The square root of 2 is irrational because assuming it is a fraction leads to a contradiction. Suppose √2 = p/q, a fraction of whole numbers in lowest terms. Squaring gives p² = 2q², so p² is even, and that forces p itself to be even. Write p = 2k. Then 4k² = 2q², so q² = 2k², and q is even too. But a fraction in lowest terms cannot have an even top and an even bottom. The assumption collapses: no fraction, however large its numbers, equals √2. Its decimal, 1.41421356…, never ends and never repeats.
That short argument broke a belief at the heart of Pythagorean mathematics.
A diagonal no fraction can measure
Draw a square with side 1. By the Pythagorean theorem its diagonal d satisfies d² = 1² + 1² = 2, so the diagonal is √2. It is right there on the page, an ordinary segment you could cut from paper.
A good approximation is not an exact ratio, though, and exactness is where the trouble began. Pythagoras, whose school was at Croton in southern Italy, believed that all relations could be reduced to relations between numbers, and by a number he meant a ratio of two whole numbers. In geometric terms, any two lengths should be commensurable: some small unit should fit a whole number of times into each. Euclid later wrote the idea down precisely:
Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure. — Euclid, Elements, Book X, Definition 1, c. 300 BC
If the side and the diagonal of a square shared a common measure, the side would be q units long and the diagonal p units long, and √2 would equal p/q. The diagonal is exactly the case where no common measure exists. The discovery is credited to the Pythagoreans as a group; it seems unlikely to have been Pythagoras himself.
Why the square root of 2 is irrational, step by step
The proof needs one small fact: the square of an odd number is odd. An odd number has the form 2m + 1, and (2m + 1)² = 4m² + 4m + 1, which is one more than an even number. So if a square is even, the number being squared must be even.
- Assume √2 = p/q, where p and q are whole numbers with no common factor. Every fraction can be reduced to this form, so the assumption costs nothing.
- Square both sides: 2 = p²/q², so p² = 2q².
- The right side is twice a whole number, so p² is even. By the small fact, p is even. Write p = 2k.
- Substitute: 4k² = 2q², so q² = 2k².
- Now q² is even, so q is even.
- Both numbers are even, so they share the factor 2. That contradicts step 1.
The only assumption was that √2 is a fraction, so that assumption is false. Aristotle, writing about proofs by contradiction, compressed the whole thing into one clause:
the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate — Aristotle, Prior Analytics I.23
Odd equals even is the contradiction: the same number would have to be both. A full written proof appears as Proposition 117 of Book X of Euclid’s Elements, although historians have agreed since the early nineteenth century that this proposition was added by a later hand, not written by Euclid.
Squares inside squares
The same impossibility has a geometric face, which the video shows as a descent of shrinking squares. One classic version is credited to Stanley Tennenbaum, who found it as a student in the early 1950s.
Suppose again that p² = 2q² for whole numbers, and pick the smallest such pair. Then a square of side p has exactly the area of two squares of side q. Place the two smaller squares inside the big one, in opposite corners. Since 2q is more than p, they overlap in a small square in the middle, with side 2q − p, and they leave two corner squares uncovered, each with side p − q.
Now count area. The two smaller squares together have the same area as the big one, so the area covered twice (the overlap) must equal the area not covered at all (the two empty corners): (2q − p)² = 2(p − q)². That is a new whole-number solution of the same equation, and it is smaller. Repeat the trick and you get a smaller one still, forever. But positive whole numbers cannot shrink forever; the staircase has a bottom step. The only way out is that the first pair never existed.
The legend of Hippasus
Who first saw all this? Little is known for certain. The name usually attached to it is Hippasus of Metapontum, a Pythagorean of the fifth century BC, and the story attached to the name is dramatic. The story goes that the Pythagoreans kept the discovery secret, that Hippasus revealed it, and that he drowned at sea for it.
The ancient evidence is thinner and messier than the story. Iamblichus, writing in the third century AD, gives inconsistent reports. In one, a Pythagorean is merely expelled for revealing the nature of the irrational. In another, Hippasus perishes at sea for betraying how to construct a dodecahedron and taking the credit himself, and the drowning is a punishment from the gods. Pappus, in the fourth century AD, says only that the member who first divulged the secret of the irrational perished by drowning, and gives no name. No ancient writer specifically credits Hippasus with the discovery. The versions in which his shipmates throw him overboard are modern retellings. It is a legend, and the paired Short tells it in under a minute: The diagonal that broke the Pythagoreans.
What the record does show is that mathematicians kept going. In Plato’s dialogue Theaetetus, set in 399 BC, the young Theaetetus describes how his teacher Theodorus of Cyrene showed that the roots of 3, 5 and so on up to 17 are not commensurable with the unit, and then, for some reason, stopped. Plato does not credit Theodorus with √2, which suggests it was already settled. Theaetetus went on to generalize, and Book X of Euclid’s Elements, written around 300 BC, is almost certainly a description of his work: a long theory of incommensurable lengths that sorts irrational lines into thirteen kinds.
Filling the gaps in the number line
The Greeks handled √2 as a magnitude, a length to be compared with other lengths. The deeper problem shows up on the number line. Fractions are everywhere on it: between any two fractions sits another, their average. Yet √2 sits on the line and is not one of them. The fractions are packed tightly and still full of holes. So what, exactly, is the number in the hole? For more than two thousand years, mathematics used such numbers without saying what they are.
Richard Dedekind found a definition on 24 November 1858, while working out how to teach calculus for the first time at the Polytechnikum in Zürich. His idea was that every real number divides the fractions into two sets, those below it and those above it. For √2, the lower set holds every fraction that is negative or zero, plus every positive fraction whose square is less than 2; the upper set holds every positive fraction whose square is more than 2. No fraction sits on the boundary, because no fraction squares to exactly 2. Dedekind let the cut itself be the number. He published the idea in 1872, in Stetigkeit und irrationale Zahlen (Continuity and Irrational Numbers):
we create a new, irrational number a, which we regard as completely defined by this cut — Richard Dedekind, Stetigkeit und irrationale Zahlen, 1872
With cuts, every gap between the fractions is filled by a number, and the real number line is complete.
The six-step argument remains a model of proof by contradiction, and its idea travels: a similar argument shows that the square root of any whole number that is not a perfect square is irrational. Other famous numbers needed far harder proofs. π was shown to be irrational only in 1761, and it turned out to be stranger than √2, which at least solves the simple equation x² − 2 = 0; the story of π as the unit of turning picks up there.
A square with side 1 hides a length that no fraction can name, and it took mathematics more than two thousand years to say what kind of number it is.
