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 in the next chapter.

Use the Commutative and Associative Properties

Think about adding two numbers, such as 5 and 3 .

$\begin{array}{cc} 5+3 & 3+5 \\ 8 & 8 \end{array}$

The results are the same. $5+3=3+5$

Notice, the order in which we add does not matter. The same is true when multiplying 5 and 3.

$\begin{array}{cc} 5 \cdot 3 & 3 \cdot 5 \\ 15 & 15 \end{array}$

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.

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

$\begin{array}{llc}7-3 & & 3-7 \\4 & & -4 \\& 4 \neq-4 &\end{array}$

$\text { The results are not the same.} 7−3 ≠ 3−7$

Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.

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

$\begin{array}{ccc}12 \div 4 & & 4 \div 12 \\\dfrac{12}{4} & & \dfrac{4}{12} \\3 & & \dfrac{1}{3} \\& 3 \neq \dfrac{1}{3} &\end{array}$

$\text { The results are not the same. So } 12 \div 4 \neq 4 \div 12$

Since changing the order of the division did not give the same result, division is not commutative.

Addition and multiplication are commutative. Subtraction and division are not commutative.

Suppose you were asked to simplify this expression.

$7+8+2$

How would you do it and what would your answer be?

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, as shown in Figure 7.3. (Remember that parentheses are grouping symbols that indicate which operations should be done first).

Figure 7.3

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 as well. Suppose we want to find the value of the following expression:

$5 \cdot \frac{1}{3} \cdot 3$

Changing the grouping of the numbers gives the same result, as shown in Figure 7.4.

Figure 7.4.

When multiplying three numbers, changing the grouping of the numbers does not change the result. 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, but the terminology may be new to you. These examples illustrate the Associative Properties.

Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.

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, as the next several examples illustrate.

Source: Rice University, https://openstax.org/books/prealgebra/pages/7-2-commutative-and-associative-properties
This work is licensed under a Creative Commons Attribution 4.0 License.

EXAMPLE 7.5

Use the commutative properties to rewrite the following expressions:

(a) $-1+3=\$ _________

(b) $4 \cdot 9=$ _________

EXAMPLE 7.6

Use the associative properties to rewrite the following:

(a) $(3+0.6)+0.4=$ ________

(b) $\left(-4 \cdot \frac{2}{5}\right) \cdot 15=$ ________

EXAMPLE 7.7

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

TRY IT 7.9

Use the commutative properties to rewrite the following:

(a) $-4+7=$ ______

(b) $6 \cdot 12=$ ______

TRY IT 7.10

Use the commutative properties to rewrite the following:

(a) $14+(-2)=$ ______

(b) $3(-5)=$ ______

Solution to Example 7.5

Solution to Example 7.6

Notice that $0.6+0.4$ is $1$, so the addition will be easier if we group as shown on the right.

Notice that $\frac{2}{5} · 15$ is $6$. The multiplication will be easier if we group as shown on the right.

Solution to Example 7.7

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

Try It 7.9

(a) $-4+7=7+(-4)$

(b) $6 \cdot 12=12 \cdot 6$

Try It 7.10

(a) $14+(-2)=-2+14$

(b) $3(-5)=(-5) 3$