grabowski and Artur Korniłowicz. Introduction to Liouville numbers. Formalized Mathematics, 25(1):39-48, 2017. doi: 10.1515/forma-2017-0003.

[13] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.

[14] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.

[15] Joseph Liouville. Nouvelle d´emonstration d’un th´eor`eme sur les irrationnelles alg´ebriques, ins´er´e dans le Compte Rendu de la derni`ere s´eance. Compte Rendu Acad. Sci. Paris, S´er.A (18):910-911, 1844.

[16] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.

[17] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.

[18] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.

[19] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.

[20] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.

[21] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495-500, 1990.

[22] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.

[23] Yasushige Watase. Algebraic numbers. Formalized Mathematics, 24(4):291-299, 2016. doi: 10.1515/forma-2016-0025.

[24] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992.

# Formalized Mathematics

## (a computer assisted approach)

### Journal Information

SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237

Target Group

researchers in the fields of formal methods and computer-checked mathematics

### Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 87 87 13