Commutative and Associative Properties

In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations.

Use the commutative and associative properties

Think about adding two numbers, such as $5$ and $3$:

The results are the same — $5 + 3 = 3 + 5$. Notice that the order in which we add does not matter. The same is true when multiplying $5$ and $3$:

Again, the results are the same! $5 \cdot 3 = 3 \cdot 5$. The order in which we multiply does not matter. These examples illustrate the commutative properties of addition and multiplication.

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

Example. Use the commutative properties to rewrite the following expressions: (a) $-1 + 3$ (b) $4 \cdot 9$.

(a) Use the commutative property of addition to change the order:

(b) Use the commutative property of multiplication to change the order:

What about subtraction? Does order matter when we subtract numbers? Does $7 - 3$ give the same result as $3 - 7$?

Since $4 \neq -4$, the results are not the same. Changing the order of the subtraction did not give the same result, so we can say that subtraction is not commutative.

Let’s see what happens when we divide two numbers. Is division commutative?

Since $3 \neq \tfrac{1}{3}$, the results are not the same, so division is not commutative either. Addition and multiplication are commutative; subtraction and division are not.

Suppose you were asked to simplify this expression:

Some people would think $7 + 8$ is $15$, and then $15 + 2$ is $17$. Others might start with $8 + 2$ makes $10$, and then $7 + 10$ makes $17$. Both ways give the same result:

When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition. The same principle holds true for multiplication. Suppose we want to find the value of $5 \cdot \tfrac{1}{3} \cdot 3$. Changing the grouping of the numbers gives the same result:

When multiplying three numbers, changing the grouping of the numbers does not change the product. This is known as the Associative Property of Multiplication. If we multiply three numbers, changing the grouping does not affect the product. You probably know this already, but the terminology may be new to you.

Example. Use the Associative Property of Multiplication to simplify: $6(3x)$.

Use the associative property to change the grouping, then multiply in the parentheses:

Notice that we can multiply $6 \cdot 3$, but we could not multiply $3 \cdot x$ without having a value for $x$.

Evaluate expressions using the commutative and associative properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier.

Example. Evaluate each expression when $x = \tfrac{7}{8}$: (a) $x + 0.37 + (-x)$ (b) $x + (-x) + 0.37$.

(a) Substituting $\tfrac{7}{8} = 0.875$ for $x$ and adding left to right:

(b) Substituting $\tfrac{7}{8}$ for $x$ and adding the opposites first:

What was the difference between part (a) and part (b)? Only the order changed — by the Commutative Property of Addition, $x + 0.37 + (-x) = x + (-x) + 0.37$. But wasn’t part (b) much easier?

Let’s do one more, this time with multiplication. Example. Evaluate each expression when $n = 17$: (a) $\tfrac{4}{3}\left(\tfrac{3}{4}n\right)$ (b) $\left(\tfrac{4}{3} \cdot \tfrac{3}{4}\right)n$.

(a) Substituting $17$ for $n$ and multiplying left to right:

(b) Substituting $17$ for $n$, then noting the product of reciprocals is $1$:

What was the difference between part (a) and part (b) here? Only the grouping changed. By the Associative Property of Multiplication, $\tfrac{4}{3}\left(\tfrac{3}{4}n\right) = \left(\tfrac{4}{3} \cdot \tfrac{3}{4}\right)n$. By carefully choosing how to group the factors, we can make the work easier.

Simplify expressions using the commutative and associative properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first, instead of automatically following the order of operations. In the examples above, part (b) was easier to simplify each time because the opposites, or the reciprocals, were next to each other and combined to $0$ or $1$. In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

Example. Simplify: $-84n + (-73n) + 84n$.

Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms:

Example. Simplify: $\tfrac{7}{15} \cdot \tfrac{8}{23} \cdot \tfrac{15}{7}$.

Notice the first and third factors are reciprocals, so reorder to group them together:

In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

Example. Simplify: $\left(\tfrac{5}{13} + \tfrac{3}{4}\right) + \tfrac{1}{4}$.

Notice the second and third terms have a common denominator, so change the grouping:

When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

Example. Simplify: $(6.47q + 9.99q) + 1.01q$.

Notice that the sum of the second and third coefficients is a whole number, so change the grouping:

Many people have good number sense when they deal with money. Think about adding $99$ cents and $1$ cent — do you see how this applies to adding $9.99 + 1.01$?

No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be.

Example. Simplify the expression: $[1.67(8)](0.25)$.

Notice that multiplying $(8)(0.25)$ is easier than multiplying $1.67(8)$ because it gives a whole number — think about having $8$ quarters, which makes $2. Regroup, then multiply in the brackets first:

When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms.

Example. Simplify: $6(9x)$.

Use the associative property of multiplication to re-group, then multiply in the parentheses:

In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together — we simplified $3x + 7 + 4x + 5$ by rewriting it as $3x + 4x + 7 + 5$ and then simplifying it to $7x + 12$. We were using the Commutative Property of Addition.

Example. Simplify: $18p + 6q + (-15p) + 5q$.

Use the Commutative Property of Addition to re-order so that like terms are together, then combine like terms:

Key terms

commutative properties — for real numbers $a$ and $b$: $a + b = b + a$ and $a \cdot b = b \cdot a$; changing the order of addition or multiplication does not change the result. associative properties — for real numbers $a$, $b$, and $c$: $(a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = a \cdot (b \cdot c)$; changing the grouping of addition or multiplication does not change the result. Subtraction and division are not commutative or associative.

This section is adapted from Prealgebra 2e, Section 7.2: Commutative and Associative Properties by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the boxed regrouping comparisons (Figures 7.3 and 7.4) as tables; omitted the Be Prepared quiz, Links to Literacy activity, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.