Every student is familiar with the real number properties which are the basics of mathematics. These properties are taught in the beginning of every mathematics books and these properties are commutative, distributive and associative.

These properties are used whenever we are dealing with algebraic equations, functions, and formulas and have a great significance while doing any new research or mathematics proof. We can say that these properties play a role of the pillar in mathematical proofs. But in the school books, we are not given the information about their historical background and use in our daily life.

In this article, we will discuss the commutative property, its origin and we will also find out why it is important to learn. So to know basics about commutative property keep reading the article until the end.

Commutative Property

Commutative property

The commutative property defines that whenever two numbers are added together or multiplied it does not matter that what order you use. We can’t apply the commutative property to subtraction and division.

Origin of Commutative Property

Origin of Commutative property

Egyptians were the first to utilize this property and the Greek mathematician Euclid applied the commutative property in the book named Elements. The term commutative was used by the French scientist Francois Servois in 1814.

Commutative Property for Addition

Commutative property for addition

Commute means “move around” and the commutative property of addition says that whenever we apply addition on two or more real numbers it is always commutative or order doesn’t matter. For example, we have to add two numbers 4 and 8 it doesn’t matter that we add 4 into 8 or 8 into 4 the result will be same.



Commutative Property for Multiplication

Commutative property for Multiplication

Just like addition commutative property of multiplication says that whenever we are dealing with the multiplication of two numbers it is always commutative or order doesn’t matter. For example, we are dealing with multiplication operation on 8 and 9 then it doesn’t matter which number we use first.

8 x 9 = 8 x 9

72 = 72

Commutative Property in Union of Two Sets

Commutative property for Union

Union of sets is known as the combining two sets or addition of two sets together and it is represented by “U”.

Taking the union of two or more sets will always be commutative. To understand this let’s take an example.

Suppose we have two sets, one is represented by A and other is denoted by B. Then the union will be

A = {4,8}

B = {5,6,7}

A U B = {4,5,6,7,8}

We can also take union as

B U A = {4,5,6,7,8}

So we can say that

A U B = B U A

It is commutative.

Commutative Property in Intersection of Two Sets

Commutative property for intersection

The intersection of two sets is known as taking the common elements from both sets in a set and it is represented by “∩”.

Just like a union, the intersection of two sets is always commutative. Given below is an example, for better illustration. Assume we have a set which is represented by P and another set which is represented by Q. then the intersection will be

P = {4,6,8,9}

Q = { 1,2,3,4,5,7,8,10}

P∩Q = {4,8}


Q∩P = {4,8}

We can say that

P∩Q = Q∩P

It is commutative.

Examples of Commutative Property Around Us

Any proof of Math is understood better when we learn it through our daily life examples. A person learns more when he sees the practical use of any theory. So written below are the most common examples from our routine life.

Brushing your Teeth

Brushing your teeth

Brushing your teeth is one of the commutative properties. Whether you put toothpaste on a brush and then apply on teeth or you put the paste on teeth and then use a brush, the result will be same. Although the second method is not used by us it is just an example to understand the use of commutative property in our life.

Ice Cubes in Cold Drink

ice cubes in drink

If you want to drink more chilled cold drink by adding some ice cubes and you have a bowl full of ice cubes then it doesn’t matter which ice cube you picked, the result will be same.

Putting on Cloths

Putting on clothes

If you have a pair of gloves then it doesn’t matter whether you put right glove first or left glove. Similarly, the order doesn’t matter whether you put right socks first or left sock, as a result, you will have both socks on.

Paying or receiving cash

paying cash

If you have to receive 100 dollars then it doesn’t matter whether you will receive a note of 100$ or in the form of change, the result will be same i.e. receiving 100 dollars.

With the help of these examples, you are now able to better understand the application of the commutative property. If you liked my post or want to give any suggestion hit it down in the comments section.