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270
CHAPTER 6
6.3
The Real Numbers and Their Representations
Rational Numbers and Decimal Representation
Properties and Operations
The set of real numbers is composed of two
important mutually exclusive subsets: the rational numbers and the irrational numbers. (Two sets are mutually exclusive if they contain no elements in common.)
Recall from Section 6.1 that quotients of integers are called rational numbers.
Think of the rational numbers as being made up of all the fractions (quotients of
integers with denominator not equal to zero) and all the integers. Any integer can
be written as the quotient of two integers. For example, the integer 9 can be written
as the quotient 91, or 182, or 273, and so on. Also, 5 can be expressed as a quotient of integers as 51 or 102, and so on. (How can the integer 0 be written as
a quotient of integers?) Since both fractions and integers can be written as quotients
of integers, the set of rational numbers is defined as follows.
Rational Numbers
Rational numbers x x is a quotient of two integers, with denominator not 0
A rational number is said to be in lowest terms if the greatest common factor of
the numerator (top number) and the denominator (bottom number) is 1. Rational
numbers are written in lowest terms by using the fundamental property of rational
numbers.
Fundamental Property of Rational Numbers
If a, b, and k are integers with b 0 and k 0, then
ak
a
.
bk
b
EXAMPLE 1
Write 3654 in lowest terms.
Since the greatest common factor of 36 and 54 is 18,
36 2 18
2
.
54 3 18
3
The calculator reduces 3654 to
lowest terms, as illustrated in
Example 1.
36
2
In the above example, 54 3. If we multiply the numerator of the fraction on
the left by the denominator of the fraction on the right, we obtain 36 3 108.
If we multiply the denominator of the fraction on the left by the numerator of the
fraction on the right, we obtain 54 2 108. The result is the same in both cases.
One way of determining whether two fractions are equal is to perform this test. If the
product of the “extremes” (36 and 3 in this case) equals the product of the “means”
(54 and 2), the fractions are equal. This test for equality of rational numbers is called
the cross-product test.
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6.3
Rational Numbers and Decimal Representation
271
Cross-Product Test for Equality of Rational Numbers
For rational numbers ab and cd,
a
c
b
d
Benjamin Banneker
(1731 – 1806) spent the first half of
his life tending a farm in Maryland.
He gained a reputation locally for
his mechanical skills and abilities
in mathematical problem solving.
In 1772 he acquired astronomy
books from a neighbor and devoted
himself to learning astronomy,
observing the skies, and making
calculations. In 1789 Banneker
joined the team that surveyed what
is now the District of Columbia.
Banneker published almanacs
yearly from 1792 to 1802. He sent
a copy of his first almanac to
Thomas Jefferson along with an
impassioned letter against slavery.
Jefferson subsequently championed
the cause of this early AfricanAmerican mathematician.
b 0, d 0,
if and only if
a d b c.
The operation of addition of rational numbers can be illustrated by the sketches
in Figure 10. The rectangle at the top left is divided into three equal portions, with
one of the portions in color. The rectangle at the top right is divided into five equal
parts, with two of them in color.
The total of the areas in color is represented by the sum
2
1
.
3
5
To evaluate this sum, the areas in color must be redrawn in terms of a common unit.
Since the least common multiple of 3 and 5 is 15, redraw both rectangles with
15 parts. See Figure 11. In the figure, 11 of the small rectangles are in color, so
1
2
5
6
11
.
3
5
15 15 15
In general, the sum
a
c
b
d
may be found by writing ab and cd with the common denominator bd, retaining
this denominator in the sum, and adding the numerators:
c
ad bc ad bc
a
.
b
d
bd bd
bd
1_
3
2_
5
1_ + 2_
3
5
FIGURE 10
5
__
15
6
__
15
5 + __
6 = 11
__
__
15 15 15
FIGURE 11
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CHAPTER 6
The Real Numbers and Their Representations
A similar case can be made for the difference between rational numbers. A formal
definition of addition and subtraction of rational numbers follows.
Adding and Subtracting Rational Numbers
If ab and cd are rational numbers, then
a
c
ad bc
and
b
d
bd
a
c
ad bc
.
b
d
bd
This formal definition is seldom used in practice. In practical problems involving addition and subtraction of rational numbers, we usually rewrite the fractions
with the least common multiple of their denominators, called the least common
denominator.
EXAMPLE
2
(a) Add:
2
1
.
15 10
The least common multiple of 15 and 10 is 30. Now write 215 and 110 with
denominators of 30, and then add the numerators. Proceed as follows:
The results of Example 2 are
illustrated in this screen.
Since 30 15 2,
2
22
4
,
15 15 2 30
and since 30 10 3,
1
13
3
.
10 10 3 30
2
1
4
3
7
.
15 10 30 30 30
Thus,
(b) Subtract:
69
173
.
180 1200
The least common multiple of 180 and 1200 is 3600.
69
3460
207
3460 207 3253
173
180 1200 3600 3600
3600
3600
The product of two rational numbers is defined as follows.
Multiplying Rational Numbers
If ab and cd are rational numbers, then
a
c
ac
.
b
d
bd
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6.3
EXAMPLE
To illustrate the results of
Example 3, we use parentheses
around the fraction factors.
3
Rational Numbers and Decimal Representation
273
Find each of the following products.
(a)
37
21
3 7
4 10 4 10 40
(b)
5 3
53
15
1 15
1
18 10 18 10 180 12 15 12
In practice, a multiplication problem such as this is often solved by using slash
marks to indicate that common factors have been divided out of the numerator
and denominator.
1
1
5 3
1 1
18 10 6 2
6
3 is divided out of the terms 3 and 18;
5 is divided out of 5 and 10.
2
1
12
FOR FURTHER THOUGHT
The Influence of Spanish Coinage on
Stock Prices
Until August 28, 2000, when decimalization of
the U.S. stock market began, market prices were
reported with fractions having denominators
5
3
with powers of 2, such as 17 and 112 . Did
4
8
you ever wonder why this was done?
During the early years of the United States,
prior to the minting of its own coinage, the
Spanish eight-reales coin, also known as the
Spanish milled dollar, circulated freely in the
states. Its fractional parts, the four reales, two
reales, and one real, were known as pieces of
eight, and described as such in pirate and
treasure lore. When the New York Stock
Exchange was founded in 1792, it chose to use
the Spanish milled dollar as its price basis,
rather than the decimal base as proposed by
Thomas Jefferson that same year.
In the September 1997 issue of COINage, Tom
Delorey’s article “The End of ‘Pieces of Eight’”
gives the following account:
by the time the Spanish-American money
was withdrawn in 1857, pricing stocks in
eighths of a dollar—and no less—was a
tradition carved in stone. Being somewhat a
conservative organization, the NYSE saw no
need to fix what was not broken.
All prices on the U.S. stock markets are now
reported in decimals. (Source: “Stock price
tables go to decimal listings,” The Times Picayune,
June 27, 2000.)
For Group Discussion
Consider this: Have you ever heard this old
cheer? “Two bits, four bits, six bits, a dollar. All
for the (home team), stand up and holler.” The
term two bits refers to 25 cents. Discuss how this
cheer is based on the Spanish eight-reales coin.
As the Spanish dollar and its fractions
continued to be legal tender in America
alongside the decimal coins until 1857, there
was no urgency to change the system—and
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CHAPTER 6
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In a fraction, the fraction bar indicates the operation of division. Recall that, in
the previous section, we defined the multiplicative inverse, or reciprocal, of the
nonzero number b. The multiplicative inverse of b is 1b. We can now define division using multiplicative inverses.
Definition of Division
If a and b are real numbers, b 0, then
a
1
a .
b
b
Early U.S. cents and half cents
used fractions to denote their
denominations. The half cent used
1
1
and the cent used
. (See
200
100
Exercise 18 for a photo of an
interesting error coin.)
The coins shown here were
part of the collection of Louis E.
Eliasberg, Sr., that was auctioned
by Bowers and Merena, Inc.,
several years ago. Louis Eliasberg
was the only person ever to
assemble a complete collection of
United States coins. The half cent
pictured sold for $506,000 and the
cent sold for $27,500. The cent
shown in Exercise 18 went for a
mere $2970.
You have probably heard the rule, “To divide fractions, invert the divisor and
multiply.” But have you ever wondered why this rule works? To illustrate it, suppose
that you have 78 of a gallon of milk and you wish to find how many quarts you
have. Since a quart is 14 of a gallon, you must ask yourself, “How many 14s are
there in 78?” This would be interpreted as
7
1
7
8
or
.
8
4
1
4
The fundamental property of rational numbers discussed earlier can be extended
to rational number values of a, b, and k. With a 78, b 14, and k 4 (the
reciprocal of b 14),
7
7
4
4
a
ak
8
8
7 4
.
b
bk
1
1
8 1
4
4
Now notice that we began with the division problem 78 14 which, through a
series of equivalent expressions, led to the multiplication problem 78 41. So
dividing by 14 is equivalent to multiplying by its reciprocal, 41. By the definition
of multiplication of fractions,
7 4
28
7
,
8 1
8
2
and thus there are 72 or 3 12 quarts in 78 gallon.*
We now state the rule for dividing ab by cd.
Dividing Rational Numbers
If ab and cd are rational numbers, where cd 0, then
a
c
a d
ad
.
b
d
b c
bc
1
*3 2 is a mixed number. Mixed numbers are covered in the exercises for this section.
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6.3
EXAMPLE
This screen supports the results in
Example 4(b) and (c).
4
Rational Numbers and Decimal Representation
275
Find each of the following quotients.
(a)
7
3 15 45 9 5
9
3
5
15
5 7
35 7 5
7
(b)
4
3
4 14 56 8 7 8
8
7
14
7
3
21
37
3
3
1
2
1
2
4
2 1
2 1
(c)
4 9
9
1
9 4
9 4
18
2
There is no integer between two consecutive integers, such as 3 and 4. However,
a rational number can always be found between any two distinct rational numbers.
For this reason, the set of rational numbers is said to be dense.
Density Property of the Rational Numbers
If r and t are distinct rational numbers, with r t, then there exists a
rational number s such that
r s t.
To find the arithmetic mean, or average, of n numbers, we add the numbers
and then divide the sum by n. For two numbers, the number that lies halfway between them is their average.
Year
Number
(in thousands)
1995
16,360
1996
16,269
1997
16,110
First, find their sum.
1998
16,211
Now divide by 2.
1999
16,477
2000
16,258
Source: U.S. Bureau of
Labor Statistics.
EXAMPLE
5
(a) Find the rational number halfway between 23 and 56.
2
5
4
5
9 3
3
6
6
6
6 2
3 1
3
3
2 2
2 2
4
The number halfway between 23 and 56 is 34.
(b) The table in the margin shows the number of labor union or employee association members, in thousands, for the years 1995–2000. What is the average number, in thousands, for this six-year period?
To find this average, divide the sum by 6.
16,360 16,269 16,110 16,211 16,477 16,258 97,685
6
6
The computation in Example 5(b)
is shown here.
16,280.8
The average to the nearest whole number of thousands is 16,281.
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CHAPTER 6
The Real Numbers and Their Representations
Repeated application of the density property implies that between two given
rational numbers are infinitely many rational numbers. It is also true that between
any two real numbers there is another real number. Thus, we say that the set of real
numbers is dense.
Ten
s
On
es
De
cim
al p
oin
t
Ten
ths
Hu
ndr
edt
hs
Tho
usa
ndt
hs
Ten
-th
ous
and
ths
Hu
ndr
edtho
usa
Mil
ndt
lion
hs
ths
...
Hu
ndr
eds
Simon Stevin (1548 – 1620)
worked as a bookkeeper in
Belgium and became an engineer
in the Netherlands army. He is
usually given credit for the
development of decimals.
Decimal Form of Rational Numbers Up to now in this section, we have
discussed rational numbers in the form of quotients of integers. Rational numbers
can also be expressed as decimals. Decimal numerals have place values that are
powers of 10. For example, the decimal numeral 483.039475 is read “four hundred
eighty-three and thirty-nine thousand, four hundred seventy-five millionths.” The
place values are as shown here.
4
8
3
.
3
0
...
9
4
7
5
Given a rational number in the form ab, it can be expressed as a decimal most
easily by entering it into a calculator. For example, to write 38 as a decimal, enter
3, then enter the operation of division, then enter 8. Press the equals key to find the
following equivalence.
.375
83.000
24
60
56
40
40
0
.3636 . . .
114.00000 . . .
33
70
66
40
33
Of course, this same result may be obtained by long division, as shown in the
margin. By this result, the rational number 38 is the same as the decimal .375. A
decimal such as .375, which stops, is called a terminating decimal. Other examples
of terminating decimals are
1
.25 ,
4
70
66
40
.
3
.375
8
.
.
7
.7 , and
10
89
.089 .
1000
Not all rational numbers can be represented by terminating decimals. For example,
convert 411 into a decimal by dividing 11 into 4 using a calculator. The display shows
.3636363636,
or perhaps
.363636364.
However, we see that the long division process, shown in the margin, indicates that
we will actually get .3636 . . . , with the digits 36 repeating over and over indefinitely. To indicate this, we write a bar (called a vinculum) over the “block” of digits
that repeats. Therefore, we can write
While 23 has a repeating decimal
representation 23 .6 , the
calculator rounds off in the final
decimal place displayed.
4
.36 .
11
A decimal such as .36, which continues indefinitely, is called a repeating decimal.
Other examples of repeating decimals are
5
.45,
11
1
.3, and
3
5
.83.
6
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6.3
Although only ten decimal digits
are shown, all three fractions have
decimals that repeat endlessly.
Rational Numbers and Decimal Representation
277
Because of the limitations of the display of a calculator, and because some rational numbers have repeating decimals, it is important to be able to interpret calculator results accordingly when obtaining repeating decimals.
While we shall distinguish between terminating and repeating decimals in this
book, some mathematicians prefer to consider all rational numbers as repeating decimals. This can be justified by thinking this way: if the division process leads to a remainder of 0, then zeros repeat without end in the decimal form. For example, we
can consider the decimal form of 34 as follows.
3
.750
4
By considering the possible remainders that may be obtained when converting
a quotient of integers to a decimal, we can draw an important conclusion about the
decimal form of rational numbers. If the remainder is never zero, the division will
produce a repeating decimal. This happens because each step of the division process
must produce a remainder that is less than the divisor. Since the number of different
possible remainders is less than the divisor, the remainders must eventually begin to
repeat. This makes the digits of the quotient repeat, producing a repeating decimal.
To find a baseball player’s
batting average, we divide the
number of hits by the number of
at-bats. A surprising paradox
exists concerning averages; it is
possible for Player A to have a
higher batting average than Player
B in each of two successive years,
yet for the two-year period, Player
B can have a higher total batting
average. Look at the chart.
Decimal Representation of Rational Numbers
Any rational number can be expressed as either a terminating decimal
or a repeating decimal.
To determine whether the decimal form of a quotient of integers will terminate
or repeat, we use the following rule.
Criteria for Terminating and Repeating Decimals
Year Player A
Player B
20
.500
1998
40
60
.300
1999
200
90
.450
200
10
.250
40
A rational number ab in lowest terms results in a terminating decimal
if the only prime factor of the denominator is 2 or 5 (or both).
A rational number ab in lowest terms results in a repeating decimal
if a prime other than 2 or 5 appears in the prime factorization of the
denominator.
Two 80
100
year .333
.417
240
total 240
In both individual years, Player A
had a higher average, but for the
two-year period, Player B had the
higher average. This is an example
of Simpson’s paradox from
statistics.
The justification of this rule is based on the fact that the prime factors of 10 are 2
and 5, and the decimal system uses ten as its base.
EXAMPLE 6
Without actually dividing, determine whether the decimal
form of the given rational number terminates or repeats.
(a)
7
8
Since 8 factors as 2 3, the decimal form will terminate. No primes other than 2
or 5 divide the denominator.
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CHAPTER 6
The Real Numbers and Their Representations
(b)
(c)
1=
.9 9 9 9 9
99
99
Terminating or Repeating?
One of the most baffling truths of
elementary mathematics is the
following:
1 .9999 . . . .
Most people believe that .9 has to
be less that 1, but this is not the
case. The following argument
shows otherwise. Let
x .9999 . . . . Then
10x 9.9999 . . .
x .9999. . .
9x 9
x 1.
6
75
First write the rational number in lowest terms.
6
2
75 25
Since 25 52, the decimal form will terminate.
We have seen that a rational number will be represented by either a terminating
or a repeating decimal. What about the reverse process? That is, must a terminating
decimal or a repeating decimal represent a rational number? The answer is yes. For
example, the terminating decimal .6 represents a rational number.
3
6
.6 10
5
The results of Example 7 are
supported in this screen.
9 99
13
150
150 2 3 52. Since 3 appears as a prime factor of the denominator, the decimal form will repeat.
9
.
..
EXAMPLE
(a) .437 7
437
1000
Write each terminating decimal as a quotient of integers.
(b) 8.2 8 82 41
2
10 10
5
Repeating decimals cannot be converted into quotients of integers quite so
quickly. The steps for making this conversion are given in the next example. (This
example uses basic algebra.)
EXAMPLE
8
Find a quotient of two integers equal to .85.
Step 1: Let x .85, so x .858585 . . . .
Step 2: Multiply both sides of the equation x .858585 . . . by 100. (Use 100
since there are two digits in the part that repeats, and 100 10 2.)
x .858585 . . .
100x 100.858585 . . .
100x 85.858585 . . .
Step 3: Subtract the expressions in Step 1 from the final expressions in Step 2.
Subtract.
Therefore, 1 .9999 . . . .
Similarly, it can be shown that any
terminating decimal can be
represented as a repeating decimal
with an endless string of 9s. For
example, .5 .49999 . . . and
2.6 2.59999 . . . . This is a way
of justifying that any rational
number may be represented as a
repeating decimal.
See Exercises 95 and 96 for
more on .999 . . . 1.
100x 85.858585 . . .
x .858585 . . .
99x 85
(Recall that x 1x and
100x x 99x.
Step 4: Solve the equation 99x 85 by dividing both sides by 99.
99x 85
99x 85
99
99
85
x
99
85
.85 99
x .85
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6.3 Rational Numbers and Decimal Representation
279
This result may be checked with a calculator. Remember, however, that the calculator will only show a finite number of decimal places, and may round off in the final
decimal place shown.
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1
8
4
.50
.55
.6
20
30
.249
25
100
.666…
5
9
1
4
.5
9
5
1
10
400
4
5
16
48
21
28
15
35
8
48
3
8
9
10
5
7
7
12