ABSTRACT
This chapter develops the theory of continued fractions, shows how they give rational numbers approximating real numbers, and use them to find solutions of John Pell's equation. The theory for these is very similar to the theory the authors will be describing for ordinary continued fractions. The chapter describes the basic theory of continued fractions and with the fundamental recursive formulas. It suggests that the continued fractions give good rational approximations to real numbers. The chapter explains which real numbers have purely periodic or eventually periodic continued fractions. It examines any continued fraction that is eventually periodic. By rationalizing denominators when necessary, the authors always end up with a quadratic irrational, that is, a number of the form. If there were only finitely many primes, the product would be a rational number. Therefore, there must be infinitely many primes.
