Lesson 13.5

Multiplication and Division in Polar Form

Multiply moduli, add arguments. Divide moduli, subtract arguments. Polar form turns complex arithmetic into simple operations.

Introduction

Multiplying complex numbers in rectangular form requires FOIL and careful handling of . In polar form? Just multiply the moduli and add the angles.

1

Prerequisite Connection

You know how to convert complex numbers to polar form .

2

Today's Increment

We learn the product and quotient formulas for complex numbers in polar form.

3

Why This Matters

These formulas lead directly to De Moivre's Theorem for powers and roots—a cornerstone of advanced mathematics and engineering.

Polar Operations

Multiplication

Multiply moduli, add arguments

Division

Divide moduli, subtract arguments

Full Notation

If and , then:

💡 Wait, what is "cis"?

No, that's not a typo! cis is a standard mathematical shorthand used when working with complex numbers.

It stands for:

So is just a shorter way of writing .

The name comes from combining cos and i sin. Using "cis" makes equations much cleaner and faster to write!

Geometric Interpretation

  • • Multiplication: Scales by and rotates by
  • • Division: Scales by and rotates by

Worked Examples

Example 1: Multiplying Complex Numbers

Find where and .

Step 1: Multiply the moduli

Step 2: Add the arguments

Solution

Example 2: Dividing Complex Numbers

Find where and .

Step 1: Divide the moduli

Step 2: Subtract the arguments

Solution

Example 3: Converting, Multiplying, Converting Back

Find using polar form.

Step 1: Convert to polar

: , →

: , →

Step 2: Multiply in polar

Step 3: Convert back (optional)

,

Solution (in polar form)

Common Pitfalls

Multiplying angles instead of adding

The rule is add arguments for multiplication. Multiplying is a common error.

Subtracting in the wrong order for division

For , compute , not .

Forgetting to simplify the angle

If your result has or , consider reducing to the principal value.

Real-World Application

Signal Processing and Phase Shifts

In audio and radio engineering, signals are represented as complex phasors. Multiplying by a complex number with modulus 1 (a "unit phasor") rotates the signal's phase without changing its amplitude.

This is how noise-canceling headphones work: they analyze incoming sound, compute the opposite phase angle, and add a canceling signal. The polar multiplication formula is at the heart of this calculation.

Practice Quiz

Loading...