The Properties of Multiplication Explained

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: Which Property?

An equation is shown below, illustrating one of the five properties. Identify which one. If you get it wrong (or leave it blank), we'll explain it.

Which property is shown?