When you multiply or divide two numbers with the same sign, the answer is positive. When the signs differ, the answer is negative. The rule is the same for both operations.
Why it works
Build the pattern yourself. Start with a multiplication by −2 and go down the list.
| Product | Answer |
|---|---|
| 3 × (−2) | −6 |
| 2 × (−2) | −4 |
| 1 × (−2) | −2 |
| 0 × (−2) | 0 |
| −1 × (−2) | 2 |
| −2 × (−2) | 4 |
Each step down raises the answer by 2. The pattern cannot jump, so −1 × −2 has to be +2.
Counting negatives in a longer product
With several factors, count the negative signs. An even count gives a positive answer, and an odd count gives a negative one.
A worked example: (−2) × (−3) × (−4). Three negatives is odd, so the answer is negative. Multiply the sizes: 2 × 3 × 4 = 24, so the answer is −24. Check stepwise: (−2) × (−3) = 6, then 6 × (−4) = −24.
The rule that gets swapped
Some students remember “two negatives make a positive” and use it for addition and subtraction too. This gives wrong answers like −3 − 4 = 7 or −3 + (−4) = 7.
| Situation | Rule | Example |
|---|---|---|
| Multiply or divide | Same signs: positive | (−5) × (−2) = 10 |
| Add or subtract | Use the number line | −5 + (−2) = −7 |
Before you write a sign, name the operation. For “times” or “divided by”, use the sign rule. For “plus” or “minus”, use the number line.
Division uses the same idea
Check a division by multiplying back. For −48 ÷ 6, the answer must give −48 when multiplied by 6. Since 6 × (−8) = −48, the answer is −8.
Try it yourself
Work out (a) (−7) × (−3), (b) −36 ÷ 4 and (c) 12 ÷ (−3) × (−2).
Answer
(a) Same signs, so positive: 7 × 3 = 21. (b) Different signs, so negative: −9.
(c) Work left to right. 12 ÷ (−3) = −4, then −4 × (−2) = 8. As a check, the expression has two negative numbers, which is an even count, so the answer is positive. That agrees with 8.
Next steps
You can now use both rules together in using brackets and operation order. Go back to adding and subtracting negative numbers if the last table felt shaky, or try SPM Mathematics tuition for one-to-one practice.