Pythagoras and the discovery of the irrationals

History — Pythagoras, the Pythagorean school and the irrational

Pythagoras of Samos (about 570–495 BC) founded at Croton, in southern Italy, a philosophical and religious community in which mathematics played a central role: for the Pythagoreans “all is number”, that is, every natural phenomenon is governed by ratios between whole numbers. The theorem that bears his name — the relation $a^2+b^2=c^2$ for right-angled triangles — was already known, in particular cases, to the Babylonians (the celebrated tablet Plimpton 322, from the 18th century BC, contains fifteen Pythagorean triples!) and to the mathematicians of ancient India and China. The novelty of the Pythagorean school was, in all likelihood, the first general proof of the theorem (Boyer).

The same theorem, paradoxically, destroyed the Pythagorean doctrine that “all is (whole) number”: applying it to the square of side $1$, one obtains a diagonal of length $\sqrt{2}$, which tradition attributes to Hippasus of Metapontum as the first irrational number ever discovered. The legend — perhaps exaggerated — recounts that Hippasus was drowned for having disclosed this unacceptable mathematical secret. Real or not, the discovery of the irrationals is one of the decisive moments in the history of mathematical thought (Kline); for a fine account of the theorem and its proof see also Dunham.

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Topics: Equivalence and Pythagoras
Concepts: Irrational numbers · Pythagoras’ theorem · Pythagorean triples
People: Hippasus · Pythagoras