Proof in a Poem

Inspired by Hannah Hoffman, here’s a poem proof of the irrationality of the square root of 2.

Suppose that root 2 could be taken to be
The integer a over the integer b
Where a and b have no factor in common
So nothing divides both the top and the bottom

Some algebra that’s quite easy to do
Gives a2 is equal to b2 times 2
But if a2 is even, so too must be a
Now each a in a2 has a 2 in our play

So a2 in fact has a factor of 4
But since this equals 2b2 we can say more
This b2 must now have a factor of 2
And just as above we know b has one too

It appears that we are now able to say
That 2 divides b and 2 divides a
But common factors were assumed to be none
So this contradiction shows we are done

Be sure to check out the excellent efforts of Timothy Gowers and Joel David Hamkins as well.

Here’s the original tweet.


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