Números irracionais: da irracionalidade de números algébricos aos primeiros transcendentes

Abstract

In primary and secondary education, irrational numbers are treated very superficially, while natural, whole, and rational numbers cover practically all mathematics education. Already in graduation, irrational numbers are studied from the point of view of analysis and algebra. Unlike the study of rationals, there are many problems about properties of various kinds of irrational numbers, of which many still intrigue scholars today. The history of mathematics relates the impact caused by the discovery of the Pythagoreans that the set of rational numbers is not sufficient to represent measures of any length. In view of the importance of the concept of irrationality and its deeper knowledge is essential for the teaching of mathematics and the understanding of its various branches, the present work sought to broaden the reader's focus on his view on numbers, with emphasis on algebraic irrational) and the first transcendents which, according to the history of mathematics, are called Liouville numbers, thus clarifying somewhat the complexity of the behavior of these numbers, showing both its challenging side and its mathematical fertility. In this work, we approach, in different ways, the irrationalities of √2 and e (Euler's Constant), in order to awaken in the reader the fascination with mathematics, showing different "tools" that we can use to obtain the same result in mathematics . We finish the work introducing concept of algebraic and transcendent numbers