[Standback]

I may or may not have quoted this before on this forum. And even if I haven't, I don't think I've ever noticed any particular enthusiasm for Gilbert and Sullivan here. But since it took me more than three seconds to relocate the lyrics, I figure this is a good place for a handy backup. With no further ado...

The Pi Proof of Penzance
A proof of the irrationality of pi, to a well-known tune

That pi must be irrational, I claim, is demonstratable:
Assume that with a quotient of whole numbers it's equatable
-- Say, m o'er n. Define a_k, by fiat dictatorial,
For every natural k to be one over k factorial
Times integral from naught to pi of (n times (t)(pi - t))
To power k, times sine (or for you Latin scholars, _sinus_) t,
dt. These a's are *positive*, with *finite sum* (indeed, it e-
Quals integral exp(n times (t)(pi - t)) sin t dt).

Chorus: It's integral exp(n times (t)(pi - t)) sin t dt!
It's integral exp(n times (t)(pi - t)) sin t dt!
It's integral exp(n times (t)(pi - t)) sin t dt, dt!

But integrate by parts -- each a's the sum of the preceding two
Times integers, a_naught is 2, a_1's 4n, thus leading to
(since *all* must then be integers) a contradiction statable,
And thus that pi's irrational, you see, is demonstratable!

Chorus: Since *all the a's are integers*, a contradiction's statable,
And thus that pi's irrational, we see, is demonstratable!