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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 NATURAL NUMBERS The set of numbers used for counting.

DEFINITION WHOLE NUMBERS The set of natural numbers and zero.

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

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

DEFINITION IRRATIONAL NUMBERS
Numbers that cannot be written as ratios of two integers.

DEFINITION REAL NUMBERS
Numbers that are either rational or irrational:

DEFINITION ADDITIVE INVERSE The additive inverse(opposite) of a is –a.

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

CAUTION The absolute value is always positive or zero.

DEFINITION TRICHOTOMY LAW
If given any two real numbers, only one of three things is true: a is equal to b, denoted by a = b, or a is less than b, denoted by a < b, or a is greater than b, denoted by a >b.

Chapter 1 The Real Numbers Section 1.1A
Practice Test Exercise #1

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

Chapter 1 The Real Numbers Section 1.1B
Practice Test Exercise #2

Chapter 1 The Real Numbers Section 1.1C
Practice Test Exercise #3

Classify the given number by making a check mark () in the appropriate row(s).
Natural number Whole number Integer Rational number Irrational number Real number

Chapter 1 The Real Numbers Section 1.1D
Practice Test Exercise #4

Chapter 1 The Real Numbers Section 1.1E
Practice Test Exercise #5

Find:

Chapter 1 The Real Numbers 1.1F
Practice Test Exercise #6

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

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.

PROCEDURE TO ADD TWO NUMBERS WITH THE SAME SIGN:
Add their absolute values and give the sum the common sign.

PROCEDURE TO ADD TWO NUMBERS WITH DIFFERENT SIGNS:
Find the absolute value. Subtract the smaller from the greater number. Use the sign of the number with the greater absolute value.

DEFINITION ADDITIVE IDENTITY For any real number a:

DEFINITION SUBTRACTION OF SIGNED NUMBERS If a and b are real numbers:

DEFINITION ADDITIVE INVERSE For any real number a:

PROCEDURE SIGNIFY MULTIPLICATION

PROCEDURE MULTIPLYING NUMBERS WITH OPPOSITE SIGNS
To multiply a positive number by a negative number, multiply their absolute values and make the product negative.

DEFINITION SIGNS OF MULTIPLICATION PRODUCTS Same signs: Positive(+)
Different signs: Negative(–)

DEFINITION IDENTITY FOR MULTIPLICATION For any real number a:

DEFINITION MULTIPLICATION OF FRACTIONS

DEFINITION DIVISION OF REAL NUMBERS
If a and b are real numbers and b is not zero:

DEFINITION SIGNS OF A FRACTION
For any real number a and nonzero real number b, there are two cases of signs:

DEFINITION ZERO IN DIVISION For a ≠ 0:

CAUTION

DEFINITION MULTIPLICATIVE INVERSE (RECIPROCAL)
Every nonzero real number a has a reciprocal such that:

DEFINITION DIVISION OF FRACTIONS

Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #7

Find.

Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #8

Find.

Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #9

Find. Least common denominator = 8. Now add numerators.

Find. Least common denominator = 12.

Now add numerators.

Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #10

Find.

Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #11

Find. 1

Find. 1 3 1 4

Chapter 1 The Real Numbers
Section 1.2B Practice Test Exercise #12

Name the property illustrated in the statement.
Commutative Property of Addition Associative Property of Addition

Chapter 1 The Real Numbers
Section 1.2B Practice Test Exercise #13

Name the property illustrated in the statement.
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.

OBJECTIVES F Convert between ordinary decimal notation and scientific notation, and use scientific notation in computations.

DEFINITION EXPONENT AND BASE
If a is a real number and n is a natural number:

Chapter 1 The Real Numbers
Section 1.3A Practice Test Exercise #14

Evaluate.

Chapter 1 The Real Numbers
Section 1.3B Practice Test Exercise #15

Write without negative exponents.

Chapter 1 The Real Numbers
Section 1.3C Practice Test Exercise #16

Perform the indicated operation and simplify.

Perform the indicated operation and simplify.

Perform the indicated operation and simplify.

Chapter 1 The Real Numbers
Section 1.3D, E Practice Test Exercise #17

Simplify.

Simplify.

Chapter 1 The Real Numbers
Section 1.3F Practice Test Exercise #18

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

Chapter 1 The Real Numbers
Section 1.3F Practice Test Exercise #19

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

Chapter 1 The Real Numbers
Section 1.3F Practice Test Exercise #20

Perform the calculation and write your answer in scientific notation.

Perform the calculation and write your answer in scientific notation.

Section 1.4 Algebraic Expressions and The Order of Operations

OBJECTIVES A Evaluate numerical expressions with grouping symbols.

OBJECTIVES 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.

PROCEDURE ORDER OF OPERATIONS Do the operations in the (). P
Evaluate exponential expressions. Perform multiplications and divisions from left to right. Perform additions and subtractions from left to right. P E (MD) (AS)

Identity for Multiplication
PROCEDURE Identity for Multiplication For any real number a:

PROCEDURE Additive Inverse For any real number a:

Additive Inverse of a Sum
PROCEDURE Additive Inverse of a Sum

Additive Inverse of a Difference
PROCEDURE Additive Inverse of a Difference

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

Chapter 1 The Real Numbers
Section 1.4A Practice Test Exercise #21

Evaluate. 11 1

Chapter 1 The Real Numbers
Section 1.4B Practice Test Exercise #22

Evaluate.

Chapter 1 The Real Numbers
Section 1.4C Practice Test Exercise #23

Evaluate. a.

Evaluate. b.

Chapter 1 The Real Numbers
Section 1.4D, E Practice Test Exercise #24

Simplify.

Chapter 1 The Real Numbers
Section 1.4F Practice Test Exercise #25

Simplify.

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