Simple Continued Fractions an Approach for High School Students
Athanasios Paraskevopoulos
TL;DR
Problem addressed: provide a compact, classroom-friendly treatment of continued fractions that distinguishes finite and infinite representations and connects rational and irrational numbers. Approach: define simple and generalized continued fractions, present convergents via recurrences, and illustrate with square roots and periodic expansions for quadratics. Findings/Contributions: explicit recurrence relations for convergents $P_n$ and $Q_n$, the best-approximation property, finite CFs for rationals, and periodic expansions for quadratic irrationals, with connections to Pell's equation. Significance: offers a pedagogical framework for introducing number theory concepts in high school and lays groundwork for deeper study of irrationality, Pell equations, and the golden ratio.
Abstract
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the fundamental properties of both Finite and Infinite Continued Fractions. It further delves into the computation of quadratic numbers using given periodic continued fractions and the concept of conjugate quadratic numbers. A significant part of the paper is dedicated to the approximation of real numbers and the convergence properties of continued fractions. The study of continued fractions offers a profound understanding of the intricate relationships within number systems, a key emphasis in contemporary mathematics education. The paper is designed to be engaging and interactive, fostering a fun and stimulating learning environment. By the end of this study, students will have gained a comprehensive understanding of continued fractions, their properties, and their applications, thus enhancing their mathematical proficiency and problem-solving skills. This paper serves as a stepping stone for students to explore more complex mathematical concepts and theories, fostering a deeper appreciation for the subject.