The Discovery of Irrational Numbers

Abstract

The discovery of these “holes” is attributed to Pythagoras, founder of the celebrated Greek school of mathematics and philosophy in the sixth century B.C. The life of Pythagoras is shrouded in mystery, and the little we know about him is more legend than fact. This is partially due to an absence of documents from his time, but also because the Pythagoreans formed a secret society, an order devoted to mysticism, whose members agreed upon strict codes of communal life. There is some doubt whether many of the contributions attributed to Pythagoras were indeed his own, but there is no question that his teaching has had an enormous influence on the subsequent history of mathematics, an influence which lasted for more than two thousand years. His name, of course, is immortally associated with the theorem relating the hypotenuse of a right triangle to its two sides, even though there is strong evidence that the theorem had already been known to the Babylonians and the Chinese at least a thousand years before him. The theorem says that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two sides: c² = a² + b² (Fig. 8.1). The Pythagorean Theorem is probably the most well know, and certainly the most widely used theorem in all of mathematics, and it appears, directly or in disguise, in almost every branch of it.

The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems—the determination of the diagonal of a square and that of the circumference of a circle—revealed the existence of new mathematical beings for which no place could be found within the rational domain.

— Tobias Dantzig (1884–1956)