A construção do conjunto dos números reais utilizando a teoria de cortes

Abstract

The present work arises from an annoyance of the author on the construction of the set of Real Numbers, more specifically in the axiomatic part. Thus, we sought new ways of thinking such a mathematical object in order to deepen knowledge about the subject. As the main objective of this work, our idea was to scrutinize the cut theory, facilitating the understanding of the demonstrations and properties for those who have an interest in knowing / studying such a theory and could be a manual for those interested (possibly teachers or future teachers of math). Searching / analyzing how the set of real numbers in high school textbooks is presented and taught is the secondary objective of our work. In relation to the main purpose, we have broken down the theory of cuts by relying almost entirely on the book "The construction of numbers", we have put our personal touch on the statements, in the comments and included some results as prerequisites. In relation to the secondary objective, the approach of five textbooks on the content of real numbers was analyzed. To do this, we create axes of analysis in order to focus our look at some elements that we consider important, namely: definition of real number; operations properties; one-to-one correspondence between the real numbers and the points of a numbered line; intervals. We believe that the construction of the set of real numbers through the theory of cuts can add more ways of thinking the object of our study. This construction is not much worked on teacher training courses, so we think that our work may be an opportunity to spread this theory to teachers and teachers to be.