AII.3 
The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers. 

Essential Knowledge and Skills 
 Recognize that the square root of –1 is represented as i.
 Determine which field properties apply to the complex number system.
 Simplify radical expressions containing negative rational numbers and express in a+bi form.
 Simplify powers of i.
 Add, subtract, and multiply complex numbers.
 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole, and natural.
 Write a real number in a+bi form.
 Write a pure imaginary number in a+bi form.

Essential Understandings 
 Complex numbers are organized into a hierarchy of subsets.
 A complex number multiplied by its conjugate is a real number.
 Equations having no real number solutions may have solutions in the set of complex numbers.
 Field properties apply to complex numbers as well as real numbers.
 All complex numbers can be written in the form a+bi where a and b are real numbers and i is the square root of 1.


Vertical Articulation 
 SOL A.2  a) apply laws of exponents to perform ops on expressions;
b) add/subtract/multiply/divide polynomials; c) factor first and second degree binomials/trinomials (1 or 2 variables)


Instructional Materials 
 VDOE ESS Lesson: Complex Numbers  Expressions and Operations (PDF)  Performing complex number arithmetic (Word)
 Warmup: Classifying Numbers (flipchart)
 Interactive Notes: Properties (flipchart)
 Properties Guided Notes (doc)

Real World Lessons 

Practice 
 VDOE ESS: Complex Numbers Worksheet (pdf)
 Worksheet: Properties (doc)

SelfAssessments 
 SOL MiniQuiz: AII.3 Properties of Complex Numbers

Skills 
 JMU Pivotal Question: Powers of i (doc)

Videos 
 Video Tutor  Complex Numbers

Explore
Learning 
 None

Websites 
 Notes: Complex Numbers Introduction from Purplemath.com
 Notes/Activity: Complex Numbers from Regentsprep.com

Common Core
Standards 
