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Converting recurring decimals - Higher

Using dot notation

A recurring decimal exists when 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.

First, convert the to a decimal by division:

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

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

Algebra skills are needed to turn 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 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:

\(x~=~0.1111111…\) then

\(10x~=~1111111…\)

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 \(0. \dot{1}~=~\frac{1}{9}\)

Question

Prove that \(0. \dot{1}\dot{8}\) is equal to \(\frac{2}{11}\).

Question

Prove that \(0.2 \dot{8}\) is equal to \(\frac{13}{45}\).

Some decimals terminate which means the decimals do not recur, they just stop. For example, 0.75.
To find out whether a fraction will have a terminating or recurring decimal, look at the prime factors of the when the fraction is in its most simple form. If they are made up of 2s and/or 5s, the decimal will terminate. If the prime factors of the denominator contain any other numbers, the decimal will recur.
Some decimals are irrational, which means that the decimals go on forever but not in a pattern (they are not recurring). An example of this would be \(\pi\) or \(\sqrt{2}\) which is also called a surd.

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