1.9 Properties of real numbers  (Page 2/10)

The associative property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.

We saw that subtraction and division were not commutative. They are not associative either.

When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the commutative property of addition to write the like terms together.

When we have to simplify algebraic expression    s, we can often make the work easier by applying the commutative or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.

Use the identity and inverse properties of addition and multiplication

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity    .

For example,

These examples illustrate the Identity Property of Addition that states that for any real number $a,$ $a+0=a$ and $0+a=a.$

What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity    .

For example,

These examples illustrate the Identity Property of Multiplication that states that for any real number $a,$ $a·1=a$ and $1·a=a.$

We summarize the Identity Properties below.

Notice that in each case, the missing number was the opposite of the number!

We call $\text{−}a.$ the additive inverse    of a . The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number $a,a+\left(\text{−}a\right)=0.$ Remember, a number and its opposite add to zero.

What number multiplied by $\frac{2}{3}$ gives the multiplicative identity, 1? In other words, $\frac{2}{3}$ times what results in 1?

What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?