Home » Mathematics » Pre Calculus » Counting

Vincent Selhorst-Jones

Vincent Selhorst-Jones

Counting

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Table of Contents

Section 1: Introduction
Introduction to Precalculus

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
Put Your Substitution in Parentheses
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
Tip To Help You Remember the Signs
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
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
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
Grades
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
The Basketball Factory
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
Some Comments
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
Leading Coefficient Test
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
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
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
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
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
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

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
An Alternate Form to Quadratics
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
Advantages of Synthetic Method
26:49
Advantages of Long Division
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
Addition and Subtraction
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
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
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
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
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
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
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
Thinking about Logs like Inverses
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
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
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
Adjacent, Opposite, Hypotenuse
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
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
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
Sign of Tangent in Quadrants
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
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
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
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
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
Shade Appropriately
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
Talking About Specific Entries
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
Matrix Addition
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
Addition Principle
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
Radius
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
Example 2: Center and Radius
11:51
Example 3: Radius
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
Diagonal Addition of Terms
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
This Game is About Limits
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
A Limit is About Figuring Out Where a Function is 'Headed'
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
Two Good Ways to Think About This
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
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
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
Deciding on Buying
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
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
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
Think About the Results
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
Adjust Viewing Window
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
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
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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Lecture Comments (1)

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Post by Orsolya Krispán on April 14, 2013

I'm in love with your hand movements :D Great work, thank you very much!

Counting

  • An event is something happening that can occur some number of different ways. If we want, we can denote an event with a symbol (such as E) and similarly for the number of ways (such as n). [Some teachers and textbooks will not use the word `event' and might use another word or nothing at all to describe this idea. That's fine. The important part is just to be aware of certain things having multiple possible outcomes or choices. All the coming ideas will work fine in any case; `event' is just a way to describe this formally.]
  • We can visualize an event (or series of events, as we'll soon see) with a branching line diagram. At each event in the diagram, it splits into its various possible outcomes.
  • Addition Principle: Let there be two events E1 and E2, where E1 can occur in m ways and E2 can occur in n. If precisely one of the events will occur (not both), the total possible ways for something to occur is
    m+n.
  • Multiplication Principle [AKA Fundamental Counting Principle]: Let there be two events E1 and E2, where E1 can occur in m ways and E2 can occur in n. If both events will occur (and they have no effect on each other), the total possible ways for something to occur is
    m·n.
  • Pigeonhole Principle: Let there be some event E that can occur in n ways. If the event E happens n+1 times, then at least one of the ways E can occur must happen twice (or more). Equivalently, given n boxes and n+1 objects, if all the objects are put into the boxes, then at least one box must have (at least) two objects.
  • When working on counting problems, it can help immensely to draw pictures and diagrams. Being able to visually follow the events and choices will make it much easier to solve problems.

Counting

You are camping in a national park. The camp is in the south part of the park. You are going to go on a hike and there are trailheads in the east, north, and west parts of the park. In the east portion, there are five trailheads. In the north portion, there are seven trailheads. In the west portion, there are four trailheads. How many different hikes are there available to you?
  • The most difficult part of this problem is probably just the English. If a word is unfamiliar to you but it seems important to what's going on, look it up! The first step to doing any problem is making sense of it, and that includes understanding the vocabulary. The word `trailhead' is somewhat uncommon, but if we look it up, we discover that it just means "the starting place for a trail". Since the problem is asking about hikes, and a hike takes place on a trail, each trailhead represents a possible hike.
  • From there, the problem is quite simple. We start off with three initial choices-east, north, or west. Depending on which direction we choose, it determines how many choices we have for a hiking trail. Since choosing one direction precludes going on hikes from the other directions, we use the addition principle: just add up the number of choices for each direction.
  • Each direction has the following number of trail choices:
    East: 5,       North: 7,       West: 4
    Since you can only choose one category (east or north or west), we add them together to find the total number of choices:
    5 + 7 + 4     =     16
    Thus there are 16 possible trails to choose, so 16 different hikes.
16
You are about to buy a new bike. At the store, they sell four different bicycle models. Each model can come in any of the following colors: black, white, red, blue, green, or yellow. How many different bikes could you buy at the store?
  • For purchasing the bike, you have two choices to make: the model of the bicycle and the color of the bicycle. Notice that choosing a different choice for either of these qualities will result in a different bike.
  • Still, the first choice of model has no effect on the second choice for color. Any color can be picked for any of the bike models. Thus, because both choices must occur and neither has any effect on the other, we can use the multiplication principle.
  • The number of ways the model can be chosen is 4. Count up the colors to find that there are a total of 6 ways to choose a color. Thus, since the two choices do not affect each other, we multiply them:
    4 ·6     =     24
24
You are at a restaurant and about to have dinner. The dinner comes as three courses, and you will order exactly one of each course. The menu has eight different appetizers, twelve different main courses, and five desserts. How many different ways are there for you to have dinner?
  • For choosing your dinner, you have three choices to make: which appetizer, which main course, and which dessert. Notice that choosing something different in any of these categories will result in a different dinner.
  • Still, each choice has no effect on the other choices. You can pick any appetizer and still go on to pick any main course and any dessert, and similarly for the other two. Thus, because each choice must occur and none of them have any effect on the others, we can use the multiplication principle.
  • The number of choices for the appetizer is 8, the number of choices for the main course is 12, and the number of choices for the dessert is 5. Since each choice does not affect the others, we multiply them:
    8 ·12 ·5     =     480
480
A family is planning a vacation to Anza-Borrego Desert State Park in California. While there, they have two options: they could stay at a campground in a tent, or they could stay at a hotel. There are a total of three different campgrounds and eight different hotels. At each hotel, they have the option of staying in a normal room or a suite. How many different ways are there for the family to stay at the park?
  • The family has to begin by making a choice between staying at a campground or a hotel. Choosing a campground means they won't choose a hotel, and choosing a hotel means they won't choose a campground. This means we can separate the two into two different categories and count them separately.
  • If they stay at a hotel, they also have to choose what type of room they stay in. Since each room type is available at all the hotels, the hotel choice does not affect the room choice, and vice-versa. Thus, we use the multiplication principle to find the total number of ways to stay in a hotel:
    Ways to stay in a hotel:    8 ·2     =     16
  • Since choosing campground or hotel means the other event will not occur, we use the addition principle: count the number of ways to camp then add it to the number of ways to stay in a hotel:
    [# campground ways] + [# hotel ways]     =     3 + 16     =     19
19
Jack Burton needs to take a class on trucking to get his Class A license. There are two different trucking schools he can attend: "Three Storms Trucking School" and "Buckaroo Trucking Institute". If he attends "Three Storms", there are three different teachers he can choose from, each of which has four different timeslots for their classes. If he attends "Buckaroo", there are six different teachers he can choose from, each of which has three different timeslots for their classes. How many different ways can Jack take a class to earn his trucking license?
  • Jack needs to take a class from some teacher during some timeslot. To begin with, he has a choice of which school he attends: "Three Storms" or "Buckaroo". Thus we can count how many ways he can take a class at each school, then add the two together to get the total number of ways:
    Total # of ways = [# ways at "Storms"] + [# ways at "Buckaroo"]
  • Let's figure out how many ways he can take a class at "Three Storms" first. At that school, there are three teachers he can choose from, and each has four timeslots. Since the choice of teacher does not affect how many timeslots he has access to, we multiply them together to find the total number of ways:
    # ways at "Storms"     =     3 ·4     =     12
    We do the same thing for the other school. At "Buckaroo" there are six teachers he can choose from, and each has three timeslots. Since the choice of teacher does not affect the number of timeslots, we multiply:
    # ways at "Buckaroo"     =     6 ·3     =     18
  • Now that we know how many choices he has at each school, we can add them together to find the total number of choices.
    Total # of ways     =     12 + 18     =     30
30
How many ways can you answer a true/false test with ten questions? (Assume that you respond to every question.)
  • Since you respond to every question and it is a true/false test, there are precisely two possible choices for each question on the test.
  • The choice of answer on each question has no effect on the other questions. For example, if you answer `true' on one question, you can still choose to answer either `true' or `false' on all the other questions. Therefore, since you will answer each question on the test and each of the questions has no effect on the others, we can calculate the total number of ways with the multiplication principle.
  • For each question, take the number of ways it can be answered, then multiply that by all the others. Since each question can be answered in two ways, we have
    2 ·2 ·2 ·2 ·2 ·2 ·2 ·2 ·2 ·2     =     210     =     1024
1024
A safe has a combination lock which has the numbers from 1 to 60 (inclusive) on the dial. The safe will open when the correct choice of four numbers (in order) are selected on the dial. How many different possible combinations can the safe have? What if it was five numbers instead of four-how many combinations would then be possible?
  • Each number choice in the combination has no effect on the other number choices. For example, if the first number is `47', the next number can be anything at all, including `47' again. Thus there are 60 possible choices for each of the four numbers in the combination.
  • Since the choice of each number has no effect on the others and all four choices must occur, we can calculate the total number of ways with the multiplication principle. For each choice, take the number of ways it can be picked, then multiply it by all the others. Since each choice can be picked in one of 60 ways, we have
    60 ·60 ·60 ·60     =     604     =     12  960  000
  • We follow the exact same logic for figuring out the total possible ways when the combination is five numbers long:
    60 ·60 ·60 ·60 ·60     =     605     =     777  600  000
Four number combination: 12  960  000,    Five number combination: 777  600  000
In California, automobile license plates (excluding trucks) are issued in the format 1ABC234, where each of the numbers can be any number from 0-9 and each of the letters can be any letter from A-Z, with the exception of I, O, and Q (which are excluded to avoid confusion with 1 and 0). With these rules in mind, how many distinct license plates can be made in this format?
  • Begin by noticing that the choice for each character has no effect on the choice of any of the others. For example, if a `7' is chosen for the first number slot, we can still choose any letter (other than the excluded ones) for the subsequent letters and any number for the subsequent numbers. Since every choice must be made and none of them affect the other choices, we use the multiplication principle.
  • For each number in the license, there are 10 possible choices (since it comes from the numbers 0-9). For each letter in the license, there are 23 possible choices (since the alphabet has 26 letters, but we were told I, O, and Q are excluded).
  • Since the choice of each character has no effect on the others, take the number of ways each can be picked, then multiply it by all the others:
    10 ·23 ·23 ·23 ·10 ·10 ·10     =     104 ·233     =     121  670  000
121  670  000
Eight people are running a race together. How many different ways can first, second, and third place be won?
  • Unlike the previous problems we've worked on, the choice for who wins each place in the race does affect the other choices. Winning first place means that person cannot also win second or third. We can not just say there are eight possibilities for each of three choices and get 8·8 ·8: each choice affects the others, so we can't naively apply the multiplication prinicple without taking that into account. However, if we think about it carefully, we can still use the multiplication principle. Start off by thinking about the different places in order. Think about first, then second, then third.
  • If we look at first place before the others, it's easy to see how many options there are: there are eight competitors, so eight people could potentially take first place. Thus we have 8 options for the first choice. With that in mind, let's now consider second place. Since we already assigned a winner to first place, notice that there are only seven competitors remaining. This will be true no matter who won first place, because winning will take that person out of the running. Thus we don't have to care about the specific winner of first, we just need to acknowledge that there are now only seven possibilities for second. Thus we have 7 options for the second choice.
  • Following the same logic, we will only have six competitors remaining for third place. Thus we have 6 options for the third choice. To find the total number of possibilities, we multiply the number of options for each choice with the others:
    8 ·7 ·6     =     336
336
Sally has a variety of socks in a drawer. They come in the following colors: black, white, tan, red, orange, and blue. How many socks must she pull out from the drawer to guarantee that she has a pair of socks with matching colors?
  • Notice that if Sally pulls two socks from the drawer, she won't necessarily have a pair of socks that match in color. For example, she could pull out one orange and one blue. Similarly, if she pulls out three socks, they still won't necessarily match: orange, blue, and tan. We are looking for how many socks she must pull out to be absolutely sure she will have a matching pair in what she has pulled out.
  • If you think it's simple to find the answer to this, you're right! We know that the socks can come in a total of six different colors (black, white, tan, red, orange, and blue): therefore the most socks she can have without pulling a matching pair is six-one of each color. As soon as she pulls the seventh sock, it absolutely must match with one of the previous socks. Thus Sally must pull out 7 socks to guarantee a match. [Notice that she might achieve a match before pulling out seven socks, it's just that she can only guarantee a match by pulling out seven.] [If you're curious to see a slightly more formal explanation/proof, check out the next step. The above does a great job of relying on intuition, but sometimes you need to prove it more carefully.]
  • To formally prove this, we can use the pigeonhole principle. Consider each of the colors as a category "box". We have a total of 6 boxes, one for each color. Each sock that is pulled is put into the matching color box. As soon as we pull 6+1 = 7 socks, we know there must be at least one box with (at least) two objects in it: our matching pair. This fact comes from the pigeonhole principle, which guarantees if something can occur in n ways and the event happens n+1 times, it must repeat at least one of the ways.
7

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Counting

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 0:08
    • Combinatorics
  • Definition: Event 1:24
    • Example
  • Visualizing an Event 3:02
    • Branching line diagram
  • Addition Principle 3:40
    • Example
  • Multiplication Principle 5:42
    • Example
  • Pigeonhole Principle 8:06
    • Example
  • Draw Pictures 11:06
  • Example 1 12:02
  • Example 2 14:16
  • Example 3 17:34
  • Example 4 21:26
  • Example 5 25:14

Transcription: Counting

Hi--welcome back to Educator.com.0000

Today, we are going to talk about counting.0002

With this, we are starting a new section; and at first, it might seem that counting is a simple thing.0004

But it actually turns out that it is extremely important in math.0009

Being able to count the number of ways that an event can turn out, a process can be performed,0012

or objects will be created is extremely useful in business, science, economics, medicine, and a huge variety of fields.0016

And at first, we are probably thinking, "Well, I could just count them: 1, 2, 3, 4, 5, 6...and be able to be done with counting."0023

Why turn it into a thing in math? Well, that is true; but what if you need to count 100 of something, or you need to count 100 of something,0029

or the thing that you are trying to count is a million large?0035

You can't count a million of the thing by hand with any reasonable amount of time.0037

That is going to take you more than a year, to count the million of something out by hand.0042

So, we need some way to be able to count on a large scale;0046

and that is why math does an actual major study of counting--how to count all sorts of different things;0049

and that is what we will be exploring in this lesson and the next two lessons.0055

The formal study of counting is called combinatorics, and this course will barely scratch the surface of combinatorics.0058

Still, we will manage to see some really important results that allow us to show all sorts of really interesting, cool things.0065

The basic ideas are intuitive--they make a lot of sense; but we can exploit them to answer really difficult questions,0070

and it won't be very hard for us to do at all; all right, let's go!0076

First, we are going to define the idea of an event; an event is simply something happening.0080

And it can occur in some number of different ways.0086

If we want, we can denote an event with a symbol, such as E; and similarly, we can talk about the number of ways that the event can occur.0089

We might use a symbol for that, as well; and we could denote it with some n.0097

So, we have some event that can occur in multiple ways.0100

For example, someone might give you a cookie, and you are allowed to choose from three types of cookie: chocolate chip, snickerdoodle, or peanut butter.0103

We have three different types of cookie that can come out of this.0112

The event, E, is getting a cookie; and the way that it can turn out is three different ways: chocolate chip, snickerdoodle, or peanut butter.0117

So, there is this event; but it doesn't necessarily have a single way of turning out,0125

because we could go with the chocolate chip version of the event,0129

the snickerdoodle version of the event, or the peanut butter version of the event.0131

But they are all within the event of getting a cookie.0135

Some teachers won't use the word event; you might see, in some textbooks, or when a teacher is teaching, that they won't use the word event.0139

And they might use a different word to describe it, or they might not use any word at all.0145

That is fine; the important part of all of this is just to be aware that certain things have multiple possible outcomes or choices involved.0149

All the coming ideas are going to work fine, whether we are using the word event or we are using some other word,0157

or we are not really using a word at all; the important thing is just to get this idea of having branching possibilities coming out of something.0161

Event is just one way to describe this formally, and that is the way we will be talking about it in this course.0169

But all of the basic ideas are going to be the same, whatever vocabulary someone else is using.0174

We can visualize an event: one way to help us visualize an event (or a series of events, as we will soon see) is with a branching line diagram.0180

So, we can have something that breaks off into multiple different directions.0188

At each event in the diagram, it will split into all of its various possible outcomes.0192

In our previous cookie example, we could represent the event with this diagram.0196

We could split into chocolate chip, snickerdoodle, and peanut butter; we have this cookie event0200

that splits into the chocolate chip version, the snickerdoodle version, and the peanut butter version.0204

We have this branching line diagram that shows us all the different ways that our event can occur,0209

because we have one event, but it splits into three possibilities.0216

Addition principle: let there be two events, E1 and E2, where E1 can occur in m ways,0221

and E2 can occur in n ways; if precisely one of the events will occur,0230

not both--we know that an event will occur, and it will be either E1 or E2,0236

but we don't know which of the two events it is going to be--we just know for sure that it will be one of them,0241

when something happens--the total possible ways for something to occur is the number of ways that our first event can occur,0246

added to the number of ways that our second event can occur: m + n.0253

For our previous example, if we were given a choice between one of the previously-mentioned cookies,0259

we can now add on this idea of a scoop of ice cream.0264

Previously, we had three different types of cookies; that would be m = 3.0267

Or we could have a scoop of ice cream; we are allowed to have either a cookie or a scoop of ice cream.0272

And the ice cream can be vanilla or chocolate; so that gives us two choices, so we have n = 2.0278

Then, the total possible choices are the m choices plus the n choices: our three choices for cookie,0285

plus the two choices that we have for ice cream, comes out to a total of 5 possible choices that we can have out of this.0294

We can see this branching like this; if we choose the dessert version of cookie0300

(we have this initial thing: we are getting a dessert; now we have the choice between the cookie event or the ice cream event;0305

so if we go down to the cookie event), we can split into chocolate chip, snickerdoodle, or peanut butter.0311

But if we went down the other event, our second event, we can split into vanilla or chocolate.0316

So, that gives us two possibilities here and three possibilities here; we count them all together, and we get 5,0323

when we combine all of the things that could happen, because we have branched off of the different events;0330

so we have to add up all of the possible outcomes.0334

Next, the multiplication principle: this is also called the fundamental counting principle, because it is just so very important to the way counting works.0337

Let there be two events, E1 and E2:0345

once again, E1 can occur in m ways, and E2 can occur in n ways.0348

If both events will occur--that is to say, Event 1 is definitely going to occur, and event 2 is definitely going to occur--0354

and we also have to have that they have no effect on each other (how Event 1 turns out0362

will have no effect on how Event 2 turns out--they are independent events;0367

what happens over here won't affect what happens over here), then the total possible ways0371

for something to occur is the number of ways that our first event can occur, times the number of ways that our second event can occur.0376

Let's see an example to help us understand why this is.0384

Going back to our previous example, once again, expanding on that, the idea of cookies and ice cream:0387

now, we are given a choice of one of the previously-mentioned cookies (that was m = 3 choices),0391

and a choice of the previously-mentioned ice cream scoops.0397

We are not just getting a cookie or a scoop of ice cream; we are getting both.0400

We are getting a cookie, and we are getting a scoop of ice cream.0404

So, we have three possibilities for the cookie choice, and then two possibilities for the scoop of ice cream.0407

That will give us a total of possible choices of 6; and let's see why that ends up working out.0414

We get 3 times 2 equals 6; that is m times n; let's see how this works out.0419

We start with our first event; our first event occurs, because we are going to have to get the cookie, and then we will get the ice cream.0424

We can do it the other way, and it would work out just as well.0430

But let's say cookie, then ice cream: so we choose our cookie choice--we have three possible ways for our cookie choice to turn out.0432

We can either have chocolate chip, snickerdoodle, or peanut butter.0438

But then, on each one of these choices, we have an expanded two more choices.0441

So, if we have chocolate chip, we now have vanilla or chocolate out of that; so that is 2 here.0445

If we took snickerdoodle, we now have vanilla or chocolate; so that is 2 here.0452

If we took peanut butter, we now have vanilla or chocolate; so that is 2 here.0457

So, because we have three possible starting branches, when the next two branches come off, we can just multiply the number for each event.0461

There are three possibilities that come out of our cookie choice, and then a further two possibilities on top of each of those.0468

So, it would be 3 times 2, for a total of 6 in the end.0474

All right, the next idea is the pigeonhole principle.0481

Let there be some event E that can occur in n ways; we have some event E, and it can happen in n ways.0485

Now, if that event happens n + 1 times--it happens one more than the number of ways that the event can turn out;0492

so it has n different possibilities for turning out, and then the event happens one more time0499

than the number of ways that it can turn out--we know that one of the ways that the event can occur must happen twice.0504

It can also happen more, but we know for sure that it will happen at least twice in one of the ways; and it might happen more.0510

But we are guaranteed at least twice in one of the ways.0516

Another way to look at this is with boxes and objects; I think this is a little bit easier to visualize.0520

Equivalently, given n boxes and n + 1 objects, if all of the objects are put into the boxes, then at least one box must have at least two objects.0526

Let's use some numbers here, so that we can see this a little better.0535

Let's say we have 3 boxes: 1, 2, 3 boxes; and we are given 4 objects: 1, 2, 3, 4.0537

If we try to fit these four objects into those three boxes, it has to be the case that, in one of those boxes, there will be at least two balls.0549

For example, if we try to keep these balls separated; so we put this ball in here first, and then we put this ball in here first,0558

and then we put this ball in here first, when we get to this ball here, it has to go into a box that has already been filled.0565

And so, whichever of the three boxes it ends up going into--it doesn't matter which one it ends up going into--0572

it is going to end up going into a box that already has another object in there.0578

And so, we will know that we had at least two objects.0582

We might have ended up putting all 4 balls in one box; that is for sure; but that would still say that at least one box has at least 2 objects.0586

And so, it will end up being true, however we decide to distribute these balls.0595

We can be guaranteed that at least one of the boxes has at least 2 objects.0599

There might be more boxes that have even more; there might be one box that has all of them.0603

But we know for certain that at least one of them has at least 2.0607

In many ways, it is just common sense; it seems a little surprising to see in a mathematical context,0610

because it just makes such perfect sense to us, intuitively.0615

But we can end up saying some pretty surprising things by using this idea, as we will see in Example 5.0618

A really quick example, just to see this applied: if we were given four cookies, and each of our cookies was,0623

once again, those three choices of chocolate chip, snickerdoodle, and peanut butter;0627

then we know, since we were given four cookies, and we only have three possibilities,0632

that we have to have at least two of one type of cookie.0637

We are going to have to have at least two chocolate chips, or at least two snickerdoodles, or at least two peanut butters.0641

We might end up having many in a different kind--we might have 4 chocolate chips or 4 snickerdoodles or 4 peanut butters,0647

or two chocolate chips and two snickerdoodles, and no peanut butters.0653

But we know for certain that there is at least one thing where we have a pairing on the cookie type.0655

All right, first: draw pictures--when working on counting problems, it can help immensely to draw pictures and diagrams.0661

Just being able to visualize this stuff can make it a lot easier to understand.0668

If you can visually follow the events and choices, if you can see what is going on--how things are branching,0671

it is going to make it that much easier to solve problems.0678

So, I really recommend drawing pictures.0680

Now, that is not to say that you need to draw accurate pictures.0682

You don't really want to draw accurate pictures.0685

Instead, what you want to do is draw pictures that are going to be pictures or diagrams0688

that just let you see how many ways an event can occur, and how it is connected to the other events,0692

so that you can see how there are things interrelated, or not interrelated, and how things can occur--0699

the number of ways, branching paths...all of those sorts of things.0705

Drawing every single possibility, drawing really careful pictures--that is not necessarily the thing.0708

You just want something so that you can visualize it and see it on paper.0712

We will see a lot of this in the examples; all of the examples will have some way of being able to visually understand what is going on.0715

All right, now we are ready for some examples.0720

The first example: we have 7 shirts, 3 pairs of pants, and 4 pairs of shoes.0722

What is the total number of outfits that you can wear from this selection?0727

The first thing to notice is: if we are going to wear an outfit, we are going to have to wear shirts, pants, and shoes,0730

because we are going to go outside with it; so we want to have definitely a shirt, definitely pants...0735

we are walking outside, so we definitely want shoes, as well.0739

So, we are going to have definitely a shirt, definitely pants, and definitely shoes.0742

Furthermore, our choice of shirt, our choice of pants, and our choice of shoes have nothing to do with each other.0744

They might not look good together, but it would be a possible output we could create.0750

Shirt is completely independent from pants, which is completely independent from shoes.0754

Each one is a choice completely in and of itself; they don't have an effect on each other.0759

So, we can think of it as three separate events.0763

First, we have the seven shirts; so we have the choices for shirts, to begin with--we have 7 possible things for shirts.0766

And then, next, we have three pairs of pants; so there are a total of...all of our pants choices,0774

our "pants event," is 3 different ways for our pants event to turn out.0782

And then finally, there are 4 pairs of shoes; there are 4 different ways for our shoes event to turn out.0787

We know that we have to end up having a shoes event turn out, and it can turn into four different possibilities.0793

So, we could think of this with branching: we dress ourselves, and so our first thing is that we branch into multiple different choices with the shirts.0800

And then, from here, we branch into multiple different choices with the pants.0808

And that is going to end up happening on each one of our blue things.0811

So then, if we end up following out any one of these, we are going to branch on this.0814

So, on every branch, it branches 7 times; and then, each of those branches 3 times, and then each of those branches 4 times.0818

That was our fundamental counting principle, the multiplication principle.0825

Since they are independent events--they don't have anything to do with each other--we just multiply the number of possibilities for each one of them.0828

So, 7 times 3 times 4: we multiply this all out together; we end up getting 84 possibilities.0834

So, we have a total of 84 different possible outfits.0845

Great; the next question: We are going out to dinner tonight--you are going out to dinner tonight (I am not coming with you).0850

Your options for the restaurant are Mexican, Japanese, or French.0856

At each restaurant, they have the following number of selections.0860

Mexican has 4 appetizers and 10 main courses; Japanese has 3 appetizers and 7 main courses; and French has 8 appetizers and 5 main courses.0863

If you will have one appetizer and one main course, how many ways are there for you to have dinner tonight?0872

The first thing that we have to realize is that, if we are going to go out to dinner, we can't go out to dinner at multiple restaurants at the same time.0878

If you go out to dinner, you have to choose one restaurant.0884

Our very first thing is that we actually have a restaurant choice; so restaurant breaks into 3 different possibilities.0886

We can either go to the Mexican, the Japanese, or the French.0895

If we go to Mexican, we will have that one here; and then, we could also have gone to Japanese;0899

and then finally, we could have gone to French.0911

And then, we were told that, when you are out at dinner, you are going to have one appetizer and one main course.0916

Now, the choice of appetizer has no effect on the choice of main course, necessarily.0925

You could choose any appetizer to go with any main course.0928

They might not fit together, but it doesn't matter; you could choose it, so they are independent events.0931

One of them does not necessarily change the way the other one can turn out.0935

For the Mexican choice, if you went with Mexican, you would have four choices for your appetizer.0939

And then, you would have 10 choices for your main course: 4 appetizers, and then 10 main courses.0946

The same thing for the Japanese, but different numbers: we have 3 choices (let me make this a little bit bigger)0953

for the appetizer, and then we have 7 choices for the main course.0962

Finally, the French restaurant: if you had gone there, you would have 8 choices for the appetizer and 5 choices for the main course.0967

So, if you had gone to Mexican, you would have 4 times 10 total possibilities at the Mexican restaurant.0976

If you had gone to Japanese, you would have 3 times 7, or 21, total possibilities if you had gone to the Japanese restaurant.0982

If you had gone to the French restaurant, you would have 8 times 5, or 40, possibilities if you had gone to the French restaurant.0989

But you don't have to go to any one of these, necessarily.0994

At the beginning, you have this branching choice: you have an exclusive choice,0998

where you can choose one of the restaurants, but you can't choose multiple of the restaurants.1002

So, you could go down Mexican and get to the Mexican set of choices.1006

You could go down Japanese and get to the Japanese set of choices.1009

Or you could go down French and get to the French set of choices.1011

This means that we have to add the number of choices at each restaurant all together to figure out what the total choices for the night are.1014

So, we end up having 40 from the Mexican, plus 21 from the Japanese, plus 40 from the French,1020

which comes out to be a total of 40 + 40 (80) + 21: 101 choices;1029

so you have 101 total choices for dinner tonight, total ways to have dinner,1035

because you could go to any of the three restaurants; and then, once you are at each of the restaurants,1041

you then have an independent choice between appetizer and main course.1045

So, at the end, we have to add them all back up together.1049

The next example: At a computer store, you are going to buy speakers, a monitor, and a computer.1052

There are four options for speakers, seven for monitors, and two for computers.1057

One of the computers gives a choice between one of two operating systems, while the other has one of three operating system options.1061

So, if you buy one of the computers, you can choose between one of two things on top of it.1068

And if you buy the other computer, you are allowed to choose one of three things on top of it.1072

How many total ways are there to make your purchase?1076

Now, of course, if you buy a computer, you have to have an operating system with it.1079

So, we are guaranteed that we are going to get some operating system.1082

With that in mind, how can we figure out how many ways to make a purchase?1085

Well, if we go in the order that this comes, we see that we have speakers; we buy speakers,1088

and we have...how many ways?...there are four options for speakers;1096

and then we have the monitors; we look at it--we have...1103

oh, oops, there are four options, not seven; I was accidentally reading monitors.1107

We have four for speakers, and then we have seven for monitors.1111

And then, the computers: you have two options for the computers.1114

But then, we have this whole thing about the OS, the operating system, of it.1119

Our operating system comes into this, as well: operating system branches, depending on which computer you had bought.1123

If you bought the first computer or the second computer, you are going to have different numbers of options here.1129

So, you could have 2 here or 3 here.1134

With that in mind, it gets a little confusing to figure out how we want to work this out.1137

Let's have our first thing be the computer choice; we will say computer choice determines everything else,1140

because speakers and monitors are completely independent of which computer you choose.1145

The operating system choice, though, does depend on which computer you chose.1149

So, we will start with choosing a computer choice.1153

We have our computer choice, which splits into one of two possibilities.1155

It splits into the two OS possibility, or it splits into the three OS possibility.1160

We have the operating system; over here, there are two possible operating systems.1168

Over here, though, there are three possible operating systems.1174

Next, we have speakers for both of them; there are going to be four choices for speaker and four choices for speaker.1178

And for monitor, we have seven choices for the monitor and seven choices for the monitor,1184

because that has no effect on which of the two computers we had initially bought.1189

Our computer--this number of choices, 2, doesn't really come into effect, because we already counted it by having it branch two different ways.1192

We have created that branching, and now we are reading both of them.1199

What we do is figure out how many options we have if we had bought the 2-operating-system computer.1201

How many options do we have if we had bought the 3-operating-system computer?1207

And then, we add those two together to figure out the total number that we have.1210

So, let's figure this out: if we have 2 times 4 times 7, we end up having 8 times 7, or 56, total options over here.1214

If we had gone with the 3-operating-system, then we have 3 times 4 times 7 possibilities, if we bought the 3-operating-system computer.1226

So, that comes out to 84 choices.1234

If we want to have the possibility of choosing between one of the two computers at first, well, that changes what happens later on.1237

So, that has to count as an either/or event; we can only choose one of the two.1243

We have the 2-operating-system possibility (56) and the 3-operating-system possibility (84).1251

If we want to be able to have an option between choosing which of those two paths we go down, we add them together.1256

So, we have the 56 options for the 2-operating-system, added to the 84 options for the 3-operating-system.1260

We add those together, and we get 140 total possibilities.1270

We have 140 total possible ways to make this purchase, when we include all of the different options that we have at our disposal.1274

The next example: On an exam, there are 25 multiple-choice questions, each one of them having 5 choices that you can answer with.1281

Now, assuming that we count not putting down any choice as an option for answering a question1289

(because it is, in a way, a choice that you have made--not putting anything down is a choice,1293

just as much as marking A or C or E)--if that is the case, we are going to assume that not putting down something1299

counts as an option for answering a question--with that in mind, how many ways total are there to mark the answer sheet?1305

There are 25 different times that we have the chance to answer.1311

For our first one, well, let's look at it as...we have 25 different slots, effectively,1316

each of our slots representing all of the choices that we have for answering.1323

This is going to keep going, going, going; there are a total of 25 slots that we are dealing with,1327

because each of the questions is a slot that gets answered.1336

OK, so with that in mind, we are going to say, "How many ways can we answer the first question?"1340

Well, the first question...we are allowed to count not putting down a choice; we count not putting a choice as an option.1345

So, if that is the case, and there are 5 choices (A, B, C, D, E) for each one of these,1354

and we count not putting down something as an additional option,1358

then that means that there is a total of 6 for one of these questions: A, B, C, D, E, nothing at all.1362

That is a total of 6 possibilities.1368

What about the next question--is the next question affected by the way that we answered the previous question?1370

Not at all; you can mark that answer any way you want.1375

Once again, we have A, B, C, D, E, nothing at all; so there are 6 choices on the next one, 6 choices on the next one, 6 choices, 6 choices.1378

There are 6 choices for each one of the 25 questions that we are answering,1386

6 times 6 times 6 times 6...because each one is an independent event,1390

so we multiply them all together to figure out how many possible things there are.1394

What we have is 6 times 6 times 6...going all the way out to more 6's.1398

We have a total of 25 6's multiplied together.1406

Well, on the bright side, we have a nice way for saying 6 multiplied by 6 25 times; that is the same thing as saying 625.1416

So, there are a total of 625 ways to answer this set of questions.1425

There are 625 ways to mark up this answer sheet.1432

How many does that come out to be?--it is a little hard to see just how large 625 is.1435

625, if we use a calculator to get an approximate value for that--that comes out to be 2.8x1019 possibilities,1439

which is an absolutely staggering number; that is a huge number.1450

If we were to count this out, it would have 19 0's: 1, 2...actually, it would have 18 zeroes,1454

because of that .8: so 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18; and then the 2.8x1019.1459

So, we would end up having this incredibly massive number of choices.1477

I can't even figure out...how many is that?1480

That is ones, thousands, millions, billions, trillions, quadrillions...28 quintillion choices for the ways that we could answer this, with only 25 questions.1482

So, that is a massive, massive number of possibilities here, which gives us an idea of just how fast things can grow.1494

This is why we don't want to have to count this stuff by hand.1500

It is because these numbers can get really big really fast in things that seem really simple; cool.1503

All right, the final example: Show that there are at least two people in London right now with the exact same number of hairs on their heads.1509

This will require a little bit of research.1516

This seems kind of like a weird thing at first; we see this use-the-pigeonhole principle after researching.1519

That is going to be a hint that we are going to want to use the pigeonhole principle.1524

A little bit before we get into this, let's consider the idea of if we were talking about birthdays.1527

That will help us understand what is about to happen in just a moment.1532

If we were talking about birthdays, if we had...let's say that there are only 365 days that you can be born on.1535

We are forgetting about leap years, because we are assuming that we slide that to either the day before or the day after.1543

There are 365 days that you can be born on.1548

That means, if we have 366 people in the same room, we are guaranteed that at least one pair of people there has the same birthday.1551

Now, there might be more birthdays that are concurrent--people that have the same birthday.1560

But we can be guaranteed that there is at least one pair, because if we have 365 effective slots1564

for a day that a birthday can be on, that is 365 boxes that a person can go into and represent "I am on that day."1570

Then, at least two people are going to end up having to be in that same box.1577

They are going to have to end up occupying the same birthday.1581

Otherwise, it just doesn't make sense; that is the pigeonhole principle in action,1584

this idea that if we are stuffing things into pigeonholes (being a little place for a message, a long time ago)--1589

if we stuffed ten messages into nine pigeonholes, there is going to have to be1595

some pigeonhole that has multiple messages in it--at least 2 messages.1601

So, if we have 366 people, we know that there are going to be at least 2 people who have the same birthday--that is the basic idea here.1605

Now, let's start thinking about this one: we are talking about hairs on heads, and we are talking about London.1613

First, let's look into hairs on heads: we look into how many hairs are on a human head.1619

A human head, assuming they have a reasonable full head of hair...a human head with full hair has around 100,000 hairs on it, about 100,000 hairs.1624

Now, of course, this number can be higher; this number can be lower; but it doesn't get much higher than 100,000.1639

200,000--pretty much no one is going to have 200,000 hairs.1645

So, it starts to limit out pretty quickly; around 100,000 hairs is about the normal amount for a human head.1648

And 150,000 would be a lot of hair; more than that just starts to become ridiculous and really hard.1656

We figured that out with just a little bit of Internet research--some cursory research.1661

We figured out pretty quickly that it looks to be about 100,000.1665

At that point...we didn't do really, really careful scientific study, so we want to make sure that we are giving some extra leeway here.1668

So, let's say, from this idea, that the absolute maximum number of hairs that can be on a human head is going to be...let's just say it is a million hairs.1674

There can't be more than a million hairs on a human head.1686

So, there have to be a million hairs or less on a human head.1690

There can be anywhere between 0 (you can't have less than 0 hairs on your head) up to a million.1696

And a million is actually a ridiculously large number; no one is going to actually have a million at all.1700

But let's set it up to a high point that is ridiculous, because if we can show that this is true,1705

even with this ridiculous amount of hair, we will have proved it for a smaller, more accurate thing--what the real world would actually have.1709

We can set a higher set.1715

The next idea: let's look at the population in London.1717

The population in London, as of the writing of this video is a little bit over 8 million.1721

It is around 8.1 million--a little more than that; so it is around 8 million; it is over 8 million.1734

Because of this, we now know, by the pigeonhole principle, that there are1739

at least two people in London who have the exact same number of hairs on their head.1742

Why? Well, let's look at it like this.1746

We can think of that absolute maximum of 1 million hairs as being a bunch of different boxes.1748

The number of hairs on a person's head...this goes all the way out to...we have 0 hairs at first, 1 hair on your head,1754

2 hairs on your head, 3 hairs on your head, 4 hairs on your head...then 999,999 hairs on your head, one million hairs on your head.1765

So, we have all of these different boxes that a person can go into.1774

We are counting all the way up from 0, up until 1 million hairs.1778

We know that, if it is a human, they are going to have to go in one of these boxes,1782

because the absolute maximum number of hairs (and this is ridiculous number) would be that a person has a million hairs on their head.1785

That is too large, but we know that the absolute maximum number of boxes1791

that we have to have here is from 0 up until a million, to mark out the number of hairs on a head.1794

Now, we know that the population of London is over 8 million.1799

That means, when we go through the first one million and one people, then unless they already end up1803

with two people having the same number of hairs on their heads, in the first million and one people,1813

then we are going to go all the way up from 0 to a million with each person having a unique number of hairs on their head.1817

But as soon as we get to the million-and-second person, they are going to end up having to go into the same box as someone else.1823

Someone else will end up having to have that same box with somebody else.1829

They will have to share; so we have the exact same number of hairs guaranteed.1833

We know that, right this moment, in London, there are somewhere in there 2 people who have the exact same number of hairs on their head.1837

It is pretty wild; in fact, because it is over 8 million, we could show that there are, in fact,1845

at least 8 people who have the exact same number of hairs on their head, right this instant--kind of surprising.1850

I won't prove why that has to be the case; but if you think about this for a little while, you will probably be able to figure it out on your own.1857

It is pretty cool; now, of course, there is no practical application to this immediately.1863

We couldn't actually find these two people with any ease.1867

But it is interesting to know that there is this guarantee that, because there are so many people in London,1870

and there are just not that many hairs on the human head--there are more people in London1875

than there can be hairs on the human head--it must end up being that two people have the same number of hairs on their heads.1879

That is one way to apply the pigeonhole principle that allows us to show these really kind of surprising, strange results with not that much difficulty.1887

All right, cool; we will see you at Educator.com later--goodbye!1893

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