complex numbers and complex plane

Exercise1.1

1. Show points $-3,2i,4 + i,2-2i$ on the complex plane 2. Prove the following theorem.

(a)

$\bar{z_{1} + z_{2}} = \bar{z_{1}} + \bar{z_{2}}$

(b)

$\bar{z_{1}z_{2}} = \bar{z_{1}}\bar{z_{2}}$

(c)

$Re(z) = \frac{z + \bar{z}}{2}$

3. Prove the following inequality.

(a)

$Re{z} \leq \vert z\vert$

(b)

$\vert z_{1} + z_{2}\vert \leq \vert z_{1}\vert + \vert z_{2}\vert$

(c)

$\vert\vert z_{1}\vert - \vert z_{2}\vert\vert \leq \vert z_{1} - z_{2}\vert$

4. Express the following complex numbers in polar form.

(a)

$-1 + i$

(b)

$3 - \sqrt{3}i$

(c)

$-1$

(d)

$2i$

5. Draw a curve that satisfies the following equation.

(a)

$\arg z =$   constant

(b)

$\vert z\vert =$   constant

(c)

$\vert z - 1\vert = \vert z - i\vert$

(d)

$\vert z - 2i\vert = 3$

(e)

$\vert z + 3\vert = 3\vert z - 1\vert$

(f)

$z - \bar{z} = 2i$