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