copyright 2013, 2010, 2007, pearson, education, inc. section 5.5 real numbers and their properties
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 5.5
Real Numbers and
Their Properties
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Properties of Real NumbersClosureCommutativeAssociativeDistributive
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Real Numbers
The set of real numbers is formed by the union of the rational and irrational numbers.The symbol for the set of real numbers is .
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Relationships Among Sets
Irrational numbers
Rational numbers
Integers
Whole numbersNatural numbers
Real numbers
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Relationships Among Sets
RealNumbers
RationalNumbers
Integers
NaturalNumbers
IrrationalNumbers
Fractions
Terminating or repeatingdecimal numbers
Zero
NegativeIntegers
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number SystemClosureIf an operation is performed on any two elements of a set and the result is an element of the set, we say that the set is closed under that given operation.
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number System
Commutative Property
for any real numbers a and b.
Addition a + b = b + a
Multiplication a • b = b • a
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number System
Associative Property
for any real numbers a, b and c.
Addition (a + b) + c = a + (b + c)
Multiplication (a • b) • c = a • (b • c)
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number SystemDistributive Property of Multiplication over Addition
for any real numbers a, b and c.
a • (b + c) = a • b + a •
c
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Identifying Properties of Real NumbersName the property illustrated.a) 2 + 5 = 5 + 2Commutative property of addition
b) (x + 3) + 5 = x + (3 + 5)Associative property of addition
c) 4 • (3 • y) = (4 • 3) • yAssociative property of multiplication
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Identifying Properties of Real NumbersName the property illustrated.d) 9(w + 3) = 9 • w + 9 • 3Distributive property of multiplication over addition
e) 5 + (z + 3) = 5 + (3 + z)Commutative property of addition
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Identifying Properties of Real NumbersName the property illustrated.f) (2p) • 5 = 5 • (2p)Commutative property of multiplication
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Simplify by Using the Distributive PropertySimplify.
a) 5 2 + 3( )
= 5 ⋅2( ) + 5 ⋅ 3( ) =10 + 5 3
b) 7 4 + 2( )
= 7 ⋅4( ) + 7 ⋅ 2
=4 7 + 14
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 4: Distributive PropertyUse the distributive property to multiply 2(x + 5). Then simplify the result.
2x + 5 =2 ⋅x + 2 ⋅5
=2 x +10
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