Um estudo sobre o tratamento formal dos números racionais

The objective of this paper is to present a study about the formal treatment of some concepts related to rational numbers, with discussions based on demonstrations. Our interest in the subject stems from own experience in the field of mathematics education both as teachers and researchers. For the present work, we assumed a phenomenological stance. This means that our considerations are not generalities, but interpretations and understandings of what we sought to study. The justification for writing this paper stems from the complexity of the theme, as there is no consensus among educators regarding when to start formal work on mathematics in a classroom context. The choice of theme is further justified by its importance, since these are mathematical demonstrations which underpin the theory of this science, thus ensuring its validity. We thread the path of the history of mathematics from the beginning, when mathematical truths were intuitively constituted, to be later intertwined by mathematical logic, to  the present day, when they are defined by their logical-epistemological, rhetoric and heuristic characteristics. In this context, we were attracted to the study of rational numbers, due to the mathematical richness involved in the constitution of this numerical set, and their presence in elementary education. We focus on the formal treatment of rational numbers while, as educators, we try not to lose sight of the educational context. We state that it is not our intention to arbitrate on the formalization of mathematics or the teaching of this science in elementary education, but rather, we aim to introduce a theoretical study regarding some mathematical ideas involving rational numbers that synthesize our understandings.