**A picture proof that √2 is irrational**This Friday I was hanging out with some philosophy professors. This is always fun, because they think sort of like me, but different. They seem more optimistic about our ability to solve all sorts of puzzles just by talking.

To annoy them a bit, I said that philosophers are great at

*verbal* reasoning, but mathematicians should be good at three kinds of reasoning:

*verbal*,

*symbolic* and

*visual* reasoning.

In response, one of them showed me this picture proof that √2 is irrational.

We just need to show that it's impossible to have

a² = b² + b²

for whole numbers a and b. So let's do a proof by contradiction. We can assume a is the

*smallest* whole number that obeys this equation for some whole number b. We'll get a contradiction, by finding an even smaller one.

We do it by drawing a picture.

The big square in this picture is an a × a square. The two light blue squares, which overlap in the middle, are b × b squares.

The area of the big square is the sum of the areas of the light blue squares. But there are two problems. First, the light blue squares overlap. Second, they don't cover the whole big square! These two problems must exactly cancel out.

So, the area of the overlap - the dark blue squares - must exactly equal the area that's not covered - the two pink squares.

So, the area of the dark blue square is the sum of the areas of the pink squares! But the lengths of the sides of these must be whole numbers, say c and d. So we have

c² = d² + d²

But c is smaller than a. So, we get a contradiction!

Actually this proof uses a mix of verbal and visual reasoning, with just a tiny touch of symbolic reasoning. I wrote the formulas like a² = b² + b² just to speed things up a bit and reassure you that this was math. I didn't really do anything with them.

The philosophers who told me about this are Mike Pelczar and Ben Blumson. The picture here comes from a website Mike pointed me to:

• Dave Richeson, Tennenbaum’s proof of the irrationality of the square root of 2,

http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-irrationality-of-the-square-root-of-2/Richeson says:

**Apparently the proof was discovered by Stanley Tennenbaum in the 1950’s but was made widely known by John Conway around 1990. The proof appeared in Conway’s chapter “The Power of Mathematics” of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).**On the other hand, Ben says John Bigelow published the proof in his book

*The Reality of Rumbers* in 1988, without citing anyone.

We wondered if it was known to the ancient Greeks.

You can do similar proofs of the irrationality of √3, √5, √6 and √10:

• Stephen J. Miller and David Montague, Irrationality from the book,

http://arxiv.org/abs/0909.4913.

And this particular style of proof by contradiction is famous! It's called

**proof by infinite descent**. You assume you have the

*smallest* whole number that's a counterexample to something you want to prove, and then you cook up an even

*smaller* one. It's really just mathematical induction in disguise, but it's more fun. It was developed by Pierre Fermat - who, by the way, was a lawyer.

If you want to take all the fun out of the proof I just gave, you can do it like this.

Assume a is the smallest whole number for which there's a whole number b with

a² = b² + b²

Let

c = 2b - a

and

d = a - b

Then c and d are whole numbers and

c² = d² + d²

(You can do some algebra to check this.) But c < a, so we get a contradiction.

Wikipedia shows you how to prove by infinite descent that whenever n is a whole number, either √n is a whole number or it's irrational:

https://en.wikipedia.org/wiki/Proof_by_infinite_descentFermat did a lot more interesting stuff with this method, too!

#spnetwork arXiv:0909.4913

#geometry