# AII.3 and *AII.2 - Complex Numbers

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.

AII.2  The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i.

Bloom's Level:  Apply, Analyze, Understand

### BIG IDEAS

• I can express the laws of electricity and work as an electrical engineer to design electromagnetic fields and electronic circuits.
• I will be able to solve quadratic equations, use the quadratic formula for all cases even when the discriminant is a negative number, and simplify even roots of negative numbers.

### UNDERSTANDING THE STANDARD

• 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 √-1.

·  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.

·  Algebraic 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 imaginary unit that satisfies the equation (e.g.,

### ESSENTIALS

Complex Number System

All.31  Recognize that the square root of -1 is represented as i.

·  AII.21  Recognize that the square root of –1 is represented as i.

All.36  Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole, and natural.

All.37  Write a real number in a + bi form.

All.38  Write a pure imaginary number in a + bi form.

Simplify powers of i
All.34  Simplify power of i.

·  AII.23  Simplify powers of i.

All.33  Simplify radical expressions containing negative rational numbers and express in a + bi form

·  AII.22  Simplify radical expressions containing negative rational numbers and express in a + bi form

Operations
All.35  Add, subtract, and multiply complex numbers.

·  AII.24  Add, subtract, and multiply complex numbers.

Properties
AII.32  Determine which field properties apply to the complex number system.

### KEY VOCABULARY

complex numbers, powers, i, field properties, square root, simplify, radical expressions, a + bi, hierarchy of subsets, complex, pure imaginary, real, rational, irrational, integers, whole, and natural numbers

Updated: Oct 27, 2017