Negative Numbers: Adding, Subtracting and Multiplying Without Sign Errors
Sign errors are the most common way to lose marks in algebra, and almost all of them start with a shaky grip on negative numbers. The rules are few, and each one has a simple picture behind it. Learn the picture and you can stop memorizing "two negatives make a positive" as a slogan and start understanding why it is true.
The number line is the whole story
Picture a horizontal line with zero in the middle, positive numbers to the right and negative numbers to the left. Adding a positive number means moving right; subtracting a positive number means moving left. A negative number is just a position to the left of zero, so −5 is the same distance from zero as 5, on the other side.
Adding and subtracting with the number line
Example: −3 + 8
Start at −3 and move 8 steps right. You pass 0 after 3 steps and land at 5.
Example: 2 − 7
Start at 2 and move 7 steps left. You pass 0 after 2 steps and land at −5.
A handy rule: when you add numbers with different signs, subtract the smaller size from the larger size and keep the sign of the larger. For −9 + 4, the sizes are 9 and 4, the difference is 5, and since the 9 was negative, the answer is −5. When the signs are the same, add the sizes and keep the sign: −6 + (−4) = −10.
Subtracting a negative is adding
This is the rule students most often resist, so here are three ways to see that it is true.
- Debt. If you owe someone $5 and they cancel that debt, you are $5 better off. Removing a negative (−5) increases your total.
- Patterns. 5 − 2 = 3, 5 − 1 = 4, 5 − 0 = 5. Each time the subtracted number drops by one, the answer rises by one. Continue the pattern: 5 − (−1) = 6, 5 − (−2) = 7.
- Distance. The difference between two numbers is the distance between them. From −2 to 5 is 7 steps, so 5 − (−2) = 7.
Example: −4 − (−9)
Subtracting −9 is the same as adding 9, so this is −4 + 9 = 5.
A practical habit: whenever you see two signs side by side, such as "− −" or "+ −", rewrite them as a single sign first. "− −" becomes "+", and "+ −" becomes "−".
Multiplying and dividing: count the negatives
Multiplication and division follow one rule: the product or quotient is positive if the number of negative factors is even, and negative if the number of negative factors is odd.
| Signs | Result | Example |
|---|---|---|
| positive × positive | positive | 3 × 4 = 12 |
| positive × negative | negative | 3 × (−4) = −12 |
| negative × negative | positive | (−3) × (−4) = 12 |
| negative ÷ positive | negative | −12 ÷ 4 = −3 |
| negative ÷ negative | positive | −12 ÷ (−4) = 3 |
Why is negative times negative positive? Use the same pattern idea: 3 × (−4) = −12, 2 × (−4) = −8, 1 × (−4) = −4, 0 × (−4) = 0. Each step up in the first factor adds 4, so (−1) × (−4) must be 4, and (−2) × (−4) must be 8. The rule is the only choice that keeps arithmetic consistent.
With a chain such as (−2) × 3 × (−5) × (−1), count the negatives: there are three, which is odd, so the answer is negative. Then multiply the sizes: 2 × 3 × 5 × 1 = 30, giving −30.
Where sign errors really come from
- Distributing a minus sign. 8 − (3 + 2) means 8 − 3 − 2 = 3, not 8 − 3 + 2. The minus applies to every term inside the bracket. Likewise, −(x − 4) = −x + 4.
- Confusing −3² with (−3)². The first is −9 and the second is 9. See our order of operations guide for why.
- Moving terms across an equals sign. To solve x + 5 = 2, subtract 5 from both sides to get x = −3. Forgetting that 2 − 5 is negative is a classic slip.
- Absolute value vs. negation. |−7| is 7 (the distance from zero), which is different from −7 itself.
A short practice set
- −7 + 12 = ? (5)
- 4 − 11 = ? (−7)
- −5 − (−8) = ? (3)
- (−6) × 7 = ? (−42)
- (−48) ÷ (−6) = ? (8)
- 10 − (4 − 9) = ? (15)
Mixed expressions with negative numbers appear throughout the Arithmetic Blitz mode and the higher algebra topics in Progression and Marathon.