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.

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

3.

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

4.

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