1. The Intuition (The Token Model)
Imagine green tokens for $+1$ and red tokens for $-1$. When one green token meets one red token, they cancel each other out completely to form a zero pair: $$(+1) + (-1) = \mathbf{0}$$
2. The Two Addition Rules
- Same Signs: Add their magnitudes and keep the common sign. $$\mathbf{(+4) + (+3) = +7}, \quad \mathbf{(-4) + (-3) = -7}$$
- Different Signs: Subtract the smaller magnitude from the larger magnitude, and attach the sign of the number with the larger magnitude. $$\mathbf{(-9) + (+4) = -5}, \quad \mathbf{(+9) + (-4) = +5}$$
Subtraction Rule: To subtract an integer, add its additive inverse!
$$\mathbf{a - b = a + (-b)}, \quad \mathbf{a - (-b) = a + b}$$
3. Concrete Worked Example
Example: Evaluate: $(-18) - (-25) + (-12)$
Step 1: Convert subtraction to addition of opposite: $(-18) + 25 + (-12)$
Step 2: Group negatives together: $[(-18) + (-12)] + 25 = (-30) + 25$
Step 3: Different signs → $30 - 25 = 5$, sign of $30$ is negative: $\mathbf{-5}$.
4. Pitfall & Examiner Trap
⚠️ Trap: Two Negatives in Addition vs. Multiplication
Students often say "two negatives make a positive" and write $(-5) + (-3) = +8$.
Reality: Two debts combined make a BIGGER debt: $(-5) + (-3) = \mathbf{-8}$! The phrase "two negatives make a positive" applies ONLY to multiplication and subtraction of a negative.
Students often say "two negatives make a positive" and write $(-5) + (-3) = +8$.
Reality: Two debts combined make a BIGGER debt: $(-5) + (-3) = \mathbf{-8}$! The phrase "two negatives make a positive" applies ONLY to multiplication and subtraction of a negative.
5. Why This Matters in Life
In high-altitude mountaineering, if night temperature is $-15^\circ\text{C}$ and daytime temperature is $+4^\circ\text{C}$, the temperature change is $4 - (-15) = 19^\circ\text{C}$.