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2.10
Rational numbers and
reciprocals
This section will show you how to:
● recognise rational numbers, reciprocals,
terminating decimals and recurring decimals
● convert terminal decimals to fractions
● convert fractions to recurring decimals
● find reciprocals of numbers or fractions
Key words
rational number
reciprocal
recurring decimal
terminating
decimal
Rational numbers
––3 .
A rational number is a number that can be written as a fraction, for example, –41 or 10
When a fraction is converted to a decimal it will be either:
• a terminating decimal or
• a recurring decimal
A terminating decimal has a finite number of digits. For example, –41 = 0.25, –81 = 0.125.
A recurring decimal has a digit, or block of digits, that repeat. For example, –31 = 0.3333 …,
2
–11
– = 0.181818 …
Recurring digits can be shown by putting a dot over the first and last digit of the group that
repeats.
·
0.3333 … becomes 0.3
··
0.181818 … becomes 0.18
· ·
0.123123123 … becomes 0.123
·
0.58333 becomes 0.583
··
0.6181818 … becomes 0.618
· ·
0.4123123123 … become 0.4123
Converting fractions into recurring decimals
A fraction that does not convert to a terminating decimal will give a recurring decimal. You may
·
already know that –31 = 0.333 … = 0.3 This means that the 3s go on for ever and the decimal
never ends.
To convert the fraction, you can usually use a calculator to divide the numerator by the
denominator.
Note that calculators round off the last digit so it may not always be a true recurring decimal in
the display.
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