Why is $e$ irrational? I'm not looking for a proof, but rather an explanation, because I know there's something wrong with my thinking.
So, I know that
$$ e = \lim_{n\to\infty} (1+\dfrac{1}{n})^n $$
And also
$$ e = \sum_{k=0}^{\infty} \dfrac{1}{k!} $$
And I'm confused as to why $e$ can be irrational, since both of those definitions are a rational number(?).
I know that addition under rationals is closed, so I'm confused as both of these can be rearranged to something with rationals (e.g. sum of rationals, or $(\dfrac{n+1}{n})^n$), so I guess my question is, is why these rationals converge to an irrational?
Thanks.

Just because every partial (finite) sum is a rational number, that does not mean the limit is.
Take the well known constant as an example: $\pi = 3+0.1+0.04+...$
Every partial sum is rational, but the limit is a well known irrational number.
Similarly, for every finite natural number $n$, $1/n > 0$ but the limit $\lim_{n\to\infty} 1/n \not > 0$.
