What Is the Associative Property of Mathematics?

To understand how the associative property is applied, we must first make sure we know the commutative property.

The commutative property states the order of variables or numbers for basic mathematical operations like addition and multiplication will not change the outcome of a solution.

However, this same rule is not applicable to division or subtraction, because changing the order of number in those two operations will most likely change the solution that we get.

When the Commutative Property Is Applicable

To get a better understanding of addition and multiplication operations, let's go through some basic commutative property formulas. In each following formula, the result remains the same regardless of how we arrange the variables we are adding or multiplying.

We can show the property in action by filling in real numbers for our variables and making practice problems.

The result remains the same irrespective of order. The same goes for the following multiplication example.

This property applies no matter how many numbers are being added or multiplied together, so long as the types of operation are not mixed. For example, let's try an equation of three variables.

Replacing these variables with arbitrary rational numbers, we get:

or

This property is applicable to theoretically infinite numbers of numerals and infinite orders of operations. As long as we are dealing with addition and multiplication by themselves, the operations satisfy the commutative property.

When the Commutative Property Is Not Applicable

The commutative property is not applicable to division or subtraction, since changing the order in which we solve those problems. Let's show this off by reworking our problems from above instead using subtraction and division.

The same goes for division.

Inserting three numbers or more into subtraction and division problems will cause these values to change in other ways.

Again, we see this with division.

The associative property states that in problems which purely concern addition and multiplication, the location of parentheses in the operation do not change the result. Next, we'll go over problems that make the associative property applicable.

Associative Property Examples

How does the associative property apply? Depending on context, the associative property is applicable to addition, multiplication, subtraction, and division. However, combining two or more types of operation may make the associative property no longer applicable.

In most problems where we can apply the associative property formula or law, the parentheses can also be removed entirely and we'll get the same answer.

Associative Property of Addition

The associative property of addition can be expressed as follows:

Now, with real numbers:

As you can see, the location of the parentheses and how the numbers are grouped has no bearing on our result. The same rule can also be applied to subtraction problems, or addition and subtraction together, as long as the order of rational numbers is not changed.

In these types of equations where the placement of parentheses is unimportant to the result, leaving them out of your work altogether will likely be the best practice. However, math textbooks may include them in order to test your understanding of the associative property.

Associative Property of Multiplication

When multiplying numbers, the associative property of multiplication states that the product of numbers remains the same no matter how our numbers are grouped.

When Not to Apply the Associative Property

The associative law states that it will not be applicable in equations where moving parentheses will change our answer.

The most common examples we'll encounter in math are problems that mix addition and multiplication or any combination of these. For these following equations, the distributive property is in effect, meaning operations inside parentheses must be solved first.

Using real numbers:

To see why these two versions of the problem are not the same, we can simplify them by solving the numbers inside parenthetical brackets first.

While the associative property of addition can also be applicable to subtraction, this associative property of multiplication cannot be used for division equations involving more than two numbers. Therefore, division problems containing at least three numbers will not be applicable to associative law.

While the associative law and commutative property are closely related to one another, and may both apply to certain equations, it's important to understand that both may not be applicable at the same time. In a basic equation that falls under the commutative property will almost always be an associative property equation as well.

However, an equation where the associative property applies may or not be applicable when it comes to the commutative property.

As a good rule of thumb, an equation that contains only addition or only multiplication will be both associative and commutative. An equation that contains subtraction and division may follow associative law, but it will not be subject to the commutative property.

Solved Examples of Equations that are Associative and Commutative

Example 1:

Example 2:

When we deal with purely addition, or purely multiplication, neither the order of rational numbers nor the placement of parenthetical brackets will affect our final sum or product answer.

Solved Example of Equation That Is Associative But Not Commutative

Problems involving subtraction are the most common example of math that follows the associative property definition but not the commutative property.