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):
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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!
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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_
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