Added division example

src/1_complex_numbers.md

@@ -59,6 +59,12 @@ complex conjugate of $z_2$:
 $$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
 Check this!
 
+**Example** 
+$$\begin{align} 
+\frac{1 + 2{\rm i}}{1 - 2{\rm i}} & = (\frac{(1 + 2{\rm i})(1 + 2{\rm i})}{1^2 + 2^2}) &= \frac{1+8{\rm i} -4}{5}\\
+&= -\frac{3}{5} + {\rm i} \frac{8}{5}
+\end{align}$$
+
 ## The complex plane
 
 Complex numbers can be rendered on a two-dimensional (2D) plane, the
@@ -115,8 +121,11 @@ $$\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
 
  It turns out that using this magnitude $|z|$ and phase $\varphi$, we can write any complex number as
  $$z = |z| e^{{\rm i} \varphi}$$
-When increasing $\varphi$ with $2 \pi$, we reach the same point on the complex plane. In other words, when adding $2 \pi$ to our argument, we get the same complex number!
-As a result, the argument is defined up to $2 \pi$, and we are free to make any choice we like:
+When increasing $\varphi$ with $2 \pi$, we make a full circle and reach the same point on the complex plane. In other words, when adding $2 \pi$ to our argument, we get the same complex number!
+As a result, the argument $\varphi$ is defined up to $2 \pi$, and we are free to make any choice we like:
+$$\begin{align}
+-\pi < \varphi < \pi $ {\rm (left image)} \\
+-\frac{\pi}{2} < \varphi < \frac{3 \pi}{2} $ {\rm (right image)} \end{align} $$
 
 ![image](figures/complex_numbers_11_0.svg)
 
@@ -124,11 +133,10 @@ As a result, the argument is defined up to $2 \pi$, and we are free to make any
 Real functions can (most of the times) be written in terms of a Taylor series:
 $$f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (x-x_{0})^{n}$$
 We can write something similar for complex functions, 
-when replacing $x \rightarrow z$:
+when replacing the *real* variable $x$ with its *complex* counterpart $z$:
 $$f(z) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (z-x_{0})^{n}$$
 
-We have already seen that we can write any complex number $z$ in polar form as, $z = |z| e^{{\rm i} \varphi}$. Apparently, we can write any complex number using the complex
-exponential function, at which we will have a look below.
+For this course, the most important function is the *complex exponential function*, at which we will have a look below.
 
 ### The complex exponential function
 The complex exponential is used *extremely often*.