Recurring Decimals

Recurring decimals
BTEOTSSBAT:

Write recurring decimals using the dot notation
Change recurring decimals into exact fractions

Key terms
Decimals
Fractions
Recurring
Exact

Recurring decimals
= 0.3333333333333……
3
1

Recurring decimals
… it keeps going on, it will never stop – we say it recurs.
= 0.3333333333333……
3
1

Recurring decimals
= 0.3333333333333……
3
We can write it in mathematical shorthand as:
1

Recurring decimals
= 0.3333333333333……
3
We can write it in mathematical shorthand as:
1
.
0.3333333333333…… = 0.3

0.12121212121212… Is another recurring decimal

This can be written as 0.12
.
.

.
.
In the same way
0.12341234123412341234 =

.
.
In the same way
0.12341234123412341234 =
.
.
0.1234

Recurring decimals as exact fractions
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)

For example
Write 0.3 as an exact decimal

For example
Here only one 3 is recurring
So the numerator is 3
3

For example
3
3
There is one digit in the numerator so we have one 9 in the denominator
9

For example
3
3
1
There is one digit in the numerator so we have one 9 in the denominator
9
This can then be cancelled down to be
3

.
.
0.12

.
.
0.12
12

.
.
0.12
2 digits
12

.
.
0.12
2 digits
99
12

.
.
0.1234

.
.
0.1234
1234

.
.
0.1234
1234
4 digits

.
.
0.1234
1234
4 digits
9999

.
.
0.12
2 digits
99
12
.
.
0.1234
1234
4 digits
9999

Now try these
Write the following using the dot notation then convert them into exact fractions.
0.111111111….
0.4343434343 ….
0.765765765 ….
0.237823782378237…

Now try these
0.111111111….
0.4343434343 ….
0.765765765 ….
0.237823782378237…
.
0.1
0.43
0.765
0.2378
.
.
.
.
.
.