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Arithmetic With Whole Numbers Arithmetic Calculations With Signed Numbers What is The Order of Operations in Math Definition Examples Order of Operations Definition Problems Examples How to Find The Prime Factorization of a Number How to Find The Greatest Common Factor How to Find The Least Common Multiple Commutative Property Definition Review

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GMAT: What Are The Different Types of Numbers?
Commutative Property Definition Review

In this lesson, you're going to learn the meaning of the commutative property as well as in which mathematical operations it is applicable and in which ones it is not.

Commutative Property Definition Review

Commutative

If you add a fourth of a bucket of water to a half bucket of water, you have three-fourths of a bucket of water. If instead you add half a bucket of water to a fourth of a bucket of water, how much do you have total? It's still just the same three-fourths of a bucket of water. It doesn't matter which bucket is added to which one; the total of both is always going to be the same. This is one example of something called the commutative property, where commutative is a word that describes something that is independent of order. Let's go over mathematical operations that are commutative and those that are not.

Three Important Properties

In general, I want you to know that there are three basic properties which describe how operations may work. The commutative property, which we'll get to shortly, the associative property, which deals with adding and multiplying numbers regardless of how they are grouped, and the distributive property, which deals with multiplying a single term with other terms inside a parentheses. These three properties are characteristics of some mathematical operations as opposed to rules. They describe how certain operations work but are not rules that state how all operations work.

The Commutative Property

With that out of the way, let's get to the important gist of the commutative property. Again, commutative means order doesn't matter. If you do an operation in one order, then do it in another order, you'll get the same answer. Remember that bucket example from the intro? It doesn't matter what order we pour the water in, from the first bucket to the second or vice versa the total amount will always be the same.

Also, like commuting to work, you take the same roads, it doesn't matter which direction you are driving. The distance you drive to work will be the same distance traveled, coming home from work. The order doesn't matter.

These are all metaphors for commutation. So why don't we get to some actual examples? There are two main examples of operations that are commutative. They are addition and multiplication. In addition, it doesn't matter if you add 4 + 5 or 5 + 4 -- you still get 9. The order doesn't matter. The commutative property holds up even when there are more than two numbers. 1 + 2 + 3 = 2 + 3 + 1. As long as you are adding the same numbers, it doesn't matter what order you add them in, you'll get the same result. It boils down to this: a + b = b + a.

In multiplication, if you multiply 2 x 3 or 3 x 2, you get 6. As with addition, you can multiply more than two numbers and the commutative property still works. 2 x 3 x 4 = 4 x 3 x 2 = 3 x 2 x 4 = 2 x 4 x 3. All this really means, in multiplication, is that ab = ba. But I want you to be careful. Not all operations are commutative, namely subtraction and division. In subtraction, order does matter. If you subtract 5 - 4 you'd get 1. But if you subtract 4 - 5 you'd get -1. Since you get different answers if you subtract the same numbers in different orders, subtraction is not commutative.

In division, the order you divide the numbers in will change your answer. For example, while 4/2 is 2. If you divide in the reverse order and do 2/4, then you get an answer of 1/2.