#### Math - 2017-18

# 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.2The student willperformoperations on complex numbers andexpressthe results in simplest form using patterns of the powers of i.

**Bloom's Level: ** Apply, Analyze, Understand

*Adopted: 2009*

### 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+b
*i*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.3_{1} **Recognize** that the square root of -1 is
represented as i.

·
AII.2_{1} **Recognize**
that the square root of –1 is represented as *i*.

All.3_{6} **Place** the following sets of numbers in
a hierarchy of subsets: complex, pure imaginary, real, rational, irrational,
integers, whole, and natural.

All.3_{7} **Write** a real number in a + bi form.

All.3_{8} **Write**
a pure imaginary number in a + bi form.

** Simplify powers of i**All.3

_{4}

**Simplify**power of i.

·
AII.2_{3} **Simplify**
powers of *i*.

** Simplify Radicals with -1**All.3

_{3}

**Simplify**radical expressions containing negative rational numbers and express in a + bi form

·
AII.2_{2} **Simplify**
radical expressions containing negative rational numbers and **express** in *a *+ *bi* form

** Operations**All.3

_{5}

**Add**,

**subtract**, and

**multiply**complex numbers.

·
AII.2_{4} **Add**,
**subtract**, and **multiply** complex numbers.

** Properties**AII.3

_{2}

**Determine**which field properties apply to the complex number system.

### KEY VOCABULARY

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

*Updated: Oct 27, 2017*