## Example 1

Let $$z = 2-3i$$. Find $$\overline{z}$$, $$|~z~|$$, and $$z^{-1}$$.

## Example 2

Let $$z = 1-2i$$ and let $$w = 1-i$$. Simplify the expression $$z + 4w^{-2}$$ as much as possible.

## Example 3

Convert each of the following complex number into polar form. (Give the modulus and argument rounded to five decimal places.)

1. $$2-7i$$

2. $$-5-3i$$

3. $$-6+i$$

## Example 4

Convert $$3 \operatorname{cis} \left(\frac{3}{5}\right)$$ into rectangular form with real and imaginary parts rounded to four decimal places.

## Example 5

Give all the fourth roots of $$4+3i$$ in rectangular form with real and imaginary parts accurate to four decimal places.

## Example 6

One of the cube roots of $$8-5i$$, with real and imaginary parts rounded to 5 decimal places, is $$-0.69947 + bi$$. What is $$b$$?

## Example 7

Does the equation $$(z-i)^2 + 1 = 0$$ have a real solution?

## Example 8

Let $$u = i$$ and $$w = -2-i$$. Find all real numbers $$a$$ such that $$|a+u| = |w|$$.

## Example 9

Find all complex numbers $$z$$ satisfying $$iz^2 - 2z +1 = 0.$$

## Example 10

Find all real numbers $$a$$ such that $$|a+(2-3i)| = 5$$.