## Multiplying and Dividing in Polar Form

Multipling and dividing complex numbers in rectangular form was covered in topic 36.

In polar form, the multiplying and dividing of complex numbers is made easier once the formulae have been developed. The following development uses trig.formulae you will meet in Topic 43.

### Multiplication

Let z1 = r1 cis θand z2 = r2 cis θ2 be any two complex numbers.

z1z2   = r1 cis θ. r2 cis θ2

= r1 r2(cis θ. cis θ2)

= r1 r2[(cos θ1 + i sin θ1).(cos θ2 + i sin θ2)]

= r1 r2(cos θ1cos θ2 + i2sin θ1sin θ2 + i sin θ1cos θ2 + i cos θ1 sin θ2)

= r1 r2[(cos θ1cos θ2 - sin θ1sin θ2) + i( sin θ1cos θ2 + cos θ1 sin θ2)]

= r1 r2[cos (θ1 + θ2) + i sin (θ1 + θ2)]

To summarise:

 z1z2 = r1r2cis (θ 1 + θ 2)

In words: When multiplying two complex numbers in polar form the modulii are multiplied and the arguments are added. This proof does not need to be memorised!

Example

5cis (0.3) x 7cis (0.2) = 35 cis (0.5)           Easy, once the formula was developed!

### Division

Let z1 = r1 cis θand z2 = r2 cis θ2 be any two complex numbers. To summarise: In words: When dividing two complex numbers in polar form the modulii are divided and thearguments are subtracted.

Example

10cis (0.8) ÷ 5cis (0.5) = 2 cis (0.3)           Just as easy, once the formula was developed!