How was the number e discovered and why is it special?

The number $e$ grew out of problems where change happens continuously. One route is compound interest. If you invest 1 unit at 100 percent interest: - compounded once, you get $2$, - compounded twice, you get $\left(1+\frac12\right)^2$, - compounded $n$ times, you get: $$\left(1+\frac{1}{n}\right)^n.$$ As $n$ gets very large, this approaches: $$e\approx 2.71828.$$ That is why $e$ is tied to continuous growth. In calculus, $e^x$ is special because its derivative is itself: $$\frac{d}{dx}e^x=e^x.$$ So $e$ is not just a random constant. It is the natural base for growth, decay, interest, probability distributions, and differential equations where the rate of change is proportional to the current amount.