Long division
Rational quotients

If the quotient is not constrained to be an integer, then the algorithm does not terminate for i > k − l {\displaystyle i>k-l}. Instead, if i > k − l {\displaystyle i>k-l} then α i = 0 {\displaystyle \alpha _{i}=0} by definition. If the remainder r i {\displaystyle r_{i}} is equal to zero at any iteration, then the quotient is a b {\displaystyle b} -adic fraction, and is represented as a finite decimal expansion in base b {\displaystyle b} positional notation. Otherwise, it is still a rational number but not a b {\displaystyle b} -adic rational, and is instead represented as an infinite repeating decimal expansion in base b {\displaystyle b} positional notation.
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