TITLE: Is it known if $\pi + e$ is transcendental over the rational numbers?
QUESTION [7 upvotes]: I recall reading a comment on reddit that had stated that it is not known if $\pi + e$, (nor $\pi e$) is transcendental over $\mathbb{Q}$, nor even if it is irrational. Is this true? It strikes me as something which is very easy to prove, in my head I could figure out the rough links quite quickly. I'm not necessarily asking $\textit{for}$ a proof, just confirmation that it's not actually some mathematical mystery that people have been trying to solve for centuries, as that comment implied.
EDIT: Okay I've been humbled, there was a gap in my logic, I'd assumed that $\mathbb{Q}(\pi) \neq \mathbb{Q}(e)$, and then realised after reading all the skepticism that that may not be a simple thing to prove....... after a good amount of time spent over it, I realise that it really, really isn't.  
I also admit that this sounded really cocky as a question, sorry. I genuinely believed that it couldn't be as difficult a question as the comment indicated, it didn't seem like it could possibly be an open problem.

REPLY [8 votes]: You might want to consider the following two irrational numbers:
$$\sqrt{2}=1.414213562373095048801688724209698.....$$
$${23481838282\over 245689351}-\sqrt{2}=94.1611061917175085769126386338...$$
Just from a quick look at those two (apparently random) decimal expansions, would you have guessed that the sum of these two irrational numbers is equal to the rational number $23481838281/245689351$?
Does your proof rule out a similar relation between $\pi$ and $e$?  How so?