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. ---------------------------------------------------------------- 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!