Download Operations, Properties, and Applications of Real Numbers

6.2
Operations, Properties, and Applications of Real Numbers
69. Comparing Employment Data Refer to the table in
Example 4. Of the household audio/video equipment
industry and computer/data processing services,
which will show the greater change (without regard
to sign)?
257
70. Students often say “Absolute value is always positive.” Is this true? If not, explain why.
Give three numbers between 6 and 6 that satisfy each given condition.
71. Positive real numbers but not integers
72. Real numbers but not positive numbers
73. Real numbers but not whole numbers
74. Rational numbers but not integers
75. Real numbers but not rational numbers
76. Rational numbers but not negative numbers
6.2
Operations, Properties, and Applications of
Real Numbers
Operations
The result of adding two numbers is called their sum.
Adding Real Numbers
Like Signs Add two numbers with the same sign by adding their ab-
Practical Arithmetic From the
time of Egyptian and Babylonian
merchants, practical aspects of
arithmetic complemented mystical
(or “Pythagorean”) tendencies.
This was certainly true in the time
of Adam Riese (1489 – 1559), a
“reckon master” influential when
commerce was growing in
Northern Europe. Riese’s likeness
on the stamp above comes from
the title page of one of his popular
books on Rechnung (or
“reckoning”). He championed new
methods of reckoning using
Hindu-Arabic numerals and quill
pens. (The Roman methods then in
common use moved counters on a
ruled board.) Riese thus fulfilled
Fibonacci’s efforts 300 years
earlier to supplant Roman
numerals and methods.
solute values. The sign of the sum (either or ) is the same as the sign
of the two numbers.
Unlike Signs Add two numbers with different signs by subtracting the
smaller absolute value from the larger. The sum is positive if the positive number has the larger absolute value. The sum is negative if the
negative number has the larger absolute value.
For example, to add 12 and 8, first find their absolute values:
12 12 and
8 8 .
Since 12 and 8 have the same sign, add their absolute values: 12 8 20. Give
the sum the sign of the two numbers. Since both numbers are negative, the sum is
negative and
12 8 20 .
Find 17 11 by subtracting the absolute values, since these numbers have
different signs.
17 17 and 11 11
17 11 6
Give the result the sign of the number with the larger absolute value.
17 11 6
a~ Negative since 17 11
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CHAPTER 6
The Real Numbers and Their Representations
EXAMPLE
1
Find each of the following sums.
(a) 6 3 6 3 9
(b) 12 4 12 4 16
(c) 4 1 3
The calculator supports the results
of Example 1(a), (c), and (e).
(d) 9 16 7
(e) 16 12 4
The result of subtraction is called the difference. Thus, the difference between
7 and 5 is 2. Compare the two statements below.
752
7 5 2
In a similar way, 9 3 9 3 .
That is, to subtract 3 from 9, add the additive inverse of 3 to 9. These examples
suggest the following rule for subtraction.
Definition of Subtraction
For all real numbers a and b,
a b a b .
(Change the sign of the second number and add.)
EXAMPLE
2
b
Perform the indicated operations.
Change to addition.
b Change sign of second number.
(a) 6 8 6 8 2
b
The calculator supports the results
of Example 2(a), (b), and (c).
Notice how the negative (negation)
sign differs from the minus
(subtraction) sign. There are
different keys on the calculator for
these purposes.
Change to addition.
Sign
changed.
b
(b) 12 4 12 4 16
(c) 10 7 10 7
10 7
3
(d) 15 3 5 12
This step can be omitted.
Perform the additions and subtractions in order from left to right.
15 3 5 12 15 3 5 12
18 5 12
13 12
1
The product is the result of a multiplication problem. Any rules for multiplication with negative real numbers should be consistent with the usual rules for
multiplication of positive real numbers and zero. To inductively obtain a rule
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6.2
Early ways of writing the four
basic operation symbols were
quite different from those used
today. The addition symbol shown
below was derived from the Italian
word piú (plus) in the sixteenth
century. The sign used today is
shorthand for the Latin et (and).
The subtraction symbol shown
below was used by Diophantus in
Greece sometime during the
second or third century A.D. Our
subtraction bar may be derived
from a bar used by medieval
traders to mark differences in
weights of products.
Operations, Properties, and Applications of Real Numbers
259
for multiplying a positive real number and a negative real number, observe the pattern of products below.
4 5 20
4 4 16
4 3 12
428
414
400
4 1 ?
What number must be assigned as the product 4 1 so that the pattern is
maintained? The numbers just to the left of the equality signs decrease by 1 each time,
and the products to the right decrease by 4 each time. To maintain the pattern, the
number to the right in the bottom equation must be 4 less than 0, which is 4, so
4 1 4 .
The pattern continues with
4 2 8
4 3 12
4 4 16 ,
and so on. In the same way,
In the seventeenth century, Leibniz
used the symbol below for
multiplication to avoid as too
similar to the “unknown” x. The
multiplication symbol is based
on St. Andrew’s Cross.
The division symbol shown below
was used by Gallimard in the
eighteenth century. The familiar symbol may come from the
fraction bar, embellished with the
dots above and below.
4 2 8
4 3 12
4 4 16 ,
and so on. A similar observation can be made about the product of two negative real
numbers. Look at the pattern that follows.
5 4 20
5 3 15
5 2 10
5 1 5
5 0 0
5 1 ?
The numbers just to the left of the equality signs decrease by 1 each time. The
products on the right increase by 5 each time. To maintain the pattern, the product
5 1 must be 5 more than 0, so
5 1 5 .
Continuing this pattern gives
5 2 10
5 3 15
5 4 20 ,
and so on. These observations lead to the following rules for multiplication.
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CHAPTER 6
The Real Numbers and Their Representations
Multiplying Real Numbers
Like Signs Multiply two numbers with the same sign by multiplying
their absolute values. The product is positive.
Unlike Signs Multiply two numbers with different signs by multiplying
their absolute values. The product is negative.
EXAMPLE
3
Find each of the following products.
(a) 9 7 63
(b) 14 5 70
(c) 8 4 32
The result obtained by dividing real numbers is called the quotient. For real
numbers a, b, and c, where b 0, ab c means that a b c. To illustrate this,
consider the division problem
An asterisk (*) represents
multiplication on this screen. The
display supports the results of
Example 3.
10 .
2
The value of this quotient is obtained by asking, “What number multiplied by 2
gives 10?” From our discussion of multiplication, the answer to this question must
be “5.” Therefore,
10
5 ,
2
because 2 5 10. Similar reasoning leads to the following results.
10
5
2
and
10
5
2
These facts, along with the fact that the quotient of two positive numbers is positive,
lead to the following rule for division.
Dividing Real Numbers
Like Signs Divide two numbers with the same sign by dividing their ab-
solute values. The quotient is positive.
Unlike Signs Divide two numbers with different signs by dividing their
absolute values. The quotient is negative.
EXAMPLE
The division operation is
represented by a slash (/). This
screen supports the results of
Example 4.
4
Find each of the following quotients.
(a)
15
3
5
This is true because 5 3 15.
(b)
100
4
25
(c)
60
20
3
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6.2
Operations, Properties, and Applications of Real Numbers
261
If 0 is divided by a nonzero number, the quotient is 0. That is,
0
0
a
for a 0.
This is true because a 0 0. However, we cannot divide by 0. There is a good reason for this. Whenever a division is performed, we want to obtain one and only one
quotient. Now consider the division problem
7.
0
Dividing by zero leads to this
message on the TI-83 Plus.
We must ask ourselves “What number multiplied by 0 gives 7?” There is no such
number, since the product of 0 and any number is zero. On the other hand, if we consider the quotient
0,
0
there are infinitely many answers to the question, “What number multiplied by 0
gives 0?” Since division by 0 does not yield a unique quotient, it is not permitted. To
summarize these two situations, we make the following statement.
Division by Zero
What result does the calculator
give? The order of operations
determines the answer. (See
Example 5(a).)
Division by 0 is undefined.
Given a problem such as 5 2 3, should 5 and 2 be
added first or should 2 and 3 be multiplied first? When a problem involves more than
one operation, we use the following order of operations. (This is the order used by
computers and many calculators.)
Order of Operations
The sentence “Please excuse my
dear Aunt Sally” is often used to
help remember the rule for order of
operations. The letters P, E, M, D,
A, S are the first letters of the
words of the sentence, and they
stand for parentheses, exponents,
multiply, divide, add, subtract.
(Remember also that M and D
have equal priority, as do A
and S. Operations with equal
priority are performed in order
from left to right.)
Order of Operations
If parentheses or square brackets are present:
Step 1: Work separately above and below any fraction bar.
Step 2: Use the rules below within each set of parentheses or square
brackets. Start with the innermost set and work outward.
If no parentheses or brackets are present:
Step 1: Apply any exponents.
Step 2: Do any multiplications or divisions in the order in which
they occur, working from left to right.
Step 3: Do any additions or subtractions in the order in which they
occur, working from left to right.
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CHAPTER 6
The Real Numbers and Their Representations
When evaluating an exponential expression that involves a negative sign, be
aware that an and an do not necessarily represent the same quantity. For example, if a 2 and n 6,
26 222222 64
while
Notice the difference in the two
expressions. This supports
26 26.
The base is 2.
26 2 2 2 2 2 2 64 .
EXAMPLE
5
The base is 2.
Use the order of operations to simplify each of the following.
(a) 5 2 3
First multiply, and then add.
52356
11
Multiply.
Add.
(b) 4 32 7 2 8
Work inside the parentheses first.
4 32 7 2 8 4 32 7 10
(c)
The calculator supports the results
in Example 5(a), (d), and (f).
4 9 7 10
Apply the exponent.
36 7 10
Multiply.
43 10
Add.
33
Subtract.
28 12 114 24 114
52 3
52 3
8 44
10 3
52
4
13
(d) 44 4 4 4 4 256
Work separately above and below fraction bar.
Base is 4.
(e) (4)4 4 444 256
Base is 4.
(f) 83 4 3 6 83 4 3
83 4 3
83 7
24 7
17
Properties of Addition and Multiplication of Real Numbers Several properties of addition and multiplication of real numbers that are essential to our
study in this chapter are summarized in the following box.
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6.2
Operations, Properties, and Applications of Real Numbers
263
Properties of Addition and Multiplication
For real numbers a, b, and c, the following properties hold.
Closure Properties
a b and ab are real numbers.
Commutative Properties
abba
Associative Properties
(a b) c a (b c)
(ab)c a(bc)
Identity Properties
ab ba
There is a real number 0 such that
a0a
and 0 a a .
There is a real number 1 such that
a1a
Inverse Properties
and 1 a a .
For each real number a, there is a
single real number a such that
a (a) 0 and
(a) a 0 .
For each nonzero real number a,
there is a single real number 1a
such that
a
Distributive Property of
Multiplication with Respect
to Addition
1
1 and
a
1
a 1.
a
a(b c) ab ac
(b c)a ba ca
The set of real numbers is said to be closed with respect to the operations of addition and multiplication. This means that the sum of two real numbers and the product
of two real numbers are themselves real numbers. The commutative properties state
that two real numbers may be added or multiplied in either order without affecting
the result. The associative properties allow us to group terms or factors in any manner
we wish without affecting the result. The number 0 is called the identity element for
addition, and it may be added to any real number to obtain that real number as a sum.
Similarly, 1 is called the identity element for multiplication, and multiplying a real
number by 1 will always yield that real number. Each real number a has an additive
inverse, a, such that their sum is the additive identity element 0. Each nonzero real
number a has a multiplicative inverse, or reciprocal, 1a, such that their product is
the multiplicative identity element 1. The distributive property allows us to change
certain products to sums and certains sums to products.
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CHAPTER 6
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EXAMPLE 6
Some specific examples of the properties of addition and multiplication of real numbers are given here.
(a) 5 7 is a real number.
Closure property of addition
(b) 5 6 8 5 6 8
(c) 8 0 8
No matter what values are stored in
X, Y, and Z, the commutative,
associative, and distributive
properties assure us that these
statements are true.
(d) 4 Associative property of addition
Identity property of addition
1
1
4
Inverse property of multiplication
(e) 4 3 9 4 9 3
(f) 5x y 5x 5y
Commutative property of addition
Distributive property
Applications of Real Numbers The usefulness of negative numbers can be
seen by considering situations that arise in everyday life. For example, we need negative numbers to express the temperatures on January days in Anchorage, Alaska,
where they often drop below zero. See Exercise 80, which explains the phrases “in
the red” and “in the black.” And, of course, haven’t we all experienced a checking
account balance below zero, with hopes that our deposit will make it to the bank before our outstanding checks “bounce”?
Problem Solving
When problems deal with gains and losses, the gains may be interpreted as positive numbers and the losses as negative numbers. Temperatures below 0° are negative, and those above 0° are positive. Altitudes above sea level are considered
positive and those below sea level are considered negative.
EXAMPLE 7
The Producer Price Index is the oldest continuous statistical
series published by the Bureau of Labor Statistics. It measures the average changes in
prices received by producers of all commodities produced in the United States. The bar
graph in Figure 8 gives the Producer Price Index (PPI) for construction materials
between 1993 and 2000.
Producer Price Index
(PPI)
CONSTRUCTION MATERIALS
150
145
139.6
140
135
130
128.6
142.1 141.4 142.8
144.1
133.8 133.8
125
0
1993 1994 1995 1996 1997 1998 1999 2000
Year
Source: U.S. Bureau of Labor Statistics, Producer Price
Indexes, monthly and annual.
FIGURE 8
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6.2 Operations, Properties, and Applications of Real Numbers
265
(a) Use a signed number to represent the change in the PPI from 1993 to 1994.
To find this change, we start with the index number from 1994 and subtract from
it the index number from 1993.
133.8
128.6







The 1993 index














The 1994 index
5.2
A positive number
indicates an increase.
(b) Use a signed number to represent the change in the PPI from 1997 to 1998.
Use the same procedure as in part (a).
141.4
FIGURE 9
The 1998 index
The 1997 index
141.4 142.1 .7







–80°







0°




 Difference is

 134° – (–80°).




142.1







134°
A negative number
indicates a decrease.
EXAMPLE 8
The record high temperature in the United States was 134°
Fahrenheit, recorded at Death Valley, California, in 1913. The record low was 80°F,
at Prospect Creek, Alaska, in 1971. See Figure 9. What is the difference between
these highest and lowest temperatures? (Source: The World Almanac and Book of
Facts, 2002.)
We must subtract the lower temperature from the higher temperature.
134 80 134 80
214
Use the definition of subtraction.
Add.
The difference between the two temperatures is 214°F.
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