Long Division Calculator

What the algorithm is actually doing

Long division is repeated subtraction organized by place value. At each step you ask how many times the divisor fits into the part of the dividend you have brought down so far, write that digit above the line, subtract, and bring down the next digit. Working left to right means each digit of the quotient is settled before you move on, and the running remainder is always smaller than the divisor. That last property is the invariant that makes the method work — if a remainder ever comes out equal to or larger than the divisor, the previous quotient digit was too small.

Remainder, decimal, or fraction

There are three legitimate ways to express the same result. 4728 divided by 23 is 205 remainder 13, or 205.565…, or 205 and 13/23. Which one is right depends on the question: remainders suit counting problems where fractions of a thing make no sense, decimals suit measurement, and fractions are exact where decimals may not be. Continuing past the decimal point is the same algorithm with zeros brought down instead of digits.

Why some decimals repeat

At every step the remainder must be one of the divisor's possible remainders — 0 to divisor minus one. Once a remainder repeats, the whole sequence of digits after it must repeat too, because the algorithm is deterministic. That is why every fraction either terminates or falls into a repeating cycle, and why the cycle can never be longer than the divisor minus one. A fraction terminates only when the denominator's prime factors are just 2 and 5, the primes that divide ten — which is why thirds repeat in decimal but would terminate in base three.

Checking your own work

Multiply the quotient by the divisor and add the remainder; you must get the dividend back exactly. That check is shown above and it catches almost every arithmetic slip, because an error anywhere in the chain breaks the identity. It is worth the ten seconds, particularly in exams where the method carries most of the marks and a single subtraction error would otherwise cost all of them.

Why learn it at all

The immediate answer is that it builds a feel for place value that calculators cannot. The longer answer is that the same algorithm reappears in polynomial long division, which is structurally identical with powers of x taking the role of powers of ten, and that turns up throughout algebra and calculus. The arithmetic version is the easy case where the idea is learned.

Frequently Asked Questions