254
CHAPTER 6
The Real Numbers and Their Representations
Applications
A table of data provides a concise way of relating information.
EXAMPLE 4
The projected annual rates of employment change (in percent)
in some of the fastest growing and most rapidly declining industries from 1994
through 2005 are shown in the table.
Industry (1994 –2005)
Percent Rate
of Change
Health services
5.7
Computer and data processing services
4.9
Child day care services
4.3
Footware, except rubber and plastic
6.7
Household audio and video equipment
4.2
Luggage, handbags, and leather products
3.3
Source: U.S. Bureau of Labor Statistics.
What industry in the list is expected to see the greatest change? the least change?
We want the greatest change, without regard to whether the change is an increase or a decrease. Look for the number in the list with the largest absolute value.
That number is found in footware, since 6.7 6.7. Similarly, the least change is
in the luggage, handbags, and leather products industry: 3.3 3.3.
6.1
EXERCISES
In Exercises 1 – 6, give a number that satisfies the given condition.
1. An integer between 3.5 and 4.5
2. A rational number between 3.8 and 3.9
3. A whole number that is not positive and is less than 1
4. A whole number greater than 4.5
5. An irrational number that is between 11 and 13
6. A real number that is neither negative nor positive
In Exercises 7 – 10, decide whether each statement is true or false.
7. Every natural number is positive.
8. Every whole number is positive.
9. Every integer is a rational number.
10. Every rational number is a real number.
In Exercises 11 and 12, list all numbers from each set that are (a) natural numbers; (b) whole numbers;
(c) integers; (d) rational numbers; (e) irrational numbers; (f ) real numbers.
11.
9, 7, 1
3
1
, , 0, 5, 3, 5.9, 7
4
5
13. Explain in your own words the different sets of numbers introduced in this section, and give an example
of each kind.
12.
5.3, 5, 3, 1, 1
, 0, 1.2, 1.8, 3, 11
9
14. What two possible situations exist for the decimal
representation of a rational number?
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6.1
Real Numbers, Order, and Absolute Value
255
Use an integer to express each number representing a change in the following applications.
15. Population of Michigan Between 1990 and 1999,
the population of Michigan increased by 568,488.
(Source: U.S. Bureau of the Census.)
16. Height of the Sears Tower The Sears Tower in
Chicago is 1450 feet high. (Source: Council on Tall
Buildings and Urban Habitat.)
17. Boiling Point of Chlorine The boiling point of chlorine is approximately 30 below 0 Fahrenheit.
18. Height of Mt. Arenal The height of Mt. Arenal, an
active volcano in Costa Rica, is 5436 feet above sea
level. (Source: The Universal Almanac, 1997, John
W. Wright, General Editor.)
19. Population of District of Columbia Between 1990
and 2000, the population of the District of Columbia
decreased by 34,841. (Source: U.S. Bureau of the
Census.)
20. Auto Production in Taiwan In a recent year, the
country of Taiwan produced 159,376 more passenger
cars than commercial vehicles. (Source: American
Automobile Manufacturers Association.)
21. Elevation of New Orleans The city of New Orleans
lies 8 feet below sea level. (Source: U.S. Geological
Survey, Elevations and Distances in the United
States, 1990.)
22. Windchill When the wind speed is 20 miles per
hour and the actual temperature is 25 Fahrenheit,
the windchill factor is 3 below 0 Fahrenheit. (Give
three responses.)
23. Depths and Heights of Seas and Mountains The chart gives selected depths and heights of bodies of water
and mountains.
Bodies of Water
Pacific Ocean
Average Depth in Feet
(as a negative number)
Mountains
Altitude in Feet
(as a positive number)
12,925
McKinley
20,320
South China Sea
4802
Point Success
14,150
Gulf of California
2375
Matlalcueyetl
14,636
Caribbean Sea
8448
Ranier
14,410
12,598
Steele
16,644
Indian Ocean
Source: The World Almanac and Book of Facts.
(a) List the bodies of water in order, starting with the deepest and ending with the shallowest.
(b) List the mountains in order, starting with the lowest and ending with the highest.
(c) True or false: The absolute value of the depth of the Pacific Ocean is greater than the absolute value of the
depth of the Indian Ocean.
(d) True or false: The absolute value of the depth of the Gulf of California is greater than the absolute value of
the depth of the Caribbean Sea.
(a) Which country has the largest average
monthly savings? the smallest?
(b) Which countries have average monthly
savings greater than $200?
(c) Estimate the average monthly savings
for Japan and Italy.
(d) Find the difference between average
monthly savings in Japan and Italy.
SAVINGS AROUND THE WORLD
Monthly Savings
(in dollars)
24. Savings for Retirement Plans The bar graph
compares average monthly savings, including retirement plans, for five countries.
400
300
200
100
0
U.S.
Germany
Italy
Country
Japan
Source: Taylor Nelson-Sofres for American Express.
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India
256
CHAPTER 6
The Real Numbers and Their Representations
Graph each group of numbers on a number line.
25. 2, 6, 4, 3, 4
27.
26. 5, 3, 2, 0, 4
1 1
4
5
, 2 , 3 , 4, 1
4 2
5
8
28. 5
29. Match each expression in Column I with its value in
Column II. Some choices in Column II may not be
used.
I
(a)
(b)
(c)
(d)
7
7
7
7
II
A.
B.
C.
D.
7
7
neither A nor B
both A and B
1
5
1
2
, 4 , 2 , 0, 3
4
9
3
5
30. Fill in the blanks with the correct values: The opposite
of 2 is
, while the absolute value of 2 is
. The additive inverse of 2 is
, while
the additive inverse of the absolute value of 2 is
.
Find (a) the additive inverse (or opposite) of each number and (b) the absolute value of each number.
31. 2
32. 8
33. 6
34. 11
35. 7 4
36. 8 3
37. 7 7
38. 3 3
39. Look at Exercises 35 and 36 and use the results to
complete the following: If a b 0, then the absolute value of a b in terms of a and b is
.
40. Look at Exercises 37 and 38 and use the results to
complete the following: If a b 0, then the absolute value of a b is
.
Select the lesser of the two given numbers.
41. 12, 4
42. 9, 14
43. 8, 1
44. 15, 16
45. 3, 4
46. 5, 2
47. 3, 4
48. 8, 9
49. 6, 4
50. 2, 3
51. 5 3, 6 2
52. 7 2, 8 1
Decide whether each statement is true or false.
53. 6 2
54. 8 2
55. 4 5
56. 6 3
57. 6 9
58. 12 20
59. 8 9
60. 12 15
61. 5 9
62. 12 15
63. 6 5 6 2
64. 13 8 7 4
Producer Price Index To answer the questions in Exercises 65–68, refer to the table, which gives the changes in
producer price indexes for two recent years.
65. What commodity for which years represents the greatest decrease?
66. What commodity for which years represents the least change?
67. Which has the smaller absolute value, the
change for video/audio equipment from
1995 to 1996 or from 1996 to 1997?
68. Which has greater absolute value, the
change for apparel from 1995 to 1996 or
from 1996 to 1997?
Commodity
Change from
1995 to 1996
Change from
1996 to 1997
Food
4.9
4.0
Transportation
3.9
1.3
.5
.9
2.6
2.2
5.3
5.3
Apparel
Video/Audio equipment
Shelter
Source: U.S. Bureau of Labor Statistics.
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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
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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