Download real numbers

Section 5.5
Real Numbers
and Their
Properties
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Properties of Real Numbers
Closure
Commutative
Associative
Distributive
5.5-2
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 .
5.5-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Relationships Among Sets
Real numbers
Rational numbers
Integers
Whole numbers
Natural numbers
5.5-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Irrational
numbers
Relationships Among Sets
Natural
Numbers
Rational
Numbers
Real
Numbers
Integers
Zero
Fractions
Negative
Integers
Terminating or repeating
decimal numbers
Irrational
Numbers
5.5-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number
System
Closure
If 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.
5.5-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number
System
Commutative Property
Addition
a+b=b+a
Multiplication
a • b = b •a
for any real numbers a and b.
5.5-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number
System
Associative Property
Addition
(a + b) + c = a + (b + c)
Multiplication
(a • b) • c = a • (b • c)
for any real numbers a, b and c.
5.5-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of the Real Number
System
Distributive Property of Multiplication
over Addition
a • (b + c) = a • b + a • c
for any real numbers a, b and c.
5.5-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Identifying Properties
of Real Numbers
Name the property illustrated.
a) 2 + 5 = 5 + 2
Commutative property of addition
b) (x + 3) + 5 = x + (3 + 5)
Associative property of addition
c) 4 • (3 • y) = (4 • 3) • y
Associative property of multiplication
5.5-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Identifying Properties
of Real Numbers
Name the property illustrated.
d) 9(w + 3) = 9 • w + 9 • 3
Distributive property of multiplication
over addition
e) 5 + (z + 3) = 5 + (3 + z)
Commutative property of addition
5.5-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Identifying Properties
of Real Numbers
Name the property illustrated.
f) (2p) • 5 = 5 • (2p)
Commutative property of multiplication
5.5-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Simplify by Using the
Distributive Property
Simplify.
   5  2 5  3  10  5 3
7 4  2   7  4 7  2
a) 5 2  3
b)
 4 7  14
5.5-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 4: Distributive Property
Use the distributive property to multiply
2(x + 5). Then simplify the result.
2x  5  2  x  2  5
 2x  10
5.5-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.