Commutative Property of Addition

One of the most basic math skills is adding numbers. After a basic introduction to natural and whole numbers, you move on to a study of mathematical operations carried out on numbers. Besides learning to add, you will also learn how to multiply, divide and subtract them from each other. To be able to precisely understand how addition is done, you need to learn all the associated properties of addition operation. One such important feature is the commutative property of addition.

About Addition

I am pretty sure that you are already familiar with what is addition of two numbers, which is the most basic of operations in mathematics. Still, I will only brush up some basics here. Addition is summing up the value of two numbers, to get a bigger number. It is denoted by the '+' sign. Adding up a zero to any number gives back the same number. This is known as the identity property of addition.

Definition

To be able to understand the commutative property, you need to know what 'commutative' means. The exact meaning of commutative is an entity which is 'independent of order'. In this case, the commutative property of addition is stated as follows, the sum of two or numbers or variables is the same, irrespective of the order in which they are added. In math terms, in addition, the order of the addends (terms being added) is immaterial, as the sum remains the same. To put it in equation form:

m + n = n + m

where n and m are variables or numbers. Thus, the gist of this property is that it doesn't matter in what order two or more variables are added. Even multiplication of two numbers is commutative in nature. Since multiplication is a form of addition, this property is inherited by multiplication too.

It is an important property of addition, which allows for splitting of a number, into a sum of numbers in any order, as each of them is equivalent to the other. This commutative property is restricted to multiplication and addition operations. It does not extend to division and subtraction.
Examples

Here are some examples that will help you grasp the concept.

• a + b + c = c + b + a
• x + y = y + x
• 3 + 4+ 5 = 5 + 4+ 3 = 4 + 3 + 5 = 4 + 5 + 3
• 2+ 0 = 0 + 2
• 1 + 0 = 0 + 1
• a + 9 = 9 + a

Though it seems quite intuitive later, it is one of the most fundamental properties of addition, which needs to be defined clearly. This property helps in simplifying addition of terms in an equation. In the simplest of words, the commutative property means that no matter, in what order you add numbers, the end sum is always going to remain the same. Not the order of numbers being added, but only the total sum is what matters! Though addition and multiplication have a commutative property, it does not extend to division or subtraction. This point needs to be noted, as it underlines a basic difference in these two sets of mathematical operations.