**7.4 Properties of Identity, Inverses, and Zero**

The topics covered in this section are:

- Recognize the identity properties of addition and multiplication
- Use the inverse properties of addition and multiplication
- Use the properties of zero
- Simplify expressions using the properties of identities, inverses, and zero

**7.4.1 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,

$13+0$ $13$ | $-14+0$ $-14$ | $0+(-3x)$ $-3x$ |

What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call 11 the **multiplicative identity**.

For example,

$43 \cdot 1$ $43$ | $-27 \cdot 1$ $-27$ | $1 \cdot \frac{6y}{5}$ $\frac{6y}{5}$ |

**IDENTITY PROPERTIES**

The **identity property of addition**: for any real number $a$,

$a+0=a$ and $0+a=a$

$0$ is called the **additive identity**

The **identity property of multiplication**: for any real number $a$,

$a \cdot 1 = a$ and $1 \cdot a = a$

$1$ is called the **multiplicative identity**

**Example 1**

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

- $7+0=7$
- $-16(1)=-16$

**Solution**

Part 1. | |

$7+0=7$ | |

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

Part 2. | |

$-16(1)=-16$ | |

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

**7.4.2 Use the Inverse Properties of Addition and Multiplication**

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

$5+$_____$=0$ | We know $5+(-5)=0$ |

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

$-6+$_____$=0$ | We know $-6+6=0$ |

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

$\frac{2}{3} \cdot $_____$=1$ | We know $\frac{2}{3} \cdot \frac{3}{2} = 1$ |

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

$2 \cdot $_____$=1$ | We know $2 \cdot \frac{1}{2} = 1$ |

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

We call $\frac{1}{a}$ the **multiplicative inverse** of $a(a \neq 0)$. 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 $a$,

$a+(-a)=0$

$-a$ is the **additive inverse** of $a$.

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

$a \cdot \frac{1}{a} = 1$

$\frac{1}{a}$ is the **multiplicative inverse** of $a$.

**Example 2**

Find the additive inverse of each expression:

- $13$
- $- \frac{5}{8}$
- $0.6$

**Solution**

To find the additive inverse, we find the opposite.

- The additive inverse of $13$ is its opposite, $-13$.
- The additive inverse of $- \frac{5}{8}$ is its opposite, $\frac{5}{8}$.
- The additive inverse of $0.6$ is its opposite, $-0.6$.

**Example 3**

Find the multiplicative inverse:

- $9$
- $- \frac{1}{6}$
- $0.9$

**Solution**

To find the multiplicative inverse, we find the reciprocal.

- The multiplicative inverse of $9$ is its reciprocal, $\frac{1}{9}$.
- The multiplicative inverse of $- \frac{1}{9}$ is its reciprocal, $-9$.
- To find the multiplicative inverse of $0.9$, we first convert $0.9$ to a fraction, $\frac{9}{10}$. Then we find the reciprocal, $\frac{10}{9}$.

**7.4.3 Use the Properties of Zero**

We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.

**Multiplication by Zero**

What happens when you multiply a number by $0$? Multiplying by $0$ makes the product equal zero. The product of any real number and $0$ is $0$.

**MULTIPLICATION BY ZERO**

For any real number $a$,

$a \cdot 0 = 0$ and $0 \cdot a = 0$

**Example 4**

Simplify:

- $-8 \cdot 0$
- $\frac{5}{12} \cdot 0$
- $0(2.94)$

**Solution**

Part 1. | |

$-8 \cdot 0$ | |

The product of any real number and $0$ is $0$. | $0$ |

Part 2. | |

$\frac{5}{12} \cdot 0$ | |

The product of any real number and $0$ is $0$. | $0$ |

Part 3. | |

$0(2.94)$ | |

The product of any real number and $0$ is $0$. | $0$ |

**Dividing with Zero**

What about dividing with $0$? Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are $0$ cookies to share, so each person gets $0$ cookies.

$0 \div 3 = 0$

Remember that we can always check division with the related multiplication fact. So, we know that

$0 \div 3 = 0$ because $0 \cdot 3 = 0$

**DIVISION OF ZERO**

For any real number $a$, except $0$, $\frac{0}{a}=0$ and $0 \div a = 0$.

Zero divided by any real number except zero is zero.

**Example 5**

Simplify:

- $0 \div 5$
- $\frac{0}{-2}$
- $0 \div \frac{7}{8}$

**Solution**

Part 1. | |

$0 \div 5$ | |

Zero divided by any real number, except $0$, is zero. | $0$ |

Part 2. | |

$\frac{0}{-2}$ | |

Zero divided by any real number, except $0$, is zero. | $0$ |

Part 3. | |

$0 \div \frac{7}{8}$ | |

Zero divided by any real number, except $0$, is zero. | $0$ |

Now let’s think about dividing a number *by* zero. What is the result of dividing $4$ by $0$? Think about the related multiplication fact. Is there a number that multiplied by $0$ gives $4$?

$4 \div 0 =$_____ means _____$\cdot 0 = 4$

Since any real number multiplied by $0$ equals $0$, there is no real number that can be multiplied by $0$ to obtain $4$. We can conclude that there is no answer to $4 \div 0$, and so we say that division by zero is undefined.

**DIVISION BY ZERO**

For any real number $a, \frac{a}{0},$ and $a \div 0$ are undefined.

Division *by* zero is undefined.

**Example 6**

Simplify:

- $7.5 \div 0$
- $\frac{-32}{0}$
- $\frac{4}{9} \div 0$

**Solution**

Part 1. | |

$7.5 \div 0$ | |

Division by zero is undefined. | undefined |

Part 2. | |

$\frac{-32}{0}$ | |

Division by zero is undefined. | undefined |

Part 3. | |

$\frac{4}{9} \div 0$ | |

Division by zero is undefined. | undefined |

We summarize the properties of zero.

**PROPERTIES OF ZERO**

**Multiplication by Zero:** For any real number $a$,

$a \cdot 0 = 0$ and $0 \cdot a = 0$. The product of any number and $0$ is $0$.

**Division by Zero:** For any real number $a, a \neq 0$

$\frac{0}{a} = 0$. Zero divided by any real number, except itself, is zero.

$\frac{a}{0}$ is undefined.Division by zero is undefined.

**7.4.4 Simplify Expressions using the Properties of Identities, Inverses, and Zero**

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

**Example 7**

Simplify: $3x+15-3x$.

**Solution**

$3x+15-3x$ | |

Notice the additive inverses, $3x$ and $-3x$ | $0+15$ |

Add. | $15$ |

**Example 8**

Simplify: $4(0.25q)$.

**Solution**

$4(0.25q)$ | |

Regroup, using the associative property. | $[4(0.25)]q$ |

Multiply. | $1.00q$ |

Simplify; $1$ is the multiplicative identity. | $q$ |

**Example 9**

Simplify: $\frac{0}{n+5}$, where $n \neq -5$.

**Solution**

$\frac{0}{n+5}$ | |

Zero divided by any real number except itself is zero. | $0$ |

**Example 10**

Simplify: $\frac{10-3p}{0}$.

**Solution**

$\frac{10-3p}{0}$ | |

Division by zero is undefined. | undefined |

**Example 11**

Simplify: $\frac{3}{4} \cdot \frac{4}{3} (6x+12)$.

**Solution**

We cannot combine the terms in parentheses, so we multiply the two fractions first.

$\frac{3}{4} \cdot \frac{4}{3} (6x+12)$ | |

Multiply; the product of reciprocals is $1$. | $1(6x+12)$ |

Simplify by recognizing the multiplicative identity. | $6x+12$ |

All the properties of real numbers we have used in this chapter are summarized in Table 7.1.

Property | Of Addition | Of Multiplication |
---|---|---|

Commutative Property | ||

If $a$ and $b$ are real numbers then… | $a+b=b+a$ | $a \cdot b = b \cdot a$ |

Associative Property | ||

If $a,b,$ and $c$ are real numbers then… | $(a+b)+c=a+(b+c)$ | $(a \cdot b) \cdot c = a \cdot ( b \cdot c )$ |

Identity Property | $0$ is the additive identity | $1$ is the multiplicative identity |

For any real number $a$, | $a+0=a$ $0+a=a$ | $a \cdot 1 = a$ $1 \cdot a = a$ |

Inverse Property | $-a$ is the additive inverse of $a$ | $a, a \neq 0$ $1/a$ is the multiplicative inverse of $a$ |

For any real number $a$, | $a+(-a)=0$ | $a \cdot \frac{1}{a} = 1$ |

Distributive PropertyIf $a,b,c$ are real numbers, then $a(b+c)=ab+ac$ | ||

Properties of Zero | ||

For any real number $a$, | $a \cdot 0 = 0$ $0 \cdot a = 0$ | |

For any real number $a, a \neq 0$ | $\frac{0}{a} = 0$ $\frac{a}{0}$ is undefined | |

Table 7.1 Properties of Real Numbers |

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*Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/7-4-properties-of-identity-inverses-and-zero*.*License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction*