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 $e$.

$e$ is just as important as $\pi$ in mathematics having uses in pretty much every field.
For example,
$$e=\lim\limits_{x\to \infty}\left(1+\dfrac{1}{x}\right)^x$$
One of the most beautiful examples of its importance would be relating trigonometric functions to hyperbolic functions using the identity:
$$e^{ix} = \cos(x)+i\sin(x)$$
For example: 
$$\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$$
Using these identities can greatly simplify the computation of antiderivatives of rational functions involving trigonometric functions.
One also must note the beauty of:
$$e^{i\pi}+1=0$$
$e$ is used to compute the compound interest of a bank account which is compounded continuously.
Many integral transformations such as the Fourier Transform and Laplace Transform make use of e to map a function into different domains in order to make its computation more simple.
$e$ can be used to parameterize the unit hyperbola.
$e$ also defines the factorial function or more generally the gamma function which has uses all throughout mathematics. 
The uses of $e$ are seemingly endless as the number keeps popping everywhere in mathematics in all types of problems.
