I was so glad to be involved with this project of Hannah Hoffman. She had inquired on Twitter whether mathematicians could provide a proof of the irrationality of root two that rhymes. I set off immediately, of course, to answer the challenge. My wife Barbara Gail Montero and our daughter Hypatia and I spent a day thinking, writing, revising, rewriting, rethinking, rewriting, and eventually we had a set lyrics providing the proof, in rhyme and meter. We had wanted specifically to highlight not only the logic of the proof, but also to tell the fateful story of Hippasus, credited with the discovery.

Hannah proceeded to create the amazing musical version:

The diagonal of a square is incommensurable with its side

an astounding fact the Pythagoreans did hide

but Hippasus rebelled and spoke the truth

making his point with irrefutable proof

it’s absurd to suppose that the root of two

is rational, namely, p over q

square both sides and you will see

that twice q squared is the square of p

since p squared is even, then p is as well

now, if p as 2k you alternately spell

2q squared will to 4k squared equate

revealing, when halved, q’s even fate

thus, root two as fraction, p over q

must have numerator and denomerator with factors of two

to lowest terms, therefore, it can’t be reduced

root two is irrational, Hippasus deduced

as retribution for revealing this irrationality

Hippasus, it is said, was drowned in the sea

but his proof live on for the whole world to admire

a truth of elegance that will ever inspire.

If the square root of two was rational

It then would have to be fractional.

If it was a over b

Then you must see

This could not be actual

For a-square over b-square would equal two

But this certainly would not do

With a bit of effort as we plod

We see a could be neither even nor odd

And this would be counter factual.

Typo: a proof of the irrationality of two

Oh dear! Now fixed.