Recently in class when I asked students to name five greatest mathematicians (Newton, Leibniz, Euler, Gauss, Poincaré, Hilbert — and yes, I do know how to count), someone bravely shouted Pythagoras. Pythagoras gets points for being a vegetarian, if he indeed was a vegetarian, and for having a golden thigh, but if his claim to mathematical fame rests on having proved the Pythagoras theorem, then it should be more widely known that he did not, and that his proof is known from earlier Babylonian sources.
There are many proofs of the Pythagoras theorem, which as you know says that in a right angle triangle with hypotenuse
But I also have a favourite complicated proof. This one comes from a book by Gregory Barenblatt (Scaling, CUP, Cambridge 2003) which I recommend. This proof is guaranteed to break the ice at parties. Why should one bother with a complicated proof of a simple fact? Because it introduces in a reasonably straightforward way a very useful idea which can be used for harder facts.
First let us introduce some notation. We will use
Let us see one application of thinking in units, or as it is put more frequently, “dimensionally”. Consider a pendulum with a bob of mass
So the only way to get seconds out of the ingredients of the problem (
This type of analysis, “dimensional analysis”, is powerful stuff. Let us see how to prove Pythagoras’ theorem by dimensional analysis.
First of all note that there are many ways of uniquely specifying a right-angle triangle. For example, it can be specified by the two perpendicular sides,
If we now drop a perpendicular from
So now cancel