Recurring decimals
BTEOTSSBAT:•Write recurring decimals using the dot notation
•Change recurring decimals into exact fractions
Key terms
Decimals
Fractions
Recurring
Exact
= 0.3333333333333……
Recurring decimals
1
3
= 0.3333333333333……
… it keeps going on, it
will never stop – we say
it recurs.
Recurring decimals
1
3
= 0.3333333333333……
… it keeps going on, it
will never stop – we say
it recurs.
We can write it in mathematical shorthand as:
Recurring decimals
1
3
= 0.3333333333333……
0.3333333333333…… = 0.3
… it keeps going on, it
will never stop – we say
it recurs.
We can write it in mathematical shorthand as:
Recurring decimals
.
1
3
0.12121212121212… Is another recurring decimal
0.12121212121212… Is another recurring decimal
This can be written as 0.12. .
0.12121212121212… Is another recurring decimal
This can be written as 0.12
In the same way
0.12341234123412341234 =
. .
0.12121212121212… Is another recurring decimal
This can be written as 0.12
In the same way
0.12341234123412341234 =
. .
0.1234
. .
Recurring decimals can easily be written as fractions.
To do this:
• place the recurring numbers at the top of the decimal (as the
numerator)
• write as many 9s as there are digits in the numerator as the bottom
bit (denominator)
Recurring decimals as exact fractions
Recurring decimals as exact fractions
For example
Write 0.3 as an exact decimal
Recurring decimals as exact fractions
For example
Here only one 3 is recurring
So the numerator is 3
Write 0.3 as an exact decimal
3
Recurring decimals as exact fractions
For example
Here only one 3 is recurring
So the numerator is 3
Write 0.3 as an exact decimal
3
3
There is one digit in the numerator so we
have one 9 in the denominator 9
Recurring decimals as exact fractions
For example
Here only one 3 is recurring
So the numerator is 3
Write 0.3 as an exact decimal
3
3
There is one digit in the numerator so we
have one 9 in the denominator 9
This can then be cancelled down to be 1
3
0.12..
0.12 12..
0.12 12..
2 digits
0.12 12
99
..
2 digits
0.1234. .
0.1234 1234
. .
0.1234 1234
. .
4 digits
0.1234 1234
9999
. .
4 digits
0.1234 1234
9999
. .
4 digits
0.12 12
99
0.1234 1234
9999
..
. .
2 digits
4 digits
0.12 12
99
0.1234 1234
9999
..
. .
2 digits
4 digits
Now try theseWrite the following using the dot notation then convert them into
exact fractions.
0.111111111….
0.4343434343 ….
0.765765765 ….
0.237823782378237…
Now try theseWrite the following using the dot notation then convert them into
exact fractions.
0.111111111….
0.4343434343 ….
0.765765765 ….
0.237823782378237…
0.1
0.43
0.765
0.2378
.
.
..
..
.
Now try theseWrite the following using the dot notation then convert them into
exact fractions.
0.111111111….
0.4343434343 ….
0.765765765 ….
0.237823782378237…
0.1
0.43
0.765
0.2378
.
.
..
..
.
1
943
99765
999 2378
9999