Commutative Property - Definition | Commutative Law and Examples
Last Updated :
23 Jul, 2025
The Commutative property is a basic math rule that helps make calculations easier. It states that the result of an operation between two numbers remains the same irrespective of the position of the numbers. For example, 2 + 3 is the same as 3 + 2, and 4 × 5 is the same as 5 × 4.
It is an important property of mathematics. Which is satisfied by the addition(+) and multiplication(×) operations.
In this article, we will explore the commutative property, its definition, and examples, in detail.
What is Commutative Property?
Commutative Property is the property according to which the sequence of two operands does not change the result. In commutative property the switching of the operands does not affect the result of the given expression.
The commutative property is the property in which the order of the two operands does not affect the result of the expression. The commutative word is derived from the word commute which means switching. The arithmetic operator addition and multiplication satisfies the commutative property.
Commutative Property DefinitionExamples,
- 4 + 7 = 7 + 4 = 11
- 5 + 4 = 4 + 5 = 9
- 4 × 7 = 7 × 4 = 28
- 5 × 4 = 4 × 5 = 20
Commutative Property Formula
Commutative property is satisfied by the addition and multiplication arithmetic operators only. For two numbers P and Q, the commutative property formula is given by:
- P + Q = Q + P
- P × Q = Q × P
Commutative Property of Addition
According to the Commutative property for addition the order of two operands related with addition operator does not affects the result of addition.
Commutative Property for Addition Formula
For addition of two numbers X and Y, the commutative property for addition formula is given below:
X + Y = Y + X
Commutative Property of Addition Examples
Following are some examples for the commutative property for addition:
- 4 + 9 = 9 + 4
- (-12) + 5 = 5 + (-12)
- 8/9 + 1/3 = 1/3 + 8 / 9
Commutative Property of Multiplication
The commutative property for multiplication states that the order of two operands related with multiplication operator has no affect in the result of multiplication.
Commutative Property for Multiplication Formula
For multiplication of two numbers X and Y, the commutative property for addition formula is given below:
X × Y = Y × X
Commutative Property of Multiplication Examples
Following are some examples for the commutative property for addition:
- 4 × 9 = 9 × 4
- (-12) × 5 = 5 × (-12)
- (8 / 9) × (1 /3) = (1 / 3) × (8 / 9)
Non-Commutative Operations - Division And Subtraction
Non-Commutative operations refers to those operations that do not follow the commutative property and changing the order of the numbers in the operations changes the result of the operation. The arithmetic operators subtraction and division does not satisfy the commutative property as changing the order of the operands changes the result of the expression, and this can be explained by the given examples.
For two operands R and S,
- R - S ≠ S - R
- R / S ≠ S / R
Example 1: Suppose we take two numbers 12 and 3 then dividing 12 by 3 and dividing 3 by 12 gives the separate results, i.e.
Example 2: Let's take two numbers 15 and 35 then subtracting 15 from 35 and subtracting 35 from 15 gives the separate results, i.e.
- 35 - 15 = 20
- 15 - 35 = -20
Commutative Property vs Associative Property
The differences between commutative property and associative property is explained in the table below,
Commutative Property | Associative Property |
|---|
Commutative property states that the order of the number in some operations(such as multiplication and addition) dose not effects the results of the operation. | Associative property states that grouping various number in different operations (such as multiplication and addition) does not changes the result of the operation. |
Commutative Formula: - A + B = B + A
- A × B = B × A
| Associative Formula: - A + (B + C) = (A + B) + C = (A + C) + B
- A × (B × C) = (A × B) × C = (A × C) × B
|
Example: 4 + 5 = 5 + 4 = 9 | Example: (3 + 4) + 5 = 3 + (4 + 5) = 12 |
Read More,
Solved Examples on Commutative Property
Example 1: Which of the following satisfies the commutative property:
(i) 4 × 5
(ii) 7 + 8
(iii) 9 / 2
(iv) 15 - 6
Solution:
4 × 5 = 20
5 × 4 = 20
Commutative Property Satisfies
7 + 8 = 15
8 + 7 = 15
Commutative Property Satisfies
8 / 2 = 4
2 / 8 = 1 / 4
4 ≠ 1/4
Commutative Property Does Not Satisfies
15 - 6 = 9
6 - 15 = -9
9 ≠ -9
Commutative Property Does Not Satisfies
Example 2: Prove that p + q = q + p if p = 9 and q = 7.
Solution:
p + q = 9 + 7 = 16
q + p = 7 + 9 = 16
p + q = q + p
Hence proved
Example 3: Prove that p × q = q × p if p = 10 and q = 3.
Solution:
p × q = 10 × 3 = 30
q × p = 3 × 10 = 30
p × q = q × p
Hence Proved
Practice Problems On Commutative Property
Problem 1: Which of Following Satisfies Commutative Property
42 × 15
12 + 7
23 / 3
25 - 8
Problem 2: Prove x + y = y + x if x = 10 and y = 17
Problem 3: Prove x × y = y × x if x = 2 and y = 23
Problem 4: Apply the commutative property of addition to rearrange and simplify the expression : 3a + 4b + 2a + 5b.
Conclusion
The commutative property is a helpful math rule that makes adding and multiplying numbers easier. It means you can change the order of the numbers, and the answer will stay the same! For example, 2 + 3 is the same as 3 + 2, and the same rule applies for multiplication 5 × 6 = 6 × 5. But remember, this doesn’t work for subtraction or division. Knowing this property is super useful when solving math problems and makes math feel a lot less tricky
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