My intuition tells me that it's not rational, and even not algebraic (i.e., it's transcendental).
But I'm having a hard time showing it.
Taking it slightly further, $e-\frac1e=\frac{e^2-1}{e}=\frac{(e+1)(e-1)}{e}$, but I don't feel I'm in the right direction.
Any ideas how to tackle this one?
Thank you!
If $p$ and $q$ are integers and
$$\frac{e^2-1}{e} = \frac{p}{q}$$
Then $qe^2-pe-q = 0$, so $e$ would satisfy a quadratic polynomial over the integers. Since $e$ is not algebraic, that gives a contradiction.