The Pythagoreans would say that the diagonal of a square is incommensurable with its side. That is, if you draw a square, where each side is measured 1, then what is the length of its diagonal? The answer is , an *irrational* number (one that cannot be written as a ratio, such as ).

There are many possible proofs that is irrational, but I like this one because of its historical import. For instance, according to Boyer’s *History of Mathematics*:^{1}

“Aristotle refers to a proof of the incommensurability of the diagonal of a square with respect to a side, indicating that it was based on the distinction between odd and even.”

He’s referring to Aristotle’s *Prior Analytics*, 17.65b.16-21:^{2}

For to put that which is not the cause as the cause, is just this: e.g. if a man, wishing to prove that the diagonal of the square is incommensurate with the side, should try to prove Zeno’s theorem that motion is impossible, and so established a

reductio ad impossibile: for Zeno’s false theorem has no connection at all with the original assumption.

## Proof that is Irrational

I’ll start by assuming^{3} that is *rational* and then proceed to show that this is impossible.^{4}

If is RATIONAL, then it can be written as a fraction, such as , where both and are elements of the Integers, , and . Assume that I’ve also chosen both and to be in *lowest terms* — you can’t reduce the fraction down any further because they don’t share a *common factor*.

Given that

then

which implies that

This means that is and *even* integer by definition, and because of this, must also be an even integer (because the square of an odd integer is always odd). Therefore there exists an integer such that .

We can then plug into the above equation and get

But that means that is also even, which (by the same property mentioned above) means that is even!

However, this presents a problem. If BOTH and are even, then the fraction cannot be in lowest terms — you can at least divide both the numerator and the denominator by 2, if not something else. But that’s a contradiction to our initial assumption that .

Therefore, is not a rational number, which is to say, it is irrational.

*Now go lift something heavy,*

Nick Horton

PS. The pic at the top is of a Gelada Baboon.