A repeating decimal has a decimal representation whose digits eventually repeat a finite block. An initial decimal preperiod may occur before the repeated tail: , whereas begins repeating immediately. Terminating expansions fit the same convention by repeating zeros, as in .
A real number is rational exactly when its decimal expansion is eventually periodic. For a rational number, long division has finitely many possible remainders, so it either reaches zero or repeats a previous remainder. Conversely, a repeating tail sums as a geometric series, giving a rational number. The same argument holds in every integer positional base at least two, as in eventually periodic positional expansion of a rational number.
Decimal representations need not be unique: . Both are eventually periodic, so this endpoint ambiguity does not affect the rationality criterion.
A repeating decimal is a decimal representation of a number in which a digit or a group of digits repeats indefinitely. This means that after a certain point, the same sequence of digits appears over and over again, without end. Repeating decimals can result from the division of certain fractions. For example: - The fraction \( \frac{1}{3} \) is equal to \( 0.333...\), where the digit '3' repeats forever.
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