The word ‘commutative’ was derived from ‘commute’, meaning to move around.
The commutative property states that the order of the operands does the change the outcome or the result. Thus, the variables or the numbers we operate with can be moved or swapped. However, the result remains unchanged. It applies to integers, fractions, and decimals.
Suppose we need to find the result of 12 + 18. It gives 30. Now, on interchanging the position of the two numbers with respect to the operator, 18 + 12 also gives 30. Thus, the order in which two numbers are added does not affect the sum. The same is also true for multiplication. 12 × 18 gives 216 and 18 × 12 also gives 216.
Thus, addition and multiplication are commutative and are given by the formulas:
Thus, the commutative property of addition and multiplication are alike as both state that we can change the order of adding and multiplying numbers.
However, the commutative property does not apply to subtraction and division.
For example, 4 – 2 = 2 and 2 – 4 = -2.
Thus, 4 – 2 ≠ 2 – 4
Similarly, 9 ÷ 3 = 3 and 3 ÷ 9 = 0.33
Thus, 4 ÷ 2 ≠ 2 ÷ 4
However, there is an exception. The commutative property for subtraction holds when the value of ‘a’ and ‘b’ are equal.
For example, 10 – 10 = 10 – 10, and both give 0
Let us find out how the commutative property works for addition and multiplication in more detail.
Commutative Property of Addition
According to the commutative property of addition, altering the order of the addends does not alter the outcome.
If ‘a’ and ‘b’ are 2 numbers, then according to the commutative property of addition:
a + b = b + a
To verify, consider adding two numbers, 26 and 32. If it holds the commutative property, then
26 + 32 = 32 + 26
=> 58 = 58
Thus, the commutative law is true for addition.
Given ‘p’ is 5, and ‘q’ is 7. Find the sum of the numbers and verify using the commutative property of addition.
Solution:
As we know, according to the commutative property of addition
a + b = b + a, here a = p = 5 and b = q = 7
=> 5 + 7 = 7 + 5
=> 12 = 12
Thus, the sum is 12, and the commutative property is verified.
Commutative Property of Multiplication
According to the commutative property of multiplication, the result remains the same if we interchange the position of the numbers multiplied.
If ‘a’ and ‘b’ are any two numbers, then according to the commutative property of multiplication,
a × b = b × a
To prove, multiply any two numbers. For example, 33 × 11 = 363, and 11 × 33 is also 363. Thus, 33 × 11 = 11 × 33, and the commutative property of multiplication hold.
The commutative property of multiplication is illustrated using an array. For example, the array above can be read as 3 rows of 4 or 4 columns of 3. Both give the same result.
Find the value of x in the given equation 24 × x = 17 × 24
Solution:
According to the commutative property of multiplication
a × b = b × a, here a = 24, b = 17
Comparing the given equation with the standard form, we get
x= 17
Thus, the value of x is 17
Alternative Method:
Solving for x in the given equation, we get,
${x=\dfrac{17\times 24}{24}}$
=> x = 17
Harris bought 5 packets of 3 pencils each. Roger bought 3 packets of 5 pencils each. Did they buy an equal number of pencils?
Solution:
Given,
Harris bought 5 packets of 3 pencils each
So, the total number of pencils bought by Harris = 5 × 3 = 15
Again, Roger bought 3 packets of 5 pencils each
So, the total number of pencils bought by Roger = 3 × 5 = 15
By the commutative property of multiplication,
5 × 3 = 3 × 5
So, both Harris and Roger bought an equal number of pencils.
Commutative Property vs Associative Property
Commutative and associative properties are two properties of numbers, along with closure and distributive properties. Both state that the order of numbers does not affect the outcome. However, there are some differences between them:
| Commutative Property | Associative Property |
|---|---|
| ‘Commutative’ came from ‘commute’, meaning to move around or swap. It states that the operands’ order changes the outcome or the result. | ‘Associative’ came from the word ‘associate’, meaning to connect or join. It states that the sum or product of 3 or more numbers can be performed in any order without affecting the result. |
| Commutative Property of Addition a + b = b + a Example: 8 + 2 = 2 + 8 = 10 | Associative Property of Addition a + (b + c) = (a + b) + c Example: 2 + (7 + 3) = (2 + 7) + 3 = 12 |
| Commutative Property of Multiplication a × b = b × a Example: 2 × 4 = 4 × 2 = 8 | Associative Property of Multiplication a × (b × c) = (a × b) × c Example: 5 × (6 × 8) = (5 × 6) × 8 = 240 |
