INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Imaginary and Complex Numbers

Dr. Carleen Eaton

Dr. Carleen Eaton

Imaginary and Complex Numbers

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
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
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (32)

1 answer

Last reply by: Dr Carleen Eaton
Fri Mar 8, 2019 1:46 PM

Post by Kenneth Geller on January 30, 2019

Is not  (A + B) times  (A-B) = to A squared - B squared not as you noted A sq+ B sq

1 answer

Last reply by: Dr Carleen Eaton
Sun Oct 7, 2018 2:59 PM

Post by Haiyan Zhang on September 22, 2018

In 26:20, you should have put a parentheses around -2, otherwise the answer would be -4 not 4.

1 answer

Last reply by: Dr Carleen Eaton
Sun Aug 12, 2018 9:35 PM

Post by Jerry Xu on July 31, 2018

i is not defined as sqrt(-1). It is defined as i^2 = -1, there is a small difference

1 answer

Last reply by: Dr Carleen Eaton
Sat Nov 1, 2014 4:14 PM

Post by Larry Oldham III on October 21, 2014

How would you solve i^2 + i^3

1 answer

Last reply by: Dr Carleen Eaton
Fri Oct 10, 2014 12:10 AM

Post by Sangeeta Chaudhari on September 27, 2014

Hello Dr. Carleen! I loved your lecture but I don't understand the absolute value part in example 1:Simplify Complex numbers. Can you please explain that because I watch your other videos and I just don't know when to use absolute values. Thanks!

1 answer

Last reply by: Dr Carleen Eaton
Thu Jul 31, 2014 6:52 PM

Post by Prashanti Kodali on July 16, 2014

Dr. Eaton I was doing practice problems and I am unsure if I got this question correct. The question was 3i(2i+4i-6). The answer I got was -12-24i. Can you tell me if I got this question right? Thank you so much!

1 answer

Last reply by: Dr Carleen Eaton
Sun Oct 20, 2013 11:26 AM

Post by dayan assaf on October 19, 2013

so if the question was 1+i, would the imaginary part be 0 or 1?

2 answers

Last reply by: Joshua Chen
Thu Jun 18, 2020 10:22 AM

Post by Juan Herrera on September 19, 2013

Difference of two squares:
a^2-b^2 = (a+b)(a-b)

1 answer

Last reply by: Dr Carleen Eaton
Sat Sep 14, 2013 2:54 PM

Post by Chateau Siqueira on September 3, 2013

Where can a find a Lecture about " Difference Quotient" ? Thanks

1 answer

Last reply by: Dr Carleen Eaton
Sat Sep 14, 2013 2:52 PM

Post by Tami Cummins on August 13, 2013

Dr. Eaton in the last example problem when using the short cut and squaring the "b" term which was 2 you didn't include the i's as part of the b term but in the previous sample division problem you did.  In the previous sample problem could you say that i had a coefficient of 1 and squared that instead just for consistency?  

1 answer

Last reply by: Dr Carleen Eaton
Sat Jul 27, 2013 10:11 AM

Post by Aleksander Rinaldo on July 10, 2013

Under divison, you used  (a+bi)(a-bi)=a^2+bi^2....... Why would it not be a^2-bi^2....

0 answers

Post by Victor Castillo on January 24, 2013

Who decided we need imaginary numbers Peter Pan?

0 answers

Post by Victor Castillo on January 24, 2013

Whaaaaat?

2 answers

Last reply by: Norman Cervantes
Mon Apr 29, 2013 12:24 PM

Post by Daniel Cuellar on October 17, 2012

why do you not simplify your answer at the end by dividing everything by 2? just as you would a normal fraction??? please explain.

2 answers

Last reply by: Dr Carleen Eaton
Tue Jul 3, 2012 7:20 PM

Post by David Burgoon on June 27, 2012

Is it necessary to factor out the i in the equations? It looks like you could just add/subtract them. If you could provide an example that supports why it is important, that would help.

Imaginary and Complex Numbers

  • Imaginary numbers are the square roots of negative numbers.
  • All the properties of square roots extend to radicands containing complex numbers.
  • The complex numbers satisfy the commutative and associative properties for addition and multiplication.
  • To divide one complex number by another one, write the division as a fraction. Then multiply numerator and denominator of this fraction by the complex conjugate of the denominator.

Imaginary and Complex Numbers

Simplify √{ − 50n2m4}
  • Use the Product Property of Square Roots to factor out perfect squares
  • √{ − 50n2m4} = √{ − 1*25*2*n2*m4} = √{ − 1} *√{25} *√2 *√{n2} *√{m4}
  • Simplify and look out for the Principle Square Roots.
  • √{ − 1} *√{25} *√2 *√{n2} *√{m4} = i*5*√2 *|n|*m2
i*5*√2 *|n|*m2
Simplify (4 − 5i) + (10 − 2i) − ( − 4 − 3i)
  • Distribute the negative
  • (4 − 5i) + (10 − 2i) − ( − 4 − 3i) = (4 − 5i) + (10 − 2i) + (4 + 3i)
  • Add the Real parts and add the Imaginary Parts together
  • (4 + 10 + 4) + ( − 5i + − 2i + 3i) = 18 − 4i
18 − 4i
Simplify (4 − 5i) − (1 − 3i) − ( − 2 − 5i)
  • Distribute the negative sign
  • (4 − 5i) − (1 − 3i) − ( − 2 − 5i) = (4 − 5i) + ( − 1 + 3i) + (2 + 5i)
  • Add the Real parts and add the Imaginary Parts together
  • (4 − 1 + 2) + ( − 5i + 3i + 5i) = 5 + 3i
5 + 3i
Multiply (3 − 2i)(3 + 5i)
  • Multiply using FOIL - First Outter Inner Last
  • 3*3 + 3*5i + ( − 2i)(3) + ( − 2i)(5i)
  • Simplify
  • 9 + 15i − 6i − 10i2 = 9 + 9i − 10i2
  • Recall that by definition i2 = − 1, subsitute i squared
  • 9 + 9i − 10i2 = 9 + 9i − 10( − 1) = 9 + 10 + 9i =
19 + 9i
Multiply (3 + 4i)(4 − 3i)
  • Multiply using FOIL - First Outter Inner Last
  • 3*4 + 3*( − 3i) + 4i*4 + (4i)( − 3i)
  • Simplify
  • 12 − 9i + 16i − 12i2 = 12 + 7i − 12i2
  • Recall that by definition i2 = − 1, subsitute i squared
  • 12 + 7i − 12i2 = 12 + 7i − 12( − 1) = 12 + 12 + 7i =
24 + 7i
On a Complex Plane, plot the following complex numbers a) 5i b) − 2i c)2 + 3id)5 − 3ie)4
  • Recall that complex plane is divided between the Imaginary Part ( y - axis) and Real Part(x - axis).
  • Plot each point on the corresponding location.
On a Complex Plane, plot the following complex numbers a) − i b) − 2 + 2i c) − 5 − 3id) − 1 + 0ie) − 3 − 2i
  • Recall that complex plane is divided between the Imaginary Part ( y - axis) and Real Part(x - axis).
  • Plot each point on the corresponding location.
Simplify [( − 9 − 9i)/3i]
  • This is a Complex Number division problem. When dividing complex numbers your goal is to
  • leave your answer in standard form (a + bi) where a is the real part and b your imaginary part.
  • This is a special case in which the divisor is not a binomial. What you do in this case is mutiply
  • numerator and denominator by the divisor. Remember that i2 = − 1
  • [( − 9 − 9i)/3i]*[3i/3i]
  • Distribute
  • [( − 9 − 9i)/3i]*[3i/3i] = [( − 27i − 27i2)/(9i2)]
  • Subsittue i2 = − 1
  • [( − 27i − 27( − 1))/(9( − 1))] = [(27 − 27i)/( − 9)]
  • Write in standard form
− 3 + 3i
Simplify [(4 + 2i)/(3 + 4i)]
  • This is a Complex Number division problem. When dividing complex numbers your goal is to
  • leave your answer in standard form (a + bi) where a is the real part and b your imaginary part.
  • This is the case in which the divisor is a binomial. What you do in this case is mutiply
  • numerator and denominator by the complex conjugate of the divisor. Remember that i2 = − 1
  • Multiply by complex conjugate of divisor
  • [(4 + 2i)/(3 + 4i)]*[(3 − 4i)/(3 − 4i)]
  • Recall that when multiplying a complex number by its conjugate you can use the short - cut a2 + b2
  • [(4 + 2i)/(3 + 4i)]*[(3 − 4i)/(3 − 4i)] = [((4 + 2i)(3 − 4i))/(32 + 42)] = [((4 + 2i)(3 − 4i))/25]
  • Multiply in the numerator
  • [((4 + 2i)(3 − 4i))/25] = [(12 − 16i + 6i − 8i2)/25] = [(12 − 10i − 8i2)/25]
  • Eliminate any i2
  • [(12 − 10i − 8i2)/25] = [(12 − 10i − 8( − 1))/25] = [(12 + 8 − 10i)/25] = [(20 − 10i)/25]
  • write into Standard form a + bi
  • [(20 − 10i)/25] = [20/25] − [10i/25] = [4/5] − [2/5]i
[4/5] − [2/5]i
Simplify [( − 4 + 4i)/(3 + 2i)]
  • This is a Complex Number division problem. When dividing complex numbers your goal is to
  • leave your answer in standard form (a + bi) where a is the real part and b your imaginary part.
  • This is the case in which the divisor is a binomial. What you do in this case is mutiply
  • numerator and denominator by the complex conjugate of the divisor. Remember that i2 = − 1
  • Multiply by the complex conjugate of the divisor
  • [( − 4 + 4i)/(3 + 2i)]*[(3 − 2i)/(3 − 2i)]
  • Recall that when multiplying a complex number by its conjugate you can use the short - cut a2 + b2
  • [( − 4 + 4i)/(3 + 2i)]*[(3 − 2i)/(3 − 2i)] = [(( − 4 + 4i)(3 − 2i))/(32 + 22)] = [(( − 4 + 4i)(3 − 2i))/13]
  • Multiply in the numerator
  • [(( − 4 + 4i)(3 − 2i))/13] = [( − 12 + 8i + 12i − 8i2)/13] = [( − 12 + 20i − 8i2)/13]
  • Eliminate any i2
  • [( − 12 + 20i − 8i2)/13] = [( − 12 + 20i − 8( − 1))/13] = [( − 12 + 8 + 20i)/13] = [( − 4 + 20i)/13]
  • Write into Standard form a + bi
  • [( − 4 + 20i)/13] = − [4/13] + [20i/13]
− [4/13] + [20i/13]

*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

Imaginary and Complex Numbers

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

  • Intro 0:00
  • Properties of Square Roots 0:10
    • Product Property
    • Example: Product Property
    • Quotient Property
    • Example: Quotient Property
  • Imaginary Numbers 3:12
    • Imaginary i
    • Examples: Imaginary Number
  • Complex Numbers 7:23
    • Real Part and Imaginary Part
    • Examples: Complex Numbers
  • Equality 9:37
    • Example: Equal Complex Numbers
  • Addition and Subtraction 10:12
    • Examples: Adding Complex Numbers
  • Complex Plane 13:32
    • Horizontal Axis (Real)
    • Vertical Axis (Imaginary)
    • Example: Labeling
  • Multiplication 15:57
    • Example: FOIL Method
  • Division 18:37
    • Complex Conjugates
    • Conjugate Pairs
    • Example: Dividing Complex Numbers
  • Example 1: Simplify Complex Number 24:50
  • Example 2: Simplify Complex Number 27:56
  • Example 3: Multiply Complex Numbers 29:27
  • Example 3: Dividing Complex Numbers 31:48

Transcription: Imaginary and Complex Numbers

Welcome to Educator.com.0000

Today, I will be introducing the concept of imaginary and complex numbers.0002

The concept of imaginary numbers is tied in with square roots; so we are going to begin by a review of some of the properties of square roots.0010

Recall the product property: the product property and the quotient property--both of these properties can be used to simplify square roots.0021

For numbers a and b that are greater than 0, if you have something like this, √ab, that can be broken down into √a, times √b.0039

And this allows us to simplify; for example, if you have something like √32x5,0054

it is not in simplest form, because there are still some perfect squares here under the radical.0062

So, the way I can simplify this is to rewrite this as the perfect squares times the other factors.0068

Let me rewrite this as the perfect square, which is 16, times 2; so the perfect square of 16 times the other factor, 2.0076

And then, for the variable, I have the perfect square, x4, times x.0085

Now, what this product property allows me to do is then separate out everything like this.0093

Once I have that, I can take the square roots of the perfect squares: I have the square root of 16 (is 4); the square root of x4 (is x2).0108

And then, I am left with 2x, the square root of 2 times the square root of x.0118

And by going the other way with the product property, I can put those back together and say what I end up with is 4x2 times √2x.0126

And this is now in simplest form.0134

The quotient property is the same idea, but with division.0137

If you have something such as √a/b, you can break that down into √a/√b.0147

So again, if I have something such as, let's see, √49x4/25, I know that this is equal to √49x4/√25.0155

This allows me to simplify these into 7x2/5.0178

So, we are going to be using these properties of square roots in today's (and in future) lessons.0185

OK, a new concept--imaginary numbers: in Algebra I, when we came across something such as √-4, we simply said that it wasn't defined.0192

And it is not, if you are looking only at the real number system.0206

However, there is another number system called the complex number system; and part of that includes imaginary numbers.0209

And what these do is: this allows us to find a solution to something like this. What is this?0216

Before we go on, let's look up here: what an imaginary number says is that i equals the square root of -1.0230

Looking at that another way: if i equals the square root of negative 1, let's say I square both of these: i2 is -1.0238

And this is a concept that is going to become important later on, and we are going to use this.0249

So, just recall that i2 equals -1, and that i equals √-1; both of these are very important.0254

So, let's think about an equation such as this: x2 = -4 (related to this).0262

We can find a solution to this by saying, "OK, x equals the square root of -4"; previously, we couldn't.0275

And the reason is: I can break this out into the following: x = √-1(4).0286

Using the product property that we just learned, I can say, "OK, x then equals the square root of -1 times the square root of 4."0298

This part is easy to handle: I know that the square root of 4 is 2.0313

Now, I have a way of handling this: I go back, and I say, "OK, the square root of -1 is i; I have something to define this as."0317

So, since the square root of -1 is i, I can simply say that this is i, or x equals 2i.0326

Whereas before I couldn't solve this, now I can, because I can pull out this -1 using the product property, and define that as the imaginary number i.0335

So, that is what this is saying up here.0348

For the positive real number b, the square root of -b2 equals bi.0351

For the positive real number in this case, it would be 2: b = 2; the square root of -22 is bi; and that gives me this.0364

Looking at this using another example: let's say that I am asked to find x = √-9.0383

Again, using the product property, I know that this is equal to -1 times 9.0393

And the product property allows me to break this up as such.0403

This is easy; I know that the square root of 9 is 3; now I have a way to define this as i; √-1 is i,0408

so I am going to say this is i3, which is usually written as 3i.0417

OK, so when you are in a situation where you need to find a negative square root,0425

the thing is to use the product property to pull this square root of -1 out, and then you just need to find the positive square root.0429

And the answer is the imaginary number bi.0436

Now, I mentioned that imaginary numbers are part of a different number system: we have the real number system,0444

and we also have another number system called the complex number system.0449

And a complex number is in the form a + bi; and there are two parts to this--a real part and an imaginary part.0453

So, this is the real part; bi is the imaginary part; and they are complex--they have two parts.0464

For example, I could have 5 + 4i; this is a real number; this is an imaginary number; together it is a complex number.0475

Or 8 - 3i: again, the real number is 8; the imaginary is -3i.0485

Now, looking a little deeper, let's say I have b = 0; then, what I am going to end up with is a + 0i, which is just a.0494

So, this is just a real number; so the real numbers are part of the complex number system.0509

For example, if I were to give you 8 + 0i, well, this is going to drop out; and this is just 8, and it is a real number.0515

So, when b equals 0, you just end up with a real number.0526

Conversely, let a equal 0; then you are going to get 0 + bi.0529

This is just a pure imaginary number; this is part of the complex number system, as well--just a pure imaginary number,0535

such as (if a is 0) 0 + 4i; 0 drops out, and I just have 4i; and this is just the imaginary part, the imaginary number.0544

You can have complex numbers: they have two parts, a real part and an imaginary part.0559

If, in the imaginary part, b equals 0, that leaves you with a real number; if a equals 0, it leaves you with a purely imaginary number.0564

Or you can have both parts and have a complex number.0572

OK, equality: this is pretty straightforward--this just says that, if you have a complex number a + bi,0577

it equals c + di if and only if the real parts are equal and the imaginary parts are equal.0583

For example, 4 + 7i equals 4 + 7i, because the real parts are equal (4 = 4, so a = a) and the imaginary parts are equal (b = b; 7 = 7)--straightforward.0593

Addition and subtraction: in order to add or subtract complex numbers, combine the real parts and the imaginary parts by addition or subtraction.0613

To illustrate this: if you are asked to add 6 + 2i plus 3 + 4i, well, we are told to combine the real parts;0625

so I have the real part 6, and I am going to combine that with the real part 3,0637

because this is in the form a + bi, where a is real, and then bi is my imaginary part.0643

I combine my reals; now the imaginary parts--I have 2i, and I am going to combine that with 4i,0653

treating the i like a variable; what you can do is factor it out.0663

So, let's pull that i out to give me 2 + 4.0667

OK, so this leaves me with 6 + 3 (is 9), plus i times 2 + 4 (is 6), but conventionally, we write it with the number first, and then the i.0672

So, this is 9 + 6i, just combining the real parts and the imaginary parts.0686

Now, working with subtraction: 5 + 2i minus 4 + 6i; OK, it is often simpler to just get rid of this negative sign0692

and rewrite this as adding the opposite, like we have done previously with just the real number system.0709

This is plus -4, minus 6i; actually, this should be...no, that is correct.0715

OK, 5 + 2i minus 4 + 6i: I am rewriting this as 5 + 2i plus -4 - 6i.0728

Now, I need to combine the real parts; so the real part I have here is 5 + -4.0740

OK, combining the imaginary parts: I have 2i - 6i; this gives me 5 - 4, plus (I want to factor out the i) i times (2 - 6).0748

Now, 5 minus 4 is going to give me 1; plus i--and here I have 2 - 6, so that is -4.0772

I am rewriting this as 1 - 4i.0785

Again, all I did is changed the signs so that I could just add the opposite of each, plus -4, plus -6i.0788

Then, I combined the real parts, which were 5 and -4, to give me 1, and the imaginary parts, which were 2i and -6i, to give me -4i: 1 - 4i.0797

OK, the complex plane is like the coordinate plane that we have worked with before, but with some important differences.0810

So, before, we worked with the coordinate plane, and of course, we had a horizontal and vertical axis.0819

And we had positive and negative numbers on it.0824

Well, here the horizontal axis represents the real part of a complex number; so this is the real part, or the real numbers.0826

On this vertical axis, we have the imaginary numbers--the imaginary part of the complex number.0838

For example, this could be labeled 1, 2, 3, 4, 5, and on--pretty familiar.0850

What is different here is the vertical axis: here I am going to have i, 2i, 3i, 4i, and the same on down...-i, -2i, -3i, -4i.0862

OK, so thinking about graphing complex numbers on a coordinate plane: 2 + 3i, for example:0879

well, the 2 is the real part, so that is going to give me my horizontal coordinate right here, 2.0887

Now, the vertical coordinate is 3i; so I have 2 here and 1, 2, 3i up here; so this is 2 + 3i.0896

Now, imagine I have something that is just a real number, like 5 + 0i; this is going to drop out, so it is just 5.0907

Therefore, it is going to be right on the x-axis; I am going to have 5, and then for the vertical plane it is just 0; so, this is 5 + 0i.0916

I may also have a pure imaginary number: maybe I have something like 0 + 2i, so there is no real part to it.0924

The real part is just going to be 0; the imaginary part is going to be right here at 2i.0932

OK, so to graph a complex number, you find the real part on this horizontal axis, and the imaginary part on the vertical axis.0938

Pure real numbers go on the horizontal; pure imaginary numbers go on the vertical axis.0948

To multiply complex numbers, you treat them just like any two binomials.0957

For example, if you are asked to multiply 3 + 4i and 2 - 5i, I am going to use FOIL;0962

and just treat the i's like variables, just as you have in the past, as far as multiplication goes.0970

Now, I multiply out 3 times 2 (First); then my Outers (and that is 3 times -5i); and then the Inners (+ 4i times 2), and then Last (that is 4i times -5i).0977

Now, in a minute, we are going to get into some differences.1007

But in these first steps, really, you are just treating it like binomials.1009

When you go to simplify, there are differences, though.1012

But for the multiplication part, it is familiar territory.1015

OK, so working this out: 3 times 2 is 6; 3 times -5i is -15i; plus 4i times 2--that is plus 8i;1018

and then, I have 4i times -5i; so this is going to give me -20, and i times i is going to be i2.1029

Simplify just as you always have: combine like terms.1040

I have a 6, and then I have a -15i; and I can combine that with 8i to get -7i.1044

OK, -20i2: now, you might think you are done, but you are actually not--1051

recall from before that i equals the square root of -1.1056

Well, if I square both sides, I mentioned that you would get i2 = -1.1062

This helps us to simplify with multiplication, because, since i2 equals -1,1068

I can say 6 - 7i - 20, and I am going to substitute in -1.1074

So, I get 6 - 7i; and a negative and a negative is a positive; now, I can simplify...-7i + 26.1082

So again, I proceeded with my multiplication, just as I would any two binomials.1095

The difference came when I went to simplify, because i2 is -1;1099

so that actually allowed me to further simplify, because it got rid of that imaginary number.1104

I still have an imaginary number here, though, so my result is going to be a complex number, -7i + 26.1109

OK, division is a little bit more complicated, but it calls upon some familiar concepts from before.1117

With imaginary numbers, we can have what are called complex conjugates.1125

Before I go into this, recall the idea of conjugates when we talked about radicals.1130

Remember that we said something like √3 + √x has the conjugate √3 - √x.1135

And you might recall that we used these conjugate pairs when we were dealing with situations where we had radicals in the denominators.1147

We used conjugate pairs, and we would multiply the numerator and the denominator by its conjugate to get rid of radicals in the denominator.1157

Here, what I want to do is: now, I want to get rid of complex numbers in the denominator, so that I can divide.1167

So, I need to think about conjugates for complex numbers.1180

And with complex numbers, it is the same idea: you just reverse the sign before the second term.1184

So, if I had a + bi, its conjugate is going to be a - bi.1190

For example, if I have 4 + 5i, its conjugate is 4 - 5i.1200

To divide complex numbers, multiply both the divisor and the dividend by the conjugate of the divisor.1207

In other words, if I have 1 + 2i, 3 - i, here is my dividend; now, what is my conjugate?1215

I have 3 - i; the conjugate is going to be 3 + i; so I need to multiply both the divisor and the dividend by 3 + i.1227

OK, so I am going to multiply this by 3 + i and this by 3 + i.1242

Using my techniques for multiplying two binomials, I am going to get 3, and then Outer terms--that is i;1252

the Inner terms--that is going to give me 6i; and then my Last terms: 2i2--using FOIL, just like we always have.1267

OK, now in the denominator: 3 times 3 is 9; Outer terms--that is positive 3i; Inner terms: -3i; Last: -i2.1277

So, this gives me (simplifying) 3; i + 6i is 7i; plus 2i2; over 9; 3i - 3i...that drops out; minus i2.1299

Now, just recall for a second the important concept that i2 equals -1.1316

Coming down here, this is going to give me 3 + 7i + 2; and I am going to substitute in -1 here and -1 there; that is going to give me -1.1326

In the denominator: 9 minus -1 squared; this is going to give me 3 + 7i - 2, over 9 minus -12, which is 1.1339

Correction: this is not squared--this is simply -1: 9 minus -1, so a negative and a negative is going to give me a positive.1365

Again, i squared is equal to -1, so this entire term would just be -1.1378

Simplifying 3 - 2 gives me 1 + 7i, over 9 + 1 (is 10).1385

So, you see what happened: by multiplying both the numerator and the denominator by the complex conjugate,1394

I was able to eliminate the complex number in the denominator.1402

Now, just to show you a little bit of a shortcut: when you multiply a complex number by its conjugate, you get a2 + b2.1409

So, if I multiply a + bi times a - bi, I am actually going to get a2 + b2.1420

And that would have allowed me to save a lot of work and a possible mistake down here, because the more work, the more chance of a mistake.1430

If you look at it this way, if I have 3 - i here, a equals 3, and b equals -1.1436

So, I could just say that what I am going to end up with, if I multiply 3 - i times 3 + i (the complex conjugates)1447

is a2, which is 32, plus -12, or 9 + 1, which equals 10.1453

And that is a good shortcut to use: I multiplied it out just to show you how this term drops out, and this term--1461

you get rid of the imaginary number, and you just end up with the real number down here.1467

But it is really a good idea to use shortcuts when you can; it will save you time and mistakes.1472

So again, to divide, you multiply both the divisor and the dividend by the conjugate of the divisor.1477

OK, in our first example, we are going to use some of the concepts of properties of square roots, and also of complex numbers.1489

Using the product property, I can rewrite this so that I can factor out the perfect squares and deal with the imaginary number.1498

I have a -1, times 36, times 2; so this is factoring out this -72, so that I have factored out the negative part, and I have factored out the perfect square.1508

x2 and y4 are also perfect squares.1523

The product property tells me that this is equal to this.1527

This allows me to simplify: recall that i equals the square root of -1, so instead of writing this, I am going to write it as i.1537

The square root of 36 is 6; I can't simplify √2 any further.1546

Now, let's look at x2; be careful with this, because what they are asking for is the principal, or positive, square root.1552

To make this more concrete--how you have to handle this--let's think about if I was told that x2 equals 4.1562

If I were to take the square root, well, the square roots of that are +2 and -2.1569

And the reason is because -2 squared equals 4, and 2 squared equals 4.1577

But when I use the radical sign here, what I really want--I am saying I want the principal, or positive, square root.1584

So, in order to ensure that I am expressing that, I need to use absolute value bars.1590

If I wanted the square root that is a principal square root, I could say it is the absolute value of x.1596

And since x equals +2 or -2, the absolute value of x here would just be 2.1601

OK, now y4, actually...the square root of that is y2, and I don't need absolute value bars.1606

And let's think about why: I don't know what y is; let's say y stands for -3.1614

Well, when I take y2, I would get a positive number.1621

So, it doesn't matter if y is negative; it doesn't matter if y is positive, because y2 will always be positive.1628

Since y2 is always positive, I don't need to specify absolute value,1639

whereas x could be negative, so I do need to specify an absolute value.1644

OK, I am just rewriting this as 6i√2|x|y2.1650

Simplifying this using properties of square roots and the properties of imaginary numbers...knowing that i is √-1 allowed me to simplify this.1659

OK, Example 2 involves addition and subtraction of imaginary numbers.1674

Simplifying: the first step, to keep my signs straight, is going to be to change this to addition, thus pushing this negative sign inside the parentheses.1682

I am going to take the opposite of -3, which is 3, and the opposite of -4i, which is positive 4i.1694

Now, remember that, to add complex numbers, you add the real parts to each other, and the imaginary parts.1701

So, let's look at what I have for real parts: I have 4; I have 7; and I have 3.1710

OK, I am adding the real parts; I am combining those; and I am combining the imaginary parts.1717

Here, I have -3i; I have 2i; and I have 4i.1722

Let's factor out the i, so all I have to do is add these real numbers in here.1733

7, 4, and 3 is simply 14; plus i, times -3, 2, and 4; so that is 6 minus 3, which is 3.1741

I am rewriting this as 14 + 3i; again, working with complex numbers, adding and subtracting,1751

you add the real parts to each other and the imaginary parts to each other.1762

In this example, we are multiplying some complex numbers; and you handle these just as you handle any other binomials, using FOIL.1769

Multiply out the First terms: that is 4 times 3; the Outer terms: 4 times 6i; the Inner terms--1778

this is going to give me + -5i, times 3; and then the Last terms: -5i times 6i.1787

Finish our multiplication, and then simplify: 4 times 3--that is 12, plus 24i; -5i times 3 is -15i; -5i times 6i is going to give me -30i2.1802

Simplify a bit more to get 12; I can combine these two imaginary numbers; 24i - 15i is positive 9i.1824

I can take it one step further with the simplification.1834

Recall that i2 equals -1; so, since i2 = -1, and I have i2 right here, I can substitute -1 right here.1837

12 + 9i - 30(-1) gives me 12 + 9i...a negative and a negative is a positive, so that gives me + 30.1859

Now, I can combine these two: 30 + 12 gives me 42 + 9i.1877

Multiply these out just like any two binomials.1885

Then, I got to this point; I combined like terms; and right here, I stopped and realized that i2 is equal to -1.1888

So, I substituted that here, which turned this into a real number that I added to 12; and this is my answer.1898

OK, simplify: now we are working with division.1909

Remember that, in order to divide, what I need is to multiply the divisor and the dividend by the conjugate of the divisor.1912

I am looking here at 4 + 2i; its complex conjugate is 4 - 2i.1932

I need to multiply this numerator and denominator by 4 - 2i.1941

OK, in the numerator, I am going to go ahead and do my FOIL.1953

First gives me 2(4), which is 8; Outer terms--this is 2(-2i)--that is -4i.1958

Inner terms: -3i(4) is -12i; Last terms: -3i(-2i) is + 6i2.1971

Now, in the denominator, I could use FOIL and multiply it out; or I could remember that, if I multiply complex conjugates,1984

a + bi times a - bi, what I am going to end up with is a2 + b2.1994

Now, looking at 4 + 2i here, a equals 4 and b equals 2; so let me just take that shortcut2003

and say that I then have a2 (which is 4, so 42), plus b2 (which is 22).2012

OK, now, simplifying the numerator a bit further gives me 8; -4i - 12i is -16i; plus 6i2.2023

In the denominator, 42 is 16, and 22 is 4.2036

OK, recall that i2 is -1; so I am going to substitute -1 here.2043

In the denominator, I just have 16 + 4 is 20; this gives me 8 - 16i; this is -6 over 20;2059

I can simplify a bit more, because 8 - 6 is 2; this is 2 - 16i, over 20--that is my solution.2074

OK, so in order to simplify this, I took the conjugate of the denominator (which is 4 - 2i),2083

and I multiplied both the divisor and the dividend by this complex conjugate.2089

In the denominator, it was easy, because I just said, "OK, multiplying these conjugates gives me a2 + b2."2098

So, that is 42 is 16, and 22 is 4, to get 20.2107

In the denominator, I used FOIL; I multiplied these out, just as I normally would, to get this.2112

I combined like terms to get 8 - 16i + 6i2; and then, I said, "OK, i2 is -1,"2121

allowing me to simplify this into -6 and combining 8 - 6 to get 2 - 16i over 20.2128

That concludes this session of Educator.com introducing complex numbers and imaginary numbers.2138

I will see you next time!2145

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