Counting back through zero
You already know how to subtract when the answer stays positive: 9 − 4 = 5. But what about 4 − 9? Until now, the answer was "you can't". With negative numbers, you can — and the answer is below zero.
The number-line picture
Start at the first number on the number line. Then move left the number of steps in the subtraction.
4 − 9
Start at 4. Count left 9 steps.
Step 1 → 3, step 2 → 2, step 3 → 1, step 4 → 0, step 5 → −1, step 6 → −2, step 7 → −3, step 8 → −4, step 9 → −5.
Answer: −5.
A faster trick
Counting one square at a time works, but it's slow for big jumps. There's a shortcut:
When the subtracted number is bigger than the start, the answer = −(subtracted − start).
In words: subtract the smaller from the bigger, and put a minus in front.
4 − 9: bigger − smaller = 9 − 4 = 5. Add a minus → −5.
2 − 11: 11 − 2 = 9. Add a minus → −9.
6 − 6: 6 − 6 = 0. 0 (neither positive nor negative).
This shortcut works only when the subtracted number is bigger than the start. If the start is bigger (or equal), use ordinary subtraction.
Worked examples
3 − 8 = ? The bigger is 8, the smaller is 3. 8 − 3 = 5. Answer: −5.
0 − 7 = ? You start at 0 and step 7 left. Answer: −7.
1 − 10 = ? 10 − 1 = 9. Answer: −9.
7 − 4 = ? Here the start is bigger, so it's ordinary subtraction: 3 (positive).
In real life
- A diver is 5 metres deep. She dives 8 more metres. New depth?
Counting starts at −5 and goes down 8: −5 − 8 = −13 m.
- It's 4 °C in the morning. The temperature falls 9 degrees by night. New temperature?
4 − 9 = −5 °C.
- Your account has £3. A direct debit takes £20. New balance?
3 − 20 = −£17 (overdrawn).
Common mistakes
- Forgetting the minus. "4 − 9 is just 5." That's the size, but the sign got lost.
- Stopping at zero. "4 − 9 is 4 − 4 = 0, and the other 5 disappears." No — the other 5 carries you past zero into negatives.
- Treating the subtraction wrong way round. "4 − 9 = 9 − 4 = 5." Yes, |result| = 5, but the sign should be negative because the answer is on the left of zero.