Section 7.3 and Section 7.4 – Nonterminating Decimals & Real Numbers
Express each common fraction (rational fraction) as a decimal fraction.
1
3
3
7
1 3
3
3
3
 



3 10 100 1000 10000
 0.3333
 0.3
0.428571
7 3.000000
–28
20
–14
60
–56
40
–35
50
–49
10
–7
3
Hence,
3
 0.428571 .
7
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.
Property of rational numbers. Every rational number may be expressed as either a terminating decimal fraction or a
repeating decimal fraction.
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.
Express each repeating decimal as a common fraction (rational fraction) in simplified form.
0.5
0.132
We use an algebraic method where we will subtract off the nonzero repeating numerals. By multiplying by a
power of ten equal to the number of digits repeating, the decimal shifts to the right. Yet all the repeating decimals
positions remain in the same place-value.
Let N = 0.5 .
Since one position is repeating,
multiply by 10.
10N = 5.5
– N = – 0.5
9N = 5.0
5
N
9
More Examples: 3.45
Let N = 0.132
Since two positions are repeating
multiply by 100.
100N = 13.232
– N = – 0.132
99N = 13.100
13.1 13.1 10 131
N


99
99  10 990
1.2198
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Order the list of rational numbers from the least value to the greatest value.
1
, 0.24, 0.25, 0.26, 0.25
4
Note
1
 0.250000 , 0.24 = 0.240000 0.25  0.252525... , 0.26 = 0.260000, 0.25  0.255555...
4
0.240000 < 0.250000 < 0.252525 < 0.255555 < 0.260000
Hence, 0.24 <
1
< 0.25 < 0.25 < 0.26.
4
Definition. A number whose decimal form does not terminate and does not repeat is called an irrational number. Note
that a decimal number is not a rational number, that is, an irrational number cannot be written as a common fraction.
Examples are  , 2 and 0.101001000100001… .
Definition. The set of real numbers is the union of the set of rational numbers and the set of irrational numbers.
We illustrate set relationships for the sets of numbers we have defined up to now with a Venn Diagram where the
universal set is the set of real numbers.
Real
Rational
Numbers
Integers
Irrational
Numbers
Whole
Numbers
Natural
Numbers
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Mixed Decimal and Fraction Arithmetic
When fractions and decimals both occur in an arithmetic problem, should we work the problem using fraction or decimal
arithmetic?
The method to use depends on the values used in the arithmetic problem.
Example:
14.8 + 15
3
14.5 + 15
4
14.8 + 15
3
4
= 14.8 +15.75
14.5 + 15
4
= 30.55
Example:
28 × 1
2
30 × 1
5
28 × 1
Example:
3
2
= 28 × 1.4
5
= 39.2
30 × 1
2
3
= 14  15
4
4
4
5
1
= 29  30
4
4
2
5
2
5
5.1 +
5
6
9.1 ×
2
3
5.1 +
5
= 5.1 + 0.83
6
9.1 ×
2
3
= 5.93
3
7
5
= 6 × 7 = 42
= 30 ×
=
18.2
3
= 6.06
Try the last example without using fractions.
How would you multiply 9.1 × 0.6 or 9.1 × 0.6666… to obtain an exact solution?
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