Ode to Hippasus
Ode to Hippasus
A musical proof of the irrationality of the square root of two
This project began two years ago when Hannah Hoffman had inquired on Twitter whether mathematicians could provide a proof of the irrationality of root two in rhyming verse.
Of course, I had set off immediately to answer the challenge. My wife Barbara Gail Montero and our daughter Hypatia and I spent a day in Oxford 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, afterwards drowned at sea. Was it punishment from the Gods?
Hannah used our lyrics to create the amazing musical version. Enjoy!
Ode to Hippasus
Lyrics by Joel David Hamkins, Barbara Gail Montero & their daughter Hypatia
Music by Hannah Hoffman
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 alternatively 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.
This is the first of several mathematical/philosophical music videos I have undertaken with Hannah Hoffman, all of which will eventually be posted here on Infinitely More—look for them in the music-videos section, linked on the main menu.
I wrote a paper with Conway on proofs of the irrationality of sqrt(2)
http://dev.mccme.ru/~merzon/mirror/mathtabletalks/files/irrational-conway.pdf
We classified 7 “essentially different” ones, based on how they generalized (each one has a natural set of algebraic numbers its argument can be applied to, and these sets overlap in various ways).
Recently I discovered an 8th proof which generalizes furthest of all, to all algebraic irrationals. Can you guess it?
There is also the ridiculous 9th proof using ultraproducts, which generalizes to all roots of irreducible polynomials that are unsolvable mod p for infinitely many p (which is, in fact, all algebraic irrationals), but is useless for most of them because you need the something like the Chebotarev density theorem to show that condition holds.