Hippasus drawing

How did hippasus die
How did hippasus discover irrational numbers
What did hippasus discover

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How a Secret Society Discovered Irrational Numbers

The ancient scholar Hippasus of Metapontum was punished with death for his discovery of irrational numbers—or at least that’s the legend. What actually happened in the fifth century B.C.E. is far from clear.

Hippasus was a Pythagorean, a member of a sect that dealt with mathematics and number mysticism, among other things. A core element of the Pythagoreans’ teachings related to harmonic numerical relationships, which included fractions of whole numbers.

The whole world, they believed, could be described using rational numbers, including natural numbers and fractions. Yet when Hippasus examined the length ratios of a pentagram—the symbol of the Pythagoreans—the story goes, he realized that some of the lengths of the shape’s sides could not be expressed as fractions. He thus provided the first proof of the existence of irrational numbers.

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There is a widespread notion that the discovery of irrational numberswas a thing of horror to the ancient Greeks. This went especially for the school of Pythagoras. Pythagoras is best known today for a famous theorem about right-angled triangles -- and we shall look at that theorem another day -- but in antiquity, his significance lay in the fact that he was a semi-legendary guru who founded a philosophico-religious sect in southern Italy.

No writings by Pythagoras himself survive (and it is extremely unlikely he ever wrote any). But the things we hear about the sect make it sound bizarre at times: depending on who you read, the Pythagoreans conveyed their teachings only orally and only in a cave, they had weirdly specific beliefs about reincarnation, and they venerated unexpected plants like fava beans and mallow. The vast majority of this information is reported very late, and is almost certainly false; the bits that are true (whichever ones they are) are difficult to understand out of context.

The legendary Pythagorean veneration of orderly, rational numbers is well exemplifie

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Pythagorean esotericism

In the 6^th Century BCE, Greater Greece stretched west all they way to Sicily and southern Italy. At this time its western colonies were suddenly reinvigorated by an influx of refugees chased out of Ionia by the conquering Persians. Pythagoras, who was born on the island of Samos some time before 550 BCE, at a time when the whole of eastern Greece was becoming unsafe, settled at Croton in Sicily, probably before 520 BCE. A great many legends soon sprang up around this mysterious figure. By the end of the 5^th Century BCE there was no longer any reliable source of information on the life and works of Pythagoras or his immediate successors. What is clear is the reason behind this mystery: it lies in the doctrine itself and the teaching methods of the man who was called the Master. Nothing Pythagoras taught was to be written down or divulged to the uninitiated, and even the disciples were divided into two classes, the μαθηματικοί (mathematikoi), who were students privileged to know the thoughts of the Master, and the ᾽ακουσματικοί (akousmati