Surprising rational/irrational formula

1. Aug 9, 2008

uman

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?

$$f(x)=\lim_{m\to\infty}[\lim_{n\to\infty}cos^{2n}(m!\pi x)]$$

2. Aug 9, 2008

uman

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.

3. Aug 9, 2008

arildno

Sur it's cool.
I'm sure it could be used for something, even though that would detract something from its pure beauty.

4. Aug 9, 2008

DeadWolfe

What exactly is the theorem you are referring to?

5. Aug 10, 2008

uman

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.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook