Real Numbers through the Lens of Continued Fractions

Abstract

Decimal representation of real numbers (or more generally, b-ary representation) give us a convenient way of expressing real numbers. Although it takes a lot of computation to find digits in the decimal expansion of irrational numbers (e.g., the decimal expansions of and e are known only up to a finite number of digits), these expansions help us find increasingly precise approximations to irrational numbers. In this paper, we study rational approximations of irrational numbers and their relationship to continued fractions. We discuss Liouville’s theorem and bounds on how well irrational numbers may be approximated. We then motivate the study of continued fractions, and prove their fundamental properties. These properties allow us to prove an important result about the relationship between rational approximations to irrational numbers and their continued fraction expansions. Finally, we look at the continued fraction expansions of famous irrational numbers.