The Root of the Matter: Approximating Roots with the Greeks

Introduction

There are many numerical methods for approximating the irrational number \(\sqrt{2}.\) In this paper we will explore an iterative method for approximating \(\sqrt{2}\) by rational numbers which was developed in about 100 CE by Theon of Smyrna. We will verify its accuracy and explore possible motivations for its development. We are mainly interested in viewing this historic method through more modern lenses, and so we will explore the method in the contexts of modern courses in Geometry and Linear Algebra. To conclude we'll show how to extend the method to approximate other square roots.

As early as 430 BCE, it was known that the sides and diagonal of a square are incommensurable. This means that the ratio of these lengths cannot be expressed as a rational number. If we think of a square of side length 1, the diagonal has length \(\sqrt{2},\) and this ancient result says that \(\sqrt{2}\) is irrational. This discovery came after the life of Pythagoras, but during the time of his followers, known as the Pythagoreans. We don't have a lot of information about this group. Some say that this incommensurability result caused conflict among Pythagoreans; others say it was fertile ground for new mathematics. [2]

Theon of Smyrna was an astronomer and writer who lived in what is now Izmir, Turkey from about 70 to 135 CE. Theon lived 400 years after Euclid and 600 years after Pythagoras, and is sometimes classified as a Neo-Pythagorean. [4] Neo-Pythagoreans approached mathematics through the Pythagorean tradition, including the idea of incommensurability. [3]