Division with negative numbers looks like a small extension of ordinary division, but it raises two questions students often stumble over. What sign does the answer have? And what does a remainder mean when one of the numbers is negative? Both have clear answers.
The sign rules
The rules for division match those for multiplication:
- Positive ÷ positive = positive
- Negative ÷ negative = positive
- Positive ÷ negative = negative
- Negative ÷ positive = negative
A short way to remember: if the signs are the same, the answer is positive; if they differ, it is negative.
Why two negatives make a positive
Consider −12 ÷ −3. We ask what number multiplied by −3 gives −12. Since −3 × 4 = −12, the answer is 4. In general, dividing is the inverse of multiplying, so the sign rules for division come directly from those of multiplication.
Method for long division
The easiest approach is to ignore the signs, do the long division with the absolute values, and then apply the sign rule at the end. For −156 ÷ 12, divide 156 by 12 to get 13. The signs differ, so the answer is −13.
Remainders with negatives
When there is a remainder, different conventions exist. For −17 ÷ 5:
- Truncation toward zero: quotient −3, remainder −2, because 5 × (−3) + (−2) = −17.
- Floor division: quotient −4, remainder 3, because 5 × (−4) + 3 = −17.
Both satisfy dividend = divisor × quotient + remainder, but they differ in how the quotient is rounded. In school arithmetic, the simple approach is to divide the absolute values and attach the sign to the quotient. In programming, languages vary: some use truncation and some use floor division, and the remainder or modulo operator can therefore produce different signs.
Decimal results
Sometimes it is clearer to continue into decimals. −17 ÷ 5 = −3.4. The decimal answer avoids the remainder question entirely, since there is only one value.
Check with multiplication
To verify −156 ÷ 12 = −13, compute −13 × 12 = −156. If you get the original dividend, the answer is right, including the sign.
Common mistakes
- Forgetting the sign entirely after the long division.
- Applying "two negatives make a positive" to subtraction problems where it does not apply.
- Dividing by zero, which is undefined regardless of sign.
- Mixing up the quotient and remainder conventions.
Real contexts
Negative numbers appear in temperatures, bank balances and elevations. A temperature that falls by 15 degrees over 5 hours changes by −15 ÷ 5 = −3 degrees per hour. An overdraft of 240 shared equally among 8 accounts is −240 ÷ 8 = −30 each. In each case, the sign carries meaning.
A tool for checking
A long division calculator for negative numbers can show the sign, the quotient and the remainder at the same time, which is handy when you are learning how conventions differ.
A worked example with a real-life context
Suppose the temperature drops from 4 degrees to −11 degrees over 5 hours. The change is −11 − 4 = −15 degrees, so the average hourly change is −15 ÷ 5 = −3 degrees per hour. Now suppose a company records a loss of 2,400 over 8 equal months. Each month averages −2,400 ÷ 8 = −300. In both cases, dividing a negative quantity by a positive count gives a negative rate, and the sign tells you the direction of change, which is often more important than the size.
Practice questions
Try these and check the signs first, then the sizes: −84 ÷ 7, −90 ÷ −15, 144 ÷ −12, and −250 ÷ 8. The answers are −12, 6, −12 and −31.25. Notice that the last one does not divide evenly, so you can give it as a decimal or as a quotient and remainder.
Summary
Divide the absolute values, apply the sign rules, and check by multiplying. Be aware that remainders with negatives depend on convention, and always state which one you are using.