The ancient Greek discovery of irrational numbers
Professor Julius Barbanel
April 19, 2001
Bailey Hall (Room 207)
A real number is irrational if and only if it cannot be expressed as the ratio of two integers. Ancient Greek mathematicians are credited with the discovery of irrational numbers, but their approach and presentation are very different from anything that you have probably seen before. Their work was geometric. It focussed on the notion of commensurable line segments: two line segments are commensurable if and only if each is a (whole number) multiple of some common line segment. The Greeks showed that there are line segments that are not commensurable. We shall consider various conjectures regarding how this was first accomplished and a related historical puzzle that is presented to us in one of Plato's dialogues, the Theaetetus. We shall also see how the idea of incommensurable line segments connects with the modern notion of irrational number. The Greek mathematicians and philosophers whose work we shall consider are Pythagoras, Eudoxus, Plato, Theodorus, Theaetetus, and Euclid.
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