Why is $e$ the number that it is? Why is $e$ the number that it is?  Most of the irrational number that we learn about in school have something to do with geometry, like $\pi$ is the ratio of a circle's diameter to its circumference.  Those numbers can also be derived using maths ($\pi = \lim_{n\to\infty}4\sum_{k = 1}^{n}\frac{(-1)^{k+1}}{2k - 1}$).  On the other hand, $e$ is derived only from math, and finds no other immediate geometric basis.  Why then, is $e$ the number that it is ($2.7182\ldots$) and not some other number?
EDIT: My question is more along the lines of why $e$ is $2.7182\ldots$ On an alien world where there is a different system of numbers, could there be a similar constant that has all the properties of $e$, but is not $2.7182\ldots$?

$e$ is like matrices. Or sets. When you first learn them, you think, "Err...so what?" As you learn more mathematics, you find yourself using these things more and more, until you can't imagine doing mathematics without them.
