Computing with Complex Numbers

Simplify the following expressions. Give your final answer in rectangular coordinates. You are encouraged, but not required, to illustrate all complex numbers graphically.

1.

$$z_1 = \frac{1-j}{1+j}$$

2.

$$z_2 = \frac{1}{5} \cdot (2+j)\cdot (2-j)$$

3.

$$z_3 = e^{j\frac{3\pi}{4}} + e^{j\frac{-3\pi}{4}}$$

4.

$$z_4 = e^{1-j\frac{\pi}{2}} + e^{1+j\frac{\pi}{6}} + e^{1+j\frac{5\pi}{6}}$$