Negative numbers follow simple sign rules once you separate addition and subtraction from multiplication and division. For adding and subtracting, think about direction on a number line. For multiplying and dividing, one rule covers everything: same signs give a positive result; different signs give a negative result.
How do you add and subtract with negative numbers?
Think of a number line running from left (negative) to right (positive):
- Adding moves you to the right (up).
- Subtracting moves you to the left (down).
Worked examples:
| Calculation | Thought process | Answer |
|---|---|---|
| −3 + 5 | Start at −3, move 5 right | 2 |
| 4 + (−6) | Start at 4, move 6 left | −2 |
| −2 + (−5) | Start at −2, move 5 left | −7 |
| 3 − 8 | Start at 3, move 8 left | −5 |
| −1 − 4 | Start at −1, move 4 left | −5 |
What happens when you subtract a negative number?
Subtracting a negative is the same as adding its positive opposite. Two negatives next to each other in a subtraction become a positive.
"Minus a minus = plus"
Worked examples:
- 7 − (−3) = 7 + 3 = 10
- −4 − (−9) = −4 + 9 = 5
- −6 − (−6) = −6 + 6 = 0
Why? Removing a debt is the same as gaining money. If you owe £6 (−6) and that debt is cancelled (subtract −6), you are £6 better off: 0 − (−6) = +6.
What are the sign rules for multiplication?
When multiplying two numbers, only the signs matter for deciding the sign of the answer.
| Signs | Result sign | Example |
|---|---|---|
| (+) × (+) | + | 4 × 5 = 20 |
| (−) × (−) | + | (−4) × (−5) = 20 |
| (+) × (−) | − | 4 × (−5) = −20 |
| (−) × (+) | − | (−4) × 5 = −20 |
Memory shortcut: Same signs → positive. Different signs → negative.
Worked examples:
- (−3) × (−7) = 21 (both negative → positive)
- 6 × (−4) = −24 (different signs → negative)
- (−2) × (−5) × (−1) = 10 × (−1) = −10 (multiply left to right; final sign is negative because three negatives overall is odd)
For three or more factors, count the negatives: an even number of negative signs gives a positive result; an odd number of negative signs gives a negative result.
What are the sign rules for division?
The sign rules for division are identical to those for multiplication.
| Signs | Result sign | Example |
|---|---|---|
| (+) ÷ (+) | + | 20 ÷ 4 = 5 |
| (−) ÷ (−) | + | (−20) ÷ (−4) = 5 |
| (+) ÷ (−) | − | 20 ÷ (−4) = −5 |
| (−) ÷ (+) | − | (−20) ÷ 4 = −5 |
Worked examples:
- (−36) ÷ (−9) = 4 (same signs → positive)
- 45 ÷ (−5) = −9 (different signs → negative)
- (−18) ÷ 6 = −3 (different signs → negative)
What are the most common mistakes?
-
Confusing addition with multiplication rules. "Minus a minus is a plus" applies to subtraction (removing a negative) and to multiplying/dividing. It does not mean (−3) + (−5) = +8 — adding two negatives gives a more negative result: (−3) + (−5) = −8.
-
Forgetting to apply the sign separately from the magnitude. Work out the size of the answer first (ignore signs), then apply the sign rule. For example: (−7) × (−8) → size = 56, signs: (−)(−) = +, answer = 56.
-
Losing track of signs in multi-step calculations. Write each sign explicitly rather than carrying it in your head.
Frequently asked questions
Why does a negative times a negative give a positive?
One explanation: multiplication by −1 flips a number to the opposite side of zero on the number line. Flipping once gives a negative; flipping again (multiplying by another −1) brings you back to positive. So (−1) × (−1) = +1, and this generalises to all negative pairs.
Does the order of numbers matter when multiplying negatives?
No. Multiplication is commutative: (−4) × 5 = 5 × (−4) = −20. The sign rule only cares about how many negative factors there are, not their position.
How do I handle negative numbers in BIDMAS?
Treat negative numbers as exactly that — negative values. Apply BIDMAS as normal, but be careful with signs at each step. For example: −3² means −(3²) = −9, not (−3)² = 9. If you want to square −3, use brackets: (−3)².
Where do negative numbers appear in other KS3 topics?
Negative numbers appear throughout KS3 maths: temperatures and sea level in real-life contexts, coordinates in all four quadrants (negative x and y values), substituting negative values into formulae, straight-line graphs with negative gradients, and sequences that decrease below zero.
For Socratic KS3 number practice including negative number operations with Professor Pi, see aitutors.me.