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.