Sullivan%20Algebra%20and%20Trigonometry:%20Section%20R.1%20Real%20Numbers

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Sullivan Algebra and Trigonometry Section
R.1Real Numbers

• Objectives of this Section
• Classify Numbers
• Evaluate Numerical Expressions
• Work with Properties of Real Numbers

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Describing Sets of Numbers The Roster Method
The roster method is used to list the elements in
a set. For example, we can describe the set of
even digits as follows E 0, 2, 4, 6, 8
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Describing Sets of Numbers Set Builder Notation
Set Builder notation is used to describe a set of
numbers by defining a property that the numbers
share. For example, we can describe the set of
odd digits as follows O x x is an odd
digit
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Subsets of the Real Numbers
The Rational Numbers A rational number is a
number that can be expressed as a quotient a/b .
The integer a is called the numerator, and the
integer b, which cannot be 0, is called the
denominator. All rational numbers can be written
as a decimal that either terminates or repeats.
For example 1/3 0.3333
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Subsets of the Rational Numbers
Subsets of the Rational Numbers can be found by
letting the denominator equal 1.

• The Natural Numbers 1, 2, 3, 4,
• The Whole Numbers 0, 1, 2, 3, 4,
• The Integers -3, -2, -1, 0, 1, 2, 3,

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Subsets of the Real Numbers The Irrational Numbers
Numbers in which the decimal neither terminates
nor repeats are called irrational numbers.
Examples of irrational numbers include
The set of all rational and irrational numbers
form the set of real numbers.
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Approximations In practice, the decimal
representation of an irrational number is given
as an approximation.
Truncation Drop all of the digits that follow
the specified final digit in the
decimal. Rounding Identify the specified digit
in the decimal. If the next digit is 5 or more,
add one to the final digit. If the next digit
is 4 or less, leave the final digit as is.
Then, truncate following the final digit.
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Order of Operations
1. Begin with the innermost parenthesis and work
outward. Remember that in dividing two
expressions the numerator and denominator are
treated as if they were in parenthesis. For
example
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Order of Operations
2. Perform multiplication and divisions, working
left to right. For example
3. Perform additions and subtractions, working
from left to right.
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Properties of Real Numbers
Commutative Properties
Addition a b b a
Multiplication ab ba
Associative Properties
Addition a (b c) (a b) c
Multiplication a(bc) (ab)c
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Properties of Real Numbers
Distributive Property
a(b c) ab ac (ab)c acbc
Example
4(x 2)
4(x) 4(2)
4x 8
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Properties of Real Numbers
Identity Properties
0 a a 0 a a(1) 1(a) a
Additive Inverse Property
a (- a) - a a 0
Multiplicative Inverse Property
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Properties of Real Numbers
Multiplication by Zero
a (0) 0
Division Properties
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Properties of Real Numbers
Rules of Signs
a(-b) -(ab) (-a)b (-a)(-b) ab - ( -a)
a
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Properties of Real Numbers
Cancellation Properties
Zero Product Property
If ab 0, then a 0 or b 0 or, both.
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Properties of Real Numbers
Arithmetic of Quotients
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Arithmetic of Quotients Example 1
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Arithmetic of Quotients Example 2