Kevin Wald: The Pi Proof of Penzance.
A proof of the irrationality of pi, to a well-known tune.
(First presented to my calculus class a short while ago,
in a slightly different version.)
Note: the actual mathematical content of this work is based
on a proof I saw posted to sci.math a while back.
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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!