The rule
dividend = divisor × quotient + remainder
Why it works
Long division breaks a large division into a series of small ones. You take the digits of the dividend one at a time, see how many times the divisor fits, write that down, and carry the remainder forward.
How to do it by hand
- Take enough digits from the left for the divisor to fit
- Write how many times it fits above
- Subtract the product and bring down the next digit
- Repeat, adding a decimal point when the digits run out
What is worth knowing
The check is worth doing every time and takes two seconds: divisor × quotient + remainder must equal the dividend. It catches essentially every arithmetic slip. Long division has fallen out of favour in some curricula, on the grounds that calculators exist, and the counterargument is that it is the first algorithm most people meet — a procedure that is guaranteed to terminate with the right answer if followed exactly. That idea is worth more than the division itself.