Mathematics — Numbers: INTEGERS

Age: 14 (Kenyan curriculum) — This page explains what integers are, how to use them, and gives rules, examples and practice questions you can use for revision.

1. What are integers?

Integers are whole numbers that can be positive, negative or zero. They do not include fractions or decimals. Examples: ... −3, −2, −1, 0, 1, 2, 3, ...

Related sets

• Natural numbers (N): 1, 2, 3, ...
• Whole numbers (W): 0, 1, 2, 3, ...
• Integers (Z): ... −2, −1, 0, 1, 2, ...

2. Number line (visual)

Integers are placed on a number line. Left = smaller (more negative); Right = larger (more positive).

3. Important ideas

• Opposite (additive inverse): The opposite of 7 is −7. Their sum is 0: 7 + (−7) = 0.
• Absolute value |a|: Distance from 0. Example: |−5| = 5, |3| = 3.
• Even and odd: Integers divisible by 2 are even (−4, 0, 2); others are odd (−3, 1, 5).
• Order: If a is left of b on the number line, a < b. Example: −2 < 1.

4. Operations with integers — rules and examples

Addition

- Same sign: add absolute values, keep the sign. Example: (−4) + (−6) = −(4+6) = −10.
- Different signs: subtract the smaller absolute value from the larger; take the sign of the larger absolute value. Example: 7 + (−3) = 4 (because 7>3, sign +).

Subtraction

a − b = a + (−b). Change the sign of the number being subtracted, then add. Example: 5 − (−2) = 5 + 2 = 7. Example: (−3) − 4 = (−3) + (−4) = −7.

Multiplication

- Same signs → positive. Example: (−3) × (−4) = 12.
- Different signs → negative. Example: (−2) × 5 = −10.

Division

Same sign → positive; different signs → negative. Division by zero is not allowed. Example: (−12) ÷ (−3) = 4; 12 ÷ (−4) = −3.

Quick sign rules summary

+ × + = +, − × − = +, + × − = −, − × + = −. Same for ÷.

5. Properties of integers (useful in exams)

• Closure: Integers are closed under addition, subtraction and multiplication (results stay integers). Not closed under division (e.g. 1 ÷ 2 = 0.5).
• Commutative: a + b = b + a; a × b = b × a.
• Associative: (a + b) + c = a + (b + c); (a × b) × c = a × (b × c).
• Distributive: a(b + c) = ab + ac.

6. Solving simple integer equations

Example 1: x − 5 = −2 → add 5 to both sides → x = 3.
Example 2: −3x = 12 → divide both sides by −3 → x = −4.

7. Worked examples

1. Calculate: (−8) + 13 − (−5).
   Solution: (−8) + 13 + 5 = (13 + 5) − 8 = 18 − 8 = 10.
2. Calculate: (−4) × (−3) + 10 ÷ (−5).
   Solution: (−4)×(−3)=12; 10÷(−5)=−2; 12+(−2)=10.

8. Practice (try these)

1. Write the opposites of: 6, −9, 0.
2. Order the integers from smallest to largest: 3, −2, 0, −7, 5.
3. Calculate: (−7) + 4 + (−2).
4. Calculate: 9 − (−6) + (−3).
5. Calculate: (−5) × 6 ÷ (−3).
6. Solve: x + (−8) = 2.
7. Solve: −2y = 14.
8. Which of these are even? −5, −6, 0, 7, 12.
9. Is 3 ÷ 4 an integer? Explain.
10. Use the number line idea: which is greater, −1 or −10? Explain briefly.

Answers (click to reveal)

1. Opposites: −6, 9, 0.
2. From smallest to largest: −7, −2, 0, 3, 5.
3. (−7)+4+(−2) = (−7−2)+4 = −5.
4. 9 − (−6) + (−3) = 9 + 6 − 3 = 12.
5. (−5)×6÷(−3) = (−30)÷(−3) = 10.
6. x + (−8) = 2 → x = 10.
7. −2y = 14 → y = −7.
8. Even: −6, 0, 12. (−5 and 7 are odd.)
9. No. 3 ÷ 4 = 0.75 is not an integer (it is a fraction / decimal).
10. −1 is greater than −10 because it is to the right of −10 on the number line (−1 > −10).

9. Exam tips

• Always watch signs carefully when adding or subtracting. When in doubt, change subtraction to addition of the opposite.
• Use the number line in your head: moving right means increasing, left means decreasing.
• Check your arithmetic by estimating: e.g. (−8)+13≈5 then add 5 → 10 (quick sanity check).

Good luck — practise with many examples to gain confidence. For more topics, study rational numbers and integers applied in equations and inequalities.