CHAPTER V - CONTINUED FRACTIONS

Abstract

Algorithms similar to continued fractions were employed by the mathematicians of antiquity. An ordinary continued fraction with positive integral partial denominators is said to be regular. Only regular continued fractions are considered in the theory of numbers. A continued fraction—in which partial numerators are equal to unity and partial denominators are positive integers—is described as regular or arithmetical. Every periodic continued fraction represents an irrational square root. An important role is played in number theory by a theorem of Lagrange, which states that every irrational square root can be expanded as a periodic regular continued fraction. The convergence of a series or infinite product is not affected by the removal of a finite set of terms. In continued fractions, the removal of a finite set of partial quotients can turn a convergent fraction into a non-essentially divergent fraction.