Recurring Decimals (or Repeating Decimals)

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Recurring Decimals (or Repeating Decimals)1. Recurring DecimalIf a figure or a set of figures is repeated continuously in a decimal fraction, it is called a recurring decimal (also known as repeating decimal and circulating decimal).

A repeating decimal is denoted by placing a horizontal line above the repeated numerals. Another way to denote a recurring decimal is placing dots over the first and last digits.

Examples

(a) $\frac{1}{3}=\mathrm{0.333...}=0.\overline{3}=0.\dot{3}$

(b) $\frac{1}{7}=\mathrm{0.142857142857142857...}$ $=0.\overline{142857}=0.\dot{1}4285\dot{7}$

(c) $\frac{7}{12}=\mathrm{0.58333...}$ $=0.58\overline{3}=0.58\dot{3}$

2. Type of Recurring DecimalsRecurring Decimals can be classified into two categories - Pure Recurring Decimals and Mixed Recurring Decimals
(i) Pure Recurring DecimalsPure Recurring Decimal is a decimal fraction in which all the figures after the decimal point are repeated.

Examples

(a) $\frac{1}{3}=\mathrm{0.333...}=0.\overline{3}=0.\dot{3}$

(b) $\frac{4}{11}=\mathrm{0.36363636...}=0.\overline{36}=0.\dot{3}\dot{6}$
(ii) Mixed Recurring DecimalsMixed Recurring Decimal is a decimal fraction in which some figures are not repeated and some figures are repeated.

Examples

(a) $\frac{7}{12}=\mathrm{0.58333...}$ $=0.58\overline{3}=0.58\dot{3}$

(b) $\frac{87}{66}=\mathrm{1.3181818...}$ $=1.3\overline{18}=1.3\dot{1}\dot{8}$