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

$e$ 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.

$e$ 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.

Eular's identity pops up everywhere, in calculus, differential equations and even probabilities. For example in elementary probability theory, it shows up in the Poisson distribution general formula which is used to calculate the probability of an event occurring given that we know the rate at which it happens $$P(X=k)=\frac{\lambda^k}{k!} e^{-\lambda}$$

Furthermore, the hyperbolic identities are defined in terms of e.

There are so many applications to e, that I can pretty much write an entire book on it (and someone probably has). It is very important to get comfortable with it as if you're doing math, it will show up everywhere.