Whole numbers are the set of natural numbers together with zero: 0, 1, 2, 3, 4, ... They are used for counting and ordering. Natural numbers are 1, 2, 3, 4, ... — whole numbers with zero added.
The Number Line
Whole numbers can be shown on a number line. Starting from 0, each step to the right adds 1. Addition means moving right; subtraction means moving left.
Properties of Whole Numbers
- 1.Closure property: The sum or product of any two whole numbers is always a whole number. (But subtraction and division may not be closed.)
- 2.Commutative property: a + b = b + a and a x b = b x a.
- 3.Associative property: (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).
- 4.Distributive property: a x (b + c) = a x b + a x c.
- 5.Identity elements: 0 is the additive identity (a + 0 = a). 1 is the multiplicative identity (a x 1 = a).
- 6.Multiplication by zero: a x 0 = 0.
Show that addition is commutative for whole numbers: 47 + 83 = 83 + 47.
LHS = 130, RHS = 130. Equal. ✓
Use the distributive property to find 25 x 102.
25 x 102 = 25 x (100 + 2) = 25 x 100 + 25 x 2 = 2500 + 50 = 2550.
Find the successor and predecessor of 99,999.
Successor = 99,999 + 1 = 1,00,000.
Predecessor = 99,999 - 1 = 99,998.
Which property is used? 7 x (8 x 12) = (7 x 8) x 12.
This is the associative property of multiplication.
Find 136 x 9 + 136 x 1 using the distributive property.
= 136 x (9 + 1) = 136 x 10 = 1,360.
Is the difference of two whole numbers always a whole number? Give an example.
No. 3 - 7 = -4, which is not a whole number. So whole numbers are NOT closed under subtraction.
On a number line, find which whole number is 5 steps to the right of 8.
8 + 5 = 13. Answer: 13.
- Patterns in Whole Numbers
- Whole numbers can form interesting patterns:
- Triangular numbers: 1, 3, 6, 10, 15, ... (each is n(n+1)/2)
- Square numbers: 1, 4, 9, 16, 25, ...
Common mistakes
- Zero is a whole number but NOT a natural number.
- Division by zero is undefined — it is not allowed.
- Subtraction and division are NOT commutative: 8 - 3 ≠ 3 - 8.
Summary
Whole numbers (0, 1, 2, 3, ...) extend natural numbers by including 0. Key properties include commutativity, associativity, and distributivity. These properties make mental calculations easier. The number line is a powerful visual tool for understanding whole numbers.