Overview

Integrated Mathematics III

48. CC.HSN.CN.A.1

48.2 Introduction to imaginary numbers

Introduction to Imaginary Numbers

Find out what the imaginary unit i really means, why negative numbers can have square roots, and how powers of i repeat in a cycle.

What You'll Learn

Define the imaginary unit i as the number whose square equals negative one.

Simplify square roots of negative numbers by rewriting them using i.

Recognize that no real number squared can ever produce a negative result.

Apply the repeating four-step cycle of powers of i to simplify any power quickly.

Connect imaginary numbers to complex numbers as the building block for the complex plane.

What You'll Practice

Simplifying roots of negative numbers like (-16) and (-40)

Evaluating powers of i from i³ to i²

Simplifying complex expressions with fractional exponents involving negative bases

Converting imaginary number expressions into simplified real or imaginary results

Why This Matters

Imaginary numbers are essential for solving equations that have no real solutions, like x² = -1. You'll use them throughout advanced algebra, precalculus, calculus, engineering, and physics to work with complex numbers, electrical circuits, and wave functions.

Before You Start — Make Sure You Can:

Converting radicals to mixed radicals Product rule of exponents Power of a product rule

This Unit Includes

2 Video lessons

Practice exercises

Learning resources

Skills

Imaginary Numbers

Complex Numbers

Radicals

Exponents

Simplification

Powers of i

OH Curriculum Aligned