7.4 Properties of Identity, Inverses, and Zero

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,

$\begin{matrix} 13 + 0 & −14 + 0 & 0 + (−3 x) \\ 13 & −14 & −3 x \end{matrix}$

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,

$\begin{matrix} 43 \cdot 1 & −27 \cdot 1 & 1 \cdot \frac{6 y}{5} \\ 43 & −27 & \frac{6 y}{5} \end{matrix}$

Example 1

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

ⓐ $7 + 0 = 7$
ⓑ $−16 open parenthesis 1 close parenthesis equals −16$

ⓐ
	$7 + 0 = 7$
We are adding 0.	We are using the identity property of addition.

ⓑ
	$−16 open parenthesis 1 close parenthesis equals −16$
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

The text reads 'We know 5 + (-5) = 0' illustrating the concept of additive inverses where a number and its negative sum to zero.

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

−6 +_____= 0

The image displays the equation 'We know -6 + 6 = 0', written in a dark teal color, with the second '6' in the equation highlighted in red to emphasize its role in the sum.

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

The image displays the equation 'We know (2/3) * (3/2) = 1', demonstrating that a fraction multiplied by its reciprocal results in 1.

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

2 \cdot___= 1

The equation illustrates that two multiplied by one half equals one.

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

We call $\frac{1}{a}$ the multiplicative inverse of $a open parenthesis a not equal to 0 close parenthesis.$ 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:

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 $0 .$

Example 4

Simplify: ⓐ $−8 times 0$ ⓑ $\frac{5}{12} \cdot 0$ ⓒ $0 open parenthesis 2.94 close parenthesis$ .

ⓐ
	$−8 times 0$
The product of any real number and 0 is 0.	$0$

ⓑ
	$\frac{5}{12} \cdot 0$
The product of any real number and 0 is 0.	$0$

ⓒ
	$0 open parenthesis 2.94 close parenthesis$
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

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

Zero divided by any real number except zero is zero.

Example 5

Simplify: ⓐ $0 divided by 5$ ⓑ $\frac{0}{−2}$ ⓒ $0 \div \frac{7}{8}$ .

ⓐ
	$0 divided by 5$
Zero divided by any real number, except 0, is zero.	$0$

ⓑ
	$\frac{0}{−2}$
Zero divided by any real number, except 0, is zero.	$0$

ⓒ
	$0 \div \frac{7}{8}$
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 $4 divided by 0,$ and so we say that division by zero is undefined.

Division by zero is undefined.

We summarize the properties of zero.

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 11

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

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

	$\frac{3}{4} \cdot \frac{4}{3} (6 x + 12)$
Multiply; the product of reciprocals is 1.	$1 open parenthesis 6 x plus 12 close parenthesis$
Simplify by recognizing the multiplicative identity.	$6 x plus 12$

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

**Table 7.1** Properties of Real Numbers
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,	$\begin{array}{l} a + 0 = a \\ 0 + a = a \end{array}$	$\begin{array}{l} a \cdot 1 = a \\ 1 \cdot a = a \end{array}$
Inverse Property	$- a$ is the additive inverse of $a$	$a, a not equal to 0$ $1 / a$ is the multiplicative inverse of $a$
For any real number a,	$a + (− a) = 0$	$a \cdot \frac{1}{a} = 1$
Distributive Property If $a, b, c$ are real numbers, then $a (b + c) = a b + a c$
Properties of Zero
For any real number a,	$\begin{array}{l} a \cdot 0 = 0 \\ 0 \cdot a = 0 \end{array}$
For any real number $a, a not equal to 0$	$\frac{0}{a} = 0$ $\frac{a}{0}$ is undefined

Key Concepts

Identity Properties
- Identity Property of Addition: For any real number a: $a + 0 = a 0 + a = a$ 0 is the additive identity
- Identity Property of Multiplication: For any real number a: $a \cdot 1 = a 1 \cdot a = a$ 1 is the multiplicative identity
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: $(a \neq 0) a \cdot \frac{1}{a} = 1 \frac{1}{a}$ is the multiplicative inverse of a
Properties of Zero
- Multiplication by Zero: For any real number a, $\begin{matrix} a \cdot 0 = 0 & 0 \cdot a = 0 & The product of any number and 0 is 0. \end{matrix}$
- Division of Zero: For any real number a, $\begin{matrix} \frac{0}{a} = 0 & Zero divided by any real number, except itself, is zero. \end{matrix}$
- Division by Zero: For any real number a, $\frac{a}{0}$ is undefined and $a divided by 0$ is undefined. Division by zero is undefined.

Practice Makes Perfect

Recognize the Identity Properties of Addition and Multiplication

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

$101 + 0 = 101$

$\frac{3}{5} (1) = \frac{3}{5}$

identity property of multiplication

$−9 \cdot 1 = −9$

$0 + 64 = 64$

identity property of addition

Use the Inverse Properties of Addition and Multiplication

In the following exercises, find the multiplicative inverse.

$8$

$14$

$\frac{1}{14}$

$−17$

$−19$

$- \frac{1}{19}$

$\frac{7}{12}$

$\frac{8}{13}$

$\frac{13}{8}$

$- \frac{3}{10}$

$- \frac{5}{12}$

$- \frac{12}{5}$

$0.8$

$0.4$

$\frac{5}{2}$

$−0.2$

$−0.5$

−2

Use the Properties of Zero

In the following exercises, simplify using the properties of zero.

$48 times 0$

$\frac{0}{6}$

$\frac{3}{0}$

$22 times 0$

$0 \div \frac{11}{12}$

$\frac{6}{0}$

undefined

$\frac{0}{3}$

$0 \div \frac{7}{15}$

$0 \cdot \frac{8}{15}$

$open parenthesis −3.14 close parenthesis open parenthesis 0 close parenthesis$

$5.72 divided by 0$

$\frac{\frac{1}{10}}{0}$

undefined

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

In the following exercises, simplify using the properties of identities, inverses, and zero.

$19 a + 44 - 19 a$

$27 c + 16 - 27 c$

$38 + 11 r - 38$

$92 + 31 s - 92$

31s

$10 open parenthesis 0.1 d close parenthesis$

$100 open parenthesis 0.01 p close parenthesis$

$5 open parenthesis 0.6 q close parenthesis$

$40 open parenthesis 0.05 n close parenthesis$

$\frac{0}{r + 20}$ , where $r not equal to −20$

$\frac{0}{s + 13}$ , where $s not equal to −13$

$\frac{0}{u - 4.99}$ , where $u not equal to 4.99$

$\frac{0}{v - 65.1}$ , where $v not equal to 65.1$

$0 \div (x - \frac{1}{2})$ , where $x \neq \frac{1}{2}$

$0 \div (y - \frac{1}{6})$ , where $y \neq \frac{1}{6}$

$\frac{32 - 5 a}{0}$ , where $32 minus 5 a not equal to 0$

$\frac{28 - 9 b}{0}$ , where $28 minus 9 b not equal to 0$

undefined

$\frac{2.1 + 0.4 c}{0}$ , where $2.1 plus 0.4 c not equal to 0$

$\frac{1.75 + 9 f}{0}$ , where $1.75 plus 9 f not equal to 0$

undefined

$(\frac{3}{4} + \frac{9}{10} m) \div 0$ , where $\frac{3}{4} + \frac{9}{10} m \neq 0$

$(\frac{5}{16} n - \frac{3}{7}) \div 0$ , where $\frac{5}{16} n - \frac{3}{7} \neq 0$

undefined

$\frac{9}{10} \cdot \frac{10}{9} (18 p - 21)$

$\frac{5}{7} \cdot \frac{7}{5} (20 q - 35)$

20q − 35

$15 \cdot \frac{3}{5} (4 d + 10)$

$18 \cdot \frac{5}{6} (15 h + 24)$

225h + 360

Everyday Math

Insurance copayment Carrie had to have $5$ fillings done. Each filling cost $$80.$ Her dental insurance required her to pay $20 %$ of the cost. Calculate Carrie’s cost

ⓐ by finding her copay for each filling, then finding her total cost for $5$ fillings, and
ⓑ by multiplying $5 open parenthesis 0.20 close parenthesis open parenthesis 80 close parenthesis.$

Cooking time Helen bought a $24-pound$ turkey for her family’s Thanksgiving dinner and wants to know what time to put the turkey in the oven. She wants to allow $20$ minutes per pound cooking time.

ⓐ Calculate the length of time needed to roast the turkey by multiplying $24 times 20$ to find the number of minutes and then multiplying the product by $\frac{1}{60}$ to convert minutes into hours.
ⓑ Multiply $24 (20 \cdot \frac{1}{60}) .$
ⓒ Which of the Properties of Real Numbers allows you to multiply $24 (20 \cdot \frac{1}{60})$ instead of $(24 \cdot 20) \frac{1}{60} ?$

ⓐ 8 hours
ⓑ 8

Writing Exercises

In your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.

How can the use of the properties of real numbers make it easier to simplify expressions?

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Math self-assessment grid for properties of addition and multiplication (identity, inverse, zero). Learners can indicate if they understand 'Confidently', 'With some help', or 'No-I don't get it!' — **Figure 7.8**

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

	$3 x + 15 - 3 x$
Notice the additive inverses, $3 x$ and $−3 x$ .	$0 plus 15$
Add.	$15$

	$4 open parenthesis 0.25 q close parenthesis$
Regroup, using the associative property.	$[ 4 open parenthesis 0.25 close parenthesis ] q$
Multiply.	$1.00 q$
Simplify; 1 is the multiplicative identity.	$q$