Converting recurring decimals - Higher

A recurring decimal exists when decimal numbers repeat forever. For example, 0. \dot{3} 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 0.5 \dot{7} is equal to 0.5777777... and 0. \dot{2} \dot{7} is equal to 0.27272727...

If two dots are used, they show the beginning and end of the recurring group of numbers: 0. \dot{3} 1 \dot{2} 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 0. \dot{5} \dot{7}.

Example

Convert \frac{5}{6} 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:

Diagram showing how to converting 5/6 into a recurring decimal

\frac{5}{6} = 0.8333 ... = 0.8\dot{3}

Algebra can be used to convert recurring decimals into fractions.

Example

Convert 0. \dot{1} to a fraction.

0. \dot{1} has 1 digit recurring.

Firstly, write out 0. \dot{1} as a number, using a few iterations (repeats) of the decimal.

0.111111111...

Call this number x. We have an equation x = 0.1111111...

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

So if:

x = 0.11111111...then

10x = 1.11111111…

Notice that after the decimal points the recurring digits match up. So subtracting these equations gives:

10x~–~x = 1.111111… – 0.111111…

so 9x = 1

Dividing both sides by 9 gives:

x = \frac{1}{9}

so ~ 0. \dot{1} = \frac{1}{9}

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 0. \dot{1} \dot{8} is equal to \frac{2}{11}.

0.\dot{1}\dot{8} 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).

0. \dot{1} \dot{8} = 0.181818 \dotsc

Call this number x.

We have an equation x = 0.181818 \dotsc

If we multiply this number by 100 it will give a different number with the same digits recurring. So if:

x = 0.181818 \dotsc then

100x = 18.181818 \dotsc

Notice that after the decimal points the recurring digits match up. So subtracting these equations gives:

100x - x = 18.181818 \dotsc - 0.181818 \dotsc

So, 99x = 18

Dividing both sides by 99 gives:

x = \frac{18}{99}

9 is a common factor of 18 and 99

so \frac{18}{99} simplifies to \frac{2}{11}

so 0. \dot{1} \dot{8} converts to \frac{2}{11}

Question

Show that 0.2 \dot{8} is equal to \frac{13}{45}.

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

0.2 \dot{8} = 0.288888 \dotsc

Give this number a name ( x):

x = 0.288888 \dotsc

1 digit recurs, so multiply by 10.

So x = 0.288888...

So 10x = 2.88888...

Subtracting these equations gives:

10x-x=2.888888… − 0.288888

So, 9x = 2.6

Multiplying both sides by 10 gives:

90x = 26

Dividing both sides by 90, (to get the value of x):

x = \frac{26}{90}

2 is a common factor of 26 and 90, so:

x = \frac{26}{90} = \frac{13}{45}

0.2 \dot{8} as a fraction is \frac{13}{45}.

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