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R
R/2
(b) Segment of a circle of radius
R, depth Ry2:
2
A 5 ~p3 2 Ï3
4 !R
R
h
(c) Frustum of cone:
V 5 13p h~R 2 1 Rr 1 r 2!
r
Conversion between fluid ounces and cubic inches:
1 quart 5 32 ounces 5 57.75 cubic inches
46. The height h and diameter d of a cylindrical can of
pineapple juice are measured: h 5 6 34 inches, d 5 4 18
inches. Find the volume in cubic inches and its equivalent in fluid ounces. Use the formula for frustum of a
cone with r 5 R. The label on the can indicates 46
ounces of pineapple juice. What is the difference between your answer and 46 ounces? Explain.
47. For a soft drink cup that is supposed to hold 44 ounces,
the top diameter is 4 38 0 and the bottom diameter is 3 380.
The height of the cup is measured as 6 340. If all measurements are accurate to the nearest 180, find the largest and
smallest possible values for the volume. Is it reasonable
to call the cup as 44-ounce cup?
48. A soft drink cup is made in the shape of a frustum of a
cone. If the cup is to have an upper diameter of 40 and
1.2
the lower diameter of 30, what should the height be if it
is to hold 32 ounces?
49. A direct mail catalog features an Oriental wok in the
shape of a section of a sphere. The catalog gives dimensions that indicate R 5 6 in., d 5 3 in. and claims that
the wok holds 2 12 qts. Assuming that the measurements
are accurate to the nearest 18 in., find the volume corresponding to
(a) R 5 5 78 in.
d 5 2 78 in.
d 5 3 18 in.
(b) R 5 6 18 in.
On the basis of your results in parts (a) and (b), is the
catalog claim of 2 12 qts reasonable? Explain.
50. A metal barrel 180 in diameter and 300 long is cut in
half to make a trough 90 deep and 300 long.
(a) Find the volume (in cubic inches) of the resulting
trough.
(b) If the diameter and length are measured accurate to
the nearest quarter-inch, find the largest and
smallest possible values for the volume (see Example 1).
51. Suppose the trough in Exercise 50 is cut down to make
a trough of depth 4.50. What percent of the volume of
the original is now in the shallower trough?
52. The box “Decimal Parts of a Mile” gives some familiar
comparison measurements for decimal parts of a mile.
Complete a similar chart for decimal parts of a kilometer.
0.1 km
0.01 km
0.001 km
0.0001 km
0.00001 km
0.000001 km
REAL NUMBERS
The complexities of modern science and modern society have created a need
for scientific generalists, for men (and women as well) trained in many fields
of science. The habits of mind and not the subject matter are what distinguish
the sciences.
Mosteller, Bode, Tukey, Winsor
Numbers occur in every phase of life. It is impossible to imagine how anyone could
function in a civilized society without having some familiarity with numbers. We
recognize that you have had considerable experience working with numbers, and
we also assume that you know something about the language and notation of sets.
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Real Numbers
11
Subsets of Real Numbers
We denote the set of real numbers by R. We make no attempt to develop the
properties and operations of R; this is reserved for more advanced courses. Several
subsets of the set of real numbers are used so frequently that we give them names.
Most of these sets are familiar. The set of natural numbers is also called the set
of positive integers or counting numbers. A prime is a positive integer greater
than 1 that is divisible only by 1 and itself. The table lists the most commonly
encountered subsets of R.
Subsets of R
Symbol and Elements
Subset
Natural numbers
Whole numbers
Integers
Even integers
Odd integers
Prime numbers
Rationals
Irrationals
I had such an amazingly
deprived high school
education. There wasn’t a
useful math book in the
library.
Bill Gosper
N
W
I
E
O
P
Q
H
5
5
5
5
5
5
5
5
$1, 2, 3, . . . %
$0, 1, 2, 3, . . . %
$ . . . , 21, 0, 1, 2, 3, . . . %
$ . . . , 22, 0, 2, 4, 6, . . . %
$ . . . , 23, 21, 1, 3, 5, . . . %
$2, 3, 5, 7, 11, 13, . . . %
p
/ 0%
$ q _ p, q [ I, q 5
$x _ x [ R and x [
/ Q%
Figure 4 indicates schematically that some of the sets listed are subsets of
others. For example, P , N, N , W, and W , I. The sets E and O together make
p
up I, so we can write E < O 5 I. Further, for any p [ I, since p 5 1, every integer
is also a rational number, so I , Q.
The existence of some irrational numbers has been known since at least the
time of the ancient Greeks, who discovered that the length of the diagonal of a
R Real numbers
Q Rational numbers
H Irrational numbers
E Even integers
I Integers
O Odd integers
W Whole numbers
N Natural numbers
P Primes
FIGURE 4
Subsets of the real numbers.
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square is not a rational multiple of the length of the sides (see Develop Mastery
Exercise 38). The length of the diagonal of a unit square is the irrational number
3
Ï2, and we recognize many others such as Ï3 2 1 and 2 1 Ï7 and p . The ratio
of the circumference of any circle to its diameter is the number p (pi), approximately 3.1416. (See the earlier Historical Note, “The Number Pi.”)
Although most of this book (and most of calculus as well) involves only real
numbers, we also make use of the set of complex numbers (see Section 1.3),
especially in Chapters 3 and 7.
cEXAMPLE 1
Set notation
(a) N , Q
(b) I > H 5 y0
(d) Ï64 [ H
(e) 41 [ P
Determine whether the statement is true.
(c) Ï5 [ Q
(f) 87 [
/ P
Solution
(a)
(b)
(c)
(d)
(e)
(f)
True; every natural number is rational.
True; every integer is rational and hence not in H.
False; Ï5 is an irrational number.
False; Ï64 5 8 and is not irrational.
True; 41 is a prime number.
True; 87 5 3 · 29, so 87 is not a prime number. b
cEXAMPLE 2
(a) P > N
Union and intersection
(b) W > Q
Simplify:
(c) Q < H
Solution
Strategy: Think about the
(a) P > N 5 P; every prime number is also a natural number.
meaning of each set (in
(b) W > Q 5 W; every whole number is also a rational number.
words). For given numbers,
(c) Q < H 5 R; every real number is rational or irrational. b
decide if each fits the description of the indicated set.
Decimal Representation of Numbers
Every real number also has a decimal “name.” For instance, the rational number
3
4 can also be written as 0.75, which is called a terminating decimal. To get the
decimal representation for the rational number 115 , we divide 5 by 11 and get the
repeating (nonterminating) decimal 0.454545. . . , which we write as 0.45. The
bar notation indicates that the block under the bar, in this instance 45, repeats
forever. A terminating decimal can also be considered as repeating. For instance,
3
4 can be named by 0.75, or by 0.750, or even by 0.749 (see Example 3).
An irrational number such as Ï2 has a nonterminating and nonrepeating
decimal representation. The distinction between repeating and nonrepeating decimals distinguishes the rational numbers from the irrationals.
Approximating Pi
As indicated in Section 1.1, the important number p occurs in problem-solving
applications as well as theoretical mathematics. In recent years sophisticated techniques have allowed computer evaluation of p to billions of decimal places, but
there is still no way to express the decimal representation of p exactly. See the
Historical Note, “Approximating the Number p .”
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Real Numbers
13
APPROXIMATING THE NUMBER p
People continued to be fascinated
p 5 3.14159 26535 89793 23846 26433
universe with an error
83279 50288 41971 69399 37510
by p even after it was shown to be
less than the radius of a single
58209 74944 59230 78164 06286
20899 86280 34825 34211 70679
irrational. In 1844 Johann Dase,
electron. People have found many
82148 08651 32823 06647 09384
who could multiply 100 digit
other reasons, in addition to the
46095 50582 23172 53594 08128
48111 74502 84102 70193 85211
numbers in his head, took months
sheer fascination of knowing, for
05559 64462 29489 54930 38196
to compute p to 205 digits. The
computing the digits of p .
44288 10975 66593 34461 28475
64823 37867 83165 27120 19091
champion at hand calculating must
Computers brought a new era.
45648 56692 34603 48610 45432
be William Shanks, who spent 20
In
1949,
a machine called ENIAC,
66482 13393 60726 02491 41273
years to grind out 707 digits. His
composed of rooms full of vacuum
record stood until 1945, when
tubes and wires, in 70 hours
A computer can calculate
D. W. Ferguson used a mechanical
computed 2037 digits of p .
these first 300 digits of p in a
More recent milestones are
calculator to find an error in
fraction of a second. The
listed
below. Remarkably, the last
Shanks’ 528th digit.
same calculation by hand
record was achieved on a
No further search for accuracy
requires months of work.
home-built super computer. You
can be justified for practical
can read more in “Ramanujan and Pi,” Scientific
purposes of distance or area computation. An
American (Feb. 1988), and in “The Mountains of
approximation to 45 digits would measure the
Pi,” The New Yorker (Mar. 12, 1992).
circumference of a circle encompassing the entire
1973
1985
1986
1987
1989
1990
1991
Jean Guilloud, M. Bouyer
R. William Gosper
David H. Bailey
Yasumasa Kanada
D. and G. Chudnovsky
Yasumasa Kanada
D. and G. Chudnovsky
CDC7600
Symbolics
Cray-2
NEC SX-2
NEC SX-2
M Zero
1
17
29
134
480
1
2.26
million
million
million
million
million
billion
billion
digits
digits
digits
digits
digits
digits
digits
The rational number 227 is sometimes used as an approximation to p , but it is
important to understand that p is not equal to 227 . Other rational number approxi355
208, 341
mations of p include 333
106 , 113 , and 66,317 (see Develop Mastery Exercises 39 and 40).
Characterizing real numbers
A real number is rational if and only if its decimal representation repeats or
terminates.
A real number is irrational if and only if its decimal representation is
nonterminating and nonrepeating.
From Decimal Representations to Quotient Form
Finding a decimal representation for a given rational number is simply a matter of
division; going the other way is more involved but is not difficult. Some graphing
calculators have built-in routines to convert decimals to fractions. Such programs
are limited because calculators must work with truncated (cut-off, finite) decimals.
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There is no way to tell a calculator that a given decimal repeats (infinitely). If we
know that a given number x has a repeating decimal representation, these steps
will give the desired rational number as a quotient.
1. Multiply x by an appropriate power of 10 to move the decimal point to
the beginning of the repeating block.
2. Multiply x by another power of 10 to move the decimal point to the
beginning of the next block.
3. The difference between these two multiples of x is an integer, which
allows us to solve for x.
cEXAMPLE 3 From decimal to quotient
tient of integers in lowest terms.
(a) 0.74
(b) 0.74
Express each number as a quo-
(c) 0.749
Solution
74
(a) From the meaning of decimal notation, 0.74 5 100
, which reduces to 37
50 . Thus
37
0.74 represents the rational number 50 .
(b) With a repeating block, we follow the procedure outlined above. Let x 5 0.74.
The decimal point is already at the beginning of the block, so multiply by 100
to move the decimal point to the beginning of the next block.
100x 5 74.74
x 5 0.74
74
.
99x 5 74, from which x 5
99
Thus 0.74 represents the rational number 74
99 . You may wish to verify this by
dividing 74 by 99.
(c)Let y 5 0.749, multiply by 1000, then by 100, and take the difference:
1000y 5 749.9
100y 5 74.9
675 3
5 .
1900y 5 675, from which y 5
900 4
Hence 0.749 represents the rational number 34 , which says that 34 has two
different decimal names, 0.749 and 0.75. Actually, every rational number that
can be written as a terminating decimal has two representations. b
Note that the procedure outlined above involves subtracting repeating decimals
as if they were finite decimals. We justify such operations in Section 8.3.
Exact Answers and Decimal Approximations
When we use a calculator to evaluate a numerical expression, in most cases the
answer is a decimal approximation of the exact answer. When we ask for a four
decimal place approximation, we mean round off the calculator display to four
decimal places.
cEXAMPLE 4 Calculator evaluation Use a calculator to get a four decimal
place value. Is the value exact or an approximation?
(a)
3
4
1
1
8
(b)
1
5
1
2
3
(c) Ï2
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Real Numbers
15
Solution
(a) 34 1 18 5 0.8750; exact decimal value.
(b) 15 1 23 < 0.866666667 < 0.8667; approximation.
(c) Ï2 < 1.414213562 < 1.4142; approximation. b
Square Roots and the Square Root Symbol
There are two numbers whose square is 2. That is, the equation x 2 5 2 has two
roots. We reserve the symbol Ï2 for the positive root, so the roots of the equation
are Ï2 and 2Ï2, which we often write as 6Ï2. For every positive x, the
Ïx
7, and we use Ïx to
calculator will display a positive number when we press 5
denote the positive number whose square is x.
cEXAMPLE 5 Calculators and rounding off
rounded off to four decimal places.
(a) 1 1 Ï3
Find an approximation
(b) Ï1 1 Ï3
Solution
(a) Using a calculator, we get 1 1 Ï3 < 2.7321.
(b) After evaluating 1 1 Ï3, take the square root to get
Ï1 1 Ï3 < 1.6529. b
EXERCISES 1.2
Check Your Understanding
Develop Mastery
Exercises 1–5 True or False. Give reasons.
1. The number p is equal to 227 .
2. The integer 119 is a prime number.
3. The intersection of the set of rational numbers and the
set of irrational numbers is the empty set.
4. The set of prime numbers is a subset of the set of odd
numbers.
5. The sum of any two odd numbers is an odd number.
Exercises 1–8 Subsets of Real Numbers Determine
whether each statement is true or false. Refer to the subsets
of R listed in this section.
Exercises 6–10 Fill in the blank so that the resulting
statement is true.
6. The product of two odd numbers is an
number.
7. When 57 is expressed as a repeating decimal, the eighth
digit after the decimal is
.
,
Ï25
2
9,
Ï64 2 14, the one
8. Of the numbers 715 , 2 24
17
that is irrational is
.
9. Of the four numbers p , Ï64 1 16, 0.564, Ï5
2 , the one
that is rational is
.
10. Of the four numbers 118 , 57 , 0.714, 0.714 the smallest one
is
.
1.
2.
3.
4.
5.
6.
7.
8.
(a)
(a)
(a)
(a)
(a)
(a)
(a)
(a)
0[N
(b) 17 [
/ P
25 [
/ N
(b) 25 [ I
$24, 3% # I
(b) $7, 81% , P
$Ï4,Ï5% , H
(b) $0.5, 0.7% , Q
I<N5I
(b) I > W 5 W
P>I5P
(b) Q < I 5 Q
Q#H
(b) H < I 5 H
P<Q5Q
(b) I > Q 5 I
Exercises 9–10 Indicate which of the subsets P, N, I, O, E,
Q, and H contain each number. For instance, 17 belongs to
P, N, I, O, and Q.
29
25
(b) Ï16
(c) Ï32
(d) 3
9. (a)
2
3
0.13
10. (a) 3.27
(b) 29
(c)
(d) 2p 2 1
1.27
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Exercises 11–14 Fraction to Decimal Express each as a
terminating decimal, or as a repeating decimal using the
bar notation.
5
5
11. (a)
(b)
8
12
73
25
(b)
12. (a)
40
33
37
10
13. (a)
(b)
45
13
16
48
14. (a)
(b)
35
65
Exercises 15–18 Decimal to Fraction Express each as a
fraction (quotient of integers) in lowest terms.
15. (a) 0.63
(b) 0.63
16. (a) 1.45
(b) 1.45
17. (a) 0.83
(b) 0.83
18. (a) 1.36
(b) 0.621
Exercises 19–21 Give a decimal approximation rounded
off to three decimal places.
Ï17
67
19. (a)
(b)
12
195
1142
(b) Ï1 1 Ï2
20. (a)
735
343
11~4 2 Ï3!
21. (a)
(b)
110
8
Exercises 22–30 Decimal Approximations Give decimal approximations rounded off to six decimal places. Do
the numbers appear to be equal?
22. Ï8; 2Ï2
23. Ï48; 4Ï3
1
24. 1 1 Ï2;
Ï2 2 1
Ï3 1 1
1
;
2
Ï3 2 1
26. Ï6 1 Ï2; 2Ï2 1 Ï3
25.
Ï3 1 Ï5 1 Ï3 2 Ï5; Ï10
28. Ï6 1 4Ï2; 2 1 Ï2
29. Ï8 1 2Ï15; Ï5 1 Ï3
30. Ï6 2 2Ï5; 1 2 Ï5
27.
31. What is the smallest nonprime positive integer greater
than 1 that has no factors less than 12?
32. What is the smallest prime number that divides
37 1 711?
Exercises 33–34 True or False. Give reasons.
33. (a) The sum of any two odd numbers is an odd number.
(b) The product of any two odd numbers is an odd
number.
(c) The product of any two consecutive positive integers is an even number.
34. (a) The sum of three consecutive even numbers is an
odd number.
(b) If a positive even integer is a perfect square, then it
is the square of an even number.
(c) If the sum of two integers is even, then both must be
even.
35. Give an example of irrational numbers for x and y that
satisfy the given condition.
(a) x 1 y is irrational.
(b) x 1 y is rational.
(c) x · y is rational.
(d) xy is rational.
36. If x 5 Ï1.5 1 Ï2 1 Ï1.5 2 Ï2, determine
whether x is rational or irrational. (Hint: Evaluate x 2.)
37. If x 5 Ï2 1 Ï3 1 Ï2 2 Ï3, determine whether x
is rational or irrational. (Hint: Evaluate x 2.)
38. Prove that Ï2 is not a rational number. (Hint: Suppose
Ï2 5 bc , where b, c [ N and bc is in lowest terms. Then
b 2 5 2c 2. Explain why b must be even. Then also explain why c must be even. This would contradict the
assumption that bc is in lowest terms.)
Exercises 39–40 Approximations for p Refer to the
number p , whose decimal form is nonterminating and nonrepeating. Rounded off to 24 decimal places,
p < 3.1415 92653 58979 32384 62643.
39. The following rational numbers are used as approximations of p . Use your calculator to evaluate and compare
each result with the given decimal approximation of p .
22
333
355
(a)
(b)
(c)
7
106
113
is
an excellent approxima40. The rational number 208,341
66,317
tion of p . Evaluate it to at least 12 decimal places and
compare the result with the approximation given above.
41. In 1991 the Chudnovsky brothers used a supercomputer they built to compute more than 2.26 billion digits
of p . Count the number of symbols in an average line of
this book and estimate how long a line of type (measured in miles) the Chudnovsky result would give. (See
the Historical Note, “Approximating the Number p .”)