The square root of 2 broke the Pythagoreans because it is the length of an ordinary line, the diagonal of a square with sides of 1, yet it cannot be written as a ratio of whole numbers. The Pythagoreans believed every length was such a ratio. But √2 = 1.41421356…, with decimals that never end and never repeat, and a five-step proof shows that no fraction p/q can equal it. The famous sequel, that a Pythagorean named Hippasus revealed the secret and was drowned at sea, is a legend, not confirmed history.
A brotherhood built on number
Pythagoras was born on Samos about 570 BC and founded a philosophical and religious school at Croton in about 518 BC (the dates are disputed). MacTutor describes the society as half religious and half scientific, bound by secrecy. Its inner circle, the mathematikoi, lived with the society, owned no personal possessions, were vegetarians and obeyed strict rules.
At the center sat one belief: at its deepest level, reality is mathematical.
the whole cosmos is a scale and a number — Aristotle, describing the Pythagoreans (quoted by MacTutor)
By a number, Pythagoras meant the ratio of two whole numbers. So any two lengths ought to be commensurable: if one is 7 units and the other 5, a small enough ruler measures both exactly.
The diagonal that would not fit
Now draw a square with side 1. By the Pythagorean theorem, 1² + 1² = 2, so the diagonal is √2 = 1.41421356…
Approximations were old news. The Babylonian tablet YBC 7289, from around 1800–1600 BC, gives √2 in base 60 as 1;24,51,10, which is 1.41421296…, accurate to about six decimal digits. But an approximation is not an exact ratio. Someone in the Pythagorean circle found that the diagonal is incommensurable with the side: no unit, however small, measures both exactly. MacTutor notes that the discovery is credited to the Pythagoreans but seems unlikely to be due to Pythagoras himself.
Why √2 is not a fraction: the proof
The video flashes this argument in a few seconds:
- Suppose √2 = p/q, a fraction of whole numbers in lowest terms.
- Square both sides and clear the denominator: p² = 2q². So p² is even.
- Odd × odd = odd, so an odd p would give an odd p². Therefore p is even: p = 2k.
- Substitute: 4k² = 2q², so q² = 2k². Now q² is even, and so q is even too.
- Both even means a common factor of 2, contradicting lowest terms. No such fraction exists.
Aristotle hinted at this reasoning: if the diagonal were commensurable with the side, odd numbers would equal even ones. A full version appears as Proposition 117 of Book X of Euclid’s Elements, although historians agree it was inserted later and is not Euclid’s own. For every step in slow motion, read why the square root of 2 is irrational.
The legend of Hippasus
Hippasus of Metapontum was a Pythagorean whose dates are uncertain, and his drowning story is pieced together from contradictory reports. Pappus says only that the member who first divulged the secret perished by drowning, without naming him. Iamblichus says Hippasus perished at sea, a punishment from the gods for impiety, but for publishing the sphere made from twelve pentagons, the dodecahedron, not for irrational numbers. Elsewhere Iamblichus says a Pythagorean was merely expelled for divulging the nature of the irrational.
The first irrational may not even have been √2: irrationality also shows up in the golden ratio of the regular pentagon, the face of that same dodecahedron.
How badly the discovery shook the brotherhood is unknown. What it did end was the idea that ratios of whole numbers can measure every length. Plato’s dialogue Theaetetus describes the mathematician Theodorus showing, case by case, that the sides of squares of 3, 5 and more square feet, up to 17, are incommensurable with the unit. He skips 2, perhaps because that case was already known.
Today we call numbers like √2 irrational. One diagonal did all that.
