# 6. Products and Quotients of Complex Numbers

by M. Bourne

When performing addition and subtraction of complex numbers, use **rectangular form**. (This is because we just add real parts then add imaginary parts; or subtract real parts, subtract imaginary parts.)

When performing multiplication or finding powers and roots of complex numbers, use **polar** and
**exponential forms**. (This is because it is a lot easier than using rectangular form.)

We start with an example using exponential form, and then generalise it for polar and rectangular forms.

### Example 1

Find (3*e*^{4j})(2*e*^{1.7j})

## Multiplying Complex Numbers in Polar Form

We can generalise the example we just did, as follows:

`(r_1\ e^(\ theta_1j))(r_2\ e^(\ theta_2j))=r_1r_2\ e^((theta_1+\ theta_2)j`

From this, we can develop a formula for multiplying using polar form:

`r_1(cos\ theta_1+j\ sin\ theta_1)xxr_2(cos\ theta_2+j\ sin\ theta_2)`

`=r_1r_2(cos[theta_1+theta_2]+j\ sin[theta_1+theta_2])`

or with equivalent meaning:

`r_1/_theta_1xxr_2/_theta_2=r_1r_2/_[theta_1+theta_2]`

In words, all this confusing-looking algebra simply means...

To multiply complex numbers in polar form,

Multiplythepartsr

Addtheangleparts

### Example 2

Find 3(cos 120° + *j* sin 120°) × 5(cos 45° + *j* sin 45°)

## Division

As we did before, we do an example in exponential form first, then generalise it for polar form.

### Example in Exponential Form:

`{:(8\ e^(\ 3.6j)-:2\ e^(\ 1.2j),=4\ e^(\ 3.6j-1.2j)),(,=4\ e^(\ 2.4j)):}`

[We divided the number parts, and subtracted the indices, just using normal algebra.]

From the above example, we can conclude the following:

`(r_1(costheta_1+j\ sintheta_1))/(r_2(costheta_2+j\ sintheta_2))=r_1/r_2(cos[theta_1-theta_2]+j\ sin[theta_1-theta_2])`

or

`(r_1/_theta_1)/(r_2/_theta_2)=r_1/r_2/_[theta_1-theta_2]`

In words, this simply means...

To divide complex numbers in polar form,

Dividethepartsr

Subtracttheangleparts

**Example 3**

Find `(2/_90^"o")/(4/_75^"o")`

**Example 4**

Find `(3/_20^"o")/(9/_60^"o")`

### Example 5

Find `(8j)/(7+2j)` using polar form.

### Exercises:

**1. **Evaluate:
`(0.5 ∠ 140^"o")(6 ∠ 110^"o")`

2. Evaluate: `(12/_320^"o")/(5/_210^"o")`

3. (i) Evaluate the following by first converting numerator and denominator into polar form.

(ii) Then check your answer by multiplying numerator and
denominator by the **conjugate **of the
denominator.

`(-2+5j)/(-1-j)`

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