Pythagorean Proof

Every once in a while, my mind gets snagged by numbers, nowhere as often as when I was seven or eight and used to work out sums in my head (that was before I discovered masturbation).

A Financial Times recent piece by Tim Harford entitled “We must face facts — even the ones we don’t like” provides a good example.

The universe is not constructed in terms of whole numbers. A hypotenuse across a square  is prima facie evidence, a dilemma for true-believing Pythagoreans, who thought whole numbers were the basis of everything. Hence the fun in using the Pythagorean theorem to show they are not.

The square of the hypotenuse is the sum of the square of the other two sides. Formulaically:  c2 = a2 + b2 

Assume a whole number fraction, a/b, does equal √2, that is a/b  =  √2 . Let’s also assume that a/b is the simplest possible fraction, with a and b sharing no common factors.

Rearranging a/b  =  √2  gives us 2b2 = a2.

That means that a2 is an even number, which implies four thngs: a is also even. and therefore a2/2 is also even, therefore b2 is even and therefore b is even.

Alas, we began by assuming that a/b was the simplest possible whole number fraction, but we’ve just proved that a/b is the ratio of two even numbers and it follows that this fraction could be simplified by dividing both of them by 2.

This contradion shows that our original assumption — that a and b exist at all — must be wrong.

From Tim Harford, Undercover Economist, Financial Times, 10/11 July, 2021, p 18.