Lesson 13.4

Complex Numbers in Polar Form

The complex plane meets polar coordinates. Representing as unlocks powerful computational shortcuts.

Introduction

You've worked with complex numbers in rectangular form: . Now we'll express them in polar form using their distance from the origin (modulus) and angle (argument).

1

Prerequisite Connection

You've worked with complex numbers in rectangular form: .

2

Today's Increment

We learn to convert complex numbers to polar form or .

3

Why This Matters

Polar form makes multiplication, division, and powers of complex numbers dramatically simpler—essential for electrical engineering and signal processing.

Polar Form of Complex Numbers

Polar Form

Modulus (r)

The distance from to the origin in the complex plane.

Argument (θ)

The angle from the positive real axis. Check the quadrant!

Converting Back to Rectangular

The "cis" Shorthand

Many textbooks use as shorthand for . This is purely notational—they mean the same thing.

Worked Examples

Example 1: Convert to Polar Form

Write in polar form.

Step 1: Find the modulus

Step 2: Find the reference angle

Reference angle:

Step 3: Determine the quadrant

Since and , the point is in Quadrant II.

Solution

Example 2: Convert to Rectangular Form

Write in rectangular form.

Step 1: Find the real part

Step 2: Find the imaginary part

Solution

Example 3: Special Complex Numbers

Express and in polar form.

For i = 0 + 1i

, located on positive imaginary axis at

For -1 = -1 + 0i

, located on negative real axis at

Key Insight

The special result (Euler's identity) connects to .

Common Pitfalls

Wrong quadrant for the argument

only gives angles in . Always check which quadrant lies in and adjust accordingly.

Confusing modulus with argument

The modulus is a distance (always non-negative). The argument is an angle (can be any real number).

Forgetting to evaluate trig functions

When converting from polar to rectangular, you must actually compute and —don't leave them as expressions.

Real-World Application

Electrical Engineering: AC Circuits

Alternating current (AC) voltages and currents oscillate sinusoidally. Engineers represent them as complex numbers in polar form: the modulus is the amplitude, and the argument is the phase angle.

This representation, called a "phasor," makes circuit analysis much simpler. Adding voltages becomes adding complex numbers; impedance calculations use polar multiplication.

Practice Quiz

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