121 LearnersLast updated on July 22nd, 2025
Commutative Property of Multiplication
When the two numbers are multiplied in any order without changing the result, it is called the commutative property of multiplication. Mathematically, it is denoted as: a b = b a. This property simplifies calculations and is useful in mental math. In this article, the commutative property of multiplication and its applications will be discussed
What is the Commutative Property?
The commutative property in multiplication states that changing the order of two numbers does not change the result. The term “commutative” is derived from the word “commute”, which means to switch places or interchange. In arithmetic, the operations of addition and multiplication follow the commutative property.
What is Multiplication?
Multiplication is one of the basic operations of mathematics and can be understood as repeated addition. (which represents the sum of a number taken n times).
For example:
5 × 3 means add 5 three times: 5 + 5 + 5 = 15.
The numbers that are multiplied together are called factors, and the answer is called the product. Multiplication is extensively used in our daily life, such as in shopping, construction, and finance.
What is the Commutative Property of Multiplication?
According to the commutative law of multiplication, when two or more numbers are multiplied, the result remains the same even if the order of the numbers is changed. Here, the order refers to the way the numbers are arranged in the multiplication expression. For example, 5 7 = 35, and 7 5 = 35. Hence, according to the commutative property of multiplication, 5 x 7 = 7 x 5. Here, even though the order is changed, the result remains the same.
Formula for Commutative Property of Multiplication
The formula for the commutative property of multiplication indicates that the order of the numbers being multiplied does not affect the product obtained. All real numbers exhibit the commutative property of multiplication. The commutative property of multiplication can be mathematically expressed as:
A B = B A.
The commutative property applies pairwise, not over multiple elements unless re-grouped (which is associativity). For example, 5 4 3 5 = 3 5 4 5 = 300.
Real-life applications of Commutative Property of Multiplication
The commutative property of multiplication has numerous applications. Let us explore how the commutative property of multiplication is used in different areas:
- Shopping and Total Cost Calculation:
When multiplying a quantity by a price to find the total cost, the order in which they are multiplied does not affect the total cost. This is the commutative property, which helps simplify calculations while shopping.
- Arranging Objects in Rows and Columns:
For seating arrangements, or stacking items, commutative property helps in organizing objects. It ensures that whatever be the order of these objects is, it remains the same.
- Construction and Area Calculation:
We use this property when calculating area. For example, the area of a rectangular room is calculated by multiplying its length and width. This property helps in material estimation and space planning in construction projects.
Common Mistakes and How to Avoid Them in the Commutative Property of Multiplication
Students tend to make mistakes while understanding the concept of commutative property of multiplication. Let us see some common mistakes and how to avoid them, in the commutative property of multiplication:
Mistake 1
Confusion between Commutative Property, Associative or Distributive Property
Students need to understand that commutativity is only about changing the order of the numbers, not changing the numbers themselves. So the product remains unchanged. To check for commutativity, focus on problems where only two numbers swap places.
Mistake 2
Applying Commutative Property to Subtraction or Division
Students should remember that the commutative property does not apply to subtraction and division. Try swapping the numbers in subtraction or division, you will see that the result changes, confirming that these operations are not commutative.
If they do, we can conclude that they are not commutative. For example, 5 − 3 ≠ 3 − 5, and 6 ÷ 2 ≠ 2 ÷ 6.
Mistake 3
Ignoring That the Property Works for All Real Numbers
Students may overlook that the commutative property applies to all real numbers, including fractions, negative numbers.
Mistake 4
Incorrectly Swapping Numbers in Expressions
Students must recall that they should swap numbers only in multiplication. Do not swap numbers in other arithmetic operations. To rewrite the expressions, use the parentheses correctly to avoid ambiguity when rewriting expressions.
Mistake 5
Incorrectly Applying the Property in Algebraic Expressions
Students should swap only the numbers that are directly multiplied. If multiplication is combined with addition or subtraction, the commutative property should be applied only to the multiplication. For example, in 2 × x + 3, you can swap 2 and x (2 × x = x × 2), but not with the addition part ( + 3).
Solved Examples on Commutative Property of Multiplication
Problem 1
Verify the commutative property for 3 5.
3 5 = 5 3 = 15
Explanation
Compute in the given order:
3 × 5 = 15.
Reverse the order:
5 × 3 = 15.
The product remains the same even when the factors’ order is switched.
Problem 2
Verify the commutative property for 7 x 2.
7 x 2 = 2 x 7 = 14
Explanation
Calculate:
7×2=14.
Switch the order:
2×7=14.
Changing the order does not affect the result.
Problem 3
Show that (-4) 6 follows the commutative property.
(-4) 6 = 6 (-4) = -24
Explanation
Multiply:
(−4) × 6 = −24.
Reverse:
6 × (−4) = −24.
Even with a negative factor, the product remains identical regardless of order.
Problem 4
Prove the property for ⅓ 9
⅓ 9 = 9 ⅓ = 3
Explanation
Compute:
⅓ 9 = 3
Reverse the factors:
9 ⅓ = 3
The commutative property holds for fractions as well.
Problem 5
Verify the commutative property for 8 x (-3)
8 (-3) = (-3) 8 = -24
Explanation
Multiply:
8 × (−3) = −24.
Reverse the order:
(−3) × 8 = −24.
Switching the order does not change the negative result.
FAQs on Commutative Property of Multiplication
1.What do you mean by the commutative property of multiplication?
2.Does the commutative property of multiplication apply to variables?
3.What is the significance of commutative property in mathematics?
4.What common mistakes might students make regarding the commutative property?
5.Is the commutative property applicable in advanced mathematics?
6.How can children in United Arab Emirates use numbers in everyday life to understand Commutative Property of Multiplication?
7.What are some fun ways kids in United Arab Emirates can practice Commutative Property of Multiplication with numbers?
8.What role do numbers and Commutative Property of Multiplication play in helping children in United Arab Emirates develop problem-solving skills?
9.How can families in United Arab Emirates create number-rich environments to improve Commutative Property of Multiplication skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
