The commutative property is one of the three fundamental numerical and algebraic mathematics laws. The other three laws besides commutative are
- Associative property,
- Identity property, and
- Distributive property
Let us first understand what commutative means. Commutative comes from the word commute, which means moving or changing position. In mathematics, commutative means to change the order of the integers or variables in an expression.
The commutative property solely deals with the addition and multiplication of terms. This law states that while adding or multiplying two terms if their positions are interchanged, there is no change in the result (total or product of terms).
What is Commutative Property?
Let us comprehend this concept with the help of an example:
Let us suppose that we have to find the sum of 10 + 20. It will suddenly click your mind that the answer is 30. Now let us interchange the positions of the numbers, 20 + 10. We find that the sum is still 30.
What about multiplication? Let us take the same example 10 x 20 = 200. Now changing the positions of the numbers, 20 x 10 = 200, still gives us the same result.
Hence, commutative property is true for addition and multiplication. You must be wondering whether subtraction and division also follow the commutative property? We will find out soon in this article along with the commutative property of addition, commutative property of addition example, commutative property of multiplication, and commutative property of multiplication example.
Formulas Related to Commutative Property
If there are two positive integers, say K and L. Then the formula of the commutative property of these integers on different operations will look something like this:
- Commutative property of addition: K + L = L + K
- Commutative property of multiplication: K x L = L x K
- Commutative property of subtraction: K – L ≠ L – K
- Commutative property of division: K ÷ L ≠ L ÷ K
As you can see, in the commutative property of subtraction and division, the left-hand side is not equal to the right-hand side. This is because while subtraction and division, the order of the numbers is important.
What is the Commutative Property of Addition?
Until now, we have briefly discussed the commutative property of addition. Let us now discuss this section and the commutative property of additional examples.
Definition: If we change the order or orientation of added two numbers, then the result doesn’t change. This is known as the commutative property of addition. For instance, if we have two positive integers ‘X’ and ‘Y’, then the commutative property of addition is expressed as
X + Y = Y + X
Example 1: Let ‘X’ be 4, and ‘Y’ be 9. Find the sum of the numbers using the commutative property of addition.
Solution: We are given X = 4 and Y = 9. Therefore,
X + Y = 4 + 9 = 13
Y + X = 9 + 4 = 13
Example 2: Let ‘X’ be 10, and ‘Y’ be 7. Find the sum of the numbers if they change their position.
Solution: We are given X = 10 and Y = 7. We know that in addition the commutative property is true,
X + Y = 10 + 7 = 17
Therefore, if the numbers interchange their positions, the sum is still 17.
What is the Commutative Property of Multiplication?
We must now be clear about the commutative property of addition. Now let us dive deep into this section, where we shall learn the concept along with the commutative property of multiplication examples.
Definition: If we interchange the position of two numbers multiplied in an expression, the result doesn’t change. This is known as the commutative property of multiplication. For instance, if we have two positive integers ‘W’ and ‘Z’, then the commutative property of multiplication is given as
W x Z = Z x W
Example 1: Let ‘W’ be 5, and ‘Z’ be 12. Find the product of the numbers using the commutative property of multiplication.
Solution: Given that W = 5 and Y = 12. Therefore,
W x Z = 5 x 12 = 60
Z x W = 12 x 5 = 60
This proves that the commutative property of multiplication is true in every case.
Example 2: Let ‘W’ be 21, and ‘Z’ be 6. Find the product of the expression if the numbers change their position?
Solution: We are given W = 21 and Z = 6. In the earlier question, we proved that for multiplication, the commutative property is true,
W x Z = 21 x 6 = 126
Therefore, even if the numbers interchange their positions, the product will remain 126.
Commutative Property of Subtraction and Division
Earlier in this article, you must have read that the commutative property is not valid for operations like subtraction and division. Let us verify them with some examples:
- For Subtraction: The commutative property of subtraction can be depicted as K – L = L – K, where K and L are positive integers.
Example: Let the value of K be 5 and L be 9.
Putting the values in the left-hand side formula, we get
K – L => 5 – 9 = -4
Now putting the values on the right-hand side of the formula, we get
L – K => 9 – 5 = 4
We can see that by changing the order of numbers in subtraction, the value of the result is changing as -4 is not equal to 4. Therefore the commutative property of subtraction is not valid.
Exceptional case: The commutative property for subtraction holds only one condition, that is when the value of K and L is equal.
Let the value of K and L be 9. Then applying the commutative property of subtraction
K – L = 9 – 9 = 0, and
L – K = 9 – 9 = 0
The left-hand side = right-hand side, proving the commutative property of subtraction true for only 1 case.
- For Division: Similarly, the commutative property of division can be written as M ÷ N = N ÷ M.
Example: Let the value of M = 8 and N = 4.
Substituting the value on the left-hand side of the expression, we get
M ÷ N => 8 ÷ 4 = 2
Now, substituting the value on the left-hand side of the expression, we get
N ÷ M => 4 ÷ 8 = 0.5 or ½
Again we find that the values after changing the orientation of the numbers yield different results. Therefore the commutative property of division is also invalid.
Exceptional case: The commutative property for division is valid only for one condition when the value of M and N is equal.
Let the value of M and N be 8. Then applying the commutative property of division
M ÷ N = 8 ÷ 8 = 1, and
N ÷ M = 8 ÷ 8 = 1
Here the left-hand side = right-hand side = 1, proving the commutative property of division valid for only 1 case.
Commutative Property Vs Associative Property
This article covers a lot of concepts related to commutative property. But, many students are still confused between the commutative property and the associative property. There is no doubt that both these properties deal with the movement of terms inside an expression, and both are only valid for cases of addition and multiplication. Still, there are a lot of differences between the two. Look at the table below for understanding how commutative property differs from associative property.
|Commutative Property||Associative Property|
|Commutative property in mathematics means to commute or switch, swap or change the order of the numbers in any expression.||Associative property in mathematics means to associate, come together, or group the numbers in any expression.|
|Commutative property deals with two numbers.||Associative property deals with more than two numbers.|
|Commutative property is expressed as: |
K + L = L + K
K x L = L x K
|Associative property is expressed as: |
(K + L) + M = K + (L + M)
(K x L) x M = K x (L x M)
13 + 4 = 4 + 13 = 17
13 x 4 = 4 x 13 = 52
(13 + 4) + 8 = 4 + (13 + 8) = 25
(13 x 4) x 8 = 13 x (4 x 8) = 416