A tale of two algorithms

Abstract

Real numbers are often a missing link in mathematical education. The standard working assumption in calculus courses is that there exists a system of ‘numbers’, extending the rational number system, adequate for measuring continuous quantities. Moreover, that such ‘numbers’ are in one-to-one correspondence with points on a ‘number line’. But typically real ‘numbers’ are not systematically presented via any constructive method. While taken for granted, they are one of the most commonly used mathematical objects. This paper proposes a geometric algorithm, extending the long division algorithm, which leads to a constructive definition of real numbers. It proceeds to describe a direct algorithm for adding ‘real numbers’. Combined use of the two algorithms enables a smooth and meaningful presentation, offering a double image (geometric and numerical) of real numbers in decimal notation. An early such presentation is of both conceptual and practical importance.