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. 
 A: 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.
A: The irrationality measure $\mu(x)$ of numbers such as $x=\pi$ or $x=e$ has been studied in detail, but there are still plenty of conjectures. You can find the definition of this measure, here. In particular:

*

*$\mu(x)=1$ if $x$ is rational

*$\mu(x)=2$ if $x$ is algebraic of degree $> 1$

*$\mu(x)\geq 2$ if $x$ is transcendental

More results about $\mu(x)$:

Another topic of interest is non-periodic continued fractions for irrational numbers, for instance (see here):

Another surprising fact is a simple recurrence relation to compute all the digits of some quadratic irrationals (see here) which could potentially help determining whether they are normal numbers:

Resulting in

Here the $d_n$'s are the digits of $x$.
Other results

*

*If $x$ and $x'$ are irrational normal numbers linearly independent
over the set of rational numbers, then the correlation between the
binary digits of $x$ and $x'$ is equal to zero. See here.

*If $x$ is irrational, then the sequence $\{kx\}$ where
$k=0,1,2\cdots$ and the brackets represent the fractional part
function, are equidistributed modulo 1.

