Why is $e$ irrational?

Question

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.

Answer 1

The error lies in the assertion that “both of those definitions are a rational number”. Not true. Both of them express $e$ as a limit of a sequence of rational numbers. Your error lies in the hidden (and false) assumption that the limit of a convergent sequence of rational numbers must be rational too.

Answer 2

That the rationals are closed under addition means that the sum of any two rationals is a rational again. By induction you can prove that this works for any finite number of rationals. But the rationals are not closed under infinite sums.

Answer 3

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 $\lim_{n\to\infty} 1/n \not > 0$.