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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
1125=1÷125=0.008numbers_1.pngTerminating decimals
132=1÷32=0.03125numbers_3.pngTerminating decimals
16=1÷6=0.1666...=0.16¯numbers_7.pngRecurring and non-terminating decimals
152333=152÷333=0.456456456...=0.456¯numbers_11.pngRecurring 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.