Complex Number

A complex number $x+yi$ is defined to be the point in the Argand diagram with coordinates (x, y).

Properties of complex number

As can be seen in the figure, $x=r\cos\theta,y=r\sin\theta$.
Then, the polar form will be $r(\cos\theta+i\sin\theta)=rcis\:\theta$, where $cis\:\theta=\cos\theta+i\sin\theta$.
Modulus: $|z|=r=+\sqrt{x^2+y^2}$
$\tan\theta=\frac{y}{x}$
Argument: $arg\:z=\theta$

$\begin{align}zw&=r(\cos\theta+i\sin\theta)r'(\cos\phi+i\sin\phi)\\
&=rr'[(\cos\theta\cos\phi-\sin\theta\sin\phi)+i(\cos\theta\sin\phi-\sin\theta\cos\phi)]\\
&=rr'[\cos(\theta+\phi)+i\sin(\theta+\phi)]\end{align}$
$|zw|=|z||w|$
Arguments add upon multiplication.

$z^n=r^n(\cos\theta+i\sin\theta)^n=r^n(\cos n\theta+i\sin n\theta)$
This is due to De Moive's Theorem.

Root of unity (applied in number theory, the theory of group characters, the discrete Fourier transform, etc)
A complex number that gives 1 when raised to some integer power n.
$z^n=1$

$\epsilon=\cos\frac{2\pi}{n}+i\sin\frac{2\pi}{n}\\
\epsilon^k=\cos\frac{2\pi k}{n}+i\sin\frac{2\pi k}{n}\\
\epsilon^{n-1}=\cos\frac{2\pi (n-1)}{n}+i\sin\frac{2\pi (n-1)}{n}\\
\epsilon^{n}=1\\
|\epsilon|=1 \Rightarrow \epsilon^k\:\:\text{lie on the unit circle.}$



Complex Exponential Taylor Series
$$e^z=\sum_{n=0}^\infty \dfrac{z^n}{n!}\\ \cos z=\sum_{n=0}^\infty \dfrac{(-1)^nz^{2n}}{(2n)!}\\ \sin z=\sum_{n=0}^\infty \dfrac{(-1)^n z^{2n+1}}{(2n+1)!}$$
Direct multiplication of series shows $\large e^{z+w}=e^z\cdot e^w$, thus $\large e^z=e^{x+yi}=e^x\cdot e^{iy}$.
$e^{iy}=\sum_{n=0}^\infty \dfrac{(iy)^n}{n!}=\sum_{n=0}^\infty \dfrac{(-1)^nz^{2n}}{(2n)!}+\sum_{n=0}^\infty \dfrac{(-1)^n z^{2n+1}}{(2n+1)!}$
Sidenote: $e^{2\pi i}=1,e^{\pi i}=-1,e^{\frac{\pi i}{2}}=i,e^{\frac{3\pi i}{2}}=-i,e^{\frac{2\pi i}{8}}=\sqrt{i}=\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2}$

More to explore:
http://www.math.vt.edu/people/dlr/m2k_bas_cmpext.pdf