How to Enhance Student Motivations by Borrowing from Ancient Tradition: Babylonian Method of Computing the Square Root

Abstract

In the previous chapter, we showed how to use ideas from Mayan and Babylonian arithmetic when teaching math. In this chapter, we consider Babylonian mathematical ideas beyond simple arithmetic, namely, the ideas of computing the square root. When computing a square root, computers still, in effect, use an iterative algorithm developed by the Babylonians millennia ago. This is a very unusual phenomenon, because for most other computations, better algorithms have been invented – even division is performed, in the computer, by an algorithm which is much more efficient that division methods that we have all learned in school. What is the explanation for the success of the Babylonians’ method? One explanation is that this is, in effect, Newton’s method, based on the best ideas from calculus. This explanations works well from the mathematical viewpoint – it explains why this method is so efficient, but since the Babylonians were very far from calculus, it does not explain why this method was invented in the first place. In this chapter, we provide two possible explanations for this method’s origin. We show that this method naturally emerges from fuzzy techniques, and we also show that it can be explained as (in some reasonable sense) the computationally simplest technique.