GRADE 8 Mathematics NUMBERS – Integers Notes
NUMBERS — Subtopic: Integers
Topic: Mathematics | Target age: 13 (Kenya)
Quick notes on integers — what they are, how to use them and simple rules you must know.
What are integers?
Integers are whole numbers that can be:
- Positive: 1, 2, 3, ... (written with + or without sign)
- Zero: 0
- Negative: −1, −2, −3, ...
Example in daily life: If you have Ksh 50, you can say +50. If you owe Ksh 30, you could write −30.
Number line (visual)
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
7
Use the number line to compare and add/subtract integers: move right to add, left to subtract.
Adding and subtracting integers (rules)
- Same signs: add the absolute values; keep the common sign.
Example: (−5) + (−3) = −8. +6 + +2 = +8.
- Different signs: subtract the smaller absolute value from the larger; take the sign of the larger absolute value.
Example: 7 + (−10) = −3 because 10 − 7 = 3 and 10 is larger (negative).
- Subtracting: a − b = a + (−b). Change subtraction to adding the opposite.
Multiplication and division rules
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Same rules apply for division.
Examples: (−4) × (−3) = +12. (−8) ÷ 2 = −4.
Absolute value
The absolute value |a| is the distance of a from 0 on the number line (always non‑negative).
Example: |−7| = 7, |5| = 5, |0| = 0.
Comparing integers
- Larger numbers are to the right on the number line.
- Any positive integer is greater than any negative integer. Example: 1 > −100.
- Among negatives, the one with smaller absolute value is greater. Example: −2 > −5.
Order of operations with integers
Use BIDMAS/BODMAS: Brackets, Orders (powers), Division, Multiplication, Addition, Subtraction.
Example: Evaluate 3 − (−2) × 4.
First (−2)×4 = −8. Then 3 − (−8) = 3 + 8 = 11.
First (−2)×4 = −8. Then 3 − (−8) = 3 + 8 = 11.
Worked examples
- Compute: (−5) + 12 − (−3).
Step: (−5)+12 = 7. Then 7 − (−3) = 7 + 3 = 10. - Compute: (−6) × (−2) + 4.
Step: (−6)×(−2) = 12. Then 12 + 4 = 16. - Compute: 5 − 9 ÷ (−3).
Step: 9 ÷ (−3) = −3. Then 5 − (−3) = 8.
Common mistakes & tips
- When subtracting, always try to change to adding the opposite: a − b = a + (−b).
- Watch signs in multiplication/division — two negatives make a positive.
- Use a number line if unsure: moving right = adding, left = subtracting.
- For word problems, clearly assign positive and negative directions (e.g., gains +, losses −).
Practice questions (try these)
- Calculate: (−3) + 7 − 10.
- Calculate: (−4) × 5 + 9.
- If you have Ksh 20 and you owe Ksh 45, what integer shows your money?
- Which is greater: −8 or −2? Explain using a number line idea.
Answers
- (−3) + 7 − 10 = 4 − 10 = −6.
- (−4) × 5 + 9 = −20 + 9 = −11.
- You owe money, so you are at −25 (Ksh 20 − 45 = −25).
- −2 is greater because it is to the right of −8 on the number line (−2 > −8).
Quick summary: Integers include negative numbers, zero and positive numbers. Learn the sign rules for adding, subtracting, multiplying and dividing. Use the number line and change subtraction to adding the opposite when stuck.