Recurring decimal

A recurring decimal is an expression representing a real number in the decimal numeral system, in which after some point the same sequence of digits repeats infinitely many times. The repetition may begin before, at, or after the decimal point. The repeating sequence may consist of just one digit or of any finite number of digits. If the repeating sequence is merely a repeating "0", then the decimal is said to terminate because it is not necessary to explicitly write that there is a repeating "0". Such terminating decimals represent rational numbers whose fractions in lowest terms are of the form k/(2n5m).

Related Topics:
Real number - Decimal - Numeral system - Rational number - In lowest terms

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One convention to indicate a recurring decimal is to put a horizontal line above the repeated numerals (1/3 = 0.overline{3}). Another convention is to place dots above the numerals. Where these methods are impossible, the extension may be represented by an ellipsis (...) although this may introduce uncertainty as to exactly which digits should be repeated:

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• 1/9 = 0.111111111111...
• 1/7 = 0.142857142857...
• 1/3 = 0.333333333333...
• 1/81 = 0.0123456790...
• 2/3 = 0.666666666666...
• 7/12 = 0.58333333333...
 
Recurring decimal: Fractions with prime denominators ►