Decimal expansion of rational numbers — lesson. Mathematics State Board, Class 9.

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Rational numbers can be expanded in the form of decimals by doing the usual long division.

A terminating decimal is a decimal number that has the finite number of digits the decimal point.

A recurring decimal is a decimal number that has repeating number/numbers which continuous infinitely.

Now we will analyse the decimal expansions of different types of rational numbers.

Rational number	Value with the help of long division	Nature of the decimal expansion
$\begin{array}{l} \frac{1}{125} = 1 \div 125 \\ = 0.008 \end{array}$		Terminating decimals
$\begin{array}{l} \frac{1}{32} = 1 \div 32 \\ = 0.03125 \end{array}$		Terminating decimals
$\begin{array}{l} \frac{1}{6} = 1 \div 6 \\ = 0.1666... \\ = 0.1 \bar{6} \end{array}$		Recurring and non-terminating decimals
$\begin{array}{l} \frac{152}{333} = 152 \div 333 \\ = 0.456456456... \\ = 0. \bar{456} \end{array}$		Recurring and non-terminating decimals

In the above examples, examples $1$ and $2$ gives terminating quotients, and the examples $3$ and $4$ gives the recurring decimals quotients.

Thus, it can be concluded as follows:

A rational number can have either a terminating decimal expansion or a non-terminating recurring decimal expansion.

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1. Decimal expansion of rational numbers