Interesting facts/ proofs about rational and irrational numbers We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at. 

You may find the Gelfond-Schneider theorem interesting. The theorem states that:

If $a$ and $b$ are algebraic numbers with $a ≠ 0,1$ and $b$ non-rational, then any value of $a^b$ is a transcendental number.

This not only gives rise to trivial results such that $2^{\sqrt 2}$ is irrational, but to some more interesting ones as well. For instance, by taking $a=-1$ and $b=-i$, where $i=\sqrt {-1}$, we can obtain the following result:
$$
(-1)^{-i} = \left( e^{i \pi} \right)^{-i} = e^\pi,
$$
showing that $e^\pi$ is transcendental.
