Complex Number Polar Form- Converting and Using Polar Coordinates

What Polar Form Actually Is

Complex numbers live on a 2D plane. You already know the rectangular form: a + bi, where a is the horizontal position and b is the vertical position. Polar form is just a different way to describe the same point.

Instead of x and y coordinates, polar form uses r (the distance from the origin) and θ (the angle from the positive real axis). Every complex number has both representations. They're interchangeable.

The formula is straightforward:

z = r(cos θ + i sin θ)

Or the shorthand you'll see everywhere:

z = r cis θ

Where r = √(a² + b²) and θ = arctan(b/a), adjusted for which quadrant the point actually sits in.

Converting Rectangular to Polar: Step by Step

Take 3 + 4i. That's your starting point.

Step 1: Find r

r = √(3² + 4²) = √(9 + 16) = √25 = 5

Step 2: Find θ

θ = arctan(4/3) = arctan(1.333) ≈ 53.13°

Since 3 + 4i sits in the first quadrant, this angle is correct. No adjustment needed.

Result: 3 + 4i = 5 cis 53.13°

Quadrant Adjustments

This trips up almost everyone. The arctan function only gives angles in Quadrants I and IV. When your complex number lands in Quadrant II or III, you must add 180°.

Example: Convert -2 + 5i

r = √((-2)² + 5²) = √(4 + 25) = √29

θ = arctan(5/-2) = arctan(-2.5) ≈ -68.2°

But -2 + 5i is in Quadrant II. So: θ = -68.2° + 180° = 111.8°

Result: -2 + 5i = √29 cis 111.8°

Converting Polar to Rectangular

This is the reverse process. Given r cis θ, find a and b.

a = r cos θ

b = r sin θ

Example: Convert 6 cis 45° to rectangular form.

a = 6 cos 45° = 6 × 0.7071 ≈ 4.24

b = 6 sin 45° = 6 × 0.7071 ≈ 4.24

Result: 6 cis 45° ≈ 4.24 + 4.24i

Why Bother With Polar Form?

Rectangular multiplication is messy. You have to distribute everything and track the i² = -1 rule. Polar multiplication is drastically simpler.

To multiply two complex numbers in polar form:

Multiply the r values. Add the angles.

That's it.

Example: Multiply (3 cis 40°) × (5 cis 25°)

Result: (3 × 5) cis (40° + 25°) = 15 cis 65°

Try doing that in rectangular form. You'll see the difference.

Division in Polar Form

Division is equally painless:

Divide the r values. Subtract the angles.

Example: Divide (10 cis 80°) by (2 cis 30°)

Result: (10/2) cis (80° - 30°) = 5 cis 50°

De Moivre's Theorem: Powers and Roots

De Moivre's Theorem is the real power behind polar coordinates. It states:

[r cis θ]ⁿ = rⁿ cis (nθ)

For integer n, this works directly. For roots, you get multiple answers.

Computing Powers

Find (2 cis 30°)³

= 2³ cis (3 × 30°) = 8 cis 90°

Convert back to rectangular: 8 cis 90° = 8(cos 90° + i sin 90°) = 8(0 + i·1) = 8i

Finding Roots

The n-th root of a complex number gives n distinct answers. The formula:

[r cis θ]^(1/n) = r^(1/n) cis [(θ + 360°k)/n]

Where k = 0, 1, 2, ..., n-1

Example: Find the cube roots of 8 cis 60°

Three cube roots, equally spaced around a circle of radius 2.

Rectangular vs Polar: When to Use Which

Neither format is "better." They serve different purposes.

Operation Best Form Why
Addition / Subtraction Rectangular Just add/subtract real and imaginary parts separately
Multiplication / Division Polar Multiply r's, add angles. No FOIL needed.
Powers (n > 2) Polar De Moivre's Theorem eliminates nested expansion
Roots Polar Clean formula, easy to find all n solutions
Graphing / Visualization Polar Directly shows distance and direction from origin

Getting Started: Your Workflow

Step 1: Identify what you're doing. Addition? Use rectangular. Multiplication or powers? Switch to polar.

Step 2: Convert if needed. Use r = √(a² + b²) and θ = arctan(b/a) with quadrant corrections.

Step 3: Perform the operation using the simpler rules.

Step 4: Convert back if the answer needs to be in rectangular form.

Practice with these conversions until the process is automatic:

The more you practice, the faster the conversions become. After enough problems, you'll start recognizing common values—like 1 + i is √2 cis 45°, or 2i is 2 cis 90°.