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Section 1.1 Numbers and Their Properties
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OBJECTIVES A Write a set of numbers using roster or set–builder notation.
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OBJECTIVES B Write a rational number as a decimal.
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OBJECTIVES C Classify a number as natural, whole, integer, rational, irrational, or real.
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OBJECTIVES D Find the additive inverse of a number.
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OBJECTIVES E Find the absolute value of a number.
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OBJECTIVES F Given two numbers, use the correct notation to indicate equality or which is larger.
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DEFINITION NATURAL NUMBERS The set of numbers used for counting.
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DEFINITION WHOLE NUMBERS The set of natural numbers and zero.
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DEFINITION INTEGERS The set of whole numbers and their opposites(negatives).
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DEFINITION RATIONAL NUMBERS
All numbers that can be written as the ratio of two integers.
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DEFINITION IRRATIONAL NUMBERS
Numbers that cannot be written as ratios of two integers.
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DEFINITION REAL NUMBERS
Numbers that are either rational or irrational:
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DEFINITION ADDITIVE INVERSE The additive inverse(opposite) of a is –a.
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DEFINITION ABSOLUTE VALUE
The distance between a and 0 on the real-number line
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CAUTION The absolute value is always positive or zero.
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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.
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Chapter 1 The Real Numbers Section 1.1A
Practice Test Exercise #1
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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
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Chapter 1 The Real Numbers Section 1.1B
Practice Test Exercise #2
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Chapter 1 The Real Numbers Section 1.1C
Practice Test Exercise #3
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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
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Chapter 1 The Real Numbers Section 1.1D
Practice Test Exercise #4
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Chapter 1 The Real Numbers Section 1.1E
Practice Test Exercise #5
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Find:
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Chapter 1 The Real Numbers 1.1F
Practice Test Exercise #6
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Fill in the blank with <, >, or = to make the resulting statement true:
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Section 1.2 Operations and Properties of Real Numbers
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OBJECTIVES A Add, subtract, multiply, and divide signed numbers.
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OBJECTIVES B Identify uses of the properties of the real numbers.
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PROCEDURE TO ADD TWO NUMBERS WITH THE SAME SIGN:
Add their absolute values and give the sum the common sign.
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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.
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DEFINITION ADDITIVE IDENTITY For any real number a:
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DEFINITION SUBTRACTION OF SIGNED NUMBERS If a and b are real numbers:
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DEFINITION ADDITIVE INVERSE For any real number a:
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PROCEDURE SIGNIFY MULTIPLICATION
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PROCEDURE MULTIPLYING NUMBERS WITH OPPOSITE SIGNS
To multiply a positive number by a negative number, multiply their absolute values and make the product negative.
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DEFINITION SIGNS OF MULTIPLICATION PRODUCTS Same signs: Positive(+)
Different signs: Negative(–)
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DEFINITION IDENTITY FOR MULTIPLICATION For any real number a:
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DEFINITION MULTIPLICATION OF FRACTIONS
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DEFINITION DIVISION OF REAL NUMBERS
If a and b are real numbers and b is not zero:
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DEFINITION SIGNS OF A FRACTION
For any real number a and nonzero real number b, there are two cases of signs:
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DEFINITION ZERO IN DIVISION For a ≠ 0:
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CAUTION
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DEFINITION MULTIPLICATIVE INVERSE (RECIPROCAL)
Every nonzero real number a has a reciprocal such that:
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DEFINITION DIVISION OF FRACTIONS
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Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #7
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Find.
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Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #8
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Find.
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Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #9
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Find. Least common denominator = 8. Now add numerators.
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Find. Least common denominator = 12.
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Now add numerators.
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Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #10
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Find.
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Chapter 1 The Real Numbers
Section 1.2A Practice Test Exercise #11
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Find. 1
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Find. 1 3 1 4
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Chapter 1 The Real Numbers
Section 1.2B Practice Test Exercise #12
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Name the property illustrated in the statement.
Commutative Property of Addition Associative Property of Addition
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Chapter 1 The Real Numbers
Section 1.2B Practice Test Exercise #13
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Name the property illustrated in the statement.
Inverse Property of Multiplication. Inverse Property of Addition.
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Section 1.3 Properties of Exponents
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OBJECTIVES A Evaluate expressions containing natural numbers as exponents.
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OBJECTIVES B Write an expression containing negative exponents as a fraction.
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OBJECTIVES C Multiply and divide expressions containing exponents.
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OBJECTIVES D Raise a power to a power.
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OBJECTIVES E Raise a quotient to a power.
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OBJECTIVES F Convert between ordinary decimal notation and scientific notation, and use scientific notation in computations.
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DEFINITION EXPONENT AND BASE
If a is a real number and n is a natural number:
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Chapter 1 The Real Numbers
Section 1.3A Practice Test Exercise #14
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Evaluate.
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Chapter 1 The Real Numbers
Section 1.3B Practice Test Exercise #15
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Write without negative exponents.
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Chapter 1 The Real Numbers
Section 1.3C Practice Test Exercise #16
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Perform the indicated operation and simplify.
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Perform the indicated operation and simplify.
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Perform the indicated operation and simplify.
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Chapter 1 The Real Numbers
Section 1.3D, E Practice Test Exercise #17
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Simplify.
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Simplify.
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Chapter 1 The Real Numbers
Section 1.3F Practice Test Exercise #18
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The exponent of 10, (–3), means move the decimal point 3 places to the left.
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Chapter 1 The Real Numbers
Section 1.3F Practice Test Exercise #19
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The exponent of 10, (5), means move the decimal point 5 places to the right.
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Chapter 1 The Real Numbers
Section 1.3F Practice Test Exercise #20
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Perform the calculation and write your answer in scientific notation.
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Perform the calculation and write your answer in scientific notation.
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Section 1.4 Algebraic Expressions and The Order of Operations
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OBJECTIVES A Evaluate numerical expressions with grouping symbols.
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OBJECTIVES Evaluate expressions using the correct order of operations.
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OBJECTIVES C Evaluate algebraic expressions.
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OBJECTIVES D Use the distributive property to simplify expressions.
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OBJECTIVES E Simplify expressions by combining like terms.
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OBJECTIVES F Simplify expressions by removing grouping symbols and combining like terms.
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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)
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Identity for Multiplication
PROCEDURE Identity for Multiplication For any real number a:
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PROCEDURE Additive Inverse For any real number a:
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Additive Inverse of a Sum
PROCEDURE Additive Inverse of a Sum
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Additive Inverse of a Difference
PROCEDURE Additive Inverse of a Difference
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DEFINITION LIKE TERMS Constant terms or terms with exactly the same variable factors are called similar or like terms.
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Chapter 1 The Real Numbers
Section 1.4A Practice Test Exercise #21
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Evaluate. 11 1
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Chapter 1 The Real Numbers
Section 1.4B Practice Test Exercise #22
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Evaluate.
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Chapter 1 The Real Numbers
Section 1.4C Practice Test Exercise #23
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Evaluate. a.
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Evaluate. b.
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Chapter 1 The Real Numbers
Section 1.4D, E Practice Test Exercise #24
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Simplify.
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Chapter 1 The Real Numbers
Section 1.4F Practice Test Exercise #25
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Simplify.
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