A square’s diagonal breaks the rules of fractions
Draw a square whose sides are each one unit long, then connect two opposite corners. The diagonal forms the hypotenuse of a right triangle with sides 1 and 1. By the Pythagorean theorem, its length squared is 1² + 1², or 2—so the diagonal is √2.
The surprise is that √2 cannot be written as a fraction of whole numbers. Suppose it could: write it as a fraction a/b in lowest terms. Squaring gives a² = 2b², which means a must be even. But if a is even, that equation also forces b to be even. The fraction could then be reduced after all, contradicting the assumption that it was in lowest terms. No fraction works.
Why the discovery made such a story
For ancient Greek mathematicians, numbers and geometric lengths were closely connected, and whole-number ratios offered a powerful way to describe harmony and proportion. The diagonal of an ordinary square exposed a limit in that picture: a perfectly definite length could not be captured by any ratio of whole numbers. Such a length is called irrational—not because it is illogical, but because it is not a ratio.
Later accounts connect the discovery to Hippasus, a member of the Pythagorean community, and say he was drowned for revealing the result. The stories vary, and the details of what happened are uncertain; the mathematical result is not. The legend gives a dramatic shape to the deeper shock: a simple geometric figure could contain a length that the familiar language of fractions could never fully express.
