Recurring Decimals

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