# Converting recurring decimals - Higher

A recurring decimal exists when decimal numbers repeat forever. For example, means 0.333333... - the decimal never ends.

Dot notation is used with recurring decimals. The dot above the number shows which numbers recur, for example is equal to 0.5777777... and is equal to 0.27272727...

If two dots are used, they show the beginning and end of the recurring group of numbers: is equal to 0.312312312...

### Example

How is the number 0.57575757... written using dot notation?

In this case, the recurring numbers are the 5 and the 7, so the answer is .

### Example

Convert to a recurring decimal.

Divide 5 by 6.

5 divided by 6 is 0, remainder 5, so carry the 5 to the tenths column.

50 divided by 6 is 8, remainder 2.

20 divided by 6 is 3 remainder 2.

Because the remainder is 2 again, the digit 3 is going to recur: Algebra can be used to convert recurring decimals into fractions.

### Example

Convert to a fraction. has 1 digit recurring.

Firstly, write out as a number, using a few iterations (repeats) of the decimal.

0.111111111...

Call this number . We have an equation ...

If we multiply this number by 10 it will give a different number with the same digit recurring.

So if: ...then Notice that after the decimal points the recurring digits match up. So subtracting these equations gives: so Dividing both sides by 9 gives: so When 2 digits recur, multiply by 100 so that the recurring digits after the decimal point keep the same place value. Similarly, when 3 digits recur multiply by 1000 and so on.

Question

Show that is equal to . has 2 digits recurring.

First, write the recurring decimal as a long number. Use a few iterations (it doesn't matter exactly how many are used). Call this number .

We have an equation If we multiply this number by 100 it will give a different number with the same digits recurring. So if: then Notice that after the decimal points the recurring digits match up. So subtracting these equations gives: So, Dividing both sides by 99 gives: 9 is a common factor of 18 and 99

so simplifies to so converts to Question

Show that is equal to .

First, write the recurring decimal as a long number. Use a few iterations (it doesn’t matter exactly how many are used). Give this number a name ( ): 1 digit recurs, so multiply by 10.

So So Subtracting these equations gives: So, Multiplying both sides by 10 gives: Dividing both sides by 90, (to get the value of ): 2 is a common factor of 26 and 90, so:  as a fraction is .

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