Negative Numbers and Integers

0°C

Step 1: Monday's temperature = –3°C. Step 2: 'Fell by 5°C' means subtract 5: –3 – 5 = –8°C. This is Tuesday's temperature. Step 3: 'Rose by 8°C' means add 8: –8 + 8 = 0°C. This is Wednesday's temperature. Step 4: Verify: –3 – 5 + 8 = –8 + 8 = 0. ✓ Final Answer: 0°C. Common mistake: Students may add 5 when temperature 'falls', but falling means it gets colder, so we subtract.

–a < –b

Step 1: We are given that a > b. This is the key rule about negatives of integers. Step 2: Example to verify — let a = 9, b = 7. So 9 > 7. Their negatives: –9 and –7. Step 3: On a number line, –9 is further LEFT than –7. So –9 < –7, which means –a < –b. ✓ Step 4: The general rule is: if a > b, then –a < –b. The greater an integer, the smaller its negative (opposite). Final Answer: –a < –b. Common mistake: Students assume the inequality direction stays the same when negating both sides. It always REVERSES.

+40

Step 1: Let the unknown integer be x. We need: (–27) + x = +13. Step 2: To find x, rearrange: x = 13 – (–27). Step 3: Subtracting a negative is the same as adding its positive: x = 13 + 27. Step 4: Calculate: x = 40. So the integer is +40. Step 5: Verify: (–27) + (+40) = 40 – 27 = +13 ✓ Final Answer: +40. Common mistake: Students compute 13 – 27 = –14, forgetting to change the sign when subtracting a negative.

Distance = 15, and midpoint is not an integer

Step 1: Distance between P (–6) and Q (+9) on a number line = |9 – (–6)| = |9 + 6| = |15| = 15 units. Step 2: The midpoint formula = (P + Q) ÷ 2 = (–6 + 9) ÷ 2 = 3 ÷ 2 = 1.5. Step 3: 1.5 is NOT an integer — it lies between integers 1 and 2 on the number line. Step 4: So the distance is 15 but the midpoint is 1.5, which is not an integer. Final Answer: Distance = 15, midpoint is not an integer. Common mistake: Students may compute distance as 9 – 6 = 3, ignoring the negative sign of P.