Hippasus and the incommensurability of the square root of 2

Why do we need a set larger than the rationals $\mathbb{Q}$? The historical answer springs from a crisis. For the followers of Pythagoras of Samos (6th century BC) the motto was “all is number”: they were convinced that every length could be expressed as a ratio $\dfrac{p}{q}$ of two integers. Two segments are called commensurable if there exists a small common unit of measure that fits a whole number of times into both; for the Pythagoreans every pair of segments had to be so.

The discovery that shatters this belief concerns the simplest thing in the world: a square of side $1$. By the Pythagorean theorem its diagonal $d$ satisfies $d^2 = 1^2 + 1^2 = 2, \qquad \text{that is}\qquad d = \sqrt{2}.$ Tradition attributes to Hippasus of Metapontum (5th century BC), a Pythagorean from Magna Graecia, the proof that $\sqrt{2}$ is not a ratio of integers: the side and the diagonal of the square are incommensurable. This was the birth of the irrational numbers and the first great scandal of Greek mathematics (Boyer).

Why $\sqrt{2}$ is not rational

Suppose, for contradiction, that $\sqrt{2} = \dfrac{p}{q}$ with $p, q$ integers having no common factors (a fraction already reduced to lowest terms). Squaring: $2 = \frac{p^2}{q^2} \;\Longrightarrow\; p^2 = 2q^2.$ Then $p^2$ is even, hence $p$ too is even: write $p = 2k$. Substituting, $4k^2 = 2q^2$, so $q^2 = 2k^2$: therefore $q^2$ is even and hence $q$ is even. But if $p$ and $q$ are both even they share the factor $2$, against the hypothesis that the fraction was in lowest terms. The contradiction shows that $\sqrt{2}$ cannot be written as $\dfrac{p}{q}$: it is irrational.

The legend — almost certainly embellished — has it that Hippasus was punished by drowning for disclosing this unacceptable secret (Dunham). Beyond the anecdote, the significance is enormous: the rationals $\mathbb{Q}$ are not even enough to measure the diagonal of a square, and one needs the wider set of the real numbers $\mathbb{R}$, which also contains the irrationals such as $\sqrt{2}$ and $\pi$ (Kline). The rigorous construction of this extension would arrive only in the nineteenth century, but the question that makes it necessary is twenty-five centuries old.

Links

Topics: Numbers and operations
Concepts: Number sets · Rational numbers · Real numbers · Irrational numbers
People: Hippasus · Pythagoras