Um estudo sobre a cardinalidade dos números transcendentes

Abstract

In this dissertation the non-enumerability of transcendent numbers is presented. It is also described in detail the enumerability of Z, Q and the non-enumerability of R, besides being presented demonstrations of the irrationalities of the number π, the number e and its transcendence, emphasizing the importance of these concepts in deep-ening students’ understanding of real numbers. The general objective was to present the characteristic of non-enumerability of transcendent numbers and improve students’ understanding of the non-enumerability of irrational numbers. The specific objective was the construction of two activities in order to help teachers and students overcome the complexities of these concepts. To this end, this research adopts a qualitative methodol-ogy based on the literature review as the main collection method. The justification is that there may be a false common sense that the cardinality of irrational numbers is less than that of rational numbers, which can occur because of the way this content is presented in classes. As a result, two activities were developed with the aim of helping teachers to improve students’ understanding on this topic. The first activity aims to demonstrate that irrational numbers exist in greater quantity than rational ones, while the second has as purpose to introduce the idea of non-enumerability of the set of transcendent numbers.