### 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 [latex]0[/latex] the additive identity.

For example,

[latex]\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}[/latex]

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

For example,

[latex]\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}[/latex]

### Identity Properties

The I**dentity Property of Addition**: for any real number [latex]a[/latex],

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

The I**dentity Property of Multiplication**: for any real number [latex]a[/latex]

[latex]\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}[/latex]

### example

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

1. [latex]7+0=7[/latex]

2. [latex]-16\left(1\right)=-16[/latex]

Solution:

1. | |

[latex]7+0=7[/latex] | |

We are adding 0. | We are using the identity property of addition. |

2. | |

[latex]-16\left(1\right)=-16[/latex] | |

We are multiplying by 1. | We are using the identity property of multiplication. |

### try it

## Use the Inverse Properties of Addition and Multiplication

What number added to 5 gives the additive identity, 0? | |

[latex]5 + =0[/latex] | We know [latex]5+(\color {red}{-5})=0[/latex] |

What number added to −6 gives the additive identity, 0? | |

[latex]-6 + =0[/latex] | We know [latex]-6+\color {red}{6}=0[/latex] |

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

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

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

[latex]\Large\frac{2}{3}\normalsize\cdot =1[/latex] | We know [latex]\Large\frac{2}{3}\normalsize\cdot\color{red}{\Large\frac{3}{2}}\normalsize=1[/latex] |

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

[latex]2\cdot =1[/latex] | We know [latex]2\cdot\color{red}{\Large\frac{1}{2}}\normalsize=1[/latex] |

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

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

We’ll formally state the Inverse Properties here:

### Inverse Properties

**Inverse Property of Addition** for any real number [latex]a[/latex],

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

**Inverse Property of Multiplication** for any real number [latex]a\ne 0[/latex],

[latex]\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}[/latex]

### example

Find the additive inverse of each expression:

1. [latex]13[/latex]

2. [latex]-\Large\frac{5}{8}[/latex]

3. [latex]0.6[/latex]

### try it

### example

Find the multiplicative inverse:

1. [latex]9[/latex]

2. [latex]-\Large\frac{1}{9}[/latex]

3. [latex]0.9[/latex]