## ABSTRACT

We have already discussed the parallel between decimal expansions and continued fraction expansions of real numbers. We know that numbers with ﬁnite decimal expansion and ﬁnite continued fraction expansion must be rational. As we have seen, only numbers with ﬁnite continued fraction expansion are rational. However, decimal number expansions of rational numbers can be inﬁnite. We know that an inﬁnite decimal expansion of a rational number must be periodic, i.e., its terms become repeating at some point. In fact, any periodic decimal expansion represents a rational number. In this chapter we will see that a real number has inﬁnite periodic continued fraction expansion if and only if it is a quadratic irrational. We will then be able to use the theory developed in proving this result to describe a procedure for factoring using continued fractions. We will also be able to apply the theory to ﬁnd solutions to Pell’s equation.