I remember hearing an interesting theory once, I don't know the source. Since there are some numbers that are precisely expressible in decimal notation that repeat in a binary base, and vice versa, perhaps there exists a base in which irrational numbers are rational.
The more I think about it, the less likely this seems, and my guess is that whatever proof that $\pi$ or $e$ are irrational doesn't involve the decimal base notation. But perhaps there's some research into this?
As long as we are restricting ourselves to integer bases, a rational number will always have a repeating pattern in any base, and an irrational number will not repeat in any base. If $x$ has a repeating pattern to base $b$, then there are exponents $n$, $m$ such that $b^nx - b^mx = y$, where $m$ is chosen to leave nothing but the repeating pattern to the right of the radix, and $n$ does the same, but with one repetition also to the left, which means that $y$ is an integer. But then $x = {y\over b^n-b^m}$, which is rational.