Download PROPERTIES OF THE REAL NUMBERS

1.5
Amount (thousands of dollars)
Growth of a $10,000 investment
at 6.2% annual rate
120
(1-31) 31
P(1 r)n to find the actual amount of the debt at the time
the payments start. $5,500, $5,441.96
Amount of debt (dollars)
a) Use the accompanying graph to estimate the amount
of a $10,000 investment in corporate bonds after
30 years. $60,000
b) Use the given expression to calculate the value of a
$10,000 investment after 30 years of growth at 6.2%
compounded annually. $60,776.47
Properties of the Real Numbers
6,000
5,000
4,000
3,000
2,000
1,000
0
100
80
60
0
1
2
3
4
Time after making the loan (years)
FIGURE FOR EXERCISE 107
40
20
0
10
20
30
40
FIGURE FOR EXERCISE 105
106. Saving for college. The average cost of a B.A. in 2005
will be $252,000 at an Ivy League school (Fortune
Investors Guide). The principal that must be invested at
interest rate r compounded annually to have A dollars
n years in the future is given by the algebraic expression
A
.
(1 r)n
What investment in 1987 would amount to $252,000 in
2005 at 7% compounded annually? $74,557.71
107. Student loan. A college student borrowed $4,000 at 8%
compounded annually in her freshman year and did not
have to make payments until 4 years later. Use the accompanying graph to estimate the amount that she owes
at the time the payments start. Use the expression
1.5
In this
section
●
Commutative Properties
●
Associative Properties
●
Distributive Property
●
Identity Properties
●
Inverse Properties
●
Multiplication Property
of Zero
108. High cost of nursing care. The average cost for a oneyear stay in a nursing home in 1990 was $29,930 (Fortune
Investors Guide). In n years from 1990 the average cost
will be 29,930(1.05)n dollars. Find the projected cost for a
one-year stay in 2005. $62,222
109. Soaring cost of nursing care. Some economists project
that the average cost of a one-year stay in a nursing home
n years from 1990 will be 29,930(1.08)n dollars. How
much more would you pay for a one-year stay in 2005
using this expression rather than the expression in the last
exercise? $32,721
GET TING MORE INVOLVED
110. Discussion. Evaluate 5(5(5 3 6) 4) 7 and 3 53
6 52 4 5 7. Explain why these two expressions
must have the same value.
111. Cooperative learning. Find some examples of algebraic
expressions that are not mentioned in this text and explain
to your class what they are used for.
PROPERTIES OF THE REAL NUMBERS
You know that the price of a hamburger plus the price of a Coke is the same as the
price of a Coke plus the price of a hamburger. But, do you know which property of
the real numbers is at work in this situation? In arithmetic we may be unaware when
to use properties of the real numbers, but in algebra we need a better understanding
of those properties. In this section we will study the properties of the basic operations on the set of real numbers.
Commutative Properties
We get the same result whether we evaluate 3 7 or 7 3. With multiplication,
we have 4 5 5 4. These examples illustrate the commutative properties.
32
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Chapter 1
The Real Numbers
Commutative Property of Addition
For any real numbers a and b,
a b b a.
Commutative Property of Multiplication
For any real numbers a and b,
ab ba.
In writing the product of a number and a variable, it is customary to write the
number first. We write 3x rather than x3. In writing the product of two variables, it
is customary to write them in alphabetical order. We write cd rather than dc.
Addition and multiplication are commutative operations, but what about subtraction and division? Because 7 3 4 and 3 7 4, subtraction is not
commutative. To see that division is not commutative, consider the amount each
person gets when a $1 million lottery prize is divided between two people and when
a $2 prize is divided among 1 million people.
Associative Properties
study
tip
If you need help, don’t hesitate
to get it. Math has a way of
building upon the past. What
you learn today will be used
tomorrow, and what you learn
tomorrow will be used the day
after. If you don’t straighten
out problems immediately,
then you can get hopelessly
lost. If you are having trouble,
see your instructor to find
what help is available.
Consider the expression 2 3 7. Using the order of operations, we add from left
to right to get 12. If we first add 3 and 7 to get 10 and then add 2 and 10, we also get
12. So
(2 3) 7 2 (3 7).
Now consider the expression 2 3 5. Using the order of operations, we multiply
from left to right to get 30. However, we can first multiply 3 and 5 to get 15 and
then multiply by 2 to get 30. So
(2 3) 5 2 (3 5).
These examples illustrate the associative properties.
Associative Property of Addition
For any real numbers a, b, and c,
(a b) c a (b c).
Associative Property of Multiplication
For any real numbers a, b, and c,
(ab)c a(bc).
Consider the expression
4 9 8 5 8 6 13.
According to the accepted order of operations, we could evaluate this expression by
computing from left to right. However, if we use the definition of subtraction, we
can rewrite this expression as
4 (9) 8 (5) (8) 6 (13).
The commutative and associative properties of addition allow us to add these numbers in any order we choose. A good way to add them is to add the positive numbers,
1.5
Properties of the Real Numbers
(1-33) 33
add the negative numbers, and then combine the two totals:
4 8 6 (9) (5) (8) (13) 18 (35) 17
For speed we usually do not rewrite the expression. We just sum the positive numbers and sum the negative numbers, and then combine their totals.
E X A M P L E
1
Using commutative and associative properties
Evaluate.
a) 4 7 10 5
b) 6 5 9 7 2 5 8
Solution
a) 4 7 10 5 14 (12) 2
↑
Sum of the positive
numbers
↑
Sum of the negative
numbers
b) 6 5 9 7 2 5 8 18 (24) 6
■
Not all operations are associative. Using subtraction, for example, we have
(8 4) 1 8 (4 1)
because (8 4) 1 3 and 8 (4 1) 5. For division we have
(8 4) 2 8 (4 2)
because (8 4) 2 1 and 8 (4 2) 4. So subtraction and division are
not associative.
Distributive Property
helpful
hint
Imagine a parade in which
6 rows of horses are followed
by 4 rows of horses with 3
horses in each row.
There are 10 rows of 3 horses
or 30 horses, or there are 18
horses followed by 12 horses
for a total of 30 horses.
Using the order of operations, we evaluate the product 3(6 4) first by adding
6 and 4 and then multiplying by 3:
3(6 4) 3 10 30
Note that we also have
3 6 3 4 18 12 30.
Therefore
3(6 4) 3 6 3 4.
Note that multiplication by 3 from outside the parentheses is distributed over each
term inside the parentheses. This example illustrates the distributive property.
Distributive Property
For any real numbers a, b, and c,
a(b c) ab ac.
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Chapter 1
The Real Numbers
Because subtraction is defined in terms of addition, multiplication distributes over
subtraction as well as over addition. For example,
3(x 2) 3(x (2))
3x (6)
3x 6.
Because multiplication is commutative, we can write the multiplication before or
after the parentheses. For example,
(y 6)3 3(y 6)
3y 18.
The distributive property is used in two ways. If we start with the product
5(x 4) and write
5(x 4) 5x 20,
we are writing a product as a sum. We are removing the parentheses. If we start
with the difference 6x 18 and write
6x 18 6(x 3),
we are using the distributive property to write a difference as a product.
E X A M P L E
study
2
tip
Get to class early so that you
are relaxed and ready to go
when class starts. Collect your
thoughts and get your questions ready. If your instructor
arrives early, you might be
able to have your questions
answered before class. Take
responsibility for your education. Many come to learn, but
not all learn.
Using the distributive property
Use the distributive property to rewrite each sum or difference as a product and each
product as a sum or difference.
a) 9x 9
b) b(2 a)
c) 3a ac
d) 2(x 3)
Solution
a) 9x 9 9(x 1)
b) b(2 a) 2b ab Note that b 2 2b by the commutative property.
c) 3a ac a(3 c)
d) 2(x 3) 2x (2)(3)
2x (6)
2x 6
■
Identity Properties
The numbers 0 and 1 have special properties. Addition of 0 to a number does not
change the number, and multiplication of a number by 1 does not change the number.
For this reason, 0 is called the additive identity and 1 is called the multiplicative
identity.
Additive Identity Property
For any real number a,
a 0 0 a a.
Multiplicative Identity Property
For any real number a,
a 1 1 a a.
1.5
Properties of the Real Numbers
(1-35) 35
Inverse Properties
The ideas of additive inverses and multiplicative inverses were introduced in Section 1.3. Every real number a has a unique additive inverse or opposite, a, such that
a (a) 0. Every nonzero real number a also has a unique multiplicative inverse (reciprocal), written 1, such that a1 1. For rational numbers the multiplia
a
cative inverse is easy to find. For example, the multiplicative inverse of 2 is 5 because
5
2
2 5 10
1.
5 2 10
Additive Inverse Property
For any real number a, there is a unique number a such that
calculator
a (a) a a 0.
close-up
Multiplicative Inverse Property
Most scientific calculators
have a key labeled 1x, which
gives the reciprocal of the
number on the display.
Graphing calculators do not
have a reciprocal key, but
you can find reciprocals as
shown here.
For any nonzero real number a, there is a unique number 1 such that
a
1 1
a a 1.
a a
Reciprocals are used in problems involving rates. For example, if Brandon
washes one car in 1 of an hour, then he is washing cars at the rate of 11 or
3
3
3 cars
hour (3 cars per hour). If Gilda washes one car in 1 of an hour, then she is
4
washing at the rate of 11 or 4 cars
hour. In general, if one task is completed in
4
x hours, then the rate is 1 tasks
hour. If Brandon and Gilda maintain the same rates
x
when working together, then their rate together is the sum of their individual rates,
or 7 cars
hour.
E X A M P L E
3
Work rates
An old computer system can process one water bill in 0.002 hour. A new computer
system can process one water bill in 0.00125 hour. If the old system is used simultaneously with the new one, then at what rate will the processing of the water bills
be accomplished?
Solution
Since the old system does one bill in 0.002 hour, its rate is 1 bills per hour. Since
0.002
the new system does one bill in 0.00125 hour, its rate is 1 bills per hour. Their
0.00125
rate when working together is the sum of their individual rates:
1
1
1300
0.002 0.00125
They are working together at the rate of 1300 bills per hour.
■
36
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Chapter 1
The Real Numbers
Multiplication Property of Zero
Zero has a property that no other number has. Multiplication involving zero always
results in zero. It is the multiplication property of zero that prevents 0 from having
a reciprocal.
Multiplication Property of Zero
For any real number a,
0 a a 0 0.
E X A M P L E
4
Recognizing properties
Identify the property that is illustrated in each case.
1
a) 5 9 9 5
b) 3 1
3
c) 1 865 865
d) 3 (5 a) (3 5) a
e) 4x 6x (4 6)x
f) 7 (x 3) 7 (3 x)
g) 4567 0 0
h) 239 0 239
i) 8 8 0
j) 4(x 5) 4x 20
Solution
a) Commutative property of
multiplication
c) Multiplicative identity property
e) Distributive property
g) Multiplication property of zero
i) Additive inverse property
WARM-UPS
b) Multiplicative inverse property
d)
f)
h)
j)
Associative property of addition
Commutative property of addition
Additive identity property
■
Distributive property
True or false? Explain your answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Addition is a commutative operation. True
8 (4 2) (8 4) 2 False
10 2 2 10 False
5 3 3 5 False
10 (7 3) (10 7) 3 False
4(6 2) (4 6) (4 2) False
The multiplicative inverse of 0.02 is 50. True
Division is not an associative operation. True
3 2x 5x for any value of x. False
A machine that washes one car in 0.04 hour is washing at the rate of
25 cars per hour. True
1.5
1. 5
Properties of the Real Numbers
(1-37) 37
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What are the commutative properties?
The commutative property of addition says that a b b a and the commutative property of multiplication says
that a b b a.
2. What are the associative properties?
The associative property of addition says that (a b) c a (b c).
3. What is the difference between the commutative property
of addition and the associative property of addition?
The commutative property of addition says that you get the
same result when you add two numbers in either order. The
associative property of addition deals with which two
numbers are added first when adding three numbers.
4. What is the distributive property?
The distributive property says that a(b c) ab ac.
5. Why is 0 called the additive identity?
Zero is the additive identity because adding zero to a number does not change the number.
6. Why is 1 called the multiplicative identity?
One is the multiplicative identity because multiplying a
number by 1 does not change the number.
Evaluate. See Example 1.
7. 9 4 6 10 1
8. 3 4 12 9 2
9. 6 10 5 8 7 14
10. 5 11 6 9 12 2 1
11. 4 11 6 8 13 20 24
12. 8 12 9 15 6 22 3 33
13. 3.2 1.4 2.8 4.5 1.6 1.7
14. 4.4 5.1 3.6 2.3 8.1 8.7
15. 3.27 11.41 5.7 12.36 5 19.8
16. 4.89 2.1 7.58 9.06 5.34 4.03
Use the distributive property to rewrite each sum or difference
as a product and each product as a sum or difference. See
Example 2.
17. 4(x 6) 4x 24
18. 5(a 1) 5a 5
19. 2m 10 2(m 5)
20. 3y 9 3(y 3)
21. a(3 t) 3a at
22. b(y w) by bw
23. 2(w 5) 2w 10 24. 4(m 7) 4m 28
25. 2(3 y) 6 2y
26. 5(4 p) 20 5p
27. 5x 5 5(x 1)
28. 3y 3 3(y 1)
29. 1(2x y) 2x y 30. 1(4y w) 4y w
31. 3(2w 3y) 6w 9y 32. 4(x 6) 4x 24
33. 3y 15 3(y 5)
34. 5x 10 5(x 2)
35. 3a 9 3(a 3)
36. 7b 49 7(b 7)
1
37. (4x 8) 2x 4
2
1
39. (2x 4) x 2
2
1
38. (3x 6) x 2
3
1
40. (9x 3) 3x 1
3
Find the multiplicative inverse (reciprocal) of each number.
1
1
41. 2
42. 3
43. 1 1
2
3
1
1
46. 8 44. 1 1
45. 6 6
8
4
10
47. 0.25 4
48. 0.75 49. 0.7 3
7
5
10
5
50. 0.9 51. 1.8 52. 2.6 13
9
9
Use a calculator to evaluate each expression. Round
answers to four decimal places.
1
1
53. 0.6200
2.3 5.4
1
4.3
55. 0.7326
1
1
5.6 7.2
57.
58.
59.
60.
1
1
54. 0.1433
13.5 4.6
1
1
4.5 5.6
56. 0.0639
1
1
3.2 2.7
Solve each problem. See Example 3.
Fastest airliner. The world’s fastest airliner, the Concorde,
travels one mile in 0.0006897 hour and carries 128 passengers (The Doubleday Almanac). Find its rate in miles per
hour. 1,450 mph
Fastest jet plane. The U.S. Lockheed SR-71 is the world’s
fastest jet plane. The SR-71 can travel one mile in 0.000456
hour (The Doubleday Almanac). Find its rate in miles per
hour. 2,193 mph
Who’s got the button. A small clothing factory has three
workers who attach buttons. Rita, Mary, and Sam can attach a single button in 0.01 hour, 0.02 hour, and 0.015 hour,
respectively. At what hourly rate are they attaching buttons
when working simultaneously? 217 buttons per hour
Modern art. Emilio can paint the exterior of a certain
house in 36.5 hours. Alex can paint the same house in
30 hours. If they work together without interfering with
each other, then at what hourly rate will the house be
painted? 0.06 house per hour
Name the property that is illustrated in each case. See Example 4.
61. 3 x x 3 Commutative property of addition
62. x 5 5x Commutative property of multiplication
63. 5(x 7) 5x 35 Distributive property
64. a(3b) (a 3)b Associative property of multiplication
38
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
(1-38)
Chapter 1
The Real Numbers
3(xy) (3x)y Associative property of multiplication
3(x 1) 3x 3 Distributive property
4(0.25) 1 Multiplicative inverse property
0.3 9 9 0.3 Commutative property of addition
y3x xy3 Commutative property of multiplication
0 52 0 Multiplication property of zero
1 x x Multiplicative identity property
(0.1)(10) 1 Multiplicative inverse property
2x 3x (2 3)x Distributive property
8 0 8 Additive identity property
7 (7) 0 Additive inverse property
1 y y Multiplicative identity property
(36 79)0 0 Multiplication property of zero
5x 5 5(x 1) Distributive property
xy x x(y 1) Distributive property
ab 3ac a(b 3c) Distributive property
Complete each statement using the property named.
81. 5 w _____, commutative property of addition
w5
82. 2x 2 _____, distributive property
2(x 1)
83. 5(xy) ____, associative property of multiplication
(5x)y
1
84. x _____, commutative property of addition
2
1
x
2
1.6
In this
section
1
1
1
85. x _____, distributive property (x 1)
2
2
2
86. 3(x 7) _____, distributive property 3x 21
87. 6x 9 _____, distributive property 3(2x 3)
88. (x 7) 3 _____, associative property of addition
x (7 3)
89. 8(0.125) _____, multiplicative inverse property 1
90. 1(a 3) _____, distributive property a 3
91. 0 5(_____), multiplication property of zero 0
92. 8 (_____) 8, multiplicative identity property 1
93. 0.25 (_____) 1, multiplicative inverse property 4
94. 45(1) _____, multiplicative identity property 45
GET TING MORE INVOLVED
95. Discussion. Does the order in which your groceries are
placed on the checkout counter make any difference in your
total bill? Which properties are at work here?
96. Discussion. Suppose that you just bought 10 grocery items
and paid a total bill that included 6% sales tax. Would there
be any difference in your total bill if you purchased the items
one at a time? Which property is at work here?
USING THE PROPERTIES
The properties of the real numbers can be helpful when we are doing computations.
In this section we will see how the properties can be applied in arithmetic and
algebra.
●
Using the Properties in
Computation
●
Like Terms
●
Combining Like Terms
Consider the product of 36 and 200. Using the associative property of multiplication, we can write
●
Products and Quotients
(36)(200) (36)(2 100) (36 2)(100).
●
Removing Parentheses
E X A M P L E
Using the Properties in Computation
To find this product mentally, first multiply 36 by 2 to get 72, then multiply 72 by
100 to get 7200.
1
Using properties in computation
Evaluate each expression mentally by using an appropriate property.
1
c) 7 45 3 45
a) 536 25 75
b) 5 426 5