This question already has an answer here:
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?