Natural Numbers and Whole Numbers
Natural numbers: 1, 2, 3, 4, 5, … (counting numbers, start from 1)
Whole numbers: 0, 1, 2, 3, 4, 5, … (natural numbers + zero)
Key fact: Every natural number is a whole number. But 0 is a whole number that is not a natural number.
- There is no largest whole number — they are infinite.
- The smallest whole number is 0.
- The smallest natural number is 1.
The Number Line
A number line is a straight line on which whole numbers are placed at equal distances.
- Numbers increase going right.
- Numbers decrease going left.
- The successor of a number is the number to its right: successor of 5 = 6.
- The predecessor of a number is the number to its left: predecessor of 5 = 4.
- 0 has no predecessor in whole numbers.
Addition on Number Line
To add 3 + 4: start at 3, move 4 steps right → land on 7.
Subtraction on Number Line
To subtract 6 − 2: start at 6, move 2 steps left → land on 4.
Properties of Whole Numbers
1. Closure Property
| Operation | Closed? | Example |
|---|---|---|
| Addition | ✅ Yes | 3 + 5 = 8 (whole number) |
| Subtraction | ❌ No | 3 − 5 = −2 (not a whole number) |
| Multiplication | ✅ Yes | 3 × 5 = 15 (whole number) |
| Division | ❌ No | 3 ÷ 5 = 0.6 (not a whole number) |
2. Commutative Property
Order doesn't matter for addition and multiplication.
- Addition: a + b = b + a → 4 + 7 = 7 + 4 = 11 ✅
- Multiplication: a × b = b × a → 4 × 7 = 7 × 4 = 28 ✅
- Subtraction: a − b ≠ b − a → 7 − 4 ≠ 4 − 7 ❌
- Division: a ÷ b ≠ b ÷ a ❌
3. Associative Property
Grouping doesn't matter for addition and multiplication.
- Addition: (a + b) + c = a + (b + c)
- (2 + 3) + 4 = 5 + 4 = 9 = 2 + (3 + 4) = 2 + 7 = 9 ✅
- Multiplication: (a × b) × c = a × (b × c)
- (2 × 3) × 4 = 6 × 4 = 24 = 2 × (3 × 4) = 2 × 12 = 24 ✅
- Subtraction and division are NOT associative.
4. Distributive Property
Multiplication distributes over addition and subtraction.
- a × (b + c) = (a × b) + (a × c)
- 4 × (5 + 3) = 4 × 8 = 32 = (4 × 5) + (4 × 3) = 20 + 12 = 32 ✅
- a × (b − c) = (a × b) − (a × c)
5. Identity Elements
| Operation | Identity | Rule |
|---|---|---|
| Addition | 0 | a + 0 = a |
| Multiplication | 1 | a × 1 = a |
6. Multiplication by Zero
- Any number × 0 = 0 → 457 × 0 = 0
- Any number + 0 = that number → 457 + 0 = 457
Patterns in Whole Numbers
Triangular Numbers
Numbers that form triangles of dots: 1, 3, 6, 10, 15, 21, …
Each next triangular number adds one more row:
- 1 = 1
- 1+2 = 3
- 1+2+3 = 6
- 1+2+3+4 = 10
Square Numbers
Numbers that form squares: 1, 4, 9, 16, 25, …
1² = 1, 2² = 4, 3² = 9, 4² = 16
Key Vocabulary
| Term | Meaning |
|---|---|
| Whole numbers | 0, 1, 2, 3, … |
| Natural numbers | 1, 2, 3, … |
| Successor | Number just after a given number (n + 1) |
| Predecessor | Number just before a given number (n − 1) |
| Commutative | Order doesn't change result |
| Associative | Grouping doesn't change result |
| Distributive | Multiplication spreads over addition |
| Identity element | A number that leaves others unchanged (0 for +, 1 for ×) |
Exam Tips
- 0 is a whole number but NOT a natural number — a very common 1-mark question.
- For closure: subtraction and division are NOT closed for whole numbers.
- Additive identity = 0; Multiplicative identity = 1 — memorise both.
- Distributive property: used to calculate mentally — e.g., 52 × 8 = 50 × 8 + 2 × 8 = 400 + 16 = 416.
- Successor of a number n = n + 1; Predecessor of n = n − 1. Predecessor of 0 does not exist in whole numbers.