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Commutative Property

 

Commutative property deals with the addition and multiplication of numbers. It does not hold for subtraction and division as the values are not suitable for performing commutativity on subtraction and division. The order of numbers is different in subtraction and division.

The word commutative originates from the word commute, meaning move around. Hence, the commutative property deals with the numbers that can be moved around without changing the final result of those values. This move around of numbers is possible only in additions and multiplications. Additions and multiplications also follow distributive and associative properties.

The general expression for commutativity is given by: (a + b) + c = a + (b + c) and P x Y = Y x P.

Commutative property of addition

According to the commutative property of addition, the value does not change even if we change the orders of numbers that have to undergo addition. For example, the summation value of
5 + 9 + 2 = 16
9 + 5 + 2 = 16
2 + 9 + 5 = 16
This implies that even if we change the order of the numbers, the final value does not change. Therefore, addition follows commutative property.

Commutative property of multiplication

According to the commutative property of multiplication, the value does not change even though we flip the order of the elements undergoing commutative action. For example, the value of
5 x 4 x 3 = 60
4 x 3 x 5 = 60
3 x 5 x 4 = 60
This implies, even though we have flipped the elements in the above sequence, the final value remains unchanged.

Non-commutativity in subtraction and division

Subtractions and divisions do not show commutativity. This is because the values will result in different incorrect answers.

For example, let us do the following calculations: 20 – 12 = 8. We know this is correct. If we apply commutative property, we get 12 – 20 = -8, which is a different value from 8. Remember, 8 and (-)8 are distinct integers. Therefore, subtractions do not follow the commutative property.

Let us check the same on divisions

Let us divide 20 by 2, which means 20 ÷ 2 = 10. However, if we flip the order of elements to find the value, we get 2 ÷ 20 = 0.1, which is not equal to 10. Therefore, the division does not follow commutative property as well.

Example 1: Explain whether x + y = y + x is an example of the commutative property or not.

Solution:

We know from the commutative property that changing the order of the addends does not change the value of the addition. Therefore, x + y = y + x.

Example 2: Radhika was asked to check the commutative property in 3 + 5 + 7 = 15. Will you help her check the property?

Solution:

By simple rules of commutative property of addition, we know, the sum of the given numbers can be calculated as 3 + 5 + 7 = 5 + 3 + 7 = 3 + 5 + 7 = 15 This shows that even after changing the sequence of the numbers, the final value remains unchanged. Therefore, 3 + 5 + 7 follows commutative property.

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