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.
Read more about this topic: Division (digital)
Famous quotes containing the words division and/or repeated:
“O, if you raise this house against this house
It will the woefullest division prove
That ever fell upon this cursed earth.”
—William Shakespeare (1564–1616)
“Modern man likes to pretend that his thinking is wide-awake. But this wide-awake thinking has led us into the mazes of a nightmare in which the torture chambers are endlessly repeated in the mirrors of reason.”
—Octavio Paz (b. 1914)