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Math Activity
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6.1
MATH ACTIVITY 6.1
Decimal Place Value with Base-Ten Pieces and Decimal Squares
Virtual Manipulatives
Purpose: Explore decimal place value concepts using Decimal Squares and base-ten pieces.
Materials: Base-Ten Pieces and Decimal Squares in the Manipulative Kit or Virtual
Manipulatives.
www.mhhe.com/bbne
*1. When the largest base-ten piece
in your kit represents the unit, the
other base-ten pieces take on the
values shown here. Notice that
the hundredths piece is divided
into 10 equal parts to represent
thousandths, and 1 part is shaded
to represent 1 thousandth.
1
.1
.01
.001
a. Look at the four different base-ten pieces and describe five mathematical relationships between pairs of pieces.
b. Form the collection of 1 unit piece, 4 tenths pieces, and 12 hundredths pieces.
By using only your base-ten pieces and exchanging (trading) the pieces, it is
possible to represent this collection in many different ways. Record some of these
in a place value table like the one shown here.
Units
Tenths
Hundredths
1
4
12
Thousandths
2. In the Decimal Squares model the unit square is divided into 10, 100, and 1000 equal
parts to represent tenths, hundredths, and thousandths (respectively). Sort your deck
of Decimal Squares into three piles according to color.
a. Determine the smallest and largest decimal represented in each pile.
b. How do the shaded amounts of each type of Decimal Square increase?
c. List some relationships between the three types of Decimal Squares.
.6
3. The two Decimal Squares shown at the left illustrate .6 5 .60 because both squares
have the same amount of shading. In the deck of Decimal Squares there are three
squares whose decimals equal .6. Sort your deck of Decimal Squares into piles so
squares with the same shaded amount are in the same pile.
a. Find all the decimals from the Decimal Squares that equal the following: .5, .35,
.9, and .10 and write each corresponding equality statement.
b. The two-place decimal .65 is not equal to a one-place decimal, such as .6 or .7. List
all the other two-place decimals from your deck that are not equal to a one-place
decimal.
.60
c. The three-place decimal .375 is not equal to a two-place decimal. List all the other
three-place decimals from your deck that are not equal to a two-place decimal.
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Section 6.1 Decimals and Rational Numbers
SECTION 6.1
DECIMALS AND RATIONAL NUMBERS
Circular patterns of atoms in
an iridium crystal, magnified
more than 1 million times by
a field ion microscope.
PROBLEM OPENER
A carpenter agrees that during a specified 30-hour period he be paid $15.50 every
hour he works and that he pay $16.60 every hour he does not work. At the end of
30 hours, he finds he has earned $47.70. How many hours did he work?*
Each dot in the remarkable photograph above is an atom in an iridium crystal. The circular patterns show the order and symmetry governing atomic structures. The diameters
of atoms, and even the diameters of electrons contained in atoms, can be measured by
decimals. Each atom in this picture has a diameter of .000000027 centimeter, and the
diameter of an electron is .00000000000056354 centimeter.
The use of decimals is not restricted to describing small objects. The U.S. gross
national product (GNP) and the U.S. national income (NI) for selected 5-year periods
are expressed to the nearest tenth of a billion dollars in Figure 6.1.†
Figure 6.1
1985
1990
1995
2000
2005
2010
GNP (billions)
$4244.0
$5835.0
$7444.3
$9989.2
$12,735.5
$14,848.7
NI (billions)
$3696.3
$5059.5
$6522.3
$8938.9
$11,273.8
$12,828.2
In our daily lives we encounter decimals in representations of dollar amounts: $17.35,
$12.09, $24.00, etc. In elementary school, pennies, dimes, and dollars are commonly used
for teaching decimals.
DECIMAL TERMINOLOGY AND NOTATION
The word decimal comes from the Latin decem, meaning ten. Technically, any number
written in base-ten positional numeration can be called a decimal. However, decimal
most often refers only to numbers such as 17.38 and .45, which are expressed with
*“Problems of the Month,” Mathematics Teacher.
†Adapted from Statistical Abstract of the United States, 131st ed. (Washington, DC: Bureau of the Census, 2012).
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Decimals: Rational and Irrational Numbers
decimal points. A combination of a whole number and a decimal, such as 17.38, is also
called a mixed decimal.
There are currently many variations in decimal notation. In England, the decimal
point is placed higher above the line than in the United States. In other European countries, a comma is used in place of a decimal point. A comma and a raised numeral denote
a decimal in Scandinavian countries.
United States
82.17
England
82?17
Europe
82,17
Scandinavian countries
82,17
The number of digits to the right of the decimal point is called the number of decimal
places. There are two decimal places in 7.08 and one decimal place in 104.5. The positions
of the digits to the left of the decimal point represent place values that are increasing powers of 10 (1, 10, 102, 103, . . . ). The positions to the right of the decimal point represent
place values that are decreasing powers of 10 (1021, 1022, 1023, . . . ), or reciprocals of
powers of 10 ( 101 , 101 2 , 101 3, . . . ) . In the decimal 5473.286 (Figure 6.2), the 2 represents 102 , the
8
6
8 represents 100
, and the 6 represents 1000
. Notice the similarity in pairs of names to the right
and left of the units digit, for example, tens and tenths, hundreds and hundredths, etc. The
convention in the following Historical Highlight of placing a small zero under the units
digit helped to focus attention on these pairs of names.
5t
ho
u
4 h sand
un
s
dre
7t
en ds
s
3u
nit
s
2t
en
ths
8h
un
dre
6t
ho dths
us
an
dth
s
5 4 7 3 .2 8 6
Figure 6.2
HISTORICAL HIGHLIGHT
The person most responsible for our use of decimals is Simon Stevin, a Dutchman. In 1585 Stevin wrote La Disme,
the first book on the use of decimals. He not only stated the rules for computing with decimals but also pointed
out their practical applications. Stevin showed that business calculations with decimals can be performed as easily
as those involving only whole numbers. He recommended that the government adopt the decimal system and enforce
its use.
As decimals gained acceptance in the sixteenth and seventeenth centuries, a variety of notations were used.
Many writers used a vertical bar in place of a decimal point. Here are eight examples of how 27.847 was written
during this period.
27 u 847
27847 . . .
E X A M P LE A
3
s
27(847)
27,8i4ii7iii
27 u 847
27847
o
27 847
27 s
0 8 s
1 4s
2 7
3
s
Express the value of the digit marked by the arrow as a fraction whose denominator is
a power of 10.
1. 47.35
Solution 1.
2. 6.089
5
100
2.
3. 14.07
9
1000
3.
0
10
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Section 6.1 Decimals and Rational Numbers
305
Like whole numbers, decimals can be written in expanded form to show the powers
of 10 (Figure 6.3).
473.2865
Figure 6.3
Content Standards
5.NBT.3
Read and write decimals to
thousandths using base-ten
numerals, number names, and
expanded form.
1
1
1
1
4(102) 7(10) 3(1) 2 10 8 100 6 1000 5 10,000
Reading and Writing Decimals The digits to the left of the decimal point are read
as a whole number, and the decimal point is read and. The digits to the right of the point
are also read as a whole number, after which we say the name of the place value of the
last digit. For example, 1208.0925 is read “one thousand two hundred eight and nine
hundred twenty-five ten-thousandths.”
Common Core
State Standards Mathematics
⏐
⏐
⏐
⏐
↓
One thousand
two hundred
eight
E XA M P LE B
⎫
⎬
⎭
⎫
⎬
⎭
1208 . 0925
⏐
⏐
⏐
⏐
↓
and
⏐
⏐
⏐
⏐
↓
Nine hundred
twenty-five
ten-thousandths
Write the name of each decimal.
1. 3.472
2. 16.14
3. .3775
Solution 1. Three and four hundred seventy-two thousandths. 2. Sixteen and fourteen hundredths. 3. Three thousand seven hundred seventy-five ten-thousandths.
TRY IT! 6.1.1
1. Write the decimal .2865 in expanded form.
2. Add the fractions in the expanded form of .2865.
3. Write the name of the final fraction form of .2865 in words.
One place where you may see the names of numbers is on bank checks. When writing an amount of money, some people write the decimal part of a dollar in words. Notice
that on the bank check in Figure 6.4 on the next page it is unnecessary to write dollars
or cents. The amount is in terms of dollars, and this unit is printed at the end of the line
on which the amount of money is written. Some people write the decimal part of a dollar as a fraction. For example, the amount of this check might have been written “one
24
hundred seventy-seven and 100
.”
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Decimals: Rational and Irrational Numbers
54-7001/2114
20
$
PAY TO THE
ORDER OF
DOLLARS
FEDERAL SAVINGS BANK
MEMO
Figure 6.4
MODELS FOR DECIMALS
Models are important for providing conceptual understanding and insight into the use of
decimals. Often, there is a rush to begin decimal computation, before devoting time to
examine decimals using concrete models that can help children understand decimals both
for their practical applications, and as their part of the rational number system.
There are several models for decimals but we will focus on the Decimal Square
model because of its easy connection to fractions and percent.
Decimal Squares The Decimal Squares model illustrates the part-to-whole concept
of decimals and place value. Unit squares are divided into 10, 100, and 1000 equal parts
(Figure 6.5), and the decimal tells what part of the square is shaded.*
Figure 6.5
Tenths square
Hundredths square
Thousandths square
.3
.35
.375
Each decimal in Figure 6.5 can be obtained by beginning with the fraction for the shaded
amount of the square and obtaining the expanded form of the decimal. For example, the
375
.
fraction for the square representing 375 parts out of 1000 is 1000
375
300
70
5
3
7
5
5
1
1
5
1
1
5 .375
1000
1000
1000
1000
10
100
1000
Place value table
Similarly, the decimal for
Tenths
Hundredths
Thousandths
3
7
5
3
35
is .3, and the decimal for
is .35.
10
100
*Decimal Squares is a registered trademark of American Education Products, LLC.
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Section 6.1 Decimals and Rational Numbers
You can use
models to repr
esent decimal
s.
1.0
one
0.1
one tenth
0.01
one hundredt
h
Try It
There are 0 w
hole
grids shaded.
Hundredths
Ones
Tenths
Complete th
e place-value
chart that re
the fraction
presents
of the grid th
at is shaded
at the right.
There are
squares shad
ed out of a to
tal of
squares.
In words, this
is forty-two hu
ndredths.
This is the sa
me as 4 tenths
and 2 hundre
dths.
So, write 4 te
nths and 2 hu
nd
re
dt
place-value ch
hs in the
art.
4 tenths
2 hundredths
4 tenths and
2
is 42 hundredt hundredths
hs.
Talk About It
2
Use
Number Sen
Marc has 6 pe
se Paulo has
nnies. How m
6 dimes.
any times grea
of 6 dimes th
ter is the valu
an 6 pennies?
e
Explain.
632Chap
From My Math,ter 10 Fractions and Deci
mals
Grade 4, by M
cGraw-Hill Ed
of McGraw-H
ucation. Copyrig
ill Education.
ht ©2013 by M
cGraw-Hill
631_634_C10_
L01_116195.in
dd 632
Copyright © The
McGraw-Hill Com
panies, Inc.
1.
Education. Re
printed by perm
ission
10/14/11 10:
53 AM
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Chapter 6
E X A M P LE C
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Decimals: Rational and Irrational Numbers
Describe the Decimal Square that would represent each fraction, and write the decimal
for each fraction.
1.
4728
10,000
2.
6
100
Solution 1. A square with 4728 parts shaded out of 10,000
4000
700
20
8
4728
5
1
1
1
10,000
10,000
10,000
10,000
10,000
4
7
2
8
5
1
1
1
10
100
1000
10,000
5 .4728
2. A square with 6 parts shaded out of 100
6
0
6
5
1
5 .06
100
10
100
TRY IT! 6.1.2
Write the decimal numeral that matches each description.
1. A Decimal Square with 123 parts shaded out of 1000.
2. A Decimal Square with 101 parts shaded out of 10,000.
Calculator
Connection
Example C shows that it is easy to obtain the decimal for a fraction whose denominator
is a power of 10; it is just a matter of locating the decimal point. Try the example on
your calculator by dividing 4728 by 10,000 and 6 by 100. It is also instructive to enter
4728 into a calculator and then repeatedly divide by 10. Each time the decimal point
moves one digit to the left.
Keystrokes
4728
View Screen
÷
10
=
472.8
÷
10
=
47.28
÷
10
=
4.728
÷
10
=
.4728
In general, to divide an integer by a power of 10, begin with the units digit and, for each
factor of 10, count off a digit to the left to locate the decimal point.
Number Line The number line is a common model for illustrating decimals. One
method of marking off a unit from 0 to 1 is to use the edge of a Decimal Square, as
shown in Figure 6.6 on the next page. This approach shows the relationship between
a region model for a unit (the Decimal Square) and a linear model for a unit (the
edge of a square). The Decimal Square can be used repeatedly to mark off tenths on
the number line from 0 to 1, 1 to 2, etc.
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Section 6.1 Decimals and Rational Numbers
Figure 6.6
0
.3
1
2
Consider locating the point for .372 on a number line. One approach is to use the
expanded form of the decimal
.372 5
Content Standards
4.NF.6
Use decimal notation for
fractions with denominators 10 or
100. For example, rewrite 0.62
as 62y100; describe a length as
0.62 meters; locate 0.62 on a
number line diagram.
3
7
2
1
1
10
100
1000
and locate the point in several steps, as shown in Figure 6.7. First, the point for 103 (.3)
is located at the end of the third interval, as in Figure 6.6. Second, the expanded form
7
shows that we must add 100
, so the interval from .3 to .4 is divided into 10 equal parts,
7
which are hundredths. To add 100
(.07), we begin at .3 and go to the end of the seventh
interval. This is the point for .37. Finally, the interval from .37 to .38 is divided into 10
2
equal parts, which are thousandths. To add 1000
, we begin at .37 and go to the end of the
second interval. This is the point for .372.
Common Core
State Standards Mathematics
0
.1
.2
.3
.4
.5
.6
.37
.7
.8
.9
.38
.4
.3
.38
.37
.372
Figure 6.7
E XA M P LE D
Sketch a number line and mark the approximate location of each decimal.
1. .46
2. 1.75
3. 2.271
Solution
.46
0
1.75
1
2.271
2
1
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Decimals: Rational and Irrational Numbers
TRY IT! 6.1.3
Using the given points for scale, precisely mark the points for 0, 2, and 1.25 on
the number line.
1.75
2.25
Decimals can represent negative as well as positive numbers. For every decimal, whether
positive or negative, there is a corresponding decimal called its opposite (or inverse for
addition) such that the sum of the two decimals is zero. Several decimals and their
opposites are shown on the number line in Figure 6.8.
-
1.2 and 1.2 are opposites
-
.6 and .6 are opposites
Figure 6.8
-
1.4
-
1.2
-
1
-
.8
-
.6
-
.4
-
.2
0
.2
.4
.6
.8
1
1.2
1.4
EQUALITY OF DECIMALS
Equality of decimals can be illustrated visually by comparing the shaded amounts in
their Decimal Squares. Figure 6.9 shows that 4 parts out of 10, 40 parts out of 100, and
400 parts out of 1000 are all represented by the same amount of shading—in each
Decimal Square, four columns are shaded. This illustrates that
.4 5 .40 5 .400
Figure 6.9
E X A M P LE E
.4
.40
.400
Complete each equation by writing the indicated decimal, and describe the Decimal
Square representing each decimal in the equation.
1. .35 5 _______ (thousandths)
3. .600 5 _______ (tenths)
2. .670 5 _______ (hundredths)
Solution 1. .35 5 .350 (35 parts out of 100 and 350 parts out of 1000 are shaded). 2. .670 5
.67 (670 parts out of 1000 and 67 parts out of 100 are shaded). 3. .600 5 .6 (600 parts out of 1000
and 6 parts out of 10 are shaded).
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Section 6.1 Decimals and Rational Numbers
Decimal Squares also give us a visual model for place value. Consider the Decimal
Square for .475 in Figure 6.10. The 4 full columns that are shaded (400 thousandths) repre400
sent 104 or .4 ( 1000
5 104 ) ; the 7 small squares that are shaded (70 thousandths)
7
70
7
re present 100 or .07 ( 1000
5 100
) ; and the 5 small parts that are shaded (5 thousandths)
5
represent 1000 or .005. Thus, the decimal .475 can be thought of as 4 tenths, 7 hundredths,
and 5 thousandths.
.475 5 .4 1 .07 1 .005
5
1000
7
70
=
1000 100
Figure 6.10
4
400
=
1000 10
INEQUALITY OF DECIMALS
Content Standards
4.NF.7
Recognize that comparisons are
valid only when the two decimals
refer to the same whole. Record
the results of comparisons with
the symbols ., 5, or ,, and
justify the conclusions, e.g., by
using a visual model.
Research indicates that students from elementary school through college often have difficulty determining inequalities for decimals. One source of confusion is to think of the
digits in the decimal as representing whole numbers (see Example F on the following page).
Figure 6.11 shows that .47 , .6. Even though 47 is greater than 6, a smaller amount
of the square is shaded for .47 than for .6.
Common Core
State Standards Mathematics
Figure 6.11
.47
<
.6
We can also see that .47 , .6 by noting that in the Decimal Square for .47, 4 full
columns and part of another are shaded, whereas in the Decimal Square for .6, 6 full
columns are shaded. In other words, the digit in the tenths place for .47 is less than the
digit in the tenths place for .6. In general, the following place value test determines
inequalities for decimals.
Place Value Test for Inequality of Decimals The greater of two positive
decimals that are both less than 1 will be the decimal with the greater digit in the
tenths place. If these digits are equal, this test is applied to the hundredths digits, etc.
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Decimals: Rational and Irrational Numbers
The question in the next example is from a test given as part of a nationwide testing
program in schools every 4 years.* Over one-half of the 13-year-olds who took the test
selected an incorrect answer.
E X A MP LE F
Which number is the greatest?
.19
.036
.195
.2
Solution One approach is to use the place value test for inequality of decimals. Since 2 is the
greatest of the digits in the tenths place of these four decimals, .2 is the greatest number. Another
approach is to change each decimal to thousandths. This will show that 200 thousandths is the
greatest number of thousandths among these four decimals.
.190
.036
.195
.200
A visual approach with Decimal Squares illustrates that 2 full columns of shading (or 2 parts shaded
out of 10) is more than 19 parts shaded out of 100 or 195 parts shaded out of 1000.
.2
.19
.195
Note: Of the 13-year-olds tested, 47 percent selected .195 for the answer. This error may be due to
the fact that 195 is greater than 19, 36, or 2.
RATIONAL NUMBERS
Up to this point, numbers written in the form ab, where a and b are integers with b ? 0, have
been called fractions. This is the terminology commonly used in elementary and middle
schools. However, the word fraction has a more general meaning and includes the quotient
of any two numbers, integers or not, as long as the denominator is not zero. Specifically,
fractions ab, where a and b are integers and b ? 0, are called rational numbers.
Rational Number Any number that can be written in the form ab , where b ? 0
and a and b are integers, is called a rational number.
2
2
For example, 19 , 73 , 15 , 107 , and 31 are rational numbers. When the denominator of a ratio2
nal number equals 1, the rational number equals an integer: 61 5 6, 41 5 2 4, 121 5 12, etc.
Therefore, integers are also rational numbers.
*T. P. Carpenter, M. K. Corbitt, H. S. Kepner, M. M. Lindquist, and R. E. Reys, Results from the Second
Mathematics Assessment of the National Assessment of Educational Progress.
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Section 6.1 Decimals and Rational Numbers
313
Rational Numbers as Terminating or Infinite Repeating Decimals Rational
numbers can be expressed by many different number symbols or numerals. For example,
3
3
10 is a rational number and 10 5 .3, so .3 is also a rational number. In the following
paragraphs we will show that all rational numbers ab can be written as decimals.
We have seen that it is easy to convert a fraction to a decimal if the denominator is
a power of 10:
64
7283
54
5 .64 5 7.283 5 .0054
100
1000
10,000
Sometimes when the denominator is not a power of 10, the fraction can be replaced by an
25
equal fraction whose denominator is a power of 10. For example, 14 can be replaced by 100
because 100 is a multiple of 4:
1
25 3 1
25
5
5
5 .25
4
25 3 4
100
Since 10 5 2 3 5, we know by the Fundamental Theorem of Arithmetic that any power
of 10 will have only 2 and 5 as prime factors.
100 5 102 5 22 3 52
10 5 2 3 5
1000 5 103 5 23 3 53 etc.
Consider replacing 38 by a fraction whose denominator is a power of 10. Since 8 5 23, we
need to multiply the numerator and denominator of 38 by 53.
3
3
3 3 53
375
5 35 3
5 3 5 .375
3
8
2
2 35
10
E XA M P LE G
Convert each fraction to a decimal by first writing a fraction whose denominator is a
power of 10.
1.
7
20
2.
3
25
3.
11
16
4.
3
40
7
35
3
12
11
11
11 3 54
6875
5
5 .35 2.
5
5 .12 3.
5 4 5 4
5
5 .6875
20
100
25
100
16
2
2 3 54
104
3
75
3
3 3 52
4.
5 3 5 .075
5 3
5 3
40
2 35
2 3 53
10
Solution 1.
In the examples on the preceding pages, all the decimals had a finite number of
digits. Such decimals are called terminating (or finite) decimals. However, there are
decimals that are not terminating. There is no power of 10 that has 3 as a factor, so 13
cannot be written as a fraction whose denominator is a power of 10. In general, we have
the following rule.
Terminating Decimal If a nonzero rational number ab is in simplest form, it can
be written as a terminating decimal if and only if b has only 2s and/or 5s in its
prime factorization.
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E X A M P LE H
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Decimals: Rational and Irrational Numbers
Which of these rational numbers can be written as terminating decimals?
1.
5
6
2.
1
80
3.
9
15
4.
3
14
5
cannot be written as a terminating decimal. 2. 80 has only
6
9
3
1
can be written as a terminating decimal. 3.
5 , and since the denomfactors of 2 and 5, so
80
15
5
9
can be written as a terminating decimal.
inator of the fraction in simplest form has only 5 as a factor,
15
3
cannot be written as a terminating decimal.
4. 14 has a factor of 7, so
14
Solution 1. 6 has a factor of 3, so
Let’s consider finding a decimal for 13 . We know from the fraction-quotient concept
in Section 5.2 that 13 5 1 4 3. Figure 6.12 illustrates the first few steps in dividing 1 by
3. Part a shows a unit square with 10 tenths and illustrates that 1 4 3 is .3 with .1 remaining. In part b, the remaining .1 is replaced by 10 hundredths, and dividing by 3 produces
.03 with .01 remaining. In part c, the remaining 1 hundredth is replaced by 10 thousandths,
and dividing by 3 produces .003 with .001 remaining. The three steps of the division
process in Figure 6.12 produce .3, .03, and .003, which give a total shaded amount of
.333 with .001 remaining.
Dividing 10
tenths by 3
.3
Dividing 10
hundredths by 3
.03
1 tenth =
10 hundredths
Dividing 10
thousandths by 3
.003
1 hundredth =
10 thousandths
.1 remaining
Figure 6.12
(a)
.01 remaining
.001 remaining
(b)
(c)
Continuing this process of splitting the remaining part by 10 and dividing by 3 shows
that the decimal for 13 has a repeating pattern of 3s. When a decimal does not terminate and contains a repeating pattern of digits, it is called a repeating decimal or an
infinite decimal.
The step-by-step visual illustration in Figure 6.12 has a corresponding numerical
division algorithm. The first step is to divide 10 tenths by 3. The result is .3 with .1
remaining.
The second step is to divide 10 hundredths by 3. The result of the first two steps is
.33 with .01 remaining. In the third step, 10 thousandths are divided by 3. The result of
the first three steps is a quotient of .333 with .001 remaining. This process can be continued to obtain any number of 3s in the decimal approximation for 13 .
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Section 6.1 Decimals and Rational Numbers
Step 1
.3
3q1.0
9
One-tenth → 1
Content Standards
7.NS.2d
Convert a rational number to a
decimal using long division;
know that the decimal form
of a rational number terminates
in 0s or eventually
repeats.
Common Core
State Standards Mathematics
Step 2
.33
3q1.00
9
10
9
One-hundredth →1
Step 3
.333
3q1.000
9
10
9
10
9
One-thousandth → 1
The division algorithm can be used to obtain a terminating or repeating decimal for
any rational number. For example, when the numerator of 38 is divided by its denominator, the division algorithm shows that the decimal terminates after three digits.
.375
8q3.000
2400
60
56
40
40
On the other hand, when the numerator of 47 is divided by its denominator, the quotient does not terminate but repeats the same arrangement of six digits (571428) over
and over. In this case the decimal is repeating. The reason for this can be seen by looking at the remainders 5, 1, 3, 2, 6, and 4, which are circled below. These six numbers,
plus 0, are all the possible remainders when a number is divided by 7. So after six steps
in the process of dividing 4 by 7, the numbers in the decimal quotient repeat. Notice the
use of the bar above the six digits in the quotient to indicate the repeating pattern. The
block of digits that is repeated over and over is called the repetend.
.5714285
7q4.0000000
35
50
49
10
7
30
28
20
14
60
56
40
35
The preceding example illustrates why every rational number rs can be represented
by either a repeating decimal or a terminating decimal. When r is divided by s, the
remainders are always less than s (see the Division Algorithm Theorem, page 152). If a
remainder of 0 occurs in the division process, as it does when we divide 3 by 8, then
the decimal terminates. If there is no zero remainder, then eventually a remainder will
be repeated, in which case the digits in the quotient will also start repeating.
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Decimals: Rational and Irrational Numbers
Calculators are convenient for finding decimal representations of fractions. For fractions
represented by repeating decimals, such as 127 5 .5833333333 . . . , the number in the
calculator view screen is an approximation because it shows only a few of the digits.
For many applications this is sufficient accuracy.
Most calculators use several more digits than are shown in their view screens. When
the number of digits in a decimal exceeds the space in a calculator’s view screen, the
calculator may keep several hidden digits that are used internally for greater accuracy.
To determine if your calculator uses hidden digits, try the following keystrokes to obtain
a decimal approximation for 171 .
Keystrokes
1
View Screen
÷
17
=
0.0588235
×
100,000
=
5882.3529
−
5882
=
0.3529411
Notice that the final view screen shows five hidden digits 2, 9, 4, 1, and 1 that were not
in the first view screen. The purpose of multiplying by a power of 10 and subtracting
the whole number part of the product is to move the digits in the view screen to the left
to make room for digits that may be hidden. Did you “uncover” hidden digits beyond
the last digit that was initially displayed for 171 by your calculator?
E X A MP LE I
Write the decimal for each rational number. Use a bar to show the repetend (repeating
digits).
1.
3
11
2.
5
6
3.
5
12
Solution 1. .27 2. .83 3. .416
Notice in the solutions to Example I that the repeating pattern in .83 does not begin
until the hundredths digit and the repeating pattern in .416 does not begin until the
thousandths digit.
Writing Terminating and Infinite Repeating Decimals as Quotients of
Integers Every rational number is the quotient of two integers, and we have seen that
such numbers can be written as a terminating or repeating decimal. Conversely, every
terminating or repeating decimal can be written as the quotient of two integers. An
example of a terminating decimal written as a quotient of two integers is shown below.
Terminating decimals can be written as fractions whose denominators are powers of
378
10. The following equations show why .378 equals 1000
.
3
7
8
1
1
10
100
1000
300
70
8
378
5
1
1
5
1000
1000
1000
1000
.378 5
Similarly,
.47 5
47
3802
643
3.802 5
and 64.3 5
100
1000
10
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Section 6.1 Decimals and Rational Numbers
317
At first it may appear to be a difficult task to write an infinite repeating decimal, such
as the following, as a rational number in the form ab . Consider:
N 5 .37 5 .373737 . . .
However, using a technique that allows us to eliminate the infinite repeating decimal
will make the process of writing a rational number in the form ab easier. Notice that if
we multiply N by 100, the decimal portion of 100N looks the same as the decimal portion of N:
100N 5 37.37 . . .
N 5 .37 . . .
Subtract the second equation from the first equation, and the infinite repeating decimal
portion is eliminated.
100N 5 37.37 . . .
2(N 5 .37 . . .)
99N 5 37
N5
Thus,
37
99
With certain adjustments this technique will enable us to express any infinite repeating
decimals in the form ab .
E XA M P LE J
Write the infinite repeating decimal N 5 .572 5 .57272 . . . as a rational number in the
form ab .
Solution First, multiply N by 10 so the repeating decimal portion begins at the decimal point:
10N 5 5.7272 . . .
Then multiply N by 1000 so that the infinite repeating decimal portion of 10N and 1000N are the
same. Subtract:
1000N 5 572.72 . . .
2(10N 5 5.72 . . .)
990N 5 572 2 5 5 567
567
N5
990
TRY IT! 6.1.4
Write each of the following rational numbers as the quotient of integers in the form ab:
1. L 5 2.7 5 2.777 . . .
3. N 5 2.12 5 2.121212 . . .
2. M 5 1.067 5 1.0676767 . . .
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Decimals: Rational and Irrational Numbers
DENSITY OF RATIONAL NUMBERS
In Chapter 5, on page 255, we saw examples of the fact that the rational numbers, when
written as fractions, are dense on the real number line. That is, between any two rational
numbers there is always another rational number. To show this, we looked at a method
for finding a fraction between two given fractions. In the following example, we will
consider ways of finding a decimal between any two given decimals.
E X A M P LE K
Sketch a number line and mark the location of each pair of decimals. Then find another
decimal between them.
1. .124 and .125
2. .47 and .621
3. 1.1 and 1.2
Solution 1. One method of finding decimals between two given decimals is to express both
decimals with a greater number of decimal places. For example, .124 5 .1240 and .125 5 .1250, and
the 9 four-place decimals .1241, .1242, . . . , .1249 are between .124 and .125. Also, .124 5 .12400
and .125 5 .12500 and the 99 five-place decimals .12401, .12402, . . . , .12499 are between .124 and
.125. Similarly, the process can be continued as there are 999 six-place decimals between .124000
and .125000, etc. 2. Since .47 5 .470, any of the three-place decimals between .47 and .621 may
be selected, and by increasing the number of decimal places for .47 and .621, more decimals can be
found between these numbers. 3. 1.1 5 1.10 and 1.2 5 1.20, so the 9 decimals 1.11, 1.12, . . . ,
1.19 are between 1.1 and 1.2. Also, 1.1 5 1.100 and 1.2 5 1.200, so the 99 decimals 1.101,
1.102, . . . , 1.199 are between 1.1 and 1.2, etc.
(a)
.124
(b)
.125
0
.47
.5
.1243
.500
(c)
1.1
.621
1.2
1
1.15
APPROXIMATION
Often a calculator view screen will be filled with the digits of a decimal, when some
approximate value is all that is necessary. Rounding to a given place value is the most
common method of obtaining decimal approximations.
Rounding Decimals can be rounded to the nearest whole number, nearest tenth, nearest hundredth, etc. Before we look at a rule for rounding decimals, let’s consider some
visual illustrations.
Squares for three decimals are shown in Figure 6.13 on the next page. Consider
rounding these decimals to the nearest tenth. The decimal .648 rounds to .6 because 6
full columns and less than one-half of the next column are shaded; .863 rounds to .9
because 8 full columns and more than one-half of the next column are shaded; .35 can
be either rounded up to .4 or rounded down to .3, because 3 full columns and one-half
of the next column are shaded. In this text, we adopt the policy of rounding up.
Next consider rounding the decimals in Figure 6.13 to the nearest hundredth. The
square for .648 shows that 64 hundredths are shaded (6 full columns and 4 hundredths
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Section 6.1 Decimals and Rational Numbers
Figure 6.13
.648
.863
.35
in the next column); the remaining 8 thousandths are more than one-half of the next
hundredth, so .648 rounds to .65. The Decimal Square for .863 shows that 86 hundredths
are shaded (8 full columns and 6 hundredths in the next column); the remaining 3 thousandths
are less than one-half of the next hundredth, so .863 rounds to .86.
E XA M P LE L
Round each decimal to the nearest tenth and to the nearest hundredth.
1. .283
2. .068
3. 14.649
Solution 1. .283 rounded to the nearest tenth is .3 and to the nearest hundredth is .28. 2. .068
rounded to the nearest tenth is .1 and to the nearest hundredth is .07. 3. 14.649 rounded to the
nearest tenth is 14.6 and to the nearest hundredth is 14.65.
Content Standards
5.NBT.4
Use place value understanding to
round decimals to any place.
Common Core
State Standards Mathematics
Notice in Example L that 14.649 rounded to the nearest tenth is not 14.7. You can
confirm this by visualizing a Decimal Square for .649: 6 full columns are shaded and
less than one-half of the next column (49 thousandths) is shaded.
The preceding examples are special cases of the following general rule for rounding
decimals. Notice that this is similar to the rule that was stated for rounding whole numbers on page 91 of Chapter 3.
Rule for Rounding Decimals
1. Locate the place value to which the number is to be rounded, and check the
digit to its right.
2. If the digit to the right is 5 or greater, then all digits to the right are dropped
and the digit with the given place value is increased by 1.
3. If the digit to the right is 4 or less, then all digits to the right of the digit with
the given place value are dropped.
TRY IT! 6.1.5
Round 1.6825 to the given number of decimal places.
1. Two decimal places (round to hundredths).
2. One decimal place (round to tenths).
3. Three decimal places (round to thousandths).
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Decimals: Rational and Irrational Numbers
Some calculators automatically round decimals that exceed the space in the view screen.
On such calculators, if 2 is divided by 3, the decimal 0.66 . . . 667 will be displayed,
where the last digit in the view screen is rounded from 6 to 7. Almost all calculators
that round off at a digit that is followed by 5 will increase this digit, as described in the
preceding rule for rounding numbers. For example, 55
99 is equal to the repeating decimal
.5555 . . . . If 55 is divided by 99 on a calculator that rounds, 0.55 . . . 556 will show
in the display. Try this on your calculator.
PROBLEM-SOLVING APPLICATION
Problem
The price of a single pen is 39 cents. This price is reduced if pens are purchased in
quantity. The price per pen is always a whole number of cents and never less than 2
cents. If all of the pens in a box were purchased at a single reduced rate, and the total
paid was $22.91, then what is the number of pens in the box?
Understanding the Problem We don’t know the reduced price per pen nor the number
of pens. But, we do know the price is a whole number (in cents) and that the price per
pen times the number of pens is $22.91.
Question 1: What are the possible reduced prices per pen when bought in quantity?
Devising a Plan The possible price in cents-per-pen ranges from 2 to 38 cents (why?).
The number of pens is an unknown whole number that we can call N (for number of
pens). Whatever number N is, (cents-per-pen) 3 N pens 5 2291 cents. So, at least one
of the following statements is true.
2 3 N 5 2291, 3 3 N 5 2291, 4 3 N 5 2291, . . . , 37 3 N 5 2291, 38 3 N 5 2291
The cost per pen cannot be an even number because 2291 is an odd number. That reduces
our list to 18 possibilities.
3 3 N 5 2291, 5 3 N 5 2291, 7 3 N 5 2291, . . . , 37 3 N 5 2291
Because neither 3 nor 5 are factors of 2291, this further reduces the list of possible
values for the cost per pen.
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37
Question 2: Why are we able to eliminate these nine numbers from the list?
Carrying Out the Plan Because the price of each pen is less than 39 cents, we need
only check the primes 7, 11, 13, 17, 19, 23, 29, 31, and 37.
Question 3: Which of these primes divide 2291, and what is the number of
pens in the box?
Looking Back Suppose we kept the conditions of the problem the same but changed
the reduced price for the entire box to $15.17.
Question 4: How many pens would be in the box?