Repeating decimal

A decimal representation in which a digit or block of digits repeats indefinitely after the decimal point. Every repeating decimal represents a rational number.

Fraction-to-decimal conversion: A procedure for expressing a fraction as a decimal by division. Long division produces repeating remainders, which determine whether the decimal eventually repeats.
Decimal place value: A positional notation system in which each digit's value depends on its power of ten. Repeating digits occupy successive powers of one tenth, one hundredth, and smaller places.
Recurring decimal to fraction: A method for converting a repeating decimal into an exact fraction using powers of ten. Subtracting shifted copies of the decimal cancels the repeating tail.
Terminating decimal: A decimal representation that has finitely many nonzero digits after the decimal point. Its expansion ends, unlike a repeating decimal's endless recurring block.
Long division: An algorithm for dividing numbers by repeatedly finding quotient digits and remainders. A remainder that recurs during division forces the same decimal digits to recur.
Long division algorithm: A step-by-step procedure for dividing one number by another using quotient digits and remainders. Its finite set of possible remainders explains why decimal expansions eventually terminate or repeat.
Nines' complement: A digit-by-digit complement formed by subtracting each decimal digit from nine. A repeating block of nines equals one, enabling a quick fraction conversion.
Eventually periodic decimal: A decimal expansion whose digits become periodic after a finite initial segment. Some decimal expansions have a nonrepeating prefix before the repeating block begins.
Rational number: A number expressible as the ratio of two integers with a nonzero denominator. Every repeating decimal is rational, and every rational number has an eventually repeating decimal expansion.
Prime factorization: The expression of an integer as a product of prime numbers. The denominator's prime factors determine whether a reduced fraction terminates in base ten.

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Midy's theoremNarrower topic: Midy's theorem concerns the repeating block in the decimal expansion of a unit fraction.