The number that can't be a fraction

Draw a square with sides of length 1. Its diagonal has length √2, the number that gives exactly 2 when multiplied by itself. The fraction 7/5 comes close, and 99/70 comes closer. But no fraction of whole numbers equals √2 exactly, and the reason fits in five lines.

Why it is worth a look

Testing fractions cannot settle this: there are infinitely many. Richard Hammack's Book of Proof uses √2 to show the power of proof by contradiction: try to prove it any other way, he asks, and where would you begin? Assume a fraction does work, and follow the consequences until two of them clash.

The result also broke an assumption. In Kurt Von Fritz's 1945 account, the early Pythagoreans took it for granted that everything could be expressed in whole numbers — that any two lengths stand in a ratio of whole numbers. The side and diagonal of a square do not.

Three words and one fact

A number is rational if it can be written as a fraction a/b of whole numbers. A whole number is even if it is 2 times a whole number, and odd if it is one more than an even number.

The fact: an odd number squared is odd. Write the odd number as 2k + 1. Its square is 4k² + 4k + 1, which is 2(2k² + 2k) + 1: an even number plus one.

The argument

Suppose √2 = a/b. Cancel any common factors first, so the fraction is fully reduced. In particular, a and b are not both even, or you could cancel a 2.

Square both sides: 2 = a²/b², so a² = 2b². The right side is 2 times something, so a² is even.

Then a is even too. If a were odd, a² would be odd, by the fact above.

So a = 2c for some whole number c. Put that into a² = 2b²: 4c² = 2b², so b² = 2c². Now b² is even, and by the same reasoning, b is even.

Both a and b are even. But the fraction was fully reduced, so they are not both even. Every step followed from the last, so the first line is false.

Every fully reduced fraction breaks the chain somewhere. If a is odd, a² is odd and 2b² is even. If a is even then, the fraction being in lowest terms, b is odd, and a² is a multiple of 4 while 2b² is not. The miss can be as small as 1: 99² is 9,801; 2 × 70² is 9,800.

The same argument as a staircase

Drop the words "fully reduced" and the proof still works. It shows that any pair solving a² = 2b² is a pair of even numbers. Halve both. The fraction is unchanged, so the halved pair solves the equation too, so it is even too. Halve again, and again, forever. Whole numbers cannot do that: halve 1,024 ten times and you reach 1, which is odd. "Fully reduced" is a shortcut that starts at the bottom of this staircase. The proof shows the staircase has no bottom.

Where it comes from, and a legend

Von Fritz points to an old version of the proof preserved in an appendix to Book X of Euclid's Elements, and argues from Aristotle that it is, in outline, the original. It takes the smallest numbers in the ratio and sets out to show "the same number is at the same time even and odd." He argues the discovery was made no later than the middle of the fifth century BC.

Ancient tradition credits the discovery to a Pythagorean named Hippasus of Metapontum, and what it credits him with is incommensurability: the fact that some pairs of lengths have no common measure at all. Von Fritz notes that this tradition survives "only in the works of very late authors" and comes wrapped in "stories of obviously legendary character." He argues the discovery was probably first made in the regular pentagon — the face of the dodecahedron, the "sphere of 12 regular pentagons" the tradition ties to Hippasus — and that the square proof came later, once incommensurability was already known. One of these stories, told by Iamblichus centuries afterwards, has Hippasus drown in the sea as a punishment from the gods for making the Pythagoreans' secret mathematics public. It is a legend, not a record.

In short

If √2 were a fully reduced fraction a/b, then a² = 2b² would force a to be even, then b to be even, so the fraction could be reduced again. So √2 is irrational, and no search will ever find a fraction that works.