Commutative Property of Multiplication
Commutative Property of Multiplication establishes that the order of two factors does not affect their product: a · b = b · a. From OpenStax Prealgebra 2E: 6 · 8 = 48 and 8 · 6 = 48. A room that is 10 feet wide by 12 feet long has the same area whether computed as 10 × 12 or 12 × 10. This property is especially useful for mental arithmetic — choosing the order that is easiest to compute — and underpins the commutative patterns that appear throughout algebra, from polynomial multiplication to matrix contexts where the property does not hold.
Key Concepts
Property Changing the order of the factors does not change their product. $$a \cdot b = b \cdot a$$.
Examples We know that $6 \cdot 8 = 48$. Because of the Commutative Property, we also know that $8 \cdot 6 = 48$. To find the product of 15 and 3, you can calculate $15 \cdot 3 = 45$ or $3 \cdot 15 = 45$. Both give the same result. A floor plan shows a room that is 10 feet wide and 12 feet long. Its area is $10 \times 12 = 120$ square feet, which is the same as $12 \times 10 = 120$.
Explanation Just like with addition, you can swap the numbers you're multiplying and still get the same answer. For example, having 3 rows with 5 apples each is the same total amount as having 5 rows with 3 apples each.
Common Questions
What is the Commutative Property of Multiplication?
The order of factors does not change the product: a · b = b · a. For example, 6 · 8 = 8 · 6 = 48.
Does 15 · 3 equal 3 · 15?
Yes. By the Commutative Property, 15 · 3 = 3 · 15 = 45.
How does the commutative property apply to a room's area?
A 10 × 12 ft room has area 120 sq ft whether calculated as 10 × 12 or 12 × 10.
Is division commutative?
No. 12 ÷ 4 = 3, but 4 ÷ 12 = 1/3. Order matters for division.
What is the difference between commutative and associative properties of multiplication?
Commutative: order of two factors — a·b = b·a. Associative: grouping of three factors — (a·b)·c = a·(b·c).
How does this property help with multiplication facts?
If you know 7 × 9 = 63, you automatically know 9 × 7 = 63, cutting the number of multiplication facts you need to memorize in half.