1.3 Complex Numbers

Miller & Gerken, College Algebra 2e — Pages 118-127

Learning Objectives

  • Understand the imaginary unit \(i\)

  • Add and subtract complex numbers

  • Multiply complex numbers

  • Divide complex numbers using conjugates

Key Concepts

The Imaginary Unit

\[i = \sqrt{-1}, \quad i^2 = -1\]

Complex Number Standard Form

\[a + bi \quad \text{where } a, b \in \mathbb{R}\]

  • \(a\) = real part

  • \(b\) = imaginary part

Complex Conjugate

The conjugate of \(a + bi\) is \(a - bi\).

\[(a + bi)(a - bi) = a^2 + b^2\]

Powers of i

\(i^1 = i\)

\(i^2 = -1\)

\(i^3 = -i\)

\(i^4 = 1\)

The pattern repeats every 4 powers.

Examples

Example 1. Example 1 (p. XX)

Work through textbook example…​

Practice Problems

Problem 1 (p. XX, #YY)

Problem:

My Work:

Answer:

Section Summary

  1. Key point 1

  2. Key point 2

  3. Key point 3

Study Notes

Personal observations and things to remember…​

Next: 1.4 Quadratic Equations