## Using the Identity and Inverse Properties of Addition and Subtraction

### Learning Outcomes

• Identify the identity properties of multiplication and addition
• Use the inverse property of addition and multiplication to simplify expressions

## Recognize the Identity Properties of Addition and Multiplication

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

For example,

$\begin{array}{ccccc}\hfill 13+0\hfill & & \hfill -14+0\hfill & & \hfill 0+\left(-3x\right)\hfill \\ \hfill 13\hfill & & \hfill -14\hfill & & \hfill -3x\hfill \end{array}$

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

For example,

$\begin{array}{ccccc}\hfill 43\cdot 1\hfill & & \hfill -27\cdot 1\hfill & & \hfill 1\cdot \frac{6y}{5}\hfill \\ \hfill 43\hfill & & \hfill -27\hfill & & \hfill \frac{6y}{5}\hfill \end{array}$

### Identity Properties

The Identity Property of Addition: for any real number $a$,

$\begin{array}{}\\ \hfill a+0=a(0)+a=a\hfill \\ \hfill \text{0 is called the}\mathbf{\text{ additive identity}}\hfill \end{array}$

The Identity Property of Multiplication: for any real number $a$

$\begin{array}{c}\hfill a\cdot 1=a(1)\cdot a=a\hfill \\ \hfill \text{1 is called the}\mathbf{\text{ multiplicative identity}}\hfill \end{array}$

### example

Identify whether each equation demonstrates the identity property of addition or multiplication.

1. $7+0=7$
2. $-16\left(1\right)=-16$

Solution:

 1. $7+0=7$ We are adding 0. We are using the identity property of addition.
 2. $-16\left(1\right)=-16$ We are multiplying by 1. We are using the identity property of multiplication.

## Use the Inverse Properties of Addition and Multiplication

 What number added to 5 gives the additive identity, 0? $5 + =0$ We know $5+(\color {red}{-5})=0$ What number added to −6 gives the additive identity, 0? $-6 + =0$ We know $-6+\color {red}{6}=0$

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

We call $-a$ the additive inverse of $a$. The opposite of a number is its additive inverse. A number and its opposite add to $0$, which is the additive identity.

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

 $\Large\frac{2}{3}\normalsize\cdot =1$ We know $\Large\frac{2}{3}\normalsize\cdot\color{red}{\Large\frac{3}{2}}\normalsize=1$

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

 $2\cdot =1$ We know $2\cdot\color{red}{\Large\frac{1}{2}}\normalsize=1$

Notice that in each case, the missing number was the reciprocal of the number.

We call $\Large\frac{1}{a}$ the multiplicative inverse of $a\left(a\ne 0\right)\text{.}$ The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to $1$, which is the multiplicative identity.

We’ll formally state the Inverse Properties here:

### Inverse Properties

Inverse Property of Addition for any real number $a$,

$\begin{array}{}\\ \hfill a+\left(-a\right)=0\hfill \\ \hfill -a\text{ is the}\mathbf{\text{ additive inverse }}\text{of }a.\hfill \end{array}$

Inverse Property of Multiplication for any real number $a\ne 0$,

$\begin{array}{}\\ \\ \hfill a\cdot \frac{1}{a}=1\hfill \\ \hfill \frac{1}{a}\text{is the}\mathbf{\text{ multiplicative inverse }}\text{of }a.\hfill \end{array}$

### example

Find the additive inverse of each expression:
1. $13$
2. $-\Large\frac{5}{8}$
3. $0.6$

### example

Find the multiplicative inverse:
1. $9$
2. $-\Large\frac{1}{9}$
3. $0.9$