# 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

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.