Chapter 1 Real Numbers and Introduction to Algebra

Chapter 1

Real Numbers and Introduction to

Algebra

Martin-Gay, Introductory Algebra, 3ed 2Martin-Gay, Introductory Algebra, 3ed 2

1.1 – Tips for Success in Mathematics

1.2 – Symbols and Sets of Numbers

1.3 – Exponents, Order of Operations, and Variable Expressions

1.4 – Adding Real Numbers

1.5 – Subtracting Real Numbers

1.6 – Multiplying and Dividing Real Numbers

1.7 – Properties of Real Numbers

1.8 – Simplifying Expressions

Chapter Sections

§ 1.1

Tips for Success in Mathematics


Positive Attitude Believe you can succeed.

SchedulingMake sure you have time for your classes.

Be PreparedHave all the materials you need, like a lab manual, calculator, or other supplies.

Getting Ready for This Course


General Tips for Success

Tip Details

Get a contact person.Exchange names, phone numbers or e-mail addresses with at least one other person in class.

Attend all class periods.Sit near the front of the classroom to make hearing the presentation, and participating easier.

Do you homework.The more time you spend solving mathematics, the easier the process becomes.

Check your work.Review your steps, fix errors, and compare answers with the selected answers in the back of the book.

Learn from your mistakes.

Find and understand your errors. Use them to become a better math student.

Continued


General Tips for Success

Tip Details

Get help if you need it.Ask for help when you don’t understand something. Know when your instructors office hours are, and whether tutoring services are available.

Organize class materials.Organize your assignments, quizzes, tests, and notes for use as reference material throughout your course.

Read your textbook.Review your section before class to help you understand its ideas more clearly.

Ask questions.Speak up when you have a question. Other students may have the same one.

Hand in assignments on time.

Don’t lose points for being late. Show every step of a problem on your assignment.


Using This Text

Resource Details

Practice Problems.Try each Practice Problem after you’ve finished its corresponding example.

Chapter Test Prep Video CD.Chapter Test exercises are worked out by the author, these are available off of the CD this book contains.

Lecture Video CDs.Exercises marked with a CD symbol are worked out by the author on a video CD. Check with your instructor to see if these are available.

Symbols before an exercise set.

Symbols listed at the beginning of each exercise set will remind you of the available supplements.

Objectives.The main section of exercises in an exercise set is referenced by an objective. Use these if you are having trouble with an assigned problem.

Continued


Using This Text

Resource Details

Icons (Symbols).

A CD symbol tells you the corresponding exercise may be viewed on a video segment. A pencil symbol means you should answer using complete sentences.

Integrated Reviews.Reviews found in the middle of each chapter can be used to practice the previously learned concepts.

End of Chapter Opportunities.Use Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews to help you understand chapter concepts.

Study Skills Builder.Read and answer questions in the Study Skills Builder to increase your chance of success in this course.

The Bigger Picture.This can help you make the transition from thinking “section by section” to thinking about how everything corresponds in the bigger picture.


Getting Help

Tip Details

Get help as soon as you

need it.

Material presented in one section builds on your understanding of the previous section. If you don’t understand a concept covered during a class period, there is a good chance you won’t understand the concepts covered in the next period.

For help try your instructor, a tutoring center, or a math lab. A study group can also help increase your understanding of covered materials.


Preparing for and Taking an Exam

Steps for Preparing for a Test

1. Review previous homework assignments.

2. Review notes from class and section-level quizzes you have taken.

3. Read the Highlights at the end of each chapter to review concepts and definitions.

4. Complete the Chapter Review at the end of each chapter to practice the exercises.

5. Take a sample test in conditions similar to your test conditions.

6. Set aside plenty of time to arrive where you will be taking the exam.

Continued


Preparing for and Taking an Exam

Steps for Taking Your Test

1. Read the directions on the test carefully.

2. Read each problem carefully to make sure that you answer the question asked.

3. Pace yourself so that you have enough time to attempt each problem on the test.

4. Use extra time checking your work and answers.

5. Don’t turn in your test early. Use extra time to double check your work.


Managing Your Time

Tips for Making a Schedule

1. Make a list of all of your weekly commitments for the term.

2. Estimate the time needed and how often it will be performed, for each item.

3. Block out a typical week on a schedule grid, start with items with fixed time slots.

4. Next, fill in items with flexible time slots.

5. Remember to leave time for eating, sleeping, and relaxing.

6. Make changes to your workload, classload, or other areas to fit your needs.

§ 1.2

Symbols and Sets of Numbers


Set of Numbers

• Natural numbers – {1, 2, 3, 4, 5, 6 . . .}• Whole numbers – {0, 1, 2, 3, 4 . . .}• Integers – {. . . –3, -2, -1, 0, 1, 2, 3 . . .}• Rational numbers – the set of all numbers that can be expressed as a quotient of integers, with denominator 0

• Irrational numbers – the set of all numbers that can NOT be expressed as a quotient of integers

• Real numbers – the set of all rational and irrational numbers combined


Equality and Inequality Symbols

Symbol Meaning

a = b

a b

a < b

a > b

a b

a b

a is equal to b.

a is not equal to b.

a is less than b.

a is greater than b.

a is less then or equal to b.

a is greater than or equal to b.


The Number Line

A number line is a line on which each point is associated with a number.

2– 2 0 1 3 4 5– 1– 3– 4– 5

Negative numbers

Positive numbers

– 4.8 1.5


For any two real numbers a and b, a is less than b if a is to the left of b on the number line.

• a < b means a is to the left of b on a number line.• a > b means a is to the right of b on a number line.

Order Property for Real Numbers

Insert < or > between the following pair of numbers to make a true statement.

8

7 0

Example


Absolute Value

The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line.

2– 2 0 1 3 4 5– 1– 3– 4– 5

| – 4| = 4

Distance of 4

Symbol for absolute value

|5| = 5

Distance of 5

§ 1.3

Exponents, Order of Operations, and Variable

Expressions


Using Exponential Notation

We may use exponential notation to write products in a more compact form.

22222 52can be written as

“three to the fourth power”

“three to the third power” or “three cubed”

“three to the second power” or “three squared.”

In WordsExpression

233343

Evaluate 26.62 64

222222

Example


Using the Order of Operations

Order of Operations

1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars or square roots.

2. Evaluate any expressions with exponents.

3. Multiply or divide in order from left to right.

4. Add or subtract in order from left to right.


Using the Order of Operations

Evaluate: 23

396

23

396 )9(

396

9

)3(6

9

9

1

Write 32 as 9.

Divide 9 by 3.

Add 3 to 6.

Divide 9 by 9.

Example


Definition ExampleVariable: A letter to represent all the numbers fitting a pattern.

Evaluate: 7 + 3z when z = – 3

Algebraic Expression: A combination of numbers, letters (variables), and operation symbols.

Evaluating the Expression: Replacing a variable in an expression by a number and then finding the value of the expression

z37 )3(37 )9(7

97 2

Evaluating Algebraic Expressions


Determining Whether a Number is a Solution

Definition ExampleSolving: In an equation containing a variable, finding which values of the variable make the equation a true statement.

Is -7 a solution of: a + 23 = –16?

Solution: In an equation, a value for the variable that makes the equation a true statement.

1623 a

( ) 27 3 16

– 7 is not a solution.


Translating Phrases

Addition

(+)

Subtraction

(–)

Multiplication

(·)

Division

()sum

plus

added to

more than

increased by

total

difference

minus

subtract

less than

decreased by

less

product

times

multiply

multiplied by

of

double/triple

quotient

divide

shared equally

among

divided by

divided into


Translating Phrases

Write as an algebraic expression. Use x to represent “a number.”

In words:

a.) 5 decreased by a number

b.) The quotient of a number and 12

5

Translate: 5

decreased by a number– x

a.)

In words: a number

Translate: x

and 12

12

b.)

or 12x

The quotient of

Example

§ 1.4

Adding Real Numbers


Adding Real Numbers

Adding 2 numbers with the same sign• Add their absolute values.• Use common sign as sign of sum.

Adding 2 numbers with different signs• Take difference of absolute values (smaller subtracted from larger).

• Use the sign of larger absolute value as sign of sum.


Opposites or additive inverses are two numbers the same distance from 0 on the number line, but on opposite sides of 0.

The sum of a number and its opposite is 0.

If a is a number, – (– a) = a.

Example

Add the following numbers.

(–3) + 6 + (–5) = –2

Additive Inverses

§ 1.5

Subtracting Real Numbers


Subtracting real numbers• Substitute the opposite of the number being subtracted

• Add.• a – b = a + (– b)

Example

Subtract the following numbers.

(– 5) – 6 – (– 3) = (– 5) + (– 6) + 3 = – 8

Subtracting Real Numbers


Complementary angles are two angles whose sum is 90o.

Example

Find the measure of the following complementary angles.

x 150 – 2x

x + 150 – 2x = 90

150 – x = 90

– x = – 60

x = 60° and 150 – 2x = 30°

Complementary Angles


Supplementary angles are two angles whose sum is 180o.

Example

Find the measure of the following supplementary angles.

x x + 78

x + x + 78 = 180

2x + 78 = 180

2x = 102

x = 51° and x + 78 = 129°

Supplementary Angles

§ 1.6

Multiplying and Dividing Real Numbers


Multiplying or dividing 2 real numbers with same sign

• Result is a positive number

Multiplying or dividing 2 real numbers with different signs

• Result is a negative number

Multiplying or Dividing Real Numbers


Example

Find each of the following products.

4 · (–2) · 3 = –24

(–4) · (–5) = 20

Multiplying or Dividing Real Numbers


If b is a real number, 0 · b = b · 0 = 0.

Multiplicative inverses or reciprocals are two numbers whose product is 1.

The quotient of any real number and 0 is undefined.

The quotient of 0 and any real number = 0.

0

a

a

0 a 0

Multiplicative Inverses (Reciprocals)


b

a

b

a

b

a

If a and b are real numbers, and b 0,

Example

Simplify the following.

)3(29

)2(38

)6(9

)6(8

69

68

3

14

3

14

Simplifying Real Numbers

§ 1.7

Properties of Real Numbers


Commutative and Associative Property

Associative property

• of addition: (a + b) + c = a + (b + c)

• of multiplication: (a · b) · c = a · (b · c)

Commutative property

• of addition: a + b = b + a

• of multiplication: a · b = b · a


Distributive property of multiplication over addition• a(b + c) = ab + ac

Identities• for addition: 0 is the identity since

a + 0 = a and 0 + a = a.• for multiplication: 1 is the identity since

a · 1 = a and 1 · a = a.

Distributive Property


Inverses• For addition: a and –a are inverses since

a + (– a) = 0.

• For multiplication: b and are inverses (b 0)

since b · = 1.b1

b1

Inverses

§ 1.8

Simplifying Expressions


Terms

A term is a number, or the product of a number and variables raised to powers – (the number is called a coefficient)

Examples of Terms

7 (coefficient is 7)

5x3 (coefficient is 5)

4xy2 (coefficient is 4)

z2 (coefficient is 1)


Like terms contain the same variables raised to the same powers

To combine like terms, add or subtract the numerical coefficients (as appropriate), then multiply the result by the common variable factors

You can combine like terms by adding or subtracting them (this is not true for unlike terms)

Like Terms


6x2 + 7x2

19xy – 30xy

13xy2 – 7x2y

13x2

-11xy

Can’t be combined (since the terms are not like terms)

Examples of Combining Terms

Terms Before Combining After Combining Terms

Combining Like Terms

Documents

Chapter 1 Real Numbers and Introduction to Algebra