Properties of Multiplication – Rules That Always Work
Multiplication follows a set of mathematical laws that hold true for every number. Knowing these properties lets you rearrange and simplify calculations with confidence.
1. Commutative Property
Order does not change the product.
4 × 7 = 7 × 4 = 28
2. Associative Property
Grouping does not change the product.
(2 × 3) × 5 = 2 × (3 × 5) = 30
3. Distributive Property
Multiplying a sum by a number gives the same result as multiplying each addend separately and then adding.
6 × (4 + 3) = (6 × 4) + (6 × 3) = 24 + 18 = 42
This property is the foundation of all written multiplication and algebra.
4. Identity Property
Multiplying by 1 leaves the number unchanged.
1 × 539 = 539
5. Zero Property
Multiplying by 0 always gives 0.
1,000,000 × 0 = 0
Summary Table
| Property | Rule | Example |
|---|---|---|
| Commutative | a × b = b × a | 5 × 9 = 9 × 5 |
| Associative | (a × b) × c = a × (b × c) | (2×3)×4 = 2×(3×4) |
| Distributive | a × (b+c) = ab + ac | 3(5+2) = 15+6 = 21 |
| Identity | a × 1 = a | 17 × 1 = 17 |
| Zero | a × 0 = 0 | 99 × 0 = 0 |
Key Takeaways
- These properties are universal — they work for all real numbers.
- The distributive property is used every time you do column or long multiplication.
- Commutativity halves the number of table facts you need to learn.
Practice Questions
- Use the commutative property to write an equivalent calculation to 8 × 13.
- Use the distributive property to calculate 7 × 23.
- Use the associative property to simplify: (5 × 4) × 25.
- What is 1,847 × 0?
- Prove that 6 × (10 + 2) = (6 × 10) + (6 × 2).