\(a + (-b) = a - b\) — Adding a negative moves LEFT on the number line
Adding a negative number moves you in the negative direction on a number line — it has exactly the same effect as subtracting the positive version of that number. Subtracting a negative number, by contrast, is the same as adding a positive one, so the two negative signs cancel out.
Check on number line: start at 5, move LEFT 3 → land on 2. ✓
Example 2 — Calculate \(-2 + (-4)\)
Step 1
\(-2 + (-4) = -2 - 4\)
Step 2
Start at −2, move 4 more left: \(\mathbf{-6}\)
Example 3 — The temperature is 3°C and falls by 7°C.
Step 1
A fall of 7°C: \(3 + (-7) = 3 - 7\)
Step 2
\(3 - 7 = \mathbf{-4°C}\)
⚠ Remember: Two signs next to each other: same signs → positive result direction; different signs → negative result direction. \(+(-) \to -\) \(-(-) \to +\)
✓ Quick Check 1 — Adding a Negative
Question 1 of 10
Subtracting a Negative = Adding
\(a - (-b) = a + b\) — Subtracting a negative moves RIGHT on the number line
When you subtract a negative, the two minus signs cancel each other out and you end up adding. On a number line, this means moving right (increasing). Think: "minus a minus equals a plus."
Number line: start at 5, move RIGHT 3 → land on 8. ✓
Example 5 — Calculate \(-1 - (-4)\)
Step 1
\(-1 - (-4) = -1 + 4\)
Step 2
Start at −1, move 4 right: \(\mathbf{3}\)
Example 6 — Calculate \(-6 - (-2)\)
Step 1
\(-6 - (-2) = -6 + 2\)
Step 2
Start at −6, move 2 right: \(\mathbf{-4}\)
✓ Quick Check 2 — Subtracting a Negative
Question 1 of 10
Real-Life Applications — Temperatures, Elevation and Overdrafts
The rules for adding and subtracting negatives appear constantly in real life. Recognise which operation applies, write the calculation, and apply the sign rules.
Example 7 — Temperature starts at 2°C and falls by 9°C.
Step 1
A fall of 9°C: \(2 + (-9) = 2 - 9\)
Step 2
\(2 - 9 = \mathbf{-7°C}\)
Example 8 — Account balance is −£40. A refund of £30 is credited (removing a debt of £30).
Step 1
The debt reduces: \(-40 - (-30) = -40 + 30\)
Step 2
\(-40 + 30 = \mathbf{-£10}\) (still overdrawn but less so)
✓ Quick Check 3 — Real-Life Contexts and Mixed Practice
Question 1 of 10
Practice: Mixed Positive and Negative Operations
Type your answer and click Check.
#
Question
Answer
Result
1
\(7 + (-10)\)
2
\(-3 - (-8)\)
3
\(-5 + (-2)\)
4
\(4 - (-4)\)
5
\(-1 - (-1)\)
6
\(-4 + (-3) - (-7)\)
7
\(5 - (-3) + (-8)\)
8
\(-2 - (-6) - 4\)
9
\(-6 + 9 - (-2)\)
10
\(3 - (-9) + (-5)\)
⚠ Things to Watch Out For
Two signs next to each other: + (−3) = − 3 (adding a negative = subtracting). − (−3) = + 3 (subtracting a negative = adding). Two signs combine into one.
Subtracting a negative makes it bigger: 5 − (−3) = 5 + 3 = 8. Many students write 5 − (−3) = 2. Subtracting a negative INCREASES the value.
Sign rule table: + and + = +. + and − = −. − and + = −. − and − = +. Learn this table — it applies consistently.
Using a number line: Adding moves RIGHT; subtracting moves LEFT. Adding a negative moves LEFT; subtracting a negative moves RIGHT.
Not simplifying double signs first: Before calculating 7 + (−4) − (−2), simplify signs: 7 − 4 + 2 = 5. Simplify all double signs before doing arithmetic.