We have shown how to express fractions whose denominators have prime factors two or five as decimals. These decimals are terminating.

We now look at fractions whose denominators cannot be factored as products of twos and fives. These fractions produce recurring decimals; that is, decimals with eventually repeating digits. We have met these previously in our discussion of division. (Obviously 3.0000… is not considered to be a recurring decimal.)

Numbers such as \(\dfrac{5}{27}\) when written as a decimal result in a block of digits that repeats indefinitely.

\(\dfrac{5}{27} = 0.185185185… = 0.\dot{1}8\dot{5}\)

This is a recurring decimal and the dots above the digits 1 and 5 indicate that 185 are the recurring digits. Some people prefer to write \(0.\overline{185}\) where the bar indicates that 185 are the recurring digits.

The fraction \(\dfrac{1}{6}= 0.16666… = 0.1\dot{6}.\) Only the digit 6 repeats indefinitely. This is an example of an eventually recurring decimal.

Example 1

Convert \(\dfrac{1}{7}\) to a decimal.

Solution

To convert \(\dfrac{1}{7}\) to a decimal divide 1 by 7. As 7 is a prime number, there is no multiple of 7 that is a power of 10.

\(\dfrac{1}{7}= 0.\dot{1}4285\dot{7}\)

Example 2

Express \(\dfrac{1}{99}\) as a recurring decimal.

Solution

We know that \(99 = 3^2 × 11,\) so \(\dfrac{1}{99}\) will not be a terminating decimal. To convert to a decimal, we must divide 1 by 99.

Hence \(\dfrac{1}{99}= 0.\dot{0}\dot{1}\)