New Zealand
Level 8 - NCEA Level 3

# Multiplication of Numbers in Polar Form

Lesson

Using Polar or Modulus Argument forms, multiplication can be distilled down to a very simple process.

For example, suppose that $z$z$_1=\left(r_{1,}\theta_1\right)$1=(r1,θ1) and $z_2=\left(r_2,\theta_2\right)$z2=(r2,θ2) , then

 $z_1z_2$z1​z2​ $=$= $\left(r_{1,}\theta_1\right)\left(r_2,\theta_2\right)$(r1,​θ1​)(r2​,θ2​) $=$= $r_1\operatorname{cis}\theta_1r_2\operatorname{cis}\theta_2$r1​cisθ1​r2​cisθ2​ $=$= $r_1\left(\cos\theta_1+i\sin\theta_1\right)\times r_2\left(\cos\theta_2+i\sin\theta_2\right)$r1​(cosθ1​+isinθ1​)×r2​(cosθ2​+isinθ2​) $=$= $r_1r_2\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2+i\sin\theta_2\right)$r1​r2​(cosθ1​+isinθ1​)(cosθ2​+isinθ2​) $=$= $r_1r_2\left(\left(\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\right)+\left(\sin\theta_1\cos\theta_2+\cos\theta_1\sin\theta_2\right)i\right)$r1​r2​((cosθ1​cosθ2​−sinθ1​sinθ2​)+(sinθ1​cosθ2​+cosθ1​sinθ2​)i) From here we use what is called the double angle formula, we can simplify this to $z_1z_2$z1​z2​ $=$= $r_1r_2\left(\cos\left(\theta_1+\theta_2\right)+i\sin\left(\theta_1+\theta_2\right)\right)$r1​r2​(cos(θ1​+θ2​)+isin(θ1​+θ2​))

What's happened here is that we have multiplied the moduli and added the arguments

For division, it can be shown that it simplifies to

 $\frac{z_1}{z_2}$z1​z2​​ $=$= $\left(r_{1,}\theta_1\right]\left[r_2,\theta_2\right)$(r1,​θ1​][r2​,θ2​) $=$= $\frac{r_1\operatorname{cis}\theta_1}{r_2\operatorname{cis}\theta_2}$r1​cisθ1​r2​cisθ2​​ $=$= $\frac{r_1}{r_2}\operatorname{cis}\left(\theta_1-\theta_2\right)$r1​r2​​cis(θ1​−θ2​)

What's happened here is that we have divided the moduli and subtracted the arguments

Activity

Pick any complex number, plot it on an Argand Plane.

a) Multiply it repeatedly by $i$i, plotting the result each time.

b) Multiply it repeatedly by $-i$i, plotting the result each time.

Is there a pattern?

Generalise the result.

See here for a solution

#### Worked Examples

##### Question 1

1. Which is true?

When multiplying two complex numbers in trigonometric form, we add their absolute values and add their arguments.

A

When multiplying two complex numbers in trigonometric form, we multiply their absolute values and add their arguments.

B

When multiplying two complex numbers in trigonometric form, we add their absolute values and multiply their arguments.

C

When multiplying two complex numbers in trigonometric form, we multiply their absolute values and multiply their arguments.

D

When multiplying two complex numbers in trigonometric form, we add their absolute values and add their arguments.

A

When multiplying two complex numbers in trigonometric form, we multiply their absolute values and add their arguments.

B

When multiplying two complex numbers in trigonometric form, we add their absolute values and multiply their arguments.

C

When multiplying two complex numbers in trigonometric form, we multiply their absolute values and multiply their arguments.

D
2. Which is true?

When dividing two complex numbers in trigonometric form, we subtract their absolute values and divide their arguments.

A

When dividing two complex numbers in trigonometric form, we subtract their absolute values and subtract their arguments.

B

When dividing two complex numbers in trigonometric form, we divide their absolute values and divide their arguments.

C

When dividing two complex numbers in trigonometric form, we divide their absolute values and subtract their arguments.

D

When dividing two complex numbers in trigonometric form, we subtract their absolute values and divide their arguments.

A

When dividing two complex numbers in trigonometric form, we subtract their absolute values and subtract their arguments.

B

When dividing two complex numbers in trigonometric form, we divide their absolute values and divide their arguments.

C

When dividing two complex numbers in trigonometric form, we divide their absolute values and subtract their arguments.

D

##### Question 2

Find the value $\sqrt{5}\left(\cos240^\circ+i\sin240^\circ\right)\times\sqrt{3}\left(\cos30^\circ+i\sin30^\circ\right)$5(cos240°+isin240°)×3(cos30°+isin30°).

##### Question 3

Consider the multiplication $\left(3\sqrt{3}-3i\right)\times\left(-5i\right)$(333i)×(5i).

1. Express $3\sqrt{3}-3i$333i in polar form, $r\left(\cos\theta+i\sin\theta\right)$r(cosθ+isinθ), with $r\ge0$r0 and $0\le\theta<2\pi$0θ<2π:

2. Express $-5i$5i in polar form, $r\left(\cos\theta+i\sin\theta\right)$r(cosθ+isinθ), with $r\ge0$r0 and $0\le\theta<2\pi$0θ<2π:

3. Use these results to evaluate $\left(3\sqrt{3}-3i\right)\times\left(-5i\right)$(333i)×(5i). Give your final answer in rectangular form.

### Outcomes

#### M8-9

Manipulate complex numbers and present them graphically

#### 91577

Apply the algebra of complex numbers in solving problems