Recurring decimals as fractions

A terminating decimal is a fraction with a power of ten as denominator. A recurring decimal repeats a fixed block forever. Multiplying by a suitable power of ten lines up identical recurring tails; subtraction removes the tail exactly.

Example

Let \(x=0.\overline{27}=0.272727\ldots\). Then \(100x=27.272727\ldots\), so \(100x-x=27\). Thus \(99x=27\) and \(x=\dfrac3{11}\).

For \(y=0.1\overline6\), the recurring tails line up in \(100y=16.666\ldots\) and \(10y=1.666\ldots\). Hence \(90y=15\), so \(y=\dfrac16\). The non-recurring digit changes which two powers of ten are needed.

A rounded display such as \(0.1667\) is only an approximation to \(\dfrac16\). Put recurring dots or a bar over precisely the repeated digits.

For the reverse conversion, divide the numerator by the denominator. With \(4\div11\), \(40=3\times11+7\), then \(70=6\times11+4\). The remainder 4 has returned, so the same digits repeat: \(\dfrac4{11}=0.\overline{36}\). A remainder of zero instead gives a terminating decimal.