Lesson 1.13

Complex Numbers

What happens when we mix real numbers with imaginary ones? We get the Complex Number System, the largest set of numbers in algebra.

Introduction

Just as a number line holds all real numbers, the Complex Plane holds all complex numbers. Every number you've ever known is technically a complex number!

Past Knowledge

You know is a Real number and is an Imaginary number.

Today's Goal

Combine them into Standard Form and graph them on a 2D plane.

Future Success

Understanding the "Real" and "Imaginary" parts separately is key to adding and multiplying them later.

Key Concepts

1. The Standard Form

A complex number is written as:

Real Part

Imaginary Part

If , it's a Real Number. If , it's Pure Imaginary.

2. Equality of Complex Numbers

Two complex numbers are equal if and only if:

Real Part = Real Part
Imaginary Part = Imaginary Part

You cannot mix them! It's like comparing x's and y's.

3. The Complex Plane

We graph complex numbers on a coordinate system:

Horizontal Axis (x): Real Axis
Vertical Axis (y): Imaginary Axis

Example: To graph , go Right 3, Up 4.

Worked Examples

Example 1: Identifying Parts

Basic

Identify the real and imaginary parts of .

Standard Form

Rewrite as to clearly see the coefficients.

Identify

Real Part (a)

Imaginary Part (b)

Example 2: Writing in Standard Form

Intermediate

Write in standard form.

Simplify the Radical

Rearrange

Standard form is (Real first).

Example 3: Solving for Unknowns

Advanced

Find and : .

Equate Real Parts

Equate Imaginary Parts

Solution:

Common Pitfalls

Is a variable?

No! is a number (). is a variable. Do not solve for .

Order Matters

Standard form is . Don't write .

Real-Life Applications

The most famous fractal in mathematics, the Mandelbrot Set, is created by iterating a simple function on the complex plane (). The beautiful infinite patterns you see are just graphed complex numbers!