TITLE: Why is $e$ so special?
QUESTION [5 upvotes]: The number $e$ (and the exponentiation function $e^x$) appears in so many places in mathematics and engineering. There seem to be a multitude of applications of it. I want to know why.

REPLY [2 votes]: The idea of exponential growth or decay is natural: certain continuous quantities  increase $10\%$ every year,  other quantities  get halved in the course of $10\,000$ years, etcetera. Given a unit of time there are still "slow" and "fast" exponential increases: a quantity can double in one unit of time, or it can increase by a factor of $100$ in one unit of time. Therefore we need a unit for the "speed" of exponential growth. We arrive at it in the following way: Among all exponential growth curves $t\mapsto f(t)$ with $f(0)=1$ there is exactly one which has slope $1$ at $(0,1)$. The base for this particular $f$ is a certain number $>1$.  This number is called $e$, its value turns out to be about $2.718$. In this way the standard exponential function is $f(t)=e^t$.