Frações contínuas: Abordagem histórica, aplicações e proposta didática no ensino básico.

Abstract

Continued fractions are used to represent real numbers, admitting Ąnite representation for rational numbers and inĄnite for irrational ones. Furthermore, continued fractions are closely associated with concepts of best rational approximations of real numbers. The aim of this dissertation is to present a historical overview of the topic, its main de Ąnitions and results, and the elaboration of a didactic proposal for basic education. In this context, it will be exposed the historical aspects that originated the continued fractions, from their use and theoretical development, passing through the main contributions made, such as Euler and Lagrange in the 18th century, ending with the Ąelds of knowledge where the topic is currently addressed. DeĄnitions and results on continued fractions involving notions of best rational approximation of a real number, will be exposed, presenting the fact that all the best approximations of an irrational number derive from the notion of convergent, being still possible to magnify the error committed in approximation. Additionally, applications of the theme are presented in the resolution of Diophantine equations and in the approximation of zeros of real functions. Finally, two didactic sequences are oriented: one for Elementary School and other for High School, relating the theme to different Ąelds of mathematics and using GeoGebra software as a didactic tool, showing that continued fractions have didactic possibilities and can contribute to the development of students in basic education.