$ e $ and Trigonometry: Euler's Formula
The Taylor series of $ e^x $ is
$ \begin{align} e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots \end{align} $
Using this equation, it is possible to relate $ e $ to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, $ ix $ for example. This yields:
$ \begin{align} e^{ix} &= \sum^{\infty}_{n=0}{\frac{(ix)^n}{n!}}\\ &= \sum^{\infty}_{n=0}{\frac{i^nx^n}{n!}}\\ &= 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + i\frac{x^5}{5!} - \cdots\\ &= (1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots) + i(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots)\\ &= \sum^{\infty}_{n=0}{\frac{(-1^nx^{2n}}{(2n)!}} + i\sum^{\infty}_{n=0}{\frac{(-1)^nx^{2n+1}}{(2n+1)!}}\\ &= \cos(x) + i\sin(x) \end{align} $
This surprising result was first published by Euler in 1748. It leads to a large number of significant insights regarding all of its parts. The first important corollary is that every nonzero complex number can be written as $ e^{a+bi} $. This follows from $ \{\cos(x)+i\sin(x) | x \in \mathbb{R}\} $ being the unit circle: it is obvious that every point can be written as a unit circle point multiplied by a scalar.
This leads to an insight regarding complex multiplication: First, observe that $ \left|e^{a+bi}\right| = e^a $. This indicates that the length and angle of the complex number are directly separated when the number is written as $ e $ to a complex number. Let $ z_1 = e^{a_1+b_1i} $ and $ z_2 = e^{a_2+b_2i} $. Then $ z_1\cdot z_2 = e^{a_1+b_1i}\cdot e^{a_2+b_2i} = e^{a_1+a_2} \cdot e^{i(b_1+b_2)} $.
Conceptually, this means that the length of the two points are multiplied to get the length of the new point. A similar thing applies to the points' angles: they are added rather than multiplied. This is an important idea fundamental to intuitively understanding the multiplication of complex numbers.
References:
(Reference 1)
(Reference 2)
