Surprising rational/irrational formula

[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?

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

 

[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.

 

[arildno]

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

 

[DeadWolfe]

What exactly is the theorem you are referring to?

 

[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.