Complex Numbers: (lesson 2 of 2)

## Polar representation

### Polar representation of complex numbers

In polar representation a complex number z is represented by two parameters **r** and **Θ**. Parameter **r** is the modulus of complex number and parameter **Θ** is the angle with the positive direction of x-axis.

The polar form of a complex number is:

This representation is very useful when we multiply or divide complex numbers.

### Polar to Rectangular Form Conversion

Here we know **r** and **Θ** and we need to find **a** and **b**.

Example 1:

Convert the complex number to rectangular form.

Solution:

Exercise 1: Convert to rectangular form:

### Rectangular to Polar Form Conversion

Here we know **a** and **b** and we need to find **r** and **Θ**. In this case we need to
use formulas:

Example 2:

Convert the complex number to polar form.

Solution:

In this example :

The polar form is:

Exercise 2: Convert to polar form:

### Product in polar representation

Example 3:

Let . Then:

### Quotient two complex numbers in polar representation

Example 4:

Let . Then:

Exercise 3: Find product and quotient:

### The inverse of a complex number in polar representation

### Conjugate numbers in polar representation

### Formula 'De Moivre'

Example 5:

Let z = 1 - i.

a) find polar representation

b) find z^{8}

Solution:

a)

b)

Exercise 4: Find z^{n}: