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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Complex Numbers

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
Addition
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
Addition Property of Zero
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
II. Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
Contradiction Cases
1:30
Solving Linear Equations in One Variable Cont.
2:00
Addition Property of Equality
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
Addition Property of Equality
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
III. Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
Quadrants
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
IV. Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
V. Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
VI. Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
VII. Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16
Adding & Subtracting Polynomials

18m 33s

Intro
0:00
Objectives
0:07
Adding and Subtracting Polynomials
0:25
Terms
0:33
Coefficients
0:51
Leading Coefficients
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
Adding and Subtracting Polynomials Cont.
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
VIII. Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12
IX. Quadratic Equations
Solving Quadratic Equations by Factoring

23m 38s

Intro
0:00
Objectives
0:08
Solving Quadratic Equations by Factoring
0:19
Quadratic Equations
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22
Solving Quadratic Equations

29m 27s

Intro
0:00
Objectives
0:12
Solving Quadratic Equations
0:29
Linear Factors
0:38
Not All Quadratics Factor Easily
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
The Quadratic Formula
7:28
Discriminants
8:25
Solving Quadratic Equations - Summary
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07
Equations in Quadratic Form

16m 47s

Intro
0:00
Objectives
0:08
Equations in Quadratic Form
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14
Quadratic Formulas & Applications

29m 4s

Intro
0:00
Objectives
0:09
Quadratic Formulas and Applications
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
Quadratic Formulas and Applications Cont.
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50
Graphs of Quadratics

26m 53s

Intro
0:00
Objectives
0:06
Graphs of Quadratics
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
Graphing in Quadratic Standard Form
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
Graphs of Quadratics Cont.
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
X. Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43
Adding & Subtracting Rational Expressions

20m 24s

Intro
0:00
Objectives
0:07
Adding and Subtracting Rational Expressions
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
Subtracting vs. Adding
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
Both Methods Lead to the Same Answer
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
XI. Radical Equations
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
Product Rule for Radicals
0:26
Product Rule to Simplify Square Roots
1:11
Quotient Rule for Radicals
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09
Adding & Subtracting Radicals

17m 22s

Intro
0:00
Objectives
0:07
Adding and Subtracting Radicals
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
Add Radicals' Coefficients
3:55
Example 1
4:47
Example 2
6:00
Adding and Subtracting Radicals Cont.
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10
Multiply & Divide Radicals

19m 24s

Intro
0:00
Objectives
0:08
Multiply and Divide Radicals
0:25
Rules for Working With Radicals
0:26
Using FOIL for Radicals
1:11
Don’t Distribute Powers
2:54
Dividing Radical Expressions
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
Multiply and Divide Radicals Cont.
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
Multiply and Divide Radicals Cont.
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25
Solving Radical Equations

15m 5s

Intro
0:00
Objectives
0:07
Solving Radical Equations
0:17
Radical Equations
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
Solving Radical Equations Cont.
7:04
Solving Radical Equations with More than One Radical
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
Adding and Subtracting Complex Numbers
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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Lecture Comments (9)

1 answer

Last reply by: Professor Eric Smith
Mon Aug 27, 2018 2:59 PM

Post by Leo Jiang on August 27 at 04:59:41 AM

In Example 3, why did you write the final answer as 32/37 + (7/37)i? I checked the part of the video where you teach us about dividing complex numbers, and the answer to the problem you presented was 11/17 - 7i/17. The two answers on the same category of complex number problems (dividing complex numbers) are presented in different forms.
The answer to the problems in where you explained how to divide complex numbers was presented in the form a/b + ci/b. And yet the form of the answer in Example 3 was presented in the form a/b + (c/b)i. Can you please explain this? Thank you.

0 answers

Post by David Saver on August 13, 2014

Great Course!

3 answers

Last reply by: Professor Eric Smith
Sun Jun 8, 2014 9:26 PM

Post by Mohamed Elnaklawi on April 6, 2014

what does FOIL stand for?

1 answer

Last reply by: Professor Eric Smith
Mon Dec 2, 2013 8:50 PM

Post by Emily Engle on December 1, 2013

Thanks for your lectures; they are very helpful. Are you going to do algebra 2?

Complex Numbers

  • The imaginary number i is defined as the square root of -1. This can be used to re-write square roots of any negative number.
  • A complex number is a number of the form a + bi. Here a is the real part and b the imaginary part.
  • To add or subtract complex numbers think of adding like terms.
  • To multiply complex numbers think of multiplying using FOIL. Note that any i2 simplifies to -1.
  • To divide by a complex number, multiply the top and bottom by the complex conjugate of the denominator.
  • Higher powers of i can be simplified into an expression that no longer has a power. One method involves dividing the power by four and checking the value of the remainder.
    • Remainder of 1 simplifies to i
    • Remainder of 2 simplifies to -1
    • Remainder of 3 simplifies to –i
    • Remainder of 0 simplifies to 1

Complex Numbers

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Objectives 0:06
  • Complex Numbers 1:05
    • Imaginary Numbers
    • Complex Numbers
    • Real Parts
    • Imaginary Parts
    • Commutative, Associative, and Distributive Properties
    • Adding and Subtracting Complex Numbers
    • Multiplying Complex Numbers
    • Dividing Complex Numbers
    • Complex Conjugate
    • Simplifying Powers of i
    • Shortcut for Simplifying Powers of i
  • Example 1 21:14
  • Example 2 22:15
  • Example 3 23:38
  • Example 4 26:33

Transcription: Complex Numbers

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at complex number.0002

I held off on this very special type of number so you could build a lot of things about them that you will see in the section0007

such as multiplying them together which will look like multiplying polynomials and rationalizing denominator will look a lot like the division process.0013

Here are some other things that we will cover.0022

First we will get into a little bit about that the vocabulary from the complex numbers.0024

You have heard about imaginary numbers and you will see how that works with complex numbers.0028

We will learn about the real and the imaginary part of complex numbers.0033

Many of the properties involving these complex numbers are the same properties that you have seen before for real numbers.0039

Then we will get into the nuts and bolts on how you can start combining these things.0046

That means adding, subtracting, multiplying and dividing our complex numbers.0051

Near the very end we will also see some nice handy ways that we can simplify powers of (i)0056

so you can always bring them down to the most simplest form.0061

An imaginary number (i) is defined as the √-1.0067

It seems like a small definition and it does not seem to be very useful, especially if we want talk about lots of different numbers.0075

By defining it in this way, the √-1 it can actually express the root of any negative number using this imaginary number (i).0082

Here is a quick example to show how it works.0091

Suppose I’m looking at the √-7 using some of our properties we can go ahead and break it up into it -1 × 7.0093

And then break up the root over each of those parts.0102

You will see the definition shows up right there.0105

I'm taking the √ -1.0108

It is that piece that gets turned into an (i) and now I'm representing this number as (i) × √7.0111

Think of doing this for many other numbers where you have a negative underneath the square root.0119

If I had -23 underneath the square root that will be (i)√23 and a negative comes out as the (i).0125

√ -16 would be (i) × 4 or just 4i.0137

How those imaginary numbers relate to our complex numbers?0150

Complex number is a number of the form A + B(i).0154

You can see that the imaginary number is sitting right there, right next to that B.0158

Those other values A and B are both real numbers.0164

We call A the real part of the complex number and B the imaginary part, since it is right next to the imaginary number.0168

You can represent a lot of different numbers using this nice complex form.0175

Maybe I will have a number like 3 – 5i that would be in a nice complex form.0180

I can read off the real and imaginary part.0186

We could throw in some fractions and in here as well, so maybe ½ × 2/3(i).0189

We can also have complex numbers like 0 + 3(i).0196

We would probably would not leave it like that, something like this would probably we will write as 3(i).0202

Notice how we can still say it is in a complex form where the real part is 0.0207

The good part about these complex numbers is they will obey most of the usual laws that you are familiar with.0217

Our commutative, associative, and distribution, all of those will work with our imaginary numbers as well.0223

The things that we could do with real numbers, we can do with imaginary numbers as well.0230

Some of the things that we will have to watch out for is when we get down in the simplification process, so keep a close eye on that.0236

Let us get into how you can start combining these numbers together.0247

To add or subtract complex numbers this is a lot like having like terms.0251

Be very careful to watch the signs when combining these numbers together.0257

To show you how this works I have 2 – 5(i) that complex number + 1 + 6(i).0262

Since I'm adding I'm going to go ahead and drop my parentheses here and just figure out which terms I should add together.0269

Notice how all of our real parts I can go ahead and put those together and I can go ahead and collect together the imaginary parts.0280

That is what I'm talking about when I say adding like terms.0290

The real ones 2 + 1 would be 3 and the imaginary I have -5(i) -6(i) that will give me -11(i).0294

I will write those as 3 -11(i).0309

When you are working with subtraction we have to be very careful with your signs.0314

Let me show you, suppose I have 3 + 2(i) and now I'm subtracting a second number 4 + 7(i).0319

Parentheses will come in handy because I need to distribute this negative sign to both parts of that second complex number.0334

I will look at this as 3 + 2(i) – 4 – 7(i) and now I can see the parts and the like terms that I need to go ahead and put together.0343

3 – 4 = -1 and 2 – 7(i) = -5(i)0357

This would be written as -1 – 5(i)0369

Again, combine those like terms.0373

To multiply complex numbers think of multiplying together two binomials.0379

It looks a lot like the same process.0384

When we are multiplying together two binomials we have great way of remembering that we can use the method of foil to accomplish that.0387

There is one additional thing you have to remember, when you have (i)2, we can go ahead and reduce that to -1,0394

that seems a little odd but let us see why that works.0401

If I'm looking at the √-1 and that is our (i) and suppose I take that and I square it.0404

I want to square that square root I will get -1 on the left.0413

That reveals that (i)2 = -1.0419

Watch for that to show up in the simplification process.0424

Let us go ahead and foil out these two complex numbers.0428

2 – 5(i) multiplied by 1 – 6(i)0432

We will begin by multiplying the first terms 2 × 1 = 2.0436

We will move on to those outside terms 2 × -6(i) = -12(i).0441

On the inside terms -5(i) and now let us go ahead and do the last terms -5(i) × -6(i),0453

minus × minus would be +, 5 × 6 is 30 and now there is my (i)2.0465

We will go along and we will start crunching things down like we normally would do if they were a couple binomials.0473

2 – 17(i) now comes this interesting part this (i)2 here is the same as -1.0480

We will go ahead and rewrite it as a -1.0491

You can see there is more than we can do to simplify this.0498

I’m not only multiplying the 30 × -1, but then I can go ahead and combine it with the two out front.0501

2 – 17(i) - 30 and then let us combine these guys that will give us -28 – 17(i).0508

This one is good.0524

With many of these problems where combining complex numbers you know that you are done0527

when we finally get into that nice complex form.0532

You can easily see the real and imaginary parts.0535

To go ahead and divide complex numbers, we want to multiply the top and the bottom by the complex conjugate.0542

The complex conjugate of a number will look pretty much the same but it will be different in sign.0550

If I'm looking at the complex conjugate of A + B(i) that will be –B(i).0557

Let us just do some quick example.0563

Let us see if I have 3 + 2(i) its complex conjugate would be 3 – 2(i).0564

If I had -7 – 3(i), the complex conjugate would be -3 + 3(i).0573

The only thing that is changing is that negative sign or that sign on the imaginary part.0583

This will have the feel of rationalizing the denominator that is why we had to cover a lot of that information before working on these complex numbers.0591

I’m going to look at the bottom of this particular division problem 4 + 2(i) ÷ 3 + 5(i).0599

I'm going to find the complex conjugate of 3 + 5(i), that would be 3 – 5(i).0606

Once we find that complex conjugate we will multiply on the top and on the bottom of our complex fraction.0616

Multiply on top, multiply on the bottom.0628

This one involves quite a bit of work.0631

I have two terms in my complex number, I will have to foil the top and we will also have to foil out the bottom.0634

This will make things look a lot more complicated at first but if you just go through the problem carefully you should do fine.0644

Let us start with the top.0653

Taking our first terms I have 4 × 3 and that would give me 12.0656

The outside terms would be 4 × -5 (i) = -20(i).0662

On the inside 2(i) + 3(i) then would be 6(i) and our last terms 2 × -5 = -10(i)2.0668

Do not forget to multiply those (i) together as well.0680

On the button 3 × 3 = 9, outside will be -15(i), inside would be 15(i).0684

And then we have 5(i) × -5(i) = -25(i)2.0695

Some things you want to notice after you go through that forming process.0703

We are using that complex conjugate on the bottom, the outside and inside terms will end up canceling.0707

If they do not cancel, make sure you have chosen the correct complex conjugate.0715

Notice that we have a couple of (i)2 showing up on the top and bottom.0720

That comes from the (i) being multiplied together.0725

Each of these will be need to change into -1.0728

Let us go through and do those two things and clean this problem a little bit.0733

I have 12 – 20 + 6(i) = -14(i), -10 × -1 ÷ 9 my outside and inside terms on the bottom cancel -25 × -1.0737

We can see the beauty of using that complex conjugate.0759

On the bottom we no longer have any more imaginary numbers.0762

That one I was talking about how I have the feel of rationalizing the denominator and we are getting rid of those roots in the bottom.0766

Here we got rid of the imaginary numbers in the bottom.0772

Let us continue to simplify and see where we can take this problem.0776

12 – 14(i) + 10 ÷ (9 + 25)0781

12 + 10 = 22 – 14(i) ÷ 340790

It is tempting to try and stop right there but do not do it, we want to continue doing this entire problem0801

and try to get it in that nice complex form where I can see the real and imaginary part.0806

To get in that form I’m going to use the part where I breakup the 34 and 22 and under the 14.0811

It looks a lot like when we are taking polynomials when we are dividing by a monomial.0818

This will be 22 ÷ 34 -14(i) ÷ 34.0824

This particular one, both of those fractions can be reduced.0833

I will divide the top and the bottom by 2, 11/17 – 7(i) ÷ 17.0837

This one is actually completely done.0846

I can easily see my real part over here and my imaginary part -7/17.0849

That is I know that this one is done.0855

Division is quite a tricky process remember to multiply by the complex conjugate of the bottom0859

and then work very carefully to simplify the problem from there.0864

It should be in its complex form when you are all done so, you can see the real and imaginary parts.0868

After working with doing some combining things with these various complex imaginary numbers,0876

you may have noticed that you rarely see any higher powers of (i).0881

In fact the largest part of (i) that we came across was i2 and we turned it immediately into a -1.0885

The interesting part is you can usually take much higher powers of (i) and end up simplifying them down.0892

To see out why this works for some much higher powers, let us just take a bunch and start picking them apart.0898

We see many instances while we are working with (i) and (i) was just equal √1 which of course we call (i).0905

When we have i2 and then we thought about squaring the √-1, so we got -1 as its most simplified result.0912

We will see what will happen with i3.0922

One way that you could interpret this would be i2 × (i).0927

The reason for doing that is because you can then take the i2 sitting right here and simplify that.0934

That will give us -1 × (i) the final result of that would be –i as it is simplest form.0942

Let us choose that same idea and see if we can go ahead and simplify i4.0955

That will be i2 × i2.0959

Both of the i2 = -10964

I have -1 × -1 = 10970

Take note how all of these are simplifying to something else that does not have any more powers on them.0975

Let us do a few more that hopefully we can develop a much easier or a shortcut way of doing this process.0983

Let us go on to i5.0990

This can be thought of as i4 × (i).0994

We have already done i4, it is way back here, that is equal to just 1.0999

1 × (i) I know this simplifies to (i).1003

On the next one i6, i4 × i2.1010

We have already done i4 that is just 1.1020

I will put in i2, I will just write across from it -1 so 1 × -1 = -1, very interesting.1024

I think we are starting to see a pattern of some of these (i).1035

Notice how they are looking exactly the same as the ones on the other side here.1038

Let us see if that continues.1044

I7 will be i4 × i3, 1 × i3 = -i.1046

Maybe the last one, i4 × i4, both of those are equal to one, this is equal to 1 as well.1058

Notice how we did starting to see same numbers.1069

There are some good patterns we can see, I mean we got past i4 and all of them contained an i41072

so we are getting this similar pieces right here.1079

If I was even to continue on to i9 then I will have even more groups of i4 and I could see that it would simplified down.1083

All of those much higher powers will end up simplifying down to something which does not have any more powers whatsoever.1097

We can take as much farther and develop a nice shortcut formula so we do not always have to string out into a bunch (i).1104

Let us see how to do that.1111

Looking at all of these examples you can see that there is a pattern on how they simplify.1115

In fact they repeats in blocks of 4 and you start with (i) then it goes -1, then –I, then 1, it just repeats over and over again.1120

To figure out where in that pattern you are at, you can take the exponent and simply divide it by 4 and observe what the remainder is.1129

From the remainder you will know exactly what it will simplify to.1138

I have made a handy table to figure out what that looks like1142

If I’m looking at (i) to some large power, I will take the power and divide it by 41146

and the remainder happens to be 3, according to my chart here it will simplify to –i.1152

Let us do a quick example to see how this works and why it works.1160

Let us take i13.1166

If I wanted to I could imagine a whole bunch of i4 in here.1173

i4, i4, i4 and then i1.1178

If we add up all those exponents they add up to 13.1186

You can see that most of them are gone since i4 = 1.1191

1 × 1 × 1 × (i) =(i)1196

Let us use our shortcut formula to figure out the same thing.1207

In the shortcut we take the exponent and divide that by 4.1210

13 ÷ 4, 4 goes in the 13 exactly 3 times.1216

Notice why that is important, that is exactly how many bunches of i4 I have.1227

When you are going through that division by 4 process you are counting up all of the i4 that will simply break down and simplify into 1.1233

It goes in there 3 times, 3 × 4 = 12 and I will subtract that away getting my remainder of 1.1243

That remainder tells me how many extra (i) I still have left over.1253

From there I can see I have only one (i) and I know it simplifies down to (i).1259

I get the same exact answer as the 4, i13 = (i).1267

Now that we know a lot more about these complex numbers let us go through some various examples1275

of putting them together and see how we can simplify powers of (i).1279

This first one we want to do a subtraction problem 4 + 5(i) – 6 – 3(i).1284

With the addition and subtraction we are looking to add like terms.1290

Subtraction is a tricky one, you have to remember to distribute our sign onto both parts of that second complex number.1295

I’m looking at 4 + 5(i) -6 + 3(i)1302

I can see my like terms, let us put together 4 and -6, and 5 and 3.1311

4 - 6 = -2, 5(i) + 3(i) = 8(i)1320

We can see that it is in that final complex form and we know that this one is done.1329

In this next example, we want to multiply two complex numbers.1337

Think foil.1342

Be on the watch out for additional things that we will need to simplify such as the i2.1346

Okay, starting off, multiplying our first terms together we will get 1.1351

Our outside terms we will multiply that will be 3(i) then we can take our inside terms 2(i) and then finally take our last terms 2(i) × 3(i) + 6i2.1357

We go through and start combining everything we can.1377

1 + I have my like terms here, so 3 + 2 = 5(i) + 6 and take the i2 and turn it into -1.1380

Let us continue 1 + 5(i) - 6 now we can see we have just a couple of more things that we can go ahead and combine.1397

This will be -5 + 5(i) and now I can easily see my real and imaginary part, so I know that one is done.1408

Let us divide these two complex numbers.1420

I have 2 – 5(i) ÷ 1 – 6(i)1422

This is the lengthy process where we find the complex conjugate of the bottom of if we multiply the top and bottom of our fraction by that.1426

Let us first find that complex conjugate.1434

That will be 1 + 6(i).1437

We are going to use that on the top and on the bottom.1440

We have to remember that since we are multiplying we will have to foil out the top and the bottom.1445

Foil top and foil bottom.1452

Let us see what the result of that one looks like.1458

I’m looking at the top, first terms 2 × 1 =2, outside terms would give me 12(i), inside terms -5(i), and last terms -30(i)2.1461

Lots of terms in there, be very careful and keep track of them all.1479

On the bottom, first terms 1, outside 6i, inside -6i, and our last terms -36i2.1483

It is looking much better, looking pretty good.1496

If we do this correctly, we should not have any more powers of (i) in the bottom.1498

Let us see what else we can simplify.1502

We will go ahead and combine these 2i on the top and we will combine these 2i of the bottom1505

and then we will put these i2 to make them both -1.1510

2 and then we will have 7(i) - 30 × -1.1518

On the bottom 1 – 36 -1.1527

It looks like we have indeed accomplished our goal there is no longer imaginary numbers on the bottom.1533

We continue putting things together 2 + 7(i) + 30 since the negative × negative is a positive and 1 + 36.1538

Just a few more things to put together 32 + 7(i) ÷ 37.1553

This point it is tempting to stop it but keep going until you get it into its nice complex form, we can see that real and imaginary part.1562

I’m going to give 3732 and 377.1570

7 /37(i) this one looks much better.1579

Sometimes you can go ahead and reduce those fractions but this one the way it is.1587

In this last problem we will go ahead and see if we can simplify some very large powers of (i).1595

Since they are very large powers we will go ahead and use our shortcut to take care of that process.1602

In the first one I have i107.1607

Let us start off by taking that exponent, 107 and we will divide it by 4.1611

This will figure out all the bunches of i4 that we have hiding in there.1618

4 of those in the 10, twice and we get 8.1623

Subtract them away I will end up with 27.1629

That goes in there 6 times and 6 × 4 = 24 and then we can subtract that away.1635

I'm down to 3 and that is smaller than 4 so I know that 3 is my remainder.1645

Think back, if my remainder is 3 then what does this entire thing simplify to?1653

It simplifies to –i.1660

If you ever forget the way you use that table, you can always build it very quickly by doing just the first few values of (i).1664

(i) = (i), i2 = -1, and we have –I and 1.1674

You quickly have built your table.1682

On to the next one i2013 something very large.1686

We will grab the exponent and we will divide it by 4.1692

4 goes into 20 5 times and I will get 20 exactly.1701

Let us put in our 0 placeholder over the 1 and then bring down the 13.1707

4 goes into 13 3 times and we will get 12.1713

This one shows that my remainder is 1.1720

What does this entire thing simplify down to?1725

If I get a remainder of 1, 2, 3 or 0, now I know what it will simplify down into.1729

It simplifies down just to (i).1736

There are many things you can do to work with these complex numbers but they are often mere things that you have learned before.1740

Keep track of this handy method for simplifying powers of (i) that way you can take those much higher powers1745

and bring them down to something nice and compact.1751

Thank you for watching www.educator.com.1754

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