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290
CHAPTER 6
The Real Numbers and Their Representations
From Example 3, we see that like radicals may be added or subtracted by adding
or subtracting their coefficients (the numbers by which they are multiplied) and
keeping the same radical. For example,
97 87 177 since 9 8 17
43 123 83 , since 4 12 8
and so on.
In the statements of the product and quotient rules for square roots, the radicands
could not be negative. While 2 is a real number, for example, 2 is not: there
is no real number whose square is 2. The same may be said for any negative radicand. In order to handle this situation, mathematicians have extended our number system to include complex numbers, discussed in the Extension at the end of this chapter.
6.4
EXERCISES
Identify each of the following as rational or irrational.
1.
4
9
2.
7
8
7. .41
6. .91
11. .878778777877778…
3. 10
4. 14
8. .32
9. 12. .434334333433334…
15. (a) Find the following sum.
5. .37
10. 0
22
14.
7
13. 3.14159
16. (a) Find the following sum.
.272772777277772…
.010110111011110…
.616116111611116…
.252552555255552…
(b) Based on the result of part (a), we can conclude
that the sum of two
numbers may be a(n)
number.
(b) Based on the result of part (a), we can conclude
that the sum of two
numbers may be a(n)
number.
Use a calculator to find a rational decimal approximation for each of the following irrational numbers. Give as
many places as your calculator shows.
17. 39
18. 44
19. 15.1
21. 884
22. 643
23.
20. 33.6
9
8
24.
6
5
Use the methods of Examples 1 and 2 to simplify each of the following expressions. Then, use a calculator to approximate both the given expression and the simplified expression. (Both should be the same.)
25. 50
31.
5
6
26. 32
32.
3
2
27. 75
33.
7
4
28. 150
34.
8
9
29. 288
35.
7
3
30. 200
36.
14
5
Use the method of Example 3 to perform the indicated operations.
37. 17 217
38. 319 19
39. 57 7
40. 327 27
41. 318 2
42. 248 3
43. 12 75
44. 227 300
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6.4
Irrational Numbers and Decimal Representation
Each of the following exercises deals with . Use a calculator as necessary.
45. Move one matchstick to make the equation approximately true. (Source: http://www.joyofpi.com)
291
tance, D, in miles, to the horizon from an observer’s
point of view over water or “flat” earth is given by
D 2H ,
where H is the height of the point of view, in feet. If
a person whose eyes are 6 feet above ground level is
standing at the top of a hill 28 feet above the “flat”
earth, approximately how far to the horizon will she
be able to see? Round your answer to the nearest
tenth of a mile.
6 feet
Horizon
28 feet
46. Find the square root of 214322 using a calculator.
Then find the square root of that result. Compare
your result to the decimal given for in the margin
note. What do you notice?
47. Find the first eight digits in the decimal for 355113.
Compare the result to the decimal for given in the
margin note. What do you notice?
48. You may have seen the statements “use 227 for ”
and “use 3.14 for .” Since 227 is the quotient of
two integers, and 3.14 is a terminating decimal, do
these statements suggest that is rational?
Solve each problem. Use a calculator as necessary, and
give approximations to the nearest tenth unless specified otherwise.
49. Period of a Pendulum The
period of a pendulum in seconds depends on its length, L, in
feet, and is given by the formula
P 2
2P
L
relates the coefficient of self-induction L (in henrys),
the energy P stored in an electronic circuit (in joules),
and the current I (in amps). Find I if P 120 joules
and L 80 henrys.
53. Area of the Bermuda Triangle Heron’s formula
gives a method of finding the area of a triangle if the
lengths of its sides are known. Suppose that a, b, and
c are the lengths of the sides. Let s denote one-half
of the perimeter of the triangle (called the semiperimeter); that is,
1
a b c .
2
Then the area A of the triangle is given by
A ss a s b s c .
If a pendulum is 5.1 feet long, what is its period? Use
3.14 for .
r
I
s
L
L
.
32
50. Radius of an Aluminum Can The
radius of the circular top or bottom
of an aluminum can with surface
area S and height h is given by
52. Electronics Formula The formula
r
h
h h2 .64S
.
2
What radius should be used to make a can with
height 12 inches and surface area 400 square inches?
51. Distance to the Horizon According to an article
in The World Scanner Report (August 1991), the dis-
Find the area of the Bermuda Triangle, if the “sides”
of this triangle measure approximately 850 miles,
925 miles, and 1300 miles. Give your answer to the
nearest thousand square miles.
54. Area Enclosed by the Vietnam Veterans’ Memorial
The Vietnam Veterans’ Memorial in Washington,
D.C., is in the shape of an unenclosed isosceles
triangle with equal sides of length 246.75 feet. If the
triangle were enclosed, the third side would have
length 438.14 feet. Use Heron’s formula from the
previous exercise to find the area of this enclosure to
the nearest hundred square feet. (Source: Information pamphlet obtained at the Vietnam Veterans’
Memorial.)
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292
CHAPTER 6
The Real Numbers and Their Representations
55. Diagonal of a Box The length of the diagonal of a
box is given by
D L2 W 2 H 2 ,
where L, W, and H are the length, the width, and the
height of the box. Find the length of the diagonal, D,
of a box that is 4 feet long, 3 feet high, and 2 feet
wide.
D
H
W
L
56. Rate of Return of an Investment If an investment of
P dollars grows to A dollars in two years, the annual
rate of return on the investment is given by
r
A P
P
.
First rationalize the denominator and then find the
annual rate of return (as a decimal) if $50,000 increases to $58,320.
57. Accident Reconstruction Police sometimes use the
following procedure to estimate the speed at which a
car was traveling at the time of an accident. A police
officer drives the car involved in the accident under
conditions similar to those during which the accident
took place and then skids to a stop. If the car is
driven at 30 miles per hour, then the speed at the time
of the accident is given by
s 30
a
,
p
where a is the length of the skid marks left at the
time of the accident and p is the length of the skid
marks in the police test. Find s for the following values of a and p.
(a) a 862 feet; p 156 feet
(b) a 382 feet; p 96 feet
(c) a 84 feet; p 26 feet
d
x
How far can one see to the horizon in a plane flying
at the following altitudes? Round to the nearest tenth.
(a) 15,000 feet
(b) 18,000 feet
(c) 24,000 feet
The concept of square (second) root can be extended to
cube (third) root, fourth root, and so on. If n 2 and
n
a is a nonnegative number, a represents the nonnegative number whose nth power is a. For example,
3
8 2 because 23 8,
3
1000 10 because 103 1000,
4
81 3 because 34 81,
and so on. Find each of the following roots.
3
64
59. 3
125
60. 3
61. 343
3
62. 729
3
63. 216
3
64. 512
65. 1
66. 16
67. 256
68. 625
69. 4096
70. 2401
4
4
4
4
4
4
Use a calculator to approximate each root. Give as
many places as your calculator shows.
3
71. 43
3
72. 87
3
73. 198
74. 2107
75. 10,265.2
76. 863.5
4
4
4
Solve each problem.
58. Distance to the Horizon A formula for calculating
the distance, d, one can see from an airplane to the
horizon on a clear day is
77. Threshold Weight The threshold weight, T, for a person is the weight above which the risk of death
increases greatly. In one instance, a researcher found
that the threshold weight in pounds for men aged
40–49 is related to height, h, in inches by the equation
d 1.22x ,
3
h 12.3 T.
where x is the altitude of the plane in feet and d is
given in miles.
What height, in feet, corresponds to a threshold of
180 pounds for a 43-year-old man?
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6.5 Applications of Decimals and Percents
78. Law of Tensions In the study of sound, one version
of the law of tensions is
f1 f2
F1
.
F2
Find f1 to the nearest unit if F1 300, F2 60, and
f2 260.
Exercises 79 and 80 investigate the concept of rational
number exponents.
79. In algebra the expression a1/2 is defined to be a for
nonnegative values of a. Use a calculator with an exponential key to evaluate each of the following, and
compare with the value obtained with the square root
key. (Both should be the same.)
(a) 21/2
(b) 7 1/2
(c) 13.21/2
(d) 251/2
293
80. Based on Exercises 59–76, answer the following.
(a) How would you expect the expression a1/3 to be
defined as a radical?
(b) Use a calculator with a cube root key to approx3
imate 16.
(c) Use a calculator with an exponential key to
approximate 161/3.
Exercises 81 and 82 investigate the irrational number e.
81. Use a calculator with an exponential key to find values for the following: 1.110, 1.01100, 1.0011000,
1.000110,000, and 1.00001100,000. Compare your
results to the approximation given for the irrational
number e in the margin note in this section. What do
you find?
82. Use a calculator with a key that determines powers
of e to find each of the following.
(a) e2
(b) e3
(c) e
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Perhaps the most frequent use of mathematics in everyday life concerns operations
with decimal numbers and the concept of percent. When we use dollars and cents,
we are dealing with decimal numbers. Sales tax on purchases made at the grocery
store is computed using percent. The educated consumer must have a working
knowledge of decimals and percent in financial matters. Look at any newspaper and
you will see countless references to percent and percentages.
Operations with Decimals Because calculators have, for the most part, replaced paper-and-pencil methods for operations with decimals and percent, we will
only briefly mention these latter methods. We strongly suggest that the work in this
section be done with a calculator at hand.
Addition and Subtraction of Decimals
To add or subtract decimal numbers, line up the decimal points in a column and perform the operation.
EXAMPLE
1
(a) To compute the sum .46 3.9 12.58, use the following method.
This screen supports the results in
Example 1.
.46
3.9
12.58
16.94
Line up decimal points.
k Sum