Continued Fractions and Quadratic Irrationals

Abstract

Most books on number theory contain a chapter or two on continued fractions, which really are indispensable in a number of areas. Nevertheless, they have not achieved a mainstream popularity and are often omitted in courses on number theory. Of course there are reasons for this; their basic construction strikes one as rather bizarre and they are notoriously impossible to manipulate with respect to the usual operations of arithmetic. Furthermore, they have no satisfactory generalization to fit into a more comprehensive framework. But, they are surprising and interesting. One of the key roles played by continued fractions is in the construction of units in real quadratic fields. In studying this topic we have found that the entire theory of units in such fields can in fact be derived via continued fractions. Also, the periods of the continued fractions associated with quadratic irrationals exhibit a certain pattern or structure when classified by discriminant, which has not been previously noted.