constants - What are the uses of Euler's number $e$?

People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never give me examples. Where is the number $e$ used? Also, how did Euler come up with the concept of some number $e$ that has the following property:

$$e=\sum\limits_{n=0}^\infty \dfrac{1}{n!}=\dfrac{1}{0!}+\dfrac{1}{1!}+\dfrac{1}{2!}+\cdots$$

I am just curious why people make such a big deal out of $e$.

Answer

$e$ is fundamental in mathematics. Aside from the awesome properties of $e$, such as $e^{i \pi}+1=0$ and the fact that

$$\frac{d}{dx} e^x=e^x,$$ it is also found in equations that directly relate to everyday phenomena. For instance, the normal distribution is represented by the probability density function $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}.$$ It also appears in Newton's law of cooling/heating, in the solution to the differential equation $$\frac{dT}{dt}=-k(T-T_0).$$

Alongside these, $e$ appears in the solutions of many differential equation that model anything from electric circuits to spring-mass systems. As for how Euler showed that

$$e=\sum_{n=0}^{\infty}\frac{1}{n!}$$ I am not sure.