Some properties of the set of Liouville numbers




Liouville Numbers, Transcendence, Hausdorff Measure, Irrationality Measure, Irrational Numbers


The present work consists on the introduction of the Transcendental Number Theory beginnings. First, we will present the classical theorems of the rational approximations, finishing with the theorem of Hurwtiz-Markov. The following part is about Liouville Numbers and their intrinsic properties, proving their transcendence. Also, we will talk about the measure of irrationality, an interesting way to classify the degree of irrationality of certain real number. The last (but not least) topic is about the Liouville numbers as a set and their paradoxes, this set, in the view of the topology, is the complement of a Meagre set, which this means that is "big". However, on the analysis point of view, the Liouville Numbers has null measure, considering the Lebesgue and Hausdorff measures.


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Author Biographies

Anderson Luiz Maciel, Federal University of Santa Maria, Santa Maria, RS, Brazil

PhD in Applied Mathematics from the University of São Paulo - USP. Area of concentration: dynamic systems. Professor at the Department of Mathematics at the Federal University of Santa Maria - UFSM.

Juan Manuel Silva Fervenza, Federal University of Santa Maria, Santa Maria, RS, Brazil

Graduating from the Bachelor's Degree in Mathematics at the Federal University of Santa Maria - UFSM.


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2022-04-18 — Updated on 2022-07-07


How to Cite

Maciel, A. L., & Fervenza, J. M. S. (2022). Some properties of the set of Liouville numbers. Ciência E Natura, 44, e5. (Original work published April 18, 2022)

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