1.9 Properties of real numbers

Use the commutative and associative properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?

The results are the same.

As we can see, the order in which we add does not matter!

What about multiplying $5\text{and}3?$

Again, the results are the same!

The order in which we multiply does not matter!

These examples illustrate the commutative property . When adding or multiplying, changing the order gives the same result.

The commutative property has 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 $3-7?$

The results are not the same.

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

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

The results are not the same.

Since changing the order of the division did not give the same result, division is not commutative . The commutative properties only apply to addition and multiplication!

• Addition and multiplication are commutative.
• Subtraction and Division are not commutative.

If you were asked to simplify this expression, how would you do it and what would your answer be?

Some people would think $7+8\text{is}15$ and then $15+2\text{is}17.$ Others might start with $8+2\text{makes}10$ and then $7+10\text{makes}17.$

Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.

When adding three numbers, changing the grouping of the numbers gives the same result.

This is true for multiplication, too.

When multiplying three numbers, changing the grouping of the numbers gives the same result.

You probably know this, but the terminology may be new to you. These examples illustrate the associative property .

Let’s think again about multiplying $5·\frac{1}{3}·3.$ We got the same result both ways, but which way was easier? Multiplying $\frac{1}{3}$ and $3$ first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!