Newsgroups: sci.math From: wald@ford.uchicago.edu (Kevin Wald) Subject: Re: pi and e irrtional Date: Wed, 8 Jul 1998 19:19:01 GMT In article <35A2AE1D.9C9C4FD2@rocketmail.com>, Peter Ammon wrote: >Nope, sorry, it won't work this time either. I'll let you or somebody >else work out the proof that pi is irrational. Here's one I posted to sci.math a while back (after presenting it to my calculus class in slightly different form): ---------------------------------------------------------------- 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! --------------------------------------------------------------- I based this, by the way, on a proof I saw posted to sci.math years ago (and which can be found at http://www.math.niu.edu/~rusin/known-math/95/pi_irrational ); that proof gives somewhat more detail, but is, alas, entirely in prose. Kevin Wald, wald@math.uchicago.edu | "Catalog of ships -- I'll remember that." http://www.math.uchicago.edu/~wald | -- Homer, _The Huntress and the Sphinx_ ============================================================================== [URL updated 1999/01 -- djr]