# 7.4 Properties of identity, inverses, and zero  (Page 2/7)

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Find the additive inverse: $18$ $\frac{7}{9}$ $1.2$ .

1. $\phantom{\rule{0.2em}{0ex}}-18\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-\frac{7}{9}$
3. $\phantom{\rule{0.2em}{0ex}}-1.2$

Find the additive inverse: $47$ $\frac{7}{13}$ $\phantom{\rule{0.2em}{0ex}}8.4$ .

1. $\phantom{\rule{0.2em}{0ex}}-47$
2. $\phantom{\rule{0.2em}{0ex}}-\frac{7}{13}$
3. $\phantom{\rule{0.2em}{0ex}}-8.4$

Find the multiplicative inverse: $9\phantom{\rule{0.2em}{0ex}}$ $-\frac{1}{9}\phantom{\rule{0.2em}{0ex}}$ $0.9$ .

## 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, $\frac{10}{9}.$

Find the multiplicative inverse: $\phantom{\rule{0.2em}{0ex}}5\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}-\frac{1}{7}\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}0.3$ .

1. $\phantom{\rule{0.2em}{0ex}}\frac{1}{5}$
2. $\phantom{\rule{0.2em}{0ex}}-7$
3. $\phantom{\rule{0.2em}{0ex}}\frac{10}{3}$

Find the multiplicative inverse: $\phantom{\rule{0.2em}{0ex}}18$ $\phantom{\rule{0.2em}{0ex}}-\frac{4}{5}$ $\phantom{\rule{0.2em}{0ex}}0.6$ .

1. $\phantom{\rule{0.2em}{0ex}}\frac{1}{18}$
2. $\phantom{\rule{0.2em}{0ex}}-\frac{5}{4}$
3. $\phantom{\rule{0.2em}{0ex}}\frac{5}{3}$

## 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 $a,$

$a·0=0\phantom{\rule{2em}{0ex}}0·a=0$

Simplify: $\phantom{\rule{0.2em}{0ex}}-8·0$ $\phantom{\rule{0.2em}{0ex}}\frac{5}{12}·0$ $\phantom{\rule{0.2em}{0ex}}0\left(2.94\right)$ .

## Solution

 ⓐ $-8\cdot 0$ The product of any real number and 0 is 0. $0$
 ⓑ $\frac{5}{12}·0$ The product of any real number and 0 is 0. $0$
 ⓒ $0\left(2.94\right)$ The product of any real number and 0 is 0. $0$

Simplify: $\phantom{\rule{0.2em}{0ex}}-14·0\phantom{\rule{0.2em}{0ex}}$ $0·\frac{2}{3}\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}\left(16.5\right)·0.$

1. 0
2. 0
3. 0

Simplify: $\phantom{\rule{0.2em}{0ex}}\left(1.95\right)·0$ $\phantom{\rule{0.2em}{0ex}}0\left(-17\right)$ $\phantom{\rule{0.2em}{0ex}}0·\frac{5}{4}.$

1. 0
2. 0
3. 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.

$0÷3=0$

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

$0÷3=0\phantom{\rule{0.2em}{0ex}}\text{because}\phantom{\rule{0.2em}{0ex}}0·3=0.$

## Division of zero

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

Zero divided by any real number except zero is zero.

Simplify: $\phantom{\rule{0.2em}{0ex}}0÷5\phantom{\rule{0.2em}{0ex}}$ $\frac{0}{-2}$ $\phantom{\rule{0.2em}{0ex}}0÷\frac{7}{8}$ .

## Solution

 ⓐ $0÷5$ Zero divided by any real number, except 0, is zero. $0$
 ⓑ $\frac{0}{-2}$ Zero divided by any real number, except 0, is zero. $0$
 ⓒ $0÷\frac{7}{8}$ Zero divided by any real number, except 0, is zero. $0$

Simplify: $\phantom{\rule{0.2em}{0ex}}0÷11$ $\phantom{\rule{0.2em}{0ex}}\frac{0}{-6}$ $\phantom{\rule{0.2em}{0ex}}0÷\frac{3}{10}$ .

1. 0
2. 0
3. 0

Simplify: $\phantom{\rule{0.2em}{0ex}}0÷\frac{8}{3}$ $\phantom{\rule{0.2em}{0ex}}0÷\left(-10\right)$ $\phantom{\rule{0.2em}{0ex}}0÷12.75$ .

1. 0
2. 0
3. 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 $4?$

$4÷0=___\phantom{\rule{0.2em}{0ex}}\text{means}\phantom{\rule{0.2em}{0ex}}___·0=4$

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 $4÷0,$ and so we say that division by zero is undefined.

## Division by zero

For any real number $a,\phantom{\rule{0.2em}{0ex}}\frac{a}{0},$ and $a÷0$ are undefined.

Division by zero is undefined.

Simplify: $\phantom{\rule{0.2em}{0ex}}7.5÷0$ $\phantom{\rule{0.2em}{0ex}}\frac{-32}{0}$ $\phantom{\rule{0.2em}{0ex}}\frac{4}{9}÷0$ .

## Solution

 ⓐ $7.5÷0$ Division by zero is undefined. undefined
 ⓑ $\frac{-32}{0}$ Division by zero is undefined. undefined
 ⓒ $\frac{4}{9}÷0$ Division by zero is undefined. undefined

Simplify: $\phantom{\rule{0.2em}{0ex}}16.4÷0$ $\phantom{\rule{0.2em}{0ex}}\frac{-2}{0}$ $\phantom{\rule{0.2em}{0ex}}\frac{1}{5}÷0$ .

1. undefined
2. undefined
3. undefined

Simplify: $\phantom{\rule{0.2em}{0ex}}\frac{-5}{0}$ $\phantom{\rule{0.2em}{0ex}}96.9÷0$ $\phantom{\rule{0.2em}{0ex}}\frac{4}{15}÷0$

1. undefined
2. undefined
3. undefined

We summarize the properties of zero.

## Properties of zero

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

$\phantom{\rule{4em}{0ex}}\begin{array}{c}a·0=0\phantom{\rule{2em}{0ex}}0·a=0\phantom{\rule{2em}{0ex}}\text{The product of any number and 0 is 0.}\hfill \end{array}$

Division by Zero: For any real number $a,\phantom{\rule{0.2em}{0ex}}a\ne 0$

$\phantom{\rule{4em}{0ex}}\begin{array}{c}\frac{0}{a}=0\phantom{\rule{0.5em}{0ex}}\text{Zero divided by any real number, except itself, is zero.}\hfill \end{array}$

$\phantom{\rule{4em}{0ex}}\begin{array}{c}\frac{a}{0}\phantom{\rule{0.2em}{0ex}}\text{is undefined. Division by zero is undefined.}\hfill \end{array}$

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