Download Rational numbers and their decimal representation

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Surds and indices
Rational numbers and their decimal
representation
All rational numbers can be expressed as either
&
decimals which terminate, such as 0.72, or
&
recurring decimals, such as 0:102 432 432 43 . . .
0:102 432 43 . . . can be written
_ The dots indicate
as 0:102_ 43.
the set of digits which recur.
Conversely, all decimals that terminate or recur represent rational numbers. A method
for converting a terminating decimal into a fraction is shown in the Example below.
Irrational numbers cannot be expressed as terminating or recurring decimals.
Conversely an infinite decimal that never recurs cannot be expressed as a rational
number. For example, in 0:101 001 0001 . . . the number of zeros increases by one
before each digit 1, so the decimal never recurs and it cannot be expressed as a ratio
of two integers.
To sum up: Rational numbers can be represented by terminating or recurring
decimals and vice versa. Irrational numbers are non-terminating, non-recurring
decimals and vice versa.
Example
Express decimals as fractions in their lowest terms.
a
b
43:72 ˆ
0:7_
Let
Then
4372
100
43.72 is 4372 hundredths.
0:2_ 4_
Let
Then
7
1
100
2
The fraction cannot be cancelled further. It is in its lowest terms.
x ˆ 0:777_
1
10x ˆ 7:777_
2
Divide both sides by 9.
1
100x ˆ 24:2_ 4_
2
99x ˆ 24
24
xˆ
99
8
33
Multiply both sides by 10. Since one digit
recurs, this lines up the recurring digits.
Subtract line 1 from line 2 .
x ˆ 0:2_ 4_
xˆ
1
10
Divide numerator and denominator by 4.
1093
ˆ
25
9x ˆ 7
7
xˆ
9
c
10 1
4 3
Multiply both sides by 100. Since two digits
recur, this lines up the recurring digits.
Subtract line 1 from line 2 .
Divide both sides by 99.
Cancel to lowest terms.
AS Core for OCR # Pearson Education Ltd. 2004
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Surds and indices
For an alternative method for converting recurring decimals to fractions, see Example
27 on page 324.
Example
Express fractions as decimals.
a
3
12
ˆ
ˆ 0:12
25 100
b
6
7
By multiplying numerator and denominator by 4, the denominator
can be made into a power of 10 (i.e. 10, 100, 1000, etc. . . .).
The conversion to decimal form is then simple.
When the denominator cannot be made into a power of
10, the numerator must be divided by the denominator.
Divide 6 by 7
0:8 5 7 1 4 2 8 . . .
7j6:04 05 01 03 02 06 0 . . .
The group of six digits will recur.
6
_ 142_
ˆ 0:857
7
Note
Division of the numerator by the denominator can always be used to convert a
fraction to a decimal.
Use of calculators
Many calculators have a fraction button and, with practice, all fractions and their
equivalent decimal form can be found using a calculator.
The serious student must aim, however, to be fluent with numerical work and this
means acquiring the ability to calculate speedily and accurately with and without a
calculator. Being skilful with numbers is satisfying as well as an important basis for
further mathematical development.
The Exercise below should be completed without a calculator. A calculator can be
used to check that an answer is correct or, at least, to provide supporting evidence.
For example,
p
p
2
6‡ 2
p
ˆ
17
3 2 1
Evaluating both these expressions on the calculator gives 0.4361 correct to 4 d.p.
This does not prove that the expressions are equivalent but it does provide some
support and reassurance!
Exercise
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1
Express each of these decimals as a fraction in its lowest terms.
a 0:6_
b 0:4_
c 0:1_ 2_
d
_
_
_
_
_
e 0:12
f 3:407
g 0:48
h
AS Core for OCR # Pearson Education Ltd. 2004
0:305_
4:2_
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2
Express each of these fractions in decimal form.
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b 125
c 59
a 507
e
44
7
f
1
6
g
3
8
Surds and indices
d
4
11
h
5
7
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Explore the decimal representation of all proper fractions (< 1) with these
denominators.
a 7
b 9
c 11
d 13
e 19
4
What must be true of the integers a, and/or b, if ab terminates when expressed in
decimal form (assume that ab is in its lowest terms)?
AS Core for OCR # Pearson Education Ltd. 2004
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