The Commutative property 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.
Examples,
- 4 + 7 = 7 + 4 = 11
- 5 + 4 = 4 + 5 = 9
- 4 × 7 = 7 × 4 = 28
- 5 × 4 = 4 × 5 = 20
Commutative Property of Addition
For addition of two numbers X and Y, the commutative property for addition formula is given below:
X + Y = Y + X
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
For multiplication of two numbers X and Y, the commutative property for addition formula is given below:
X × Y = Y × X
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.
- 12/3 = 4
- 3/12 = 1/4
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:
| Associative Formula:
|
Example: 4 + 5 = 5 + 4 = 9 | Example: (3 + 4) + 5 = 3 + (4 + 5) = 12 |
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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
4 × 5 = 20
5 × 4 = 20
Commutative Property Satisfies
- 7 + 8
7 + 8 = 15
8 + 7 = 15
Commutative Property Satisfies
- 8 / 2
8 / 2 = 4
2 / 8 = 1 / 4
4 ≠ 1/4
Commutative Property Does Not Satisfies
- 15 - 6
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