Integers
1. What Are Integers?
The set of integers includes all WHOLE numbers and their NEGATIVES: ..., -3, -2, -1, 0, 1, 2, 3, ...
Notation: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Positive integers: 1, 2, 3, 4, ... (move RIGHT on number line)
- Negative integers: -1, -2, -3, -4, ... (move LEFT on number line)
- Zero: Neither positive nor negative.
The INTEGERS are CLOSED under addition, subtraction, and multiplication — meaning the result is ALWAYS another integer.
2. Properties of Integer Operations
Addition and Multiplication Properties
| Property | Addition | Multiplication |
|---|---|---|
| Closure | a + b is always an integer | a × b is always an integer |
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
| Identity | a + 0 = a | a × 1 = a |
| Inverse | a + (-a) = 0 | — (only 1 and -1 have multiplicative inverses) |
| Distributive | — | a × (b + c) = a × b + a × c |
Important Observations
- SUBTRACTION is NOT commutative: 5 - 3 ≠ 3 - 5.
- DIVISION is NOT commutative: 10 ÷ 2 ≠ 2 ÷ 10.
- Division by ZERO is NOT defined.
Example 1 (ICSE 2024, 2 marks)
Simplify using properties: (-25) × 37 + (-25) × 63.
Solution: Using DISTRIBUTIVE property in reverse: a × b + a × c = a × (b + c) = (-25) × (37 + 63) = (-25) × 100 = -2500
3. Sign Rules for Operations
Rule Table for Signed Numbers
| Operation | Sign Pattern | Result Sign |
|---|---|---|
| (+) × (+) | Same signs | POSITIVE |
| (-) × (-) | Same signs | POSITIVE |
| (+) × (-) | Different signs | NEGATIVE |
| (-) × (+) | Different signs | NEGATIVE |
| (+) ÷ (+) | Same signs | POSITIVE |
| (-) ÷ (-) | Same signs | POSITIVE |
| (+) ÷ (-) | Different signs | NEGATIVE |
| (-) ÷ (+) | Different signs | NEGATIVE |
Key idea: 'Same signs → POSITIVE. Different signs → NEGATIVE.'
Subtraction as Adding the Opposite
a - b = a + (-b). This is a TRICK to avoid confusion with signs.
Example 2: (-8) - (-3) = (-8) + (+3) = -5. Example 3: 7 - (-4) = 7 + (+4) = 11.
4. Number Line
Using the Number Line
- Numbers INCREASE as you move RIGHT.
- Numbers DECREASE as you move LEFT.
- Every integer has a UNIQUE position.
Comparing Integers
- Any POSITIVE integer is GREATER than any NEGATIVE integer.
- On the number line: the number to the RIGHT is always LARGER.
Examples:
- 5 > -7 (positive is always greater than negative)
- -3 > -8 (on number line, -3 is to the right of -8)
- 0 > -1 (zero is greater than any negative integer)
Absolute Value
The DISTANCE of a number from zero on the number line. |a| = a if a ≥ 0, |a| = -a if a < 0. Examples: |7| = 7, |-7| = 7, |0| = 0.
5. BODMAS Rule with Integers
BODMAS tells the ORDER of operations:
- Brackets (solve innermost first)
- Of (power, exponents) — sometimes 'Orders'
- Division and Multiplication (left to right)
- Addition and Subtraction (left to right)
Worked Example (ICSE Focus, 3 marks)
Simplify: 15 - [8 - {4 - (6 - 8 - 3)}]
Solution: Step 1: Innermost bracket: 6 - 8 - 3 = -2 - 3 = -5 Step 2: Curly bracket: 4 - (-5) = 4 + 5 = 9 Step 3: Square bracket: 8 - 9 = -1 Step 4: 15 - (-1) = 15 + 1 = 16
Worked Example 2 (ICSE 2023, 3 marks)
Simplify: 36 ÷ 4 + 3 × (-2) - 8 ÷ (-4)
Solution: Step 1: Division: 36 ÷ 4 = 9, 8 ÷ (-4) = -2 Step 2: Multiplication: 3 × (-2) = -6 Step 3: Addition/Subtraction: 9 + (-6) - (-2) = 9 - 6 + 2 = 5
Common Mistake
'Do NOT add before dividing. Division comes BEFORE addition in BODMAS.' Wrong: 12 ÷ 3 × 2 = 12 ÷ 6 = 2 Right: 12 ÷ 3 × 2 = 4 × 2 = 8 (Division and Multiplication are done LEFT to RIGHT)
6. ICSE Exam Focus
Frequently Asked Topics (with mark distribution)
| Topic | Marks | Frequency |
|---|---|---|
| Properties of integers (name the property) | 1-2 marks | High |
| BODMAS simplification | 3-4 marks | Very High |
| Word problems with integers | 2-3 marks | Medium |
| Compare integers | 1 mark | Low |
| Absolute value | 1 mark | Medium |
Common Mistakes in ICSE Exams
- Forgetting BODMAS order — especially doing addition before division.
- Sign errors: (-2) × (-3) = -6 instead of +6.
- Wrong bracket simplification order — always start from INNERMOST bracket.
- Writing '0 is a negative integer' — NO, 0 is NEITHER positive nor negative.
Self-Test (5 Questions)
Q1. Simplify using BODMAS: 20 - [12 - {8 - (4 - 6 - 2)}]. (3 marks)
- A) 8
- B) 10
- C) 6
- D) 4
Q2. Name the property: a × (b + c) = a × b + a × c. (1 mark)
- A) Associative
- B) Commutative
- C) Distributive
- D) Closure
Q3. Which integer is greater: -17 or -5? (1 mark)
Q4. The value of (-8) × (-3) × 2 is: (2 marks)
- A) -48
- B) 48
- C) -24
- D) 24
Q5. If a = -3, b = 2, c = -5, find the value of a × (b + c). (2 marks)
- A) 9
- B) -9
- C) 21
- D) -21
Answers
A1. A) 8. Steps: 4 - 6 - 2 = -4. 8 - (-4) = 12. 12 - 12 = 0. 20 - 0 = 8. A2. C) Distributive property. A3. -5 (because -5 lies to the right of -17 on the number line). A4. B) 48. [(-8) × (-3)] × 2 = 24 × 2 = 48. A5. A) 9. b + c = 2 + (-5) = -3. a × (-3) = (-3) × (-3) = 9.
