Why the Square Root of 2 Is Irrational

The square root of 2 is irrational: it cannot be written as a fraction p/q of whole numbers. The classic proof assumes it can, in lowest terms, and shows that p and q would then both be even — impossible for a fraction in lowest terms. The question matters because √2 is the diagonal of a square with side 1.

This page explains where √2 comes from, proves its irrationality twice (once with algebra, once with a picture), separates the legend of Hippasus from the evidence, and shows how a Babylonian scribe approximated it nearly 4,000 years ago.

√2 is the diagonal of a unit square

Cut a square with side 1 along its diagonal d and you get two right triangles with legs 1 and 1. By the Pythagorean theorem,

The same number appears in every 45-45-90 triangle: the hypotenuse is always √2 times a leg. Plato's dialogue Meno contains a famous geometric version. Socrates asks an enslaved boy with no training in geometry how to draw a square with twice the area of a given one; the answer is the square built on the diagonal.

What "irrational" means

A rational number is one that can be written as a ratio p/q of whole numbers with q ≠ 0, such as 7 over 5 or 3 = 3 over 1. Its decimal expansion either stops or repeats forever. An irrational number is a real number that is not rational; its decimals never stop and never fall into a repeating pattern.

The Greeks thought in terms of lengths, not decimals. Two lengths are commensurable if some small unit fits a whole number of times into each. Saying √2 is irrational is the same as saying that the side and the diagonal of a square are incommensurable: no unit, however tiny, measures both exactly. That was a shock to a culture that assumed every pair of lengths had a common measure.

The classic proof: even and odd

We need one small fact first. If p² is even, then p is even. Reason: if p were odd, say p = 2k + 1, then p² = 4k² + 4k + 1 would be odd.

1. Assume the opposite. Suppose √2 = p over q for whole numbers p and q, with the fraction in lowest terms (no common factor).
2. Square both sides. 2 = p² over q², so p² = 2q². That makes p² even, so p is even: write p = 2m.
3. Substitute. (2m)² = 2q² gives 4m² = 2q², so q² = 2m². Now q² is even, so q is even too.
4. Reach a contradiction. Both p and q are even, so they share the factor 2 — but we chose a fraction in lowest terms. The assumption must be false.

This argument is very old. Aristotle, writing in the 4th century BCE, mentions in passing (Prior Analytics I.23) the proof that the diagonal is incommensurable with the side because otherwise odd numbers would equal even ones — clearly this proof. A full version appears as Proposition 117 of Book X in some manuscripts of Euclid's Elements, although historians have long agreed it was added after Euclid.

The same idea with prime factors

Every whole number factors into primes in exactly one way. In p² every prime appears an even number of times; in 2q² the prime 2 appears an odd number of times. So p² = 2q² is impossible. The same reasoning shows that √n is irrational for every whole number n that is not a perfect square — so √3, √5 and √8 are irrational, while √9 = 3 is not. It also explains why no Pythagorean triple has two equal legs: an isosceles right triangle with whole-number legs q would need a whole-number hypotenuse q√2. See square roots and radicals for working with these numbers, or simplify one instantly with the square root simplifier.

Tennenbaum's proof without words

There is a lovely geometric proof attributed to the American logician Stanley Tennenbaum, who found it as a student in the early 1950s; John Conway later made it famous. It turns "p² = 2q²" into a picture about squares of carpet.

1. Set up. Suppose p² = 2q² for whole numbers, and pick the smallest such p. Picture a square room of side p and two square carpets of side q; together the carpets have exactly the room's area.
2. Lay the carpets in opposite corners. Since q < p < 2q, they overlap in a central square of side 2q − p and leave two bare corner squares of side p − q.
3. Compare areas. Because the carpets' total area equals the room's area, the area covered twice must equal the area not covered at all: (2q − p)² = 2(p − q)².
4. Contradiction. That is the same equation with the smaller whole numbers 2q − p and p − q, contradicting the choice of the smallest p.

This style of argument — any solution produces a smaller one, so there can be no solution at all — is called infinite descent. It needs no facts about even and odd numbers, only areas. Pierre de Fermat used the same method on right triangles to prove the case n = 4 of Fermat's last theorem. Tom Apostol published another geometric descent proof in 2000, using a compass construction inside an isosceles right triangle with whole-number sides.

Hippasus: legend versus evidence

Popular books often say that a Pythagorean named Hippasus of Metapontum discovered that √2 is irrational, and that his horrified brotherhood drowned him for revealing it. The story is memorable. The evidence for it is thin.

What historians broadly accept is more modest. Incommensurability was discovered in the 5th century BCE, probably within or close to the Pythagorean school, and by about 400 BCE it was common knowledge among mathematicians: Plato's Theaetetus shows Theodorus proving the irrationality of √3, √5 and further roots up to √17, taking √2 for granted (see the spiral of Theodorus). Some scholars, notably Kurt von Fritz in 1945, argued that the first incommensurable pair was not the square's side and diagonal at all but the side and diagonal of the regular pentagon. Treat the drowning as a legend, and the identity of the discoverer as unknown. For more on what we really know about the Pythagoreans, see Pythagoras.

YBC 7289: a Babylonian approximation

Long before the Greeks worried about incommensurability, Babylonian scribes were computing √2 to remarkable accuracy. The clay tablet known as YBC 7289 (Yale Babylonian Collection, number 7289) dates from the Old Babylonian period, roughly 1800–1600 BCE. Its significance was recognized by Otto Neugebauer and Abraham Sachs in 1945.

The Babylonians wrote numbers in base 60. The number along the diagonal, written today as 1;24,51,10, means

The true value is 1.41421356…, so the scribe's value is too small by only about 0.0000006 — less than one part in two million. The number below it, 42;25,35 ≈ 42.4263889, is exactly 30 × 1; 24, 51, 10: the diagonal of a square of side 30. In other words, the scribe knew that diagonal = side × √2, the Pythagorean relationship for a square, more than a thousand years before Pythagoras. How the value was computed is not recorded; a table of constants with the same entry survives, which suggests it was a standard value. The same approximation still appears much later in Ptolemy's Almagest (2nd century CE). For more Babylonian mathematics, see Plimpton 322 and the history of the Pythagorean theorem.

Decimals and the continued fraction of √2

Because √2 is irrational, its decimal expansion never ends or repeats. To 40 decimal places:

√2 = 1.4142135623 7309504880 1688724209 6980785696…

A much more revealing way to write it is as a continued fraction. Start from √2 = 1 + (√2 − 1) and notice that √2 − 1 = 1 over √2 + 1, since (√2 − 1)(√2 + 1) = 1. Also √2 + 1 = 2 + (√2 − 1). Substituting again and again:

The pattern of 2s goes on forever. That gives yet another proof of irrationality: every rational number has a continued fraction that stops. Cutting the pattern off at each stage gives the convergents, the best fractions for their size:

Each new fraction comes from the previous one by the rule p over q → p + 2q over p + q, and each satisfies p² − 2q² = ±1 — a near miss for p² = 2q², and an instance of Pell's equation. The Greeks knew this sequence as the "side and diagonal numbers"; it is described by Theon of Smyrna around 100 CE. Notice that running the rule backwards, (p, q) → (2q − p,  p − q), is exactly the descent step in Tennenbaum's proof: 99/70 → 41/29 → 17/12. The decimals were rounded to nine places and the errors to three significant figures.

The ratio itself turns up in everyday life: ISO paper sizes such as A4 have sides in the ratio 1 : √2, so that folding a sheet in half gives a smaller sheet of exactly the same shape.