The Associative and Commutative Properties

Ordering and Grouping Elements of Equations

associative property formula
The associative property concerns itself with the regrouping of elements and an operation. C.K.Taylor

There are several mathematical properties that are used in statistics and probability; two of these, the commutative and associative properties, are generally associated with the basic arithmetic of integers, rationals, and real numbers, though they also show up in more advanced mathematics.

These properties—the commutative and the associative—are very similar and can be easily mixed up. For that reason, it is important to understand the difference between the two.

The commutative property concerns the order of certain mathematical operations. For a binary operation—one that involves only two elements—this can be shown by the equation a + b = b + a. The operation is commutative because the order of the elements does not affect the result of the operation. The associative property, on the other hand, concerns the grouping of elements in an operation. This can be shown by the equation (a + b) + c = a + (b + c). The grouping of the elements, as indicated by the parentheses, does not affect the result of the equation. Note that when the commutative property is used, elements in an equation are rearranged. When the associative property is used, elements are merely regrouped.

Commutative Property

Simply put, the commutative property states that the factors in an equation can be rearranged freely without affecting the outcome of the equation. The commutative property, therefore, concerns itself with the ordering of operations, including the addition and multiplication of real numbers, integers, and rational numbers.

For example, the numbers 2, 3, and 5 can be added together in any order without affecting the final result:

2 + 3 + 5 = 10
3 + 2 + 5 = 10
5 + 3 + 2 = 10

The numbers can likewise be multiplied in any order without affecting the final result:

2 x 3 x 5 = 30
3 x 2 x 5 = 30
5 x 3 x 2 = 30

Subtraction and division, however, are not operations that can be commutative because the order of operations is important. The three numbers above cannot, for example, be subtracted in any order without affecting the final value:

2 - 3 - 5 = -6
3 - 5 - 2 = -4
5 - 3 - 2 = 0

As a result, the commutative property can be expressed through the equations a + b = b + a and a x b = b x a. No matter the order of the values in these equations, the results will always be the same.

Associative Property

The associative property states that the grouping of factors in an operation can be changed without affecting the outcome of the equation. This can be expressed through the equation a + (b + c) = (a + b) + c. No matter which pair of values in the equation is added first, the result will be the same.

For example, take the equation 2 + 3 + 5. No matter how the values are grouped, the result of the equation will be 10:

(2 + 3) + 5 = (5) + 5 = 10
2 + (3 + 5) = 2 + (8) = 10

As with the commutative property, examples of operations that are associative include the addition and multiplication of real numbers, integers, and rational numbers. However, unlike the commutative property, the associative property can also apply to matrix multiplication and function composition.

Like commutative property equations, associative property equations cannot contain the subtraction of real numbers. Take, for example, the arithmetic problem (6 – 3) – 2 = 3 – 2 = 1; if we change the grouping of the parentheses, we have 6 – (3 – 2) = 6 – 1 = 5, which changes the final result of the equation.

What Is the Difference?

We can tell the difference between the associative and the commutative property by asking the question, “Are we changing the order of the elements, or are we changing the grouping of the elements?” If the elements are being reordered, then the commutative property applies. If the elements are only being regrouped, then the associative property applies.

However, note that the presence of parentheses alone does not necessarily mean that the associative property applies. For instance:

(2 + 3) + 4 = 4 + (2 + 3)

This equation is an example of the commutative property of addition of real numbers. If we pay careful attention to the equation, though, we see that only the order of the elements has been changed, not the grouping. For the associative property to apply, we would have to rearrange the grouping of the elements as well:

(2 + 3) + 4 = (4 + 2) + 3