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07. Decimals
07-1 Decimals
(1) Definition: Decimals are numbers expressed with a decimal point. The decimal system is
based on powers of ten (base ten system).
whole
number
part
 Place Value: The value of a digit depends on its place in the number. Each place has
a value of 10 times the place to its right
decimal part
ex.) 5.26  5  0.2  0.06  (5  1)  (2  0.1)  (6  0.01)
 (5  100)  (2  101)  (6  102)
 Every number to the right of the decimal point is a decimal fraction.
ex.) 0.75  “seventy-five hundredths”
ex.) 45.321  “forty-five and three hundred twenty-one thousandths”
(2) Rounding off Decimals
Rounding off is a kind of estimating. Look at the digit to the right of the “rounding digit”. If the digit is 5 or more,
round up by adding one to the rounding digit and drop all digits to the right of it; otherwise, do not change the rounding
digit but drop all digits to the right of it.
[Example]  Round off 3815.296.
Round to the nearest:
thousand: 4000
hundred: 3800
tenth: 3815.3
ten: 3820
hundredth: 3815.30
one: 3815
07-2 Classification of Decimals
Decimals
Finite decimals (or terminating)
Rational Numbers
Repeating decimals
Infinite decimals
Non-repeating decimals
Irrational Numbers
 Terminating decimals and repeating decimals are rational numbers and can be written in the form of a fraction.
 Irrational numbers cannot be expressed as fractions.
[Note]
 When a fraction is converted to a decimal, it is as either a terminating decimal or a repeating decimal.
(a) Fractions representing terminating decimals
(i) Terminating decimals can be represented by the fraction of which the denominator has a power of 10.
216
216
4
1485
1485
ex.) 0.4 
,
2.16 

,
1.485 

10
1000
100
103
102
(ii) When terminating decimals can be represented by fractions in simplest form, there are only prime factors of 2 and 5
in the denominator.
3
3
ex.)

 There are only prime factors of 2 and 5 in the denominator.  Terminating decimal
50
2  52

3
2  52

32
2  52  2

6
22  5 2

6
 0.06  The denominator has a power of 10.
100
(b) Fractions representing repeating decimals: If there is any prime factor other than 2 or 5 in the denominator, the
fraction cannot be a terminating decimal but a repeating decimal.
7
7
ex.)

 There is a prime factor of 3 other than 2 or 5 in the denominator.  Repeating decimal
15
35
 Repeating Decimals
(a) Repeating decimals: An infinite repeating decimal is one that has a specified sequence of digits that repeat.
ex.) 0.263737373737…
(b) Repeating block
(c) The repeating decimal can be represented by putting a dot over the digit or the sequence of digits that repeat.
ex.) 0.166666666 can also be written as 0.16
I. Number and Operations
(d) Fractions that can be represented by repeating decimals: Fractions of which the denominator has any prime
factor other than 2 or 5.
ex.)
7
7

 0.2333···  0.23
30
2 3 5
Repeating decimals
Repeating block
3.1232323···
23
0.555···
Representation
0.5
5
1.234234234···
3.123
1.2 34
234
 Converting Repeating Decimals to Fractions
(a) Let N be the given repeating decimal
(b) Make two equations by multiply both sides of the equation by a power of 10 so that the repeating block cancels
out
(c) Subtract N from both sides
ex.) Converting a repeating decimal of a .bcd to a fraction
(i) N  a .bcd
(ii) (i)  1000  1000 N  abcd.cdcdcd ···
(i)  10 
10 N  ab.cdcdcd ···
[Formula]
(iii) subtract both sides:
whole number without
decimal point
abcd  ab
a .bcd 
990
2 repeating
decimals
1 non-repeating decimal
after decimal point
number
without a
repeating
decimal
990 N  abcd  ab
N 
abcd  ab
990
 The number of digits in the repeating block  2
 Denominator: write 9 for every repeating decimal and 0 for every
number that does not have a repeating decimal
 Numerator: (whole number)  (number without a repeating decimal)
a
ab
(i) 0. a 
(ii) 0. a b 
9
99
ab  a
(iii) 0. a b 
90
ex.) 3.792 
3792  3
999
591  59
a b c a b
(iv) 0. a b c 
900

3789 421

999
111
532 266


90
90
45
472  4 468 26


0.472 


990
990 55
5.91 
I. Number and Operations