Complex numbers in polar form are a cornerstone of higher mathematics, offering a unique perspective on arithmetic operations. This comprehensive guide delves into the details of multiplication and division in polar form, highlighting the significance of modulus and argument in these processes. Understanding these concepts is crucial for students aiming to master complex number manipulations.

Image courtesy of Cuemath

## Multiplication of Complex Numbers in Polar Form

**Complex Number in Polar Form:**Represented as $z = r e^{i\theta}$, where:- $r$ = Modulus (distance from origin)
- $\theta$ = Argument (angle from positive x-axis)

**Multiplying Two Complex Numbers:**For $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$:**Procedure:**Multiply their moduli (r-values) and add their arguments (θ-values).**Result:**$z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$

**Geometric Interpretation:**- On Argand Plane, multiplication:
- Stretches by the modulus of one number.
- Rotates by the angle of the other number.

- Helps visualize the effect on position and size of complex numbers.

- On Argand Plane, multiplication:

## Division of Complex Numbers in Polar Form

**Complex Number in Polar Form:**Represented as $z = r e^{i\theta}$, where:- $r$ = Modulus (distance from origin)
- $\theta$ = Argument (angle from positive x-axis)

**Dividing Two Complex Numbers:**For $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$:**Procedure:**Divide their moduli (r-values) and subtract their arguments (θ-values).**Result:**$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$

**Geometric Interpretation:**- On Argand Plane, division:
- Shrinks by the modulus of the divisor.
- Rotates by the negative of the divisor's angle.

- Helps visualize the effect on position and size of complex numbers.

- On Argand Plane, division:

**Example Questions**

**Problem 1: Multiplying Complex Numbers**

**Given:**$4 e^{i\pi/3}$ and $4 e^{i\pi/3}$**Step 1: Multiply Moduli**- $4 \times 2 = 8$

**Step 2: Add Arguments**- Convert to common denominator: $\frac{\pi}{3} = \frac{4\pi}{12}, \frac{\pi}{4} = \frac{3\pi}{12}$
- Add: $\frac{4\pi}{12} + \frac{3\pi}{12} = \frac{7\pi}{12}$

**Result:**$8 e^{i(7\pi/12)}$

### Problem 2: Dividing Complex Numbers

**Given:**$6 e^{i\pi/2}$ and $3 e^{i\pi/6}$**Step 1: Divide Moduli**- $6 \div 3 = 2$

**Step 2: Subtract Arguments**- Convert to common denominator: $\frac{\pi}{2} = \frac{3\pi}{6}, \frac{\pi}{6} = \frac{1\pi}{6}$
- Subtract: $\frac{3\pi}{6} - \frac{1\pi}{6} = \frac{2\pi}{6}$, simplify to $\frac{\pi}{3}$

**Result:**$2 e^{i(\pi/3)}$

Written by: Dr Rahil Sachak-Patwa

LinkedIn

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.