How long division works
Long division turns one hard division into a sequence of easy ones. You scan the dividend from left to right, and at each position you ask a single question: how many whole times does the divisor fit into the running number you have so far?
That answer becomes the next digit of the quotient. You then multiply it back by the divisor, subtract, and carry whatever is left over down to the next digit. Repeating this until you run out of digits leaves you with the final quotient and the last remainder.
- Bring down the next digit and append it to the previous remainder.
- Divide to find the quotient digit, then multiply and subtract.
- Carry the remainder forward and repeat to the end.
Reading the result
The quotient is the whole number of times the divisor fits into the dividend, and the remainder is what is left over. Together they satisfy the identity dividend = divisor × quotient + remainder, which is a quick way to check your answer.
The decimal value continues the division past the decimal point, so it is the exact (or rounded) fraction dividend ÷ divisor. Use the quotient and remainder when you need whole-number sharing, and the decimal when you need a precise ratio.
Practical tips
Estimating before you start helps you catch mistakes: round both numbers to convenient values and predict the rough size of the quotient.
- A remainder smaller than the divisor confirms you took the quotient as far as it goes.
- Multiply the quotient by the divisor and add the remainder to verify the original dividend.
- Trailing zeros in the dividend simply carry the remainder forward unchanged.
Common mistakes
The most frequent slips happen when a digit of the dividend is too small for the divisor to fit even once. In that case the quotient digit is zero, and forgetting to write it shifts every later digit and ruins the answer.
- Do not skip a zero quotient digit when the divisor does not fit.
- Keep the columns aligned so each subtraction lines up correctly.
- A remainder equal to or larger than the divisor means the quotient digit was too small.
Formula
dividend = divisor × quotient + remainderFrequently asked questions
- Does it handle negative numbers?
- The worked steps assume non-negative inputs. The quotient, remainder and decimal value are still computed for any sign.