A recurring decimal is a decimal in which one or more digits repeat forever. Recurring decimals are exact fractions, meaning they can be written exactly as a fraction, not just approximately.

For example, the decimal \( 0.\dot{3} \) has the digit 3 repeating. This decimal is equal to the fraction:

$$
\frac{1}{3}
$$

This shows that the recurring decimal represents an exact value.

Some recurring decimals have more than one repeating digit. For example:

$$
0.\dot{1}\dot{6}
$$

This recurring decimal is equal to the fraction:

$$
\frac{1}{6}
$$

This demonstrates that recurring decimals can always be converted into exact fractions.

The reverse is also true. Some exact fractions produce recurring decimals when written in decimal form. Any fraction whose denominator has prime factors other than \( 2 \) or \( 5 \) will produce a recurring decimal.

For example:

$$
\frac{2}{3} = 0.\dot{6}
$$

$$
\frac{5}{11} = 0.\dot{4}\dot{5}
$$

These decimals never terminate, but they are still exact values because the repeating pattern continues forever.

It is important to recognise that recurring decimals are not approximations. Although they look similar to rounded decimals, they represent exact fractions. Understanding this link helps when converting between fractions and decimals and when deciding whether a value is exact or approximate.

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