Lesson 1.16

Conjugates & Division

Division in the complex world is actually just clever multiplication. We use a special tool called the "conjugate" to remove imaginary numbers from the denominator.

Introduction

Just as we don't like square roots in the bottom of a fraction (), we don't allow imaginary numbers there either.

Past Knowledge

You rationalized by multiplying by .

Today's Goal

Clean up fractions like using the Complex Conjugate.

Future Success

This technique is standard procedure for simplifying results from the Quadratic Formula.

Key Concepts

1. The Complex Conjugate

To find the conjugate, flip the sign of the imaginary part only.

Original

Conjugate

2. Product of Conjugates

When you multiply opposites, the middle terms cancel and becomes real.

Concept Check: The result is ALWAYS a real number!

Worked Examples

Example 1: Finding Conjugates

Basic

Find the conjugate of and multiply them.

Find Conjugate

Change to .

Multiply

Result:

Example 2: Dividing by Binomial

Intermediate

Simplify .

Multiply by Conjugate

The conjugate of the denominator is .

Simplify Top and Bottom

Reduce Fraction:

Example 3: Complex Division

Advanced

Simplify .

Set up Conjugate Product

FOIL Everything

Simplify Final Fraction

Standard Form:

Common Pitfalls

Sign Errors

Conjugate of is . Do NOT change the real part's sign!

Illegal Cancellation

In , you must divide BOTH terms. .

Real-Life Applications

In advanced electronics (Antenna theory), you need to "match the impedance" to get the best signal. This often involves taking the complex conjugate of the load impedance. Without this math, your Wi-Fi signal would be terrible!