# Introduction to Liouville Numbers

```@article{Grabowski2017IntroductionTL,
title={Introduction to Liouville Numbers},
journal={Formalized Mathematics},
year={2017},
volume={25},
pages={39 - 48}
}```
• Published 28 March 2017
• Mathematics
• Formalized Mathematics
Summary 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 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…
1 Citations
All Liouville Numbers are Transcendental
• Mathematics
Formaliz. Math.
• 2017
This Mizar article completes 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 and shows that all Liouvile numbers are transcendental, based onLiouville's theorem on Diophantine approximation.

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