Lesson 1.12

The Imaginary Unit

We unlock a new number system to solve equations like . Meet the imaginary unit, the building block of complex numbers.

Introduction

Until now, taking the square root of a negative number was "undefined" or "no real solution." But what if we defined a number that could extend our system? This allows us to solve every quadratic equation.

Past Knowledge

You know that because .

Today's Goal

Define and use it to simplify radicals like .

Future Success

This lets you use the Quadratic Formula for any equation, even when the parabola doesn't touch the x-axis.

Key Concepts

1. The Definition of

The imaginary unit is defined as:

Therefore:

This simple definition allows us to break apart negative roots.

2. Simplifying Negative Roots

Use the product property to separate the :

Think: "Pull out the negative as an ."

3. The Cycle of Powers

Powers of repeat in a cycle of 4:

(Important!)

Worked Examples

Example 1: Basic Simplification

Basic

Simplify .

1

Separate the Negative

2

simplify

Example 2: Not a Perfect Square

Intermediate

Simplify .

1

Pull out the

2

Simplify the Radical

. Square root of 4 is 2.

Example 3: Complex Simplification

Advanced

Simplify .

1

Handle Negatives

The negative outside stays there. The negative inside becomes .

2

Simplify Radical

.

Common Pitfalls

The "Negative Times Negative" Trap

You cannot multiply negatives inside square roots first.

Forgetting the

is , NOT .

Check your work: , not .

Real-Life Applications

It might seem "imaginary," but is fundamental to modern electronics. In AC circuits (what powers your house), electricity flows in waves. Engineers use complex numbers ( instead of ) to calculate voltage and impedance, because the math of rotating cycles perfectly matches the properties of .

Practice Quiz

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