Adding a positive moves right on the number line, subtracting a positive moves left, and subtracting a negative moves right because it reverses the direction. With a number line in your head, you do not have to memorise most sign rules.
Start with a picture
Imagine a thermometer in a cold room. It reads −4 °C and the heater raises it by 10 °C. You start at −4 and move up 10: −4 + 10 = 6, so the room is now 6 °C.
If the same room reads 5 °C and a cold front drops it by 8 °C, you start at 5 and move down 8: 5 − 8 = −3.
Subtracting a negative
Now think of a lift in a building. You are on floor 5, and you cancel a move of “down 3”, which is written 5 − (−3). Cancelling a downward move sends you up, so you end on floor 8.
- 5 − (−3) = 5 + 3 = 8
- −4 − (−6) = −4 + 6 = 2
- −7 + (−2) = −7 − 2 = −9
Once the picture is clear, you can merge two signs into one: same signs make a plus, different signs make a minus.
The mistake, contrasted
A student simplifies 5 − (−3) as 5 − 3 = 2, dropping one of the minus signs. The number line shows the error, because subtracting a negative must move you right, and 2 is to the left of 5.
| Wrong | Right |
|---|---|
| 5 − (−3) = 2 | 5 − (−3) = 5 + 3 = 8 |
| −7 + 10 = −17 | −7 + 10 = 3 |
For −7 + 10, the wrong answer adds the digits and ignores that the move crosses zero. Check: start at −7, take 7 steps to reach zero, then 3 more steps to land on 3.
Try it yourself
Work out (a) −2 − (−9) and (b) −6 + 4 − (−1).
Answer
(a) −2 − (−9) = −2 + 9 = 7. Start at −2 and move 9 right.
(b) Work left to right: −6 + 4 = −2, then −2 − (−1) = −2 + 1 = −1. The answer is −1. A common mistake is treating the final −(−1) as −1, which gives −3.
Next steps
The sign rules for products come next in multiplying and dividing signed numbers. These skills are used again when you collect like terms in algebra, and the signed numbers hub lists the rest. If you want a teacher to watch your working, try SPM Mathematics tuition.