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Surprising rational/irrational formula

  1. Aug 9, 2008 #1
    I found this interesting. Maybe a few other people here will too. I would have never thought that there is a formula based only on limits and elementary functions that tells whether a given number is rational or irrational. Anyone else think this is cool?

    [tex]f(x)=\lim_{m\to\infty}[\lim_{n\to\infty}cos^{2n}(m!\pi x)][/tex]
  2. jcsd
  3. Aug 9, 2008 #2
    Not that this is useful for any purpose, because to evaluate that limit you'd probably already have to know if the number is rational or irrational... but I still thought it was neat.
  4. Aug 9, 2008 #3


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    Sur it's cool.
    I'm sure it could be used for something, even though that would detract something from its pure beauty.
  5. Aug 9, 2008 #4
    What exactly is the theorem you are referring to?
  6. Aug 10, 2008 #5
    If x is rational, f(x)=1, because eventually m! is a multiple of the denominator of x, and so m!x is an integer. Then cos^{2n}(m!x pi) = 1, so the inner limit is 1.

    If x is irrational, no matter how high m is, m! is not an integer and so cos^2(m!x pi) is less than one, and so the inner limit is equal to zero and hence f(x)=0.
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