Commutative Property of Multiplication
2026-02-28 10:27 Diff
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Last updated on December 3, 2025

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?

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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, both addition and multiplication follow the commutative property.

Commutative property of addition example: 


5 + 3 = 3 + 5 = 8
12 + 5 = 5 + 12 = 17

6 + 8 = 8 + 6 = 14

Commutative property of multiplication example: 

5 × 3 = 3 × 5 = 15

2 × 4 = 4 × 2 = 8

35 × 6 = 6 × 35 = 210
 

What is Multiplication?

Multiplication is one of the basic operations of mathematics and can be understood as repeated addition. Multiplication 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 × 7 = 7 × 5. Here, even though the order is changed, the result remains the same.

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Commutative Property of Multiplication Formula

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 to two numbers at a time; for multiple numbers, re-grouping is required (associativity).

 For example, (5 × 4) × (3 × 5) = (3 × 5) × (4 × 5) = 300.

Tips and Tricks to master Commutative Property of Multiplication

According to the commutative property of multiplication, changing the order of numbers being multiplied will not change the product. This is a basic property that applies to whole numbers, fractions, and decimals, simplifies computation, and is consistent with algebraic thinking. Here are a handful of steps to help with your understanding:
 

  • Learn and memorize the basic formula A × B = B × A. This simple way of thinking helps you internalize the fact that the order of factors does not change the product.
     
  • Use a range of numerical values that include positive numbers, negative numbers, fractions, and decimal values. A variety of samples will help you see the commutative property in action.
     
  • To help with your visualization of multiplication, create a visual with an area model or arrays to help you see the multiplication as a rectangular area. A rectangle will help show the reorder of factors yields the same area or product.
     
  • In problem-solving, trying to strategically rearrange the order of factors is a good way to both simply problems and think in your head when calculating problems.
     
  • The commutative property only applies to addition and multiplication, so we would want to clarify what its scope is not applicable, which would be subtraction and division.
     

  • Parents can help children arrange objects such as beads, buttons, or blocks into groups. For example, make three groups of 4 buttons and four groups of 3 buttons to show they both make 12.
     

  • Teachers can help students switch the order of numbers during multiplication to make calculations easier. For example, turning 7 × 25 into 25 × 7 helps develop quick, confident mental-math skills.
     

  • Teachers can use digital tools, multiplication games, or virtual blocks that allow students to rearrange groups visually.

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:
 

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 illustrates the commutative property, simplifying calculations while shopping.

  • Arranging Objects in Rows and Columns:

    For seating arrangements, or stacking items, commutative property helps in organizing objects. It ensures that the order of objects does not impact the arrangement outcome.

  • 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, mathematically expressed as: Area = Length × Width. This property helps in material estimation and space planning in construction projects.

  • Recipes and Cooking:

    When adjusting a recipe, we often increase one or more of the ingredients in the recipe by multiplying them by a specific factor or scaling it. The order in which we do this does not matter due to the commutative property of multiplication, so it is faster and more flexible to do it in the way described above.

  • Counting and Grouping:

    Multiplication's commutative property aids counting and grouping, since it shows that the order of the factors does not influence the total. For example, 4 boxes with 6 pencils each equal the same total as 6 boxes with 4 pencils each. This shows that changing the order in which we counted (factors) did not change the total.

Problem 1

Verify the commutative property for 3 × 5.

Okay, lets begin

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.

Well explained 👍

Problem 2

Verify the commutative property for 7 × 2.

Okay, lets begin

7 × 2 = 2 × 7 = 14.

Explanation

Calculate: 

7 × 2 = 14.

Switch the order: 

2 × 7 = 14

Changing the order does not affect the result.

Well explained 👍

Problem 3

Show that (-4) × 6 follows the commutative property.

Okay, lets begin

(– 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.

Well explained 👍

Problem 4

Prove the property for ⅓ × 9

Okay, lets begin

 (\(1 \over 3\)) × 9 = 9 × (\(1 \over 3\)) = 3.

Explanation

Compute:

(\(1 \over 3\)) × 9 = 3

Reverse the factors:

9 × (\(1 \over 3\)) = 3

The commutative property holds for fractions as well.

Well explained 👍

Problem 5

Verify the commutative property for 8 × (- 3)

Okay, lets begin

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.

Well explained 👍

FAQs on Commutative Property of Multiplication

1.What do you mean by the commutative property of multiplication?

The commutative property of multiplication states that the order of factors does not affect the product.
 

2.Does the commutative property of multiplication apply to variables?

Yes, the commutative property of multiplication holds true for variables. For example, a × b = b × a.

3.What is the significance of commutative property in mathematics?

The Commutative property is significant in mathematics as it helps simplify expressions, solve equations, and perform mental arithmetic.   
 

4.What common mistakes might students make regarding the commutative property?

The most common mistake that students make is assuming that all operations are commutative. Sometimes students might apply this property to subtraction or division, which does not follow this rule.
 

5.Is the commutative property applicable in advanced mathematics?

Yes, commutative property applies to many areas of mathematics, including algebra, calculus, and other fields. However, some operations like matrix multiplication do not follow it.

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.