Commutative Property

Commutativity is a widely used term in mathematics.  It refers to the ability to change the order of something without changing the final result. It is a basic but important property in most branches of mathematics. The commutative property i.e. commutative law is associated with binary operations. Also, if the commutative property holds for a pair of elements under a certain binary operation then it is said that the two elements commute under that operation. In this article, the student will learn about the commutative property with examples. Let us begin it!

Commutative Property

What is the commutative property?

In group and set theory, many algebraic structures are known for having commutative when certain operands satisfy the commutative property. The word “commutative” comes from “commute” i.e. “move around”. So, the Commutative Property is the one which refers to moving stuff around.

Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.

This property has two main types:

(1) Commutative Property of Addition:

It says that the numbers can be added in any order, and we will get the same result. For example, if we are adding one and two together, the commutative property of addition says,

a+ p = p+a

where a and p are any two positive real numbers.

For example, we will get the same answer whether you are adding 1 + 2 or 2 + 1.

This also works for more than two numbers.

Let’s understand it using real-life examples. Think about marbles for a minute. Suppose we have two groups of marbles. One group only has one marble and the other group has three marbles. Then how many marbles do we have all together? We have four. Now, does it matter where we place our groups of marbles? Obviously No. This is a commutative paw.

Thus changing the order of addends will not change the result of the sum. The addends may be numbers or expressions.

(2) Commutative Property of Multiplication:

Changing the order of variables in multiplication will not change the product. Here factors may be numbers, variables or expressions.

Thus  (a × p ) = (p × a ).

Some exceptions of commutative laws: Two such exceptions are as follows:

1. Subtraction is not Commutative for the whole numbers, i.e. when we change the order of numbers in subtraction expression, the result also changes.
2. The division is not commutative for the whole numbers, i.e. when we change the order of numbers in the division expression, the result also changes.

Solved Examples

Q1: Consider the algebraic expression \(5 \times 8 = 8 \times 5\). Prove it using the law of commutative.

Solution:

LHS expression is \(5 \times 8\) = 40

And RHS expression is \(8 \times 5\) = 40.

As the LHS = RHS are the same, so the commutative property holds for multiplication

Example-2: Explain Commutative Property for Division of Whole Numbers is not applicable.

Solution: Suppose two whole numbers 8 & 4 are given.

Then LHS will be  \(8\div 4\) = 2

and RHS \(4\div 8= \frac{1}{2}\)

Obviously

\(LHS \neq RHS\)

Hence not applicable to division.