  Vincent Selhorst-Jones

Complex Numbers

Slide Duration:

Section 1: Introduction
Introduction to Math Analysis

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
18:50
19:04
19:21
20:04
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
9:56
11:38
Derivation
11:43
Final Form
12:23
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
26:49
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
19:24
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
8:25
9:26
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2

52m 52s

Intro
0:00
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
11:42
Example of Using a Point
12:41
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
11:51
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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• ## Related Books 2 answersLast reply by: Tiffany WarnerThu Jun 2, 2016 6:39 PMPost by Tiffany Warner on June 1, 2016Hi Professor!In one of the practice problems, we are asked to find the roots of x^2-6x+20The answer I continuously got was 3+sqrt(11)i and 3-sqrt(11)iThe answer given does not match. I looked at the steps and there’s two places where I think I’m missing something. Under the square root they show 36-100 but I got 36-80. I’m not sure where the 100 is coming from. Then in the last step I noticed the first real number is divided by the denominator, but the real number in the “imaginary part” was left alone. Can we do that?I often make small silly mistakes that lead me to strange conclusions so I’d thought it best to get clarification. Thank you very much for your time and help! 3 answers Last reply by: Professor Selhorst-JonesMon Feb 16, 2015 12:14 PMPost by Andre Strohhofer on June 19, 2013How would you graph these numbers, or how would the roots show up on a graph?

### Complex Numbers

• So far, we've found that some polynomials are irreducible in the real numbers: they cannot be broken down into smaller factors and they have no roots. For example, x2+1 has no roots and cannot be broken down any further.
• We introduce a new number to get around this: the imaginary number,
 √ −1 = i.
Notice that i2 = −1, that is, we can square it and get a negative. That's something we could never do in the real numbers!
• With this idea in mind, we can now take the square root of any negative number. The negative just pops out as i, while we do the rest of the square root normally. For example, √{−49} = 7i.
• With the idea of the imaginary number i in place, we can create a new set of numbers, which we'll call the complex numbers (denoted by ℂ). We still want to be able to talk about real numbers (ℝ), so we'll need them to appear along with i. Thus we'll make each complex number with a real part and an imaginary part:
 a   +   b i,
where a and b are both real numbers. [a is the real part, bi is the imaginary part.]
• Working with complex numbers is similar to working with real numbers.
• Two complex numbers are equal when the parts in one number are equal to the parts in the other.
• We can do addition and subtraction by having the real parts add/subtract together and the imaginary parts add/subtract together. Other than the two parts staying separate, it works like normal. Notice how this is similar to adding/subtracting like variables in algebra.
• Multiplication also works very similarly to what we're used to. Just approach it like a FOIL expansion. [Since i2 = −1, it will transform during simplification.]
• Division is a little trickier. First, we need the idea of a complex conjugate. The conjugate of (a+bi) is (a−bi) and vice-versa. Notice that whenever you multiply a complex number by its conjugate, it always results in a real number. This means if we have a fraction (division) with a complex number in the denominator, we can multiply the numerator and denominator by the complex conjugate for the denominator so that we can now have a real in the denominator. Then we just divide like usual.
• Now that we understand how complex numbers work, we can revisit the quadratic formula and use it to find the roots of "irreducible" quadratics.
x = −b ± √ b2 −4ac

2a

Previously, we couldn't use the formula if b2 − 4ac < 0, but now we know it just produces an imaginary number! Furthermore, because of the ±√{b2−4ac} part in the quadratic formula, we see that complex roots must come in conjugate pairs. That is, if p(x) is a polynomial:
 p(a+bi) = 0     ⇔     p(a−bi) = 0.

### 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
• Introduction 0:04
• A Wacky Idea 1:02
• The Definition of the Imaginary Number
• How it Helps Solve Equations
• Square Roots and Imaginary Numbers 3:15
• Complex Numbers 5:00
• Real Part and Imaginary Part
• When Two Complex Numbers are Equal
• Deal with Real and Imaginary Parts Separately
• Two Quick Examples
• Multiplication 9:07
• FOIL Expansion
• Note What Happens to the Square of the Imaginary Number
• Two Quick Examples
• Division 11:27
• Complex Conjugates 13:37
• Getting Rid of i
• How to Denote the Conjugate
• Division through Complex Conjugates 16:11
• Multiply by the Conjugate of the Denominator
• Example
• Factoring So-Called 'Irreducible' Quadratics 19:24
• Conjugate Pairs
• But Are the Complex Numbers 'Real'? 21:27
• What Makes a Number Legitimate
• Where Complex Numbers are Used
• Still, We Won't See Much of C 29:05
• Example 1 30:30
• Example 2 33:15
• Example 3 38:12
• Example 4 42:07

### Transcription: Complex Numbers

Hi--welcome back to Educator.com.0000

Today, we are going to talk about complex numbers.0002

At this point, we know a lot about factoring polynomials and finding their roots.0005

Still, there are some polynomials we can't factor.0009

There is no way to reduce them into smaller factors, so they are irreducible; and they simply have no roots.0011

There is just no way to solve something like x2 + 1 = 0.0016

There are no roots to x2 + 1, because for that to be true, we would need x2 = -1.0020

But when you square any number, it becomes a positive; if we square +2, then that becomes +4.0025

But if we square -2, then that is going to become +4, as well; the negatives hit each other, and they cancel out.0033

This pattern is going to happen for any negative number and any positive number, and 0's are just going to stay 0's.0040

So, there is no way we can square a number and have it become negative.0045

But what if that wasn't the whole story--what if there was some special number we hadn't seen, that, when squared, does not become positive?0050

That is an interesting idea; if that is the case, we had better explore--let's go!0058

We are looking for a way to solve x2 = -1; in other words, we are looking for something that is the square root of -1.0063

We are looking for something that, when you square it, gives out -1.0070

So, here is a crazy idea: why don't we just make up a new number?0074

We will try something really crazy, and we will just create a number out of whole cloth.0078

We will imagine a special number that becomes -1 when squared; so we are making a new number.0082

And since we are using our imagination to think of this new thing, we will call it an imaginary number, and we will denote it with the symbol i.0088

We know that i = √-1, whatever that means...which means that, when you square i, you get -1.0097

The square root of 4 is 2, because when you square 2, you get 4.0104

So, when you square i (since it is the square root of -1), you get -1.0108

These two ideas are how we are going to define this new thing.0113

When you are writing it, I would also recommend writing it with a little curve on the bottom, just so you don't get it confused.0117

Sometimes, if you are writing quickly, you might end up not even putting a dot,0122

at which point it would be hard to see whether you meant to put down 1 or you meant to put down i.0125

So, I recommend putting a tiny little curve at the bottom, and then a dot like that, when you are writing it by hand.0130

That way, you will have some way of being able to clearly see that you are talking about the imaginary number, and not a normal number.0134

With this new idea of i, we can solve the original equation: since i is equal to √-1, we can just take the square root of both sides.0141

And remember: when you take the square root of both sides, you have to introduce a plus/minus.0148

The square root of both sides of an equation...plus/minus shows up, no matter what.0152

So, we get x = ±i; let's check it really quickly.0155

If we have positive i, squared, then that is going to be equal to i2, which is equal to -1, as we just had here.0159

And our other possibility, if we have (-i)2...well, negative times negative becomes positive.0168

So, we have positive i squared; but i2 is, once again, -1; so it checks out--both of these are, indeed, solutions.0174

So, we have found how to solve x2 = -1.0183

We have this new idea of taking a square root of -1, which means, when we square that thing,0186

this thing that we have just created, the imaginary number, we will have -1.0191

We can now use this idea to take the square root of any negative number.0197

We are not just limited to taking the square root of -1; we can do it on anything.0199

We just separate out that √-1; that will become an i; and then we take the root as normal.0203

For example, √-25 would become...we pull out -1, so that is the same thing as 25 times -1.0207

So, we separate this out; we break it into two square roots (a rule we are allowed to do): times √-1.0218

So, √-1 becomes i; √25 is 5; so we have gotten 5i out of that.0223

We look at the square root of -98; well, that is the square root of 98, times -1 inside.0230

So, we can separate that out, and we will get √98 times √-1.0236

√-1 will become i; but what is the square root of 98?0242

I am not quite sure, so we need to break it up a bit more; look at how we can break that into its multiplicative factors.0245

98 is the same thing as 49 times 2; the square root of 49 is 7; 7 times 7 is 49; so we have 7√2i.0250

Finally, the square root of -60; we can see that as 60 times -1, so we separate out the -1 from √60, so this becomes i.0260

What is √60? How can we break that into its factors?0270

Well, we have a 6 times a 10; that is still not quite enough, though, so we break that up some more.0273

We have 2 times 3 for 6, and 2 times 5 for 10, still times i; we see we have a 2 here and a 2 here,0279

so we can pull them out, because they come as a pair.0286

2 times √(3 times 5); we can't do that, so we might as well just turn it into one number, 2√15 times i.0288

So, we can now take the square root of any negative number by having this idea of i, the imaginary number.0295

With this idea of the imaginary number in place, we can create a new set of numbers, which we will call the complex numbers.0302

And we denote it with ℂ; if you are writing that by hand, you make a normal C, and then you make a little vertical line like that.0307

We still want to be able to talk about the real numbers, which, remember, we denote with this weird ℝ symbol.0313

So, we will need them to appear along with i; so we need to have real numbers show up along with this imaginary number.0318

And so, we just saw that we can have imaginary numbers that had this real component multiplied against them.0323

We took the square root of -25, and we got 5i; so we are going to have to have some number times i;0328

and we will also want to have a real part, a; so we will have the real part that will be a.0335

a is the real part; and then we will have the imaginary part that will be bi; bi is the imaginary part.0343

And a and b are both real numbers; they are just coming out of that real set, like usual.0353

This gives us an entirely new form of number: as opposed to just being stuck with real numbers,0358

we have a way of having a real number and a complex number, both interacting with each other.0362

They will come together as a package; and this is the idea of the complex numbers.0366

Two complex numbers are equal when the parts in one number are equal to the parts in the other.0371

If we have a + bi, and we are told that that is equal to c + di, then that means the real parts have to be equal.0375

So, we know that a and c are equal.0383

Also, we know that the imaginary parts have to be equal; both parts have to be equal for this equality to hold.0387

bi and di are the same thing, which means that b and d must be the same thing, since clearly i is going to be the same thing on both sides.0393

Great; all right, so how do we do our basic arithmetic with these things?0399

Addition and subtraction first: we add and subtract complex numbers pretty much like we are used to.0404

a + bi plus c + di just means we are going to combine our real parts (they will become a + c);0408

and our imaginary parts (bi and di) will come together, and we will get (b + d) all times i; bi + di becomes (b + d)i.0415

The same thing over here; if we have a + bi minus c + di, then we have (a - c), and the bi's do the same thing with the di's.0426

So, we are going to have b - d, because remember: there is that minus symbol there; so it is b - d there.0439

a + bi minus c + di...we can also think of that as just distributing this negative sign, like we did before.0445

So, we are now adding a -c and a -di; that is one way of looking at subtraction.0451

So, real parts add and subtract together, and imaginary parts add and subtract together.0456

Other than the fact that they stay separate, it pretty much works like normal.0460

So, as long as they stay separate--they stay on their two sides--imaginaries can't interact directly with the reals;0464

reals can't interact directly with the imaginaries when we are keeping it in addition and subtraction--it is pretty much just normal.0469

Let's look at two examples: 5 + 2i plus 8 - 4i--our 5 and 8 will be able to interact together, because they are both real numbers.0475

And let's color-code that, just so we can see exactly what is going on.0483

So, with our colors before, with red representing reals again, we have 5 + 8, and then it will be +, and then our imaginaries interact as well; 2i - 4i.0486

5 + 8...we get 13; and 2i - 4i (because this was a -4i here)...we get -2i, which we could also write as 13 - 2i.0497

13 + -2i and 13 - 2i mean the same thing; great.0511

If we did it with subtraction, (5 + 2i) - (8 - 4i), then we can distribute this negative sign; and we get -8,0516

and then we will cancel out the minus there, and we get + 4i; so 5 - 8 + 2i + 4i.0524

5 - 8 becomes -3; plus 6i; great.0540

All right, multiplication is pretty much going to work very similarly, as well.0547

Now, notice: we have two things in this sort of factor-looking form; so we do it as a FOIL expansion.0551

We are going to do that same idea of distribution.0558

We will multiply again in the same way that we distributed before.0559

a will multiply on c, and a will multiply on di; so we get ac and adi.0563

And then, bi will multiply on c, and bi will multiply on di; so we will get...bi on c gets us bci, and bi on di will get us bdi2.0570

Now, remember: i2 = -1; so when we have this i2 here, it cancels out, and it is like we have subtraction.0581

So, we put together our real components now: bd and ac gets us ac - bd, because it was -bd.0590

And we have our imaginary components, adi and bci, so we have (ad + bc)i.0599

Remember, this i2 equals -1, so it will just transform during the process of simplification.0606

Now, you could memorize this formula right here, but I wouldn't recommend memorizing this formula right here,0611

because you already know the FOIL expansion, and as long as you can remember i2 = -1, that will keep it easier.0616

That is the better way to do it.0621

All right, let's see some examples: (1 + 2i)(5 - i): 1 times 5 becomes 5; 1 times -i becomes -i;0623

2i times 5 becomes 10i; 2i times -i becomes -2...and we have two i's, so 2i2.0632

Once again, i2 is equal to -1; that cancels out and becomes a plus.0639

So, 5 - i + 10i + 2: 5 and 2 combine to give us 7; and -i + 10i combine to give us + 9i.0645

Great; the next one is (6 + 10i)(5 + 3i): 6 times 5 is 30; 6 times 3i is 18i; 10i times 5 is 50i; and 10i times 3i is 30i2.0658

Once again, remember: i2 is -1, so we have -30, at which point our -30 and positive 30 cancel each other out.0674

And we are left with just 18i + 50i, so we get a total of 68i.0681

Great; the final one is division; now, division is a little more tricky.0686

Consider if we had (10 - 15i)/(1 + 2i): now, at first, we might think,0690

"Oh, we have 10 and 1; we have -15 and 2i; so we will get 10/1 and -15/2,0695

because the i's will cancel out"; but that would be wrong.0702

We have to divide by the entire denominator, not just bits and pieces.0704

For example, to see why this has to be the case, imagine if we had 5 + 5, over 3 + 2; that is really 10/5, which equals 2.0708

But we could get confused and think that that was going to be 5/3 + 5/2; but division does not distribute like that.0719

We are not allowed to do that; so we can't do the same thing here with our 1 + 2i.0729

We can't break it up and distribute the pieces, because it is nonsense in real numbers; so it is definitely going to be nonsense in the complex.0733

What if we break it up, and we put 1 + 2i onto the 10, and then we put 1 + 2i on the -15i separately?0740

We get 10/(1 + 2i), and we get + -15/(1 + 2i).0747

Well, that is true; we broke it up; we can do that with normal things.0754

We could have it, if we wanted to, going back to 5 + 5 over 3 + 2--we could have that as 5/(3 + 2) + 5/(3 + 2).0758

But that doesn't help us; we still have to divide, ultimately, by this 1 + 2i.0766

We don't know how to divide by a complex number yet; that is our problem.0774

Simply put, we have no idea how to divide by complex numbers; that is our problem.0779

Addition and subtraction made natural sense; real numbers stuck together; imaginary numbers stuck together.0786

FOIL was able to allow us to do multiplication--we just did normal distribution, and we remembered the rule that i2 becomes -1.0791

But division...we don't have a good understanding of what it means to divide by a complex number.0797

That is tough; now, what we could do, if there was some clever way to get rid of having a complex number in the denominator--0803

if we could somehow make it into an alternate form where we disappeared the complex number in the denominator-- we would be good.0810

Hmm...to figure out this clever method that we want, first notice something you might have seen while we were working on quadratics.0818

If you have (x - 2)(x + 2), you get x2, and then + 2x, but also - 2x;0824

since we have the -2 and the +2 here, they end up canceling each other out, and so we are left with just x2 - 4.0830

And there is no middle term with just x; there is no x that shows up.0837

We are able to get rid of it, and have only the doubled and then no x whatsoever.0843

We can expand this idea to complex numbers: we do a similar pattern, (1 - 2i)(1 + 2i); let's work that out.0848

We get 1 - 2i + 2i - 4i2; so +2i and -2i cancel each other out, because we have the negative here and the plus here.0855

And then, we have 4i2, so that will cancel and become a plus; and we get 1 + 4 = 5.0864

So, we have been able to figure out a way to multiply this thing and get just a real number.0870

So, if we use this pattern, (a + bi)(a - bi), you automatically get a real number.0875

When you multiply it out, it results in a number that has no imaginary part.0881

You can get that i to disappear entirely, and get something that is completely real.0884

This idea we call the complex conjugate; it comes up often enough, and it becomes important enough,0890

that we give it a special name (complex conjugate); and we also give it a special symbol.0895

We denote it with a bar over the number; so if we have a + bi as our complex number,0899

we can talk about its conjugate with a bar over it; and that is a - bi.0904

The conjugate of a + bi is a - bi; and what is the conjugate of a - bi?0909

Well, we just flip it again, back to a plus, vice versa; a + bi is the conjugate of a - bi; the conjugate of a - bi is a + bi.0913

They just end up flipping between each other, as long as we are doing conjugates.0920

Notice that, whenever you multiply a complex number by its conjugate, it always results in a real number.0924

So, by multiplying with the conjugate, we can get rid of imaginary things; we can get rid of it.0930

Multiply by a conjugate; you always get a real number out of it.0937

So, let's look at that: a + bi times a - bi: we get the a2, and we will get - abi + abi;0941

plus here, minus here; those cancel out; i2 is a -1, so that causes that to become a plus.0948

So, we will end up with a2 + b2, and no i; a and b were just real numbers, so we have something that is entirely real.0955

We started with imaginary things, but by multiplying these together,0963

choosing carefully what we had, we were able to knock out imaginaries entirely and get something that is just real.0966

With this idea of complex conjugates in mind, we can now deal with division.0972

We simply turn the denominator into a real number by multiplying top and bottom of the denominator's conjugate.0976

We want to get the bottom to turn into just a real number, because we know how to divide by reals; you just put it on a fraction.0981

So, with c + di, we need to multiply c - di.0987

Of course, we can't just multiply the bottom because we feel like it;0991

so we also have to multiply the top by c - di, as well, because something over itself is always 10994

(as long as you didn't start with 0/0; then the world explodes).1000

But as long as it is something over something, and that something isn't 0, you get 1.1003

So, c - di over c - di...we can do that; we just trust, intrinsically, that dividing something by itself is 1.1006

That is the nature of division; that is the point of it.1012

So, ac + bd + bc - ad times i over c2 + d2; that is what it will end up simplifying to.1015

And we could work this out, and we would see that this formula ends up working out.1023

I don't want you to memorize this formula; I don't even really see a good point to working through it.1027

The important thing to know is: just remember to multiply top and bottom.1030

Remember to multiply top and bottom by the denominator's conjugate.1034

This idea of being able to multiply by a conjugate--that is the really cool thing.1039

You could memorize a formula, but it is not going to help you to memorize a formula, because it is hard to recall a formula like this.1043

It is much easier to remember that I have division of a complex number.1050

I multiply by the conjugate, because I want to get rid of real numbers.1054

I have to multiply top and bottom, though, because of course, if you did otherwise, you would just be playing fantasy.1057

You have to multiply by the same thing on the top and the bottom to keep it what you started with.1061

All right, let's see an example: (10 - 59) over (1 + 2i).1064

We want to multiply by the conjugate of (1 + 2i) (if we wanted to, we could express that with a bar all over the top of it), which would be (1 - 2i).1070

We multiply by that, and we know we will have gotten to just a real number.1079

1 - 2i multiplies top and bottom; and it does have to come in parentheses, because it is a whole thing multiplying some other whole thing.1083

You don't just get multiplied bits and pieces.1090

We work that out: 10 times 1 gets us 10; 10 times -2i gets us -20i; -15i times 1 gets us -15i; -15i times -2i gets us +30i2.1093

What is on the bottom? We have (1 + 2i)(1 - 2i); 1 times 1 gets us 1; + 2i, - 2i; those will cancel out; -2i + 2i, and then -4i2.1107

Remember, i2 becomes -1; so we cancel out like that.1126

And then, we also see that -2i + 2i cancel each other out.1132

What does this become next? We combine things: 10 - 30, our real parts on the top, become -20.1135

-20i - 15i becomes -35i; what is on the bottom? 1 + 4 is 5, so we can divide -20/5, minus 35i/5; so that gets us -4 - 7i.1141

Great; all right, now that we understand the basics of how to work with complex numbers,1162

we are now at a point where we can actually see how to factor irreducible quadratics.1168

It is now possible for us to factor previously irreducible quadratics and find their roots.1172

So, x2 + 1, we see, is now factored into (x + i) and (x - i).1176

Let's check this: we get that this would be equal to x2 - ix + ix - i2.1180

Oops, ix; I didn't write that whole thing.1189

Those cancel each other out; i2 becomes + 1, so we get x2 + 1.1193

Sure enough, it checks out; and we have found a way to be able to factor this thing that, before, we could not factor.1201

It used to be irreducible, but now we see that, through the complex numbers, it is not irreducible at all.1206

It is totally factorable; we can revisit the quadratic formula and use it to find the roots of these supposedly irreducible quadratics.1210

What used to be irreducible for us is no longer, so we can use the quadratic formula.1216

Previously, we couldn't use it when b2 - 4a was less than 0, because there was no square root of a negative number.1221

But now we know that that just means an imaginary number; so if our discriminant, b2 - 4a, shows it is less than 0,1226

then that means, not that we have no answers, but that we just have imaginary answers.1232

Cool; furthermore, because of the ± √(b2 - 4ac) part in the quadratic formula,1236

we see that complex conjugates must come in conjugate pairs.1242

If b2 - 4ac was less than 0, so this gives out stuff times i, then we have this ± thing;1246

so it is going to be plus stuff(i), minus stuff(i); so we have one version that is a +i and one version that is a -i.1254

That is what happens when we are doing a conjugate pair.1260

We have a + bi; its conjugate is a - bi, so if we have stuff + stuff(i) and minus stuff(i), that is what we have right there.1264

All right, so if we have a polynomial where we know that a + bi is a root1274

(that is to say, when you plug it in you get 0), then we know that a - bi has to also be a root;1279

these things come in conjugate pairs all the time.1285

So, we talked a lot about the complex numbers; but we probably have this nagging question in the back of our head.1288

Are they real? They are clearly not real numbers, because we are saying that they are not the real numbers,1294

which are numbers like 5, 0, π, √2...we have been working with them all up until now.1299

But are they real--are they legitimate--are they something that we really can use,1306

and not be thinking that we shouldn't be using these?1310

I mean, they have the word "imaginary" in their definition; do we really want to be trying to do science or math with something that is inherently imaginary?1313

What is this idea of a number being a legitimate number that is valid for science, valid for math?1327

Now, we probably all agree that 1, 2, 3...those are totally valid.1333

You could pick up one rock; you could pick up two rocks; you could pick up three rocks.1337

We could actually have these things in our hands and say, "Look, I have that many objects."1341

And we might not be able to pick up 5 billion rocks, but we can get this idea that we could count that many rocks in front of us.1346

So, that seems pretty valid; these nice whole numbers are perfectly reasonable.1352

But what about 1/2 being a real number? 1/2 seems pretty valid, because we could take a pizza, and we could cut the pizza in half.1357

We take a pizza; we cut it down the middle; and now, all of a sudden, we have two chunks of pizza.1366

1 here; 1 here; we are left with two objects that come together to form a whole.1372

But at the same time, we could say, "Well, this is one object, and this is one object; so it is 1 and 1."1378

But we could also say it is 1/2 of what we originally started with, and it is 1/2 of what we originally started with.1384

So, it is 1/2 and 1/2; so it is a little bit more questionable that this is valid,1390

because can you actually pick up a half-object? No, it is an object in and of itself.1397

But it is connected to other things, so it is not perfectly valid--not as valid as the rocks.1401

We can grab rocks; we can hold rocks; but we can definitely believe in half-numbers.1405

We can believe in rational numbers; we can believe in fractions; it seems reasonable.1410

Well, OK, what about something even more slippery--what about the negative numbers?1414

Negatives are pretty bad; or we could go even worse, and we could talk about irrational numbers.1419

How can you possibly hold -1 rock? What does that mean?1425

Can you hold √2 rock? Can you hold π rock?1428

You can't hold these things in your hand; so are they valid?1431

We can't cut a π slice of pizza; we can't cut a √2 slice of pizza; what does this mean now?1433

Are they really valid? We certainly used them a lot before--we are used to using them.1440

So, they seem reasonable in that way; but are they things that are real in the real world?1445

Don't worry; they are not illegitimate; it doesn't mean that they are illegitimate because you can't hold them in your hand.1451

We can use them to represent things in the real world.1456

We can talk about an object falling with negative numbers; we can say it is going a negative height.1458

As opposed to a positive height, where it goes up, it goes a negative height, where it goes down.1464

Or maybe you have $100 in your bank account, but then you pull out 150; that leaves you with negative$50 in your bank account.1468

So, you have an overdrawn bank account; we talk about that with negative numbers.1474

So, that seems pretty reasonable; we could also talk about √2.1477

√2 is able to connect the sides of a square.1481

If we have a square where all of the sides are the same on our square, then the connection between one side and the diagonal is side times √2.1484

So, we can figure that out from the Pythagorean theorem.1498

That makes sense; there is some stuff going on.1500

Or if we go ahead and we look at a circle, we will see π showing up.1502

If we want to talk about the circumference of a circle (pardon my circle; it is not quite perfect), it will be π times 2 time the radius.1506

So, there is π showing up; or, if we wanted to talk about the area, it will be π times the radius squared.1515

So, there are relationships going on in circles.1520

And circles are real-life things; we see circles in lots of places; we see spheres and other circular objects in lots of places in real life.1522

So, it seems reasonable to count √2, π, -1...they are all valid numbers,1529

not because we can hold it in our hand, but because we can use it for totally reasonable things.1534

So, ultimately, these numbers are "real" (not to say real numbers, but "real" numbers, numbers that we believe in),1539

because they have meaning--because they are useful for something.1545

A number is valid, not because we can hold it in our hands, but because it is useful and/or interesting.1550

That is what makes a number a valid number that we want to work with--because we can either use it in real things,1557

or it is really interesting and fascinating--it is telling us cool stuff.1562

After all, math is a language; and in language, we can talk about things that aren't just concrete.1565

You can talk about things like "cat" and "tree"; but at the same time, you can also express abstract concepts--things like "justice" and "freedom."1571

You can walk down the street, and you can point at a cat, and you can point at a tree.1581

But you can't really hold a justice in your hand; and you can't say, "Oh, look, here is a freedom."1585

They are not things that you can hold; they are not tangible, real things.1590

They are abstract concepts that require us to think in this other way.1594

And that is how the numbers work: 1, 2, 3...they are representing concrete things that we can really hold.1598

But we can also talk about abstract ideas, like √2 or π, that are telling us relationships that are really useful.1603

We might not be able to hold it in our hand; but it is still a really useful idea.1609

So, it is just as valid; "cat" and "justice" are both valid things, because they are useful to us.1613

They represent something worthwhile; they represent something interesting.1619

It is the exact same way with the complex numbers; this is how it is with the complex numbers.1624

You can't hold i rocks in your hands; you can't hold 52i in your bank account.1628

But they still have validity; there is still meaning there; they are still valid; they still have meaning.1634

In fact, they have direct connections to the real world; so that might be our other issue:1641

"OK, I can believe in the fact that numbers get to be valid when they are interesting; but are they useful--can we use them in the real world?"1646

Sure enough, you can: complex numbers show up a lot in electrical engineering.1653

They show up in advanced physics; and they show up in other fields of science.1657

They also show up in lots of advanced mathematics.1660

If you are interested in mathematics--in the really, really high, interesting stuff--complex start to show up a lot.1663

They are totally valid; you can prove real things; they are really meaningful.1667

By using complex numbers, we can actually model real-world phenomena; and we can make accurate predictions.1671

Complex numbers are proven to be useful; we can actually use a complex number and get truth out of it that we can then measure in the real world.1677

You don't get a complex number of things; but you can have a complex number help you on your way1685

to finding an accurate measurement, to finding something and predicting something that actually works.1690

So, complex numbers are totally valid in terms of being useful in the real world, and also just as a thought construct.1694

In many ways, the name "imaginary" is unfortunate: they are not imaginary in terms of "they don't count; they aren't really there."1700

They are just imaginary because the name stuck; there is no reason that they are less valid than real numbers.1709

They aren't less valid; they are just as valid as any other number.1714

They are not real numbers, which is to say they are not ℝ; they are not those numbers that we talked about before.1717

But the complex numbers can still represent reality.1723

So, they are not real numbers, but they still show reality.1725

They are imaginary, but only in name; they are actually things that can be used to show real life.1728

They tell us all sorts of useful things, and they are pretty cool.1734

Complex numbers are legitimate and valid; they are not real numbers, but they are "real" in the sense that they are a part of the real world.1737

All that said, nonetheless, complex numbers are not going to be something that we will see a lot.1746

They are totally legitimate; they are valid; but we won't see much of them.1751

Complex numbers tend to be connected to advanced math, for the most part.1755

And so, it is really going to be more advanced math than we want to study right now.1758

So, if you keep going in math, or you keep going and see some really high-level science at some point in a few years,1763

you will probably end up seeing complex numbers be used for real things.1768

But right now, we are just sort of saying, "Oh, look--complex numbers! That is cool," and we are moving on to something else.1772

So, most math courses--especially courses at this level--will limit themselves to just the real numbers,1778

because if they go too far, it will get too complex (get the joke?).1783

Unless a question specifically asks about complex numbers, or they were directly mentioned in the lesson1791

(such as this one), just stick to the real numbers.1796

You really want to just stick to the real numbers, unless you are working specifically with the complex numbers,1798

or you have been told to work specifically with the complex numbers.1803

We will briefly play with complex numbers in a couple of lessons in this course.1805

But they are something best explored later on in a more advanced mathematics course, or an advanced science course.1808

Thus, in general, limit yourself to using just the real numbers, ℝ, for now.1814

And really, that is going to be pretty easy, because it is what you are used to doing.1819

You are used to just working with the real numbers; so it is not going to be hard to just go back to working with the real numbers,1823

because it is what you have been doing for years and years and years.1827

All right, we are ready for some examples.1830

Simplify (25 - 45i)/(-3 + 4i); remember, we need to multiply by the conjugate.1832

The conjugate to -3 + 4i, which we could denote with a bar over all the top of it, is equal to -3...1838

and then we flip the sign on the imaginary part, so it will be - 4i.1844

So, we want to multiply this by (-3 - 4i)/(-3 - 4i).1848

Now, notice: you have to put parentheses around all of this, because the whole thing is multiplying--not just bits and pieces, but the whole thing.1856

So, we work this out; 25 times -3 becomes -75; 25 times -4i becomes -100i; -45i times -3 will become positive 45...1864

that is 3 times 5 off of 150, or + 135i; -45 times -4 becomes positive, so we will get 4 times 5 off of 200, so 180i.1878

Divided by...-3 times -3 gets us positive 9; -3 times 4i gets us + 12i; +4i times -3 gets us -12i; 4i times -4i gets us -4i2.1896

So, we see that we have -12i + 12i, and also when we have i2, it becomes positive.1911

Oops, I accidentally made a typo here: -45 times -4i will become + 180i2.1916

So, cancel out that i2; we get -180; now, let's combine things.1923

-75 - 180; that will get us -255; -100i + 35i will get us +35i; what is on the bottom?1928

42...we missed that; sorry--one more mistake; 4 times 4 gets us -42i2,1938

so 42 gets us 16; 9 + 16 is in our division, so divide by 25.1949

-255 + 35i; divide by 25; we notice that we can pull out a 5 from all of these; this is 5 times 51.1957

This is 5 times 7; this is 5 times 5; so we go through and cancel one of the 5's on all of them.1964

And we are left with -51 + 7i, all over 5, which, if we wanted to, we could alternately represent as -51/5 + 7/5 i,1972

keeping our imaginary part and our real part completely separate.1987

Both of these are totally legitimate answers; we would know what we were talking about in either case.1990

All right, the second example: Given that x = -2 + i is a root to the below polynomial, find the other root and verify both.1994

Remember: if x = -2 + i is one of our roots, the conjugate is also the case.2002

So, x bar, the conjugate of x being -2 + i, is going to be...what is the conjugate of that?...-2 - i.2007

So, we know what the other root is; the other root is -2 - i, and our first root is -2 + i.2015

We are guaranteed that a complex conjugate must be the other root, from what we talked about earlier.2021

So now we are told to verify both of them.2027

There are two different ways we can verify this.2029

First, we could verify this through factors; we could show that, if we were to use these as factors...2031

because remember, knowing a root tells you a factor; remember, if we know that there is a root at k,2038

then we know that there is a factor, (x - k); so if we know that there is a root at (-2 + i),2046

then we know that there is a factor of (x - -2 + i), following that same pattern of x - k.2051

It is just that k, in this case, is two things.2058

That is times (x - (-2 - i)) for our other factor.2061

So, if we can multiply these two factors together, and we can get x2 + 4x + 5,2067

then we will have verified that those must be the roots, because they are the factors,2071

and there is this deep connection between roots and factors; you can go either way.2075

So, let's work this out: simplify the insides first: x minus a negative will become + 2 - i;2078

times x minus a minus will become + 2 + i; we can start working this out.2086

x times x becomes x2; x times 2 becomes + 2x; x times i will become + ix.2093

2 times x will become + 2x; 2 times 2 will become + 4; 2 times i will become + 2i; -i times x will become -ix;2100

-i times 2 will become -2i; -i times +i will become -i2.2109

-i2 becomes +1, because the i2 cancels out.2116

And now, let's work through and see this.2120

So, let's simplify this: x2: how many other x2's do we have?2122

That is the only one, so we get x2 + 2x; how many other x's do we have?2125

We have x there, 2x there, and no other x's; so we put those all together, and we get + 4x.2130

ix's--how many ix's do we have? We have that ix and that ix, so ix - ix.2137

They cancel each other out, and they completely nullify each other; so we don't have to put them down at all.2142

How many constants do we have? 4 there; don't forget the 1 that came out of our i2.2147

So, we have 4 + 1, because it flipped the sign; that is + 5.2152

And then 2i - 2i; once again, they nullify each other, so we get x2 + 4x + 5; it checks out; great.2156

The alternate way that we could do this is: we could do this by verifying that they are, indeed, roots.2164

So, we could do this another way by showing that they are roots; let's start by showing that x = -2 + i is a root.2171

We plug that in; x2, (-2 + i)2, plus 4(-2 + i), plus 5.2182

(-2 + i)2 becomes: -2 times -2 becomes positive 4; -2 on i, plus i on -2, becomes -4i; i on i becomes + i2.2192

And then, continue on: plus 4 on -2 becomes + -8; 4 on i becomes +4i; and pull down the 5.2204

i2 becomes -1; notice that we have + 4i - 4i, so they eliminate each other here and here.2213

4 - 1 becomes 3; -8 + 5 becomes -3; and we get 0...sure enough, that is a root, because it produces 0.2223

The other one: let's plug in x = -2 - i; we plug that one in: (-2 - i)2 + 4(-2 - i) + 5.2232

-2 times -2 is positive 4; -2 on -i and -i on -2 get us + 4i; -i on -i gets us + i2.2244

Plus -8, minus 4i, plus 5...so we see that we have a positive 4i here and a negative 4i here; they eliminate each other.2253

We have this i2; it becomes -1; so 4 and -1 gets us 3; -8 and 5 gets us -3, which, once again, equals 0; so they are both roots.2264

There are two different ways to do it: we can show that these are the factors that would be given by those roots,2276

and when you multiply those factors, you get back exactly to where you started; that checks out.2280

Or alternately, we can do it by roots and show that when you plug that in, you get the zeroes; so that checks out.2286

Great; the third example: Factor x2 - 8x + 19.2291

Well, we know that this is probably going to involve complex numbers; it is probably a little bit hard to figure it out in terms of complex numbers.2297

But can we find the roots? Sure enough, we can find the roots.2302

Let's find roots, and then we will use the roots to give us factors.2305

Remember: once you know roots, you know factors; so we find the roots first.2309

We can just use the quadratic formula, because now we can use it on anything.2313

We don't have to worry about if it is a complex or not.2317

The discriminant won't hold us back, because now we can just get imaginary answers, as well.2319

We have x =...the roots occur at [-b ± √(b2 - 4ac)]/2a.2324

And hopefully, you were able to say that out loud before I said it, to yourself, because really, you want to have that one memorized.2335

I said it the last time we talked about the quadratic formula.2342

The quadratic formula comes up enough in math and science that it is ultimately something you really want to have memorized.2344

All right, so what is our b? Our b is -8.2349

So, we plug that in: [-(-8) ± √((-8)2 - 4 (what is our a? our a is a 1) (1) (times...what is our c? c is 19)(19)...2353

let's move that square root over all of the way; 2 times...a is 1 again, so 2 times 1.2367

That equals -(-8) (gets us positive 8), plus or minus the square root of...64; what is 4 times 19? that is 76, so minus 76; all over 2.2373

We divide out the 2, so we will get 8/2; that gets us 4; plus or minus the square root of 64 - 76; that will still be over 2.2387

Let's put it over that, just so we don't forget that.2395

64 - 76 gets us -12; so we have 4 ± √-12...so we can pull that out as an i, so we will get √12 i, over 12,2397

equals 4 ± √12...what is √12? √12 we can see as √4(3), which equals 2√3,2412

so plus or minus 2√3 i, over 2; look, we have 2 and 2; those cancel out, and we are left with all of our roots.2425

They are when x is equal to 4, plus or minus the square root of 3, times i.2436

Those are our roots; however, those aren't our factors.2442

We want to find what the factors are; so let's get that in another color.2446

If we know that our roots are 4 ± √3i, remember: if you know k is a root, then that tells you x - k is a factor.2450

So, in this case, our roots are x = 4 + √3i, and x = 4 - √3i, which is good, because they came as a conjugate pairing there.2463

So, those are both of our possibilities; those are both of our factors.2477

x - k: our factors will be x minus this one right here, so minus (4 + √3i)...not that whole thing...2480

I put that parenthesis on the wrong place; i...the parentheses close there; times (x - this thing here, (4 - √3i).2494

So now, let's simplify it, so we can get the factors in a nice, slightly-simpler form to look at.2506

x - 4 - √3i and x - 4 + √3i; we have factored it by being able to do that.2510

And if we wanted to, we could also expand this and check this.2522

And we would be able to show that that is, indeed, exactly what it is; great.2524

The final example: What is i3, i4, i5, i6, i7, i8, etc.?2528

What pattern appears as we go through these powers of i?2535

Let's take a look at how we work through it.2538

If we have i1, just plain i, we have i.2541

That is just what it is; it is just i.2545

What about when we have i2? Well, by definition, that was -1.2547

So, let's see the way it keeps going as we take this up.2551

i3...we multiply the -1 by one more i, so we would get -1 times i, or just -i.2553

i4 would be equal to...i times i gets us -i2; -i2(i2) cancels, and we get positive 1.2559

-i2 cancels, and we get positive 1; so we are left at 1, just a plain + 1.2567

What if we keep going? i5 is equal to...well, we multiply by 1, so it is just i, once again.2578

i6 would be equal to i2, multiplying by one more i, which we know is -1.2584

i7 is equal to i3, which is equal to...we already figured this out; that was -i.2590

i8...well, that is going to be equal to i4, because we just multiply the one above.2597

We already figured out what i4 is; that is going to be positive 1.2602

Let me make that plus sign a little clearer.2607

i9...if we just kept going, we would have i5;2608

we already figured out what i5 was--that was i1, which is just i; and so on, and so on, and so on.2611

So, the pattern repeats every 4.2617

What we need to do is: we basically need to divide by 4 and see what we have.2627

What we can do is divide the exponent of i by 4; then, what do we do next?2632

Let's do a quick check: if we did i9, 4 goes into 9 how many times?2643

It goes in twice; so we would have 8; 9 - 8 is 1, so we would get a remainder of 1.2648

So then, you look at the remainder, and that tells you that it is equal to i to whatever-you-just-figured-out-your-remainder-is.2653

So, for example, if we wanted to figure out what i80 is (which is divisible by 4),2674

we can see that is just i to the 4 times 4 times 4 times 4 times 4; if we figure that out for i80,2681

then we can figure out that what that is equivalent to...by 4...how many times does that go into 80?2687

4 goes into 8 twice, so that gets us 8 - 0; bring down the 0; 0; we get 20, and our remainder is 0.2692

So, that would be the equivalent of i0, which is just the same thing as i4, which is +1.2700

So, that is how you want to do it if you are given a really, really, really large i.2708

It is just a question of if you divided it by 4--what would be left over? What would be the remainder?2712

And if you end up having a remainder of 0, then it fit perfectly, so it ends up coming out just as 1.2716

All right, great; we will see you at Educator.com later.2721

And we will finally see how complex numbers tell us something about polynomials, more than just quadratics.2723

We will see how they are deeply connected to everything that we have been talking about.2728

It will be so deep that it is called the fundamental theorem of algebra.2731

All right, see you later--goodbye!2735