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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Operations on Numbers

Slide Duration:

Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

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

19m 26s

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

12m 6s

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

18m 52s

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

12m 22s

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

28m 52s

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

12m 2s

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

28m 41s

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

24m 26s

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

22m 55s

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

27m 58s

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

20m 36s

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

21m 24s

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

16m 12s

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

25m 54s

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

20m 1s

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

19m 40s

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

24m 34s

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

17m 13s

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

16m 13s

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

14m 12s

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

17m 7s

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

15m 33s

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

21m 13s

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

44m 51s

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

18m 33s

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

25m 7s

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

44m 56s

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

28m 27s

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

21m 44s

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

30m 9s

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

30m 14s

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

23m 38s

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

29m 27s

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

16m 47s

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

29m 4s

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

26m 53s

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

21m 42s

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

26m 41s

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

20m 24s

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

18m 23s

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

16m 24s

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

18m 54s

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

20m 20s

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

27m 4s

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

14m 32s

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

15m 12s

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

17m 22s

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

19m 24s

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

15m 5s

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

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
Adding and Subtracting Complex Numbers
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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Lecture Comments (11)

1 answer

Last reply by: Professor Eric Smith
Sun Dec 3, 2017 5:52 PM

Post by Genevieve Carisse on October 19, 2017

You're an amazing Prof! Thank you for your lectures!

1 answer

Last reply by: Professor Eric Smith
Wed Mar 18, 2015 10:28 PM

Post by antonio cooper on March 18, 2015

I was wondering since I see some other math prof. create quizzes I was wondering if you were planning on doing this as well? Thank you very much for what you do.

1 answer

Last reply by: Professor Eric Smith
Mon Jun 16, 2014 3:21 PM

Post by C. Barnes on June 13, 2014

WHOA!! in example number 1 where you have a positive 8 -(-13) wouldn't that result in a -5 instead of a positive 21? I believe that the sign follows the larger number when there is a positive and negative combination. Ie, 2 -(-3)= - 1. If I am incorrect, please clarify>

1 answer

Last reply by: Professor Eric Smith
Thu Oct 10, 2013 2:02 PM

Post by Ezuma Ngwu on October 4, 2013

Is there any easy way to calculate exponents without a calculator?

2 answers

Last reply by: james templeton
Mon Apr 27, 2015 10:25 AM

Post by Abhijith Nair on August 20, 2013

Does this course cover everything found in an Algebra 1 course?

Related Articles:

Operations on Numbers

  • To add numbers that are the same sign, add the absolute value of each number together. The answer will have the same sign as the sign of the two numbers.
  • Adding numbers that are different in sign can be done using subtraction. To subtract numbers, find the difference of the absolute value of the two numbers. The answer will have the same sign as the number that was larger in absolute value.
  • When multiplying and dividing, if the numbers have the same sign the result will be positive. If the numbers have a different sign, the result will be negative.
  • When raising a number to a power of a natural number, we can think of it as repeated multiplication
  • When taking the square root of a number, we want the positive number that when multiplied by itself would give use the number under the square root.
  • Square roots can be split up over multiplication and division. Be careful not to split up a square root over addition or subtraction.

Operations on Numbers

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

  • Intro 0:00
  • Objectives 0:06
  • Operations on Numbers 0:25
    • Addition
    • Subtraction
    • Multiplication & Division
    • Exponents
    • Bases
    • Square Roots
    • Principle Square Roots
    • Perfect Squares
    • Simplifying and Combining Roots
  • Example 1 8:16
  • Example 2 12:30
  • Example 3 14:02
  • Example 4 16:27

Transcription: Operations on Numbers

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at operations on numbers.0002

When you hear me use that word operations, I'm talking about ways that we can combine together numbers or do something to a number.0007

It gets very familiar things such as adding, subtracting, multiplying and dividing.0015

As well as exponents and square roots.0020

To combine numbers together, we use a lot of familiar operations in order to do so.0028

Again, adding, subtracting, stuff like that.0033

Note that depending on the types of numbers being used, certain rules applies that will help us put them together.0036

The rules that I will focus on are the ones that involve positive and negative numbers.0042

How should we deal with those negative signs? Let us see what we can do with addition.0048

When you add numbers that have the same sign then you are looking at adding their absolute values together.0055

The results if they do have the same sign or have the same sign as the original numbers.0062

In other words, if you are adding the other two positive numbers its result will be positive.0068

For adding together two negative numbers, then your result will be negative.0073

Now if you happen to have two numbers that you are adding together and they are different in sign,0079

then we will actually handle these using subtraction.0084

Wait for the rules on what to do with positive and negative on my next slide.0087

When dealing with subtraction and what we want to do is subtract the number that is smaller in absolute value0095

from the number that is larger in absolute value and looking at each of their absolute values taking the smaller one from the larger one.0102

When looking at the final result, the answer will have the same sign as whichever number was larger in absolute value.0111

If your larger number was or the larger number in absolute value is negative, then the result will be negative.0119

Remember that we will also think of adding negative numbers using subtraction in this way.0128

In different way we are writing it but instead we are really using subtraction.0134

The other two familiar operations that we have are multiplication and division.0142

The way we handle a lot of signs with these ones are by thinking of this.0147

If we multiply a negative × a negative then the result will be positive.0153

A negative times a negative results positive.0158

If we take two things that are different in sign such as negative times a positive, then the result will be negative.0164

Now these are the same two rules that we end up using for division.0175

If we divide a negative by a negative, you will get a positive.0180

Negative divided by negative result positive.0184

If they are different in sign such as negative divided by positive, then the result is negative.0189

Keep these in mind when working with that your signs, multiplication and division.0198

Let us talk about exponents.0207

When we have an exponent, we can think of it as taking a number and multiplying it by itself in a certain number of times.0210

For example, maybe I have 37, 7 would be our exponent.0218

I can interpret that as multiplying that number by itself 7 times or some other key vocabulary you want to pick up on.0224

The exponent is that number raised next of the 3, exponent.0236

In the base of the number is the number that we are actually being repeatedly multiplied, base.0244

Once we see what number that is and how many times we need to multiply it, usually we can go ahead and simplify from there.0252

There are some common exponents that we usually give other names.0258

If I'm taking 5 and I'm raising it to the power of 2, 5 × 5, this is often said 52.0262

Another good common one would be something like 23, 2 × 2 × 2 that would be 8.0278

I could also say that this is 23.0290

Key on these special words for some of these other powers.0294

Another operation that we can do with numbers is taking the root of a number.0303

The principal square root of a number is the non negative number (n)0309

that you know if I were to say multiply it by itself, I would end up giving that (n).0313

This seems a little funny special worry but let us see if I can describe it using (n) as an example.0319

Let us say I wanted to find out the square root of 25.0326

What I'm looking for is what number would multiply it by itself in order to get a 25? That has to be 5.0330

It is what I'm talking about here, positive number such that when it is multiplied by itself you get that number underneath the root.0340

For this reason we have a little bit of a problem with our negatives underneath the root.0353

After all, what number would you multiply it by itself in order to get -16?0360

We learned from our rules of positive and negative numbers that -4 and +4 would work to get 160364

but unfortunately they are different in sign and we need them to be exactly the same, that is not going to work.0373

We are dealing with the principal square root because we are only interested in the positive numbers0382

that when multiplied by themselves would give us that number.0387

A perfect square is a number that is the square of a whole number and this one is usually reduced very nicely.0393

For example the square root of 9 would be the example of a perfect square, as it reduces down to a nice whole number, 3.0401

If you have a (a, b) being non negative real numbers,0416

then there are a few different ways that you can say combine or rip apart those roots.0421

3 multiplied underneath the same root and you can apply the root to each of those pieces.0426

If you are dividing and you have a root then you can put it over each of its pieces in the numerator and in the denominator.0432

You can use these rules in two different ways to simplify or combine words together.0442

We will see that a lot in some future lessons.0446

I'm pointing this out now so you do not make a common mistake.0449

Do not try and split up your roots over addition or subtraction.0453

We do not have a rule to do that yet or a good way to handle it.0458

In fact, in this example I have written below you can see that the two are not equal by simply evaluating each side.0461

9 + 16 would be a 25 and the square root of 25 is 5.0469

Looking at the right side square root of 9 is 3, square root of 16 is 4.0475

I'm putting those together I will get 7 and you can see that these things are not the same.0482

Be very careful when working with your roots.0492

Now that we know a few things about combining these, let us go through some examples and just practice with them.0498

The first one I want to add together a -3 and a -6.0504

I'm adding together two numbers that have exactly the same sign.0510

I will look at their absolute value and add those together.0514

The absolute value of 3 is 3, the absolute value of -6 is 6, if I combine those together I will get 9.0519

Since I'm adding together numbers that have exactly the same sign, the result will also have the same sign.0531

Adding two negative numbers my result is negative, -3+ -6 is a -9.0537

Moving on, 19 + a -12, I want to think of this as a subtraction problem since their signs are different.0545

How do I handle subtraction? Again I will look at their absolute value.0558

The absolute value of 19 and the absolute value of a-12, 19, 12.0562

I will subtract the smaller number from the larger number 19-12, what will that give me? I will get7.0571

I have to determine what sign this should be.0584

In subtraction we take the same sign as the larger number.0588

19 was larger in absolute value, it was positive, I know my result is 7.0594

If we do get a positive result as our answer, we do not write that positive sign up there.0604

We are just having 7, we will assume that that is a positive 7.0609

-8-11, an interesting way we can look at the problem, we could look at this as adding a -11.0614

The reason why I have looked at it in that way is that I could use my rules for addition.0626

If I'm adding things that have the same sign, I have looked at the absolute value of each of them, 8, 11.0631

Then I can simply add up those two values and get 19.0640

Since I'm adding two negative numbers, I know my result will also be negative, -19.0645

Let us try another one, 8 - -13.0651

When you subtract the negative this is another good situation that you could end up rewriting in a much simpler form.0657

When you subtract the negative, you can change it into addition.0662

This is 8 + 13 and now I'm adding together two positive numbers.0667

You made a positive 21 and one more, negative the absolute value of a -4 + 9.0673

Let us start inside those absolute values and see what we can do.0683

I'm adding together two things but they have different signs.0687

Let us look at the absolute value of a -4 and the absolute value of 9, 4, 9.0692

We want to subtract the smaller value from the larger value, 9-4 and that result would be 5.0699

I know what sign should that have, or the number that is larger in absolute value is 9.0708

And that was a positive value over here, I'm looking at a 5.0716

There are a lot of other things I have left out here so far,0721

those would be the absolute values and that leading negative sign.0724

Let us go ahead and put those in there now.0727

I want to do the absolute value of 5, all that is simply 5.0731

I still have that negative sign hanging out front and it is been there since the very beginning.0740

I can see that after done evaluating this one all the way, my answer is actually a 5.0743

Let us work on adding, multiplying and dividing the following numbers.0752

We only have one rule to take care of that is when we are multiplying together two negative numbers, we get a positive.0757

If we are multiplying together two numbers that are different in sign, the result should be negative.0766

4 × -7, I just want to think of 4 × 7 that would be 28.0775

Since they are different in sign, I know that this will be a -28.0784

Moving on, -6 × -5, 6 × 5 would give me 30 and now here I have a - × - I know the result will be positive.0789

But again we usually do not write that positive sign in there so just leave this as 30.0801

-12 ÷ 3, 12 ÷ 3 would be a 4, negative ÷ positive would be negative, our result is a -4.0806

One last one, -60 ÷ a -5, 60 ÷ 5 goes in there 12 times and negative ÷ negative is positive, my result is a 12.0820

You need to be very careful with these rules for multiplication, make sure you have these memorized.0835

Let us do a few involving our exponents.0843

Remember we can think of these as repeated multiplication.0846

2/34, it can be the same as 2/3 × 2/3 × 2/3 × 2/3, we are doing it 4 times.0850

I simply multiply it across the top and across the bottom 2 × 2 × 2 × 2 would be 16.0862

Then I have 3 × 3 × 3 × 3 = 81.0871

All the numbers here are positive so I know my result is positive.0876

Here is a tricky one, -5 ×-5 × -5, let us take this two other times so we can what is going on here.0881

Here I have a -5 × -5 the result of taking 5 × 5 would be 25.0896

Taking a negative × negative I would know that this would be 25.0904

That looks good, let us go ahead and work in this last value of -5.0910

We want to multiply 25 × -5, the result there would be 125.0918

Since I'm multiplying a positive × a negative result is -125, -125 would be my answer.0929

The next one looks very similar but it is actually very different.0939

That one is 23 and that negative sign is just out front of all that.0943

We recognize that the 2 is the base and that the negative is not included in that base0949

since there is no parenthesis given to group it in that way.0956

We want to figure this one as 2 × 2 × 2, I have multiplied it out three times.0960

And as for what to do with that negative sign, if we have not put it up front it is a long pretty ride.0966

2 × 2 × 2 that would give me an 8, all of those numbers are positive, 8.0973

Of course let us put our negative sign out front since it was out front at the very beginning.0979

We can see that our result is -8.0984

Now on to some square roots, these ones we simply just want to break them down and simplify them as much as possible.0990

The first one I have the square root of 64.0998

Think for yourself what number would you have to multiply it by itself in order to get 64?1001

I have a couple of options that could be 8 and 8, that would have given me 64.1008

Or it could be -8 × -8, that would also give me 64.1013

We are only interested in the positive values that do so, let us not worry about those -8.1018

We will say that our answer for the square of 64 is 8.1024

Moving on, the square root of 169 ÷ 81.1031

This one we can use one of our rules and break up the root over the top and over the bottom.1037

Now we can end up taking the root of each of these individually.1045

What number multiplied by itself would give us a 169? That have to be a 13.1048

A number that would be multiply by itself to get an 81? 9.1055

The answer in this one is 13/9.1060

Continuing on, this one has a negative sign out front but that is not underneath the root.1064

I would not worry about it just yet instead let us just focus on the square root of 36.1071

That would be 6, of course we will go ahead and put our negative sign and see that our final result is a -6.1079

One more, this last one involves the square root.1091

What two numbers when multiplied together would give us a -49 remember they must be the same.1095

We got a few problems, do not we? If I try and use 7 and 7 that would give me 49, that does not work.1102

If I try and use a couple of-7, that does not work, that still gives me 49.1109

I can not use one positive and one negative even though those give me a -49.1114

Those are not the same sign, one is positive and one is negative.1121

What is going on here, if you remember about your types of numbers, these are imaginary numbers.1125

I will leave that one just as it is until we learn about simplifying imaginary numbers in some later lessons.1137

Some various different ways that you can go ahead and combine numbers using some very familiar operations.1144

Remember that most of the rules that I covered will give you some tips on what to do when they are different in signs.1150

Positive, negative, negative, positive and all those will be handy in figuring out the overall sign of your answer.1155

Thank you for watching www.educator.com.1166

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