Section 7.1 – Decimal Fractions
We often abbreviate how we write common fractions that have denominators that are powers of ten by writing them
as decimal fractions. The same models that were used for common fractions may be used with decimal fractions: money,
multibase blocks, fraction strips, area models and volume models.
We begin by illustrating the concept of writing “abbreviated” fractions in another base with the multibase blocks
model. Example:
Base Ten
Base Three
Ternary
Fraction
Base Three Dienes’ Multibase Blocks
Block
Flat
Long
1
1
1
3
9
1
0.1three
0.01three
Unit
1
27
0.001three
Model
with
Blocks
Example. Consider the ternary fraction 0.212three, where a block represents one whole unit.
base three
language of the blocks
0.212three
2 flats 1 long 2 units
base ten value
1  1
 1  23
2    1   2   
3
9
   
 27  27
Convert the following "abbreviated" fractions to a base ten common fraction (rational fraction).
0.321four
0.243five
0.532six
0.637eight
0.7893
-1-
100
10
1
0.1
0.01
0.001
0.0001
100,000
10,000
1,000
100
10
1
1
10
1
100
1
1, 000
1
10, 000
105
104
103
102
101
100
10–1
10–2
10–3
10–4
Ten Thousands
Thousands
Hundreds
Tens
Ones
Ten-thousandths
Thousandths
1,000
Hundredths
10,000
Tenths
100,000
Hundred Thousands
Definition. A decimal fraction (decimal) is an abbreviation for a fraction whose denominator is a power of ten, i.e.,
rational numbers written as decimals are fractions.
dime
cent
mill
Write the following in words and as decimal fractions in expanded place-value form.
0.24
twenty-four hundredths
fraction expanded form
0.24 = 2 
1
1
 4
10
100
exponential expanded form 0.24 = 2 ∙ 10–1 + 4 ∙ 10–2
27.836
twenty-seven and eight hundred thirty-six thousandths
fraction expanded form 27.836 = 2 10  7 1  8 
1
1
1
 3
 6
10
100
1000
exponential expanded form 27.836 = 2 ∙ 101 + 7 ∙ 100 + 8 ∙ 10–1 + 3 ∙ 10–2 + 6 ∙ 10–3
800.043 versus 0.843
Note. When writing a decimal with no whole number part, writing a zero before the decimal point adds clarity. For
example: 0.312 the 0 draws attention to the decimal point; whereas, .312 a person may overlook the decimal point.
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Express each in words and as a common fraction (rational fraction) in simplified form.
0.36
0.018
thirty-six hundredths
0.36 
eighteen thousandths
36
94
9


100 25  4 25
0.018 
18
92
9


1000 500  2 500
Express each common fraction (rational fraction) as a decimal fraction.
1
2
31
50
31
31
31 2
62
62
62


 2 2  2 
 0.62
2
2
50 2  5
 2  5   2 2  5 10 100
1 1 5
5


 0.5
2 2  5 10
3
8
17
40
17
17
17  52
17  25 425 425
 3  3
 3 3  3 
 0.425
2
40 2  5  2  5   5
1000
2 5
10
3 3
3  53 3 125 375
 3  3 3 

 0.375
8 2
1000
2 5
103
Note: We will use the division algorithm to complete the above problems in the next section where we discuss the
division algorithm for decimal fractions.
You may remember that
1
 0.3333333... where the 3’s repeat without end.
3
Definition. A decimal fraction is called a terminating decimal if it can be represented with a finite number of nonzero
digits. A decimal fraction is called a repeating decimal if a finite group of digits after the decimal repeat ad infinitum,
e.g., 0.4736736736736… where 736 is the repeating group of digits, often denoted as 0.4736 with the bar grouping the
digits.
a
in simplest form can be written in terminating form if and only if
b
the prime factorization of the denominator is of the for 2m ∙ 5n.
Terminating Decimal Theorem. A rational number
Which of the following common fractions may be written as a terminating decimal?
7
80
5
38
Yes, since the prime factorization
is 80 = 24 ∙ 5.
5
35
No, since the prime factorization is 38 = 2 ∙ 19.
39
260
5 1 5 1

 the prime
35 7  5 7
factorization 7.
No, since
Yes, since
-3-
39
3 13
3
.


260 20 13 22  5
Property of rational numbers. Every rational number may be expressed as either a terminating decimal fraction or a
repeating decimal fraction.
Definition. A number that is not a rational number is called an irrational number. Note that the decimal form of an
irrational number does not terminate and does not repeat. Examples are  and 2 .
When may a fraction be represented as a terminating decimal fraction? We answer this question in later sections.
Order each list of rational numbers from the least value to the greatest value.
0.8, 0.123, 0.045, 0.03
0.8 = 0.800, 0.03 = 0.030
0.030 < 0.045 < 0.123 < 0.800
0.03 < 0.045 < 0.123 < 0.8
1
6
13
, 0.0240, , 0.27,
4
25
50
1 1  25
25
6
64
24
13 13  2 26


 0.25 ,


 0.24 ,


 0.26
4 4  25 100
25 25  4 100
50 50  2 100
0.0240 < 0.24 < 0.25 < 0.26 < 0.27
0.0240 
6 1 13
 
 0.27
25 4 50
Note. Students will often consider only the nonzero digits. How would you correct this error?
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Problems and Exercises
1.
2.
3.
4.
5.
Convert the following "abbreviated" fractions to a base ten common fraction (rational fraction) or mixed number.
(a) 0.213four
(b) 0.403five
(c) 0.423six
(d) 0.6281
(e) 32.103four
(f) 23.214five
(g) 41.253six
(h) 65.6281
Convert each base ten common fraction to an “abbreviated” fraction in the designated base.
(a)
16
to base three
27
(b)
27
to base four
64
(c)
73
to base five
125
(d)
201
to base six
216
(a) 4.35 + 27.8
(b) 0.00456 + 0.0357
(c) 873 + 4.76
(d) 93.2 + 1.43 + 0.832
(e) 456 + 45.7 + 5.38 + 70.8
(f) 59 – 3.5
(g) 5.72 – 3.687
(h) 14.7 – 9.34
(i) 23.83 – 26.7
(j) 0.625 – 0.25
(a) 54  7.3
(b) 8.4  3.6
(c) 5.08  3.2
(d) 0.67  0.0083
(e) 2.73  0.5
(f) 211.6  0.4
(g) 48  1.6
(h) 7.605  0.09
(i) 190.4  0.56
(j) 0.15249  0.039
(a)
31
 0.28
40
(b) 3.38  1
3
8
(c)
8
 0.27
15
(d) 1.45 
(e)
3
5 + 14.89
(f) 0.3 × 4.2
5
13
(g) 0.311.2
(h) 0.54  0.85
(i) 91.2 ÷ 0.6
(j) 4.8 + 9.125 + 7
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4