The Early Ages

Abstract

As I mentioned in the introduction, algorithms equivalent to the modern use of continued fractions were in use for many centuries before their real discovery. This chapter is devoted to these early attempts. The best known example is Euclid’s algorithm for the greatest common divisor of two integers, which leads to a terminating continued fraction. The approximate simplification of fractions (as practiced by the Greeks), is also related to this algorithm. The fundamental question of the irrationality of the square root of two was an important question for many years. The approximate computation of square roots led to some numerical methods which can be viewed as the ancestors of continued fractions. Another important problem related to astronomy and architecture is that of the solution of diophantine equations. The so-called Pell’s equation was also treated by the ancients (mostly by Indian mathematicians), who can be credited with the early discovery of algorithms analogous to continued fractions. The chapter will end with a short account on the history of the notations for continued fractions.