Section 1.1 Numbers and Their Properties. OBJECTIVES A Write a set of numbers using roster or set–...

Section 1.1

Numbers and Their Properties

OBJECTIVES

A Write a set of numbers using roster or set–builder notation.

OBJECTIVES

B Write a rational number as a decimal.

OBJECTIVES

C Classify a number as natural, whole, integer, rational, irrational, or real.

OBJECTIVES

D Find the additive inverse of a number.

OBJECTIVES

E Find the absolute value of a number.

OBJECTIVES

F Given two numbers, use the correct notation to indicate equality or which is larger.

DEFINITION

The set of numbers used for counting.

NATURAL NUMBERS

N = {1, 2, 3, . . . }

DEFINITION

The set of natural numbers and zero.

WHOLE NUMBERS

W = { 0, 1, 2, 3, . . .}

DEFINITION

The set of whole numbers and their opposites(negatives).

INTEGERS

I = {. . . , –2, –1, 0, 1, 2, . . .}

DEFINITION

All numbers that can be written as the ratio of two integers.

RATIONAL NUMBERS

{ and integers }aQ = r |r = , a b , b 0b

DEFINITION

Numbers that cannot be written as ratios of two integers.

IRRATIONAL NUMBERS

H = {x |x is a number that is not rational}

DEFINITION

Numbers that are either rational or irrational:

REAL NUMBERS

R = {x | x a number that is rational or irrational}

DEFINITION

The additive inverse(opposite) of a is –a.

ADDITIVE INVERSE

DEFINITION

The distance between a and 0 on the real-number line

ABSOLUTE VALUE

a if a is positive |11 | = 11

0 if a is zero |0 | = 0

-a if a is negative | -5 | = - (-5) = 5 |a | =

CAUTION

The absolute value is always positive or zero.

–|– 3| = –3, –|4.2 | = –4.2

DEFINITION

If given any two real numbers, only one of three things is true:

TRICHOTOMY LAW

1. a is equal to b, denoted by a = b, or2. a is less than b, denoted by a < b, or3. a is greater than b, denoted by a >b.

Practice Test

Exercise #1

Chapter 1The Real NumbersSection 1.1A

Use roster notation to list the natural numbers between 5 and 9.

The set of natural numbers between 5 and 9 is {6, 7,8}

Note 5 and 9 are not included

Practice Test

Exercise #2

Chapter 1The Real NumbersSection 1.1B

Write as a decimal:

= 8 3 b. 2

3 a. 3

8

= 8 3.000

= 0.375

= 3 2

= 3 2.000

= 0.666...

= 0.6

Practice Test

Exercise #3

Chapter 1The Real NumbersSection 1.1C

Set

Classify the given number by making a check mark () in the appropriate row(s).

Natural numberWhole numberIntegerRational numberIrrational numberReal number

0.5 0 – 6 27 5

Practice Test

Exercise #4

Chapter 1The Real NumbersSection 1.1D

Find the additive inverse of 4

5.

= – 4

5

Practice Test

Exercise #5

Chapter 1The Real NumbersSection 1.1E

= 9 a. | –9 |

Find:

b. | 0.5 | = 0.5

Practice Test

Exercise #6

Chapter 1The Real Numbers1.1F

14

= 4 1

– 0.66 is farther from 0 than – 0.25so – 0.25 is greater than – 0.66

a. – 1

4 _____ – 1

3

Fill in the blank with <, >, or = to make the resulting statement true:

= 4 1.00 = 0.25 so – 1

4 = – 0.25

13

= 3 1 = 3 1.00 = 0.66 so – 1

3 = – 0.66

>

b. 0.4 _____ 2

5

25

= 5 2 = 5 2.0 = 0.4

=

Section 1.2

Operations and Properties of Real Numbers

OBJECTIVES

A Add, subtract, multiply, and divide signed numbers.

OBJECTIVES

B Identify uses of the properties of the real numbers.

TO ADD TWO NUMBERS WITH THE SAME SIGN:

PROCEDURE

Add their absolute values and give the sum the common sign.

TO ADD TWO NUMBERS WITH DIFFERENT SIGNS:

PROCEDURE

1. Find the absolute value.2. Subtract the smaller from the

greater number.3. Use the sign of the number with

the greater absolute value.

DEFINITION

For any real number a:

ADDITIVE IDENTITY

a + 0 = a = 0 + a

DEFINITION

If a and b are real numbers:

SUBTRACTION OF SIGNED NUMBERS

a - b = a + (-b)

DEFINITION


ADDITIVE INVERSE

a + (-a) = (-a) + a = 0

SIGNIFY MULTIPLICATIONPROCEDURE

raised dot : a • bnext to each other : abparentheses: (a)(b), a(b), or (a)b

MULTIPLYING NUMBERS WITH OPPOSITE SIGNS

PROCEDURE

To multiply a positive number by a negative number, multiply their absolute values and make the product negative.

DEFINITION

Same signs: Positive(+)Different signs: Negative(–)

SIGNS OF MULTIPLICATION PRODUCTS

DEFINITION


IDENTITY FOR MULTIPLICATION

a • 1 = 1 • a = a

DEFINITIONMULTIPLICATION OF FRACTIONS

•••

a c a c = (b, d 0)b d b d

DEFINITION

If a and b are real numbers and b is not zero:

DIVISION OF REAL NUMBERS

ab

= q means that a = b • q

DEFINITION

For any real number a and nonzero real number b, there are two cases of signs:

SIGNS OF A FRACTION

-ab

= a-b

= - ab

or -a-b

= ab

DEFINITION

For a ≠ 0:

ZERO IN DIVISION

0a = 0 and a

0 = undefined

CAUTION

0k

is okay but n0

is a no-no!

DEFINITION

Every nonzero real number a has a reciprocal such that:

MULTIPLICATIVE INVERSE (RECIPROCAL)

a •

1a = 1

DEFINITIONDIVISION OF FRACTIONS

a c a d = (b, c and d 0)cb d b

•÷

Practice Test

Exercise #7


= –4

a. –9 + 5

Find.

–0.8 + b. –0.7

= –1.5

Practice Test

Exercise #8


a. –16 – 7

Find.

– 0.6 – b. –0.4

= –16 + –7

= –23

= –0.6 + 0.4

= –0.2

Practice Test

Exercise #9


Find.

a. – 1

8 – 3

4

Least common denominator = 8.

1 3= – + –8 4

1 6= – + –8 8

Now add numerators.

–1 + –6= 8

= –7

8

= – 7

8

3 5= – + 4 6

Least common denominator = 12.

= – 9

12 + 10

12

3 5 – – –4

b.6

Find.

= –9 +10

12

= 1

12

Now add numerators.

Practice Test

Exercise #10


6a. –9

Find.

–4 b. –1.2

= – 6 9

= –54

= + 4 12

= 4.8

Practice Test

Exercise #11


= – 1

9

1

1

a. – 1

2 2

9

Find.

= – 4

3

= – 3

2 8

9

1

31

4

Find.

b. – 3

2 ÷ 9

8

Practice Test

Exercise #12


7 + 3 + 6 = 3 + 7a. + 6

Name the property illustrated in the statement.

Commutative Property of Addition

2 + 9 + 4 = 2 + 9b. + 4

Associative Property of Addition

Practice Test

Exercise #13


Name the property illustrated in the statement.

a. 3 1

3 = 1

0.3 + –0.3b. = 0

Inverse Property of Multiplication.

Inverse Property of Addition.

Section 1.3

Properties of Exponents

OBJECTIVES

A Evaluate expressions containing natural numbers as exponents.

OBJECTIVES

B Write an expression containing negative exponents as a fraction.

OBJECTIVES

C Multiply and divide expressions containing exponents.

OBJECTIVES

D Raise a power to a power.

OBJECTIVES

E Raise a quotient to a power.

OBJECTIVESF Convert between

ordinary decimal notation and scientific notation, and use scientific notation in computations.

DEFINITION

If a is a real number and n is a natural number:

EXPONENT AND BASE

an = a • a • a • • • a

n factors

Practice Test

Exercise #14


a.

Evaluate.

b.

–3 4

= +9 +9

= 81

= – 3 3 3 3

= – 81

= –3 –3 –3 –3

–34

Practice Test

Exercise #15


a. 7–2

Write without negative exponents.

b. x –8

= 1

72 = 1

7 7 = 1

49

= 1

x8

Practice Test

Exercise #16


Perform the indicated operation and simplify.

a. (3x 4y)(–4x –8y8)

b. 48x 4

16x –8


4 –8 1 8= 3 –4 x x y y

4 + –8 1 + 8= –12 x y

= –12x –4y 9

94

1= –12 yx

= –12y 9

x 4 = – 12y 9

x 4

a. (3x 4y)(–4x –8y8)

= 48

16 x 4

x –8


b. 48x 4

16x –8

= 3 x 4 – –8

= 3x 4 + 8

= 3x12

Practice Test

Exercise #17

Chapter 1The Real NumbersSection 1.3D, E

Simplify.

a. (–2x8y –2)3

3 8 3 –2 3= (–2) ( ) ( ) x y

8 3 –2 3= –2 –2 –2 x y

= –8x 24 y –6

= –8x 24

y6 = – 8x 24

y6

–35

–3. y

b x

=

(x5)–3

(y–3)–3

= x5(–3)

y (–3)(–3)

= x –15

y 9 = 1

x15 1

y 9 = 1

x15y 9

Simplify.

Practice Test

Exercise #18

Chapter 1The Real NumbersSection 1.3F

The exponent of 10, (–3), means move the decimal point 3 places to the left.

Write in standard notation.

6.5 x 10–3

= .006.5

= 0.0065

Practice Test

Exercise #19


The exponent of 10, (5), means move the decimal point 5 places to the right.

Write as a whole number.

8.5 x 105

= 8.50000

= 850,000

Practice Test

Exercise #20


Perform the calculation and write your answer in scientific notation.

5 – 77.1 10 4 10

5 –7= 7.1 4 10 10

= 28.4 105 + (–7)

–2= 28.4 10

NOTE28.4 = 2.84 101

1 –2= 2.84 10 10

= 2.84 101 + (–2)

= 2.84 10–1

Perform the calculation and write your answer in scientific notation.

–2= 28.4 10

5 – 77.1 10 4 10

Section 1.4

Algebraic Expressions and The Order of Operations

OBJECTIVES

A Evaluate numerical expressions with grouping symbols.

OBJECTIVES

B Evaluate expressions using the correct order of operations.

OBJECTIVES

C Evaluate algebraic expressions.

OBJECTIVES

D Use the distributive property to simplify expressions.

OBJECTIVES

E Simplify expressions by combining like terms.

OBJECTIVES

F Simplify expressions by removing grouping symbols and combining like terms.

ORDER OF OPERATIONSPROCEDURE

1. Do the operations in the ().2. Evaluate exponential

expressions.3. Perform multiplications and

divisions from left to right.4. Perform additions and

subtractions from left to right.

PE

(MD)

(AS)

Identity for Multiplication

a = 1 • a

PROCEDURE


Additive Inverse

– a = –1 • a

PROCEDURE


Additive Inverse of a Sum

–(a + b) = –a – b

PROCEDURE

Additive Inverse of a Difference

–(a – b) = –a + b

PROCEDURE

DEFINITION

Constant terms or terms with exactly the same variable factors are called similar or like terms.

LIKE TERMS

Practice Test

Exercise #21


a. [ –7(4 + 3)] + 9

Evaluate.

b.

5 (131 – 32)9

= –7(7) + 9

= –49 + 9

= –40

=

5(99)9

= 55

11

1

Practice Test

Exercise #22


– 43 + 6 – 12

2 + 15 ÷ 3

Evaluate.

= – 64 + 6 – 12

2 + 15 ÷ 3

= – 64 + – 6

2 + 15 ÷ 3

= – 64 – 3 + 5

= –62

Practice Test

Exercise #23


12

(b1 + b2)h gives the area of a

trapezoid. Find the area of thetrapezoid if b1 = 8, b2 = 3, and h = 6.

Evaluate.

a.

= 1

28 + 3 6

= 1

211 6

= 33

b. 7 – x 2 + 20 ÷ y – xy;if x = –2 and y = 4.

= 7 – (–2)2 + (20 ÷ 4) – (–2 4)

= 7 – (–2)2 + 5 – (–8)

= 7 – 4 + 5 + 8

= 16

Evaluate.

Practice Test

Exercise #24

Chapter 1The Real NumbersSection 1.4D, E

a. –5 x + 7

Simplify.

b. 7x – 3x + 1 + 2x + 2

= –5x – 35

= 7x – 3x – 1 + 2x + 2

= 6x + 1

Practice Test

Exercise #25


[(5x 2 – 3) + (3x + 7)] – [(x – 3) + (2x 2 – 2)]

Simplify.

= 5x 2 – 3 + 3x + 7 – x – 3 + 2x 2 – 2

= 5x 2 + 3x + 4 – 2x 2 + x – 5

= 5x 2 + 3x + 4 – 2x 2 – x + 5

= 3x 2 + 2x + 9

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Section 1.1 Numbers and Their Properties. OBJECTIVES A Write a set of numbers using roster or set–...

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