Recurring Decimals as fractions

This chapter explores recurring decimals as fractions. It covers using recurring decimal notation and expressing recurring decimals as fractions. No prior knowledge is required for this chapter.

Recurring decimals

Recurring decimals are decimals which contain digits repeat forever, For example;

…repeats forever. To avoid writing so many 3s at the end to denote the repetition of 3 we put a dot on top of 3 as shown below.

Here is another recurring decimal;

In the example above we put the dot above 6 to show that it carries on forever as shown below.

The decimal number below has two digits that repeat 3 and 6.

To show the digits that repeat forever we put the dot on 3 and 6 as shown below.

The following decimal number contains 4 digits that repeat;

We put the dot above the 3 and the 4 to show that the group of digits starting at the 3 and ending at the 4 repeat forever as shown below;

What is 1 third as a decimal?

All recurring decimals can be written as fractions for example;

Let’s first put the recurring fraction above on a number line. The recurring decimal above is greater than and less than 0.4

Above we can see that he recurring decimal is close to the fraction of 1/3. That must mean that;

Here is another example;

First we put the decimal on the number line.

Above we can see that;

Fractions to recurring decimals

The fraction;

…can be written as a recurring decimal, we can work out the fraction using the division.

…which is;

Dividing 9 by 1 is very difficult. First we add a decimal point and a few zeros after it.

9 divide by 1 is 0 remainder 1 so we write as shown below notice where we have put the remainder;

9 divide by 10 is 1 remainder 1. So we write as shown below.

You continue through the same steps until we realise that the answer goes on forever.

…we have seen that;

…also

It has also indicated that;


…and so further…

Another example

Here is another example; Suppose we wanted to write;

…as a decimal. Then we would want to do really is divide 1 over 6.

First we add the extra zero as explained above.

…1 divide by 6 is 0 remainder 1 so we get;

10 divide by 6 is 1 remainder 4 so we write;

40 divide by 6 is 6 remainder 4. So write;

You’ll realise the 6 goes on forever that must mean that;

…also;