Division By Repeated Subtraction
The simplest division algorithm, historically incorporated into a greatest common divisor algorithm presented in Euclid's Elements, Book VII, Proposition 1, finds the remainder given two positive integers using only subtractions and comparisons:
while N ≥ D do N := N - D
end
return N
The proof that the quotient and remainder exist and are unique, described at Euclidean division, gives rise to a complete division algorithm using additions, subtractions, and comparisons:
function divide(N, D) if D == 0 then throw DivisionByZeroException end if D < 0 then (Q,R) := divide(N, -D); return (-Q, R) end if N < 0 then (Q,R) := divide(-N, D); return (-Q - 1, D - R) end // At this point, N ≥ 0 and D > 0 Q := 0; R := N while R ≥ D do Q := Q + 1 R := R - D end return (Q, R)
end
This procedure always produces R ≥ 0. Although very simple, it takes Ω(Q) steps, and so is exponentially slower than even slow division algorithms like long division. It is useful if Q is known to be small (being an output-sensitive algorithm), and can serve as an executable specification.
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