Lesson 13.6

De Moivre's Theorem and n-th Roots

Raising complex numbers to powers—and extracting roots—becomes elegant with De Moivre's Theorem. One formula, infinite applications.

Introduction

Computing in rectangular form would be brutal. De Moivre's Theorem makes it a single calculation: raise the modulus to the power, multiply the angle by the power.

1

Prerequisite Connection

You can multiply and divide complex numbers in polar form.

2

Today's Increment

We learn De Moivre's Theorem for powers and extend it to find all n-th roots of a complex number.

3

Why This Matters

This theorem is essential for solving polynomial equations, analyzing oscillations, and understanding the geometry of complex numbers.

De Moivre's Theorem

De Moivre's Theorem for Powers

Or in shorthand:

n-th Roots of a Complex Number

To find all -th roots of :

Key Facts About Roots

  • • Every nonzero complex number has exactly distinct -th roots
  • • The roots are equally spaced around a circle
  • • The angular spacing between roots is

Roots of Unity

The -th roots of 1 are called "roots of unity." They form a regular -gon centered at the origin, with one vertex at .

Worked Examples

Example 1: Computing a Power

Find .

Step 1: Convert to polar form

,

Step 2: Apply De Moivre's Theorem

Step 3: Convert to rectangular

Solution

Example 2: Finding Cube Roots

Find all cube roots of .

Step 1: Write 8 in polar form

Step 2: Apply the root formula

Step 3: Calculate for k = 0, 1, 2

:

:

:

Solution

The three cube roots of 8 are: , , and .

Example 3: Square Roots of a Complex Number

Find all square roots of .

Step 1: Write -4i in polar form

lies on the negative imaginary axis: ,

Step 2: Apply the root formula (n = 2)

Step 3: Calculate for k = 0, 1

:

:

Solution

The square roots of are , which equals and .

Common Pitfalls

Finding only one root

Every nonzero complex number has distinct -th roots. Don't stop at ; continue through .

Forgetting to take the n-th root of r

For roots, the modulus becomes , not or .

Wrong formula for root arguments

The argument is , not . The goes inside the fraction.

Real-World Application

Fractals and Computer Graphics

Fractals like the Mandelbrot set are generated by iterating complex functions. Each pixel represents a complex number, and the color depends on how behaves—a direct application of complex powers.

The -th roots of unity create the vertices of regular polygons, which is fundamental to designing symmetric patterns, gears, and tiled floors.

Practice Quiz

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