Incommensurables

The geometers immediately following Pythagoras (c. 580–c. 500 bc) shared the unsound intuition that any two lengths are “commensurable” (that is, measurable) by integer multiples of some common unit. To put it another way, they believed that the whole (or counting) numbers, and their ratios (rational numbers or fractions), were sufficient to describe any quantity. Geometry therefore coupled easily with Pythagorean belief, whose most important tenet was that reality is essentially mathematical and based on whole numbers. Of special relevance was the manipulation of ratios, which at first took place in accordance with rules confirmed by arithmetic. The discovery of surds (the square roots of numbers that are not squares) therefore undermined the Pythagoreans: no longer could a:b = c:d (where a and b, say, are relatively prime) imply that a = nc or b = nd, where n is some whole number. According to legend, the Pythagorean discoverer of incommensurable quantities, now known as irrational numbers, was killed by his brethren. But it is hard to keep a secret in science.

The ancient Greeks did not have algebra or Hindu-Arabic numerals. Greek geometry was based almost exclusively on logical reasoning involving abstract diagrams. The discovery of incommensurables, therefore, did more than disturb the Pythagorean notion of the world; it led to an impasse in mathematical reasoning—an impasse that persisted until geometers of Plato’s time introduced a definition of proportion (ratio) that accounted for incommensurables. The main mathematicians involved were the Athenian Theaetetus (c. 417–369 bc), to whom Plato dedicated an entire dialogue, and the great Eudoxus of Cnidus (c. 390–c. 340 bc), whose treatment of incommensurables survives as Book V of Euclid’s Elements.

Euclid gave the following simple proof. A square with sides of length 1 unit must, according to the Pythagorean theorem, have a diagonal d that satisfies the equation d² = 1² + 1² = 2. Let it be supposed, in accordance with the Pythagorean expectation, that the diagonal can be expressed as the ratio of two integers, say p and q, and that p and q are relatively prime, with p > q—in other words, that the ratio has been reduced to its simplest form. Thus p²/q² = 2. Then p² = 2q², so p must be an even number, say 2r. Inserting 2r for p in the last equation and simplifying, we obtain q² = 2r², whence q must also be even, which contradicts the assumption that p and q have no common factor other than unity. Hence, no ratio of integers—that is, no “rational number” according to Greek terminology—can express the square root of 2. Lengths such that the squares formed on them are not equal to square numbers (e.g., Square root of√2, Square root of√3, Square root of√5, Square root of√6,…) were called “irrational numbers.”