# 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