Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Prealgebra 2e

7.4 Properties of Identity, Inverses, and Zero

Prealgebra 2e7.4 Properties of Identity, Inverses, and Zero

Learning Objectives

By the end of this section, you will be able to:

  • 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

Be Prepared 7.10

Before you get started, take this readiness quiz.

Find the opposite of −4.−4.
If you missed this problem, review Example 3.3.

Be Prepared 7.11

Find the reciprocal of 52.52.
If you missed this problem, review Example 4.29.

Be Prepared 7.12

Multiply: 3a5·92a.3a5·92a.
If you missed this problem, review Example 4.27.

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 00 the additive identity.

For example,

13+0−14+00+(−3x) 13−14−3x 13+0−14+00+(−3x) 13−14−3x

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,

43·1−27·11·6y543−276y543·1−27·11·6y543−276y5

Identity Properties

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

a+0=a0+a=a0 is called the additive identitya+0=a0+a=a0 is called the additive identity

The identity property of multiplication: for any real number aa

a·1=a1·a=a1 is called the multiplicative identitya·1=a1·a=a1 is called the multiplicative identity

Example 7.33

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

  1. 7+0=77+0=7

  2. −16(1)=−16−16(1)=−16

Try It 7.65

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

23+0=2323+0=23 −37(1)=−37.−37(1)=−37.

Try It 7.66

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

1·29=291·29=29 14+0=14.14+0=14.

Use the Inverse Properties of Addition and Multiplication

What number added to 5 gives the additive identity, 0?
5+_____=05+_____=0 .
What number added to −6 gives the additive identity, 0?
−6+_____=0−6+_____=0 .

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

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

What number multiplied by 2323 gives the multiplicative identity, 1?1? In other words, two-thirds times what results in 1?1?

23·___=123·___=1 .

What number multiplied by 22 gives the multiplicative identity, 1?1? In other words two times what results in 1?1?

2·___=12·___=1 .

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

We call 1a1a the multiplicative inverse of a(a0).a(a0). The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to 1,1, 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+(a)=0ais the additive inverse ofa.a+(a)=0ais the additive inverse ofa.

Inverse Property of Multiplication for any real number a0,a0,

a·1a=11ais the multiplicative inverse ofa.a·1a=11ais the multiplicative inverse ofa.

Example 7.34

Find the additive inverse of each expression: 1313 5858 0.60.6.

Try It 7.67

Find the additive inverse: 1818 7979 1.21.2.

Try It 7.68

Find the additive inverse: 4747 713713 8.48.4.

Example 7.35

Find the multiplicative inverse: 99 1919 0.90.9.

Try It 7.69

Find the multiplicative inverse: 55 1717 0.30.3.

Try It 7.70

Find the multiplicative inverse: 1818 4545 0.60.6.

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?0? Multiplying by 00 makes the product equal zero. The product of any real number and 00 is 0.0.

Multiplication by Zero

For any real number a,a,

a·0=00·a=0a·0=00·a=0

Example 7.36

Simplify: −8·0−8·0 512·0512·0 0(2.94)0(2.94).

Try It 7.71

Simplify: −14·0−14·0 0·230·23 (16.5)·0.(16.5)·0.

Try It 7.72

Simplify: (1.95)·0(1.95)·0 0(−17)0(−17) 0·54.0·54.

Dividing with Zero

What about dividing with 0?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 00 cookies to share, so each person gets 00 cookies.

0÷3=00÷3=0

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

0÷3=0because0·3=0.0÷3=0because0·3=0.

Division of Zero

For any real number a,a, except 0,0a=00,0a=0 and 0÷a=0.0÷a=0.

Zero divided by any real number except zero is zero.

Example 7.37

Simplify: 0÷50÷5 0−20−2 0÷780÷78.

Try It 7.73

Simplify: 0÷110÷11 0−60−6 0÷3100÷310.

Try It 7.74

Simplify: 0÷830÷83 0÷(−10)0÷(−10) 0÷12.750÷12.75.

Now let’s think about dividing a number by zero. What is the result of dividing 44 by 0?0? Think about the related multiplication fact. Is there a number that multiplied by 00 gives 4?4?

4÷0=___means___·0=44÷0=___means___·0=4

Since any real number multiplied by 00 equals 0,0, there is no real number that can be multiplied by 00 to obtain 4.4. We can conclude that there is no answer to 4÷0,4÷0, and so we say that division by zero is undefined.

Division by Zero

For any real number a,a0,a,a0, and a÷0a÷0 are undefined.

Division by zero is undefined.

Example 7.38

Simplify: 7.5÷07.5÷0 −320−320 49÷049÷0.

Try It 7.75

Simplify: 16.4÷016.4÷0 −20−20 15÷015÷0.

Try It 7.76

Simplify: −50−50 96.9÷096.9÷0 415÷0415÷0

We summarize the properties of zero.

Properties of Zero

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

a·0=00·a=0The product of any number and 0 is 0.a·0=00·a=0The product of any number and 0 is 0.

Division by Zero: For any real number a,a0a,a0

0a=00a=0 Zero divided by any real number, except itself, is zero.

a0a0 is undefined. Division by zero is undefined.

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 7.39

Simplify: 3x+153x.3x+153x.

Try It 7.77

Simplify: −12z+9+12z.−12z+9+12z.

Try It 7.78

Simplify: −25u18+25u.−25u18+25u.

Example 7.40

Simplify: 4(0.25q).4(0.25q).

Try It 7.79

Simplify: 2(0.5p).2(0.5p).

Try It 7.80

Simplify: 25(0.04r).25(0.04r).

Example 7.41

Simplify: 0n+50n+5, where n−5n−5.

Try It 7.81

Simplify: 0m+70m+7, where m−7m−7.

Try It 7.82

Simplify: 0d40d4, where d4d4.

Example 7.42

Simplify: 103p0.103p0.

Try It 7.83

Simplify: 186c0.186c0.

Try It 7.84

Simplify: 154q0.154q0.

Example 7.43

Simplify: 34·43(6x+12).34·43(6x+12).

Try It 7.85

Simplify: 25·52(20y+50).25·52(20y+50).

Try It 7.86

Simplify: 38·83(12z+16).38·83(12z+16).

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+aa+b=b+a a·b=b·aa·b=b·a
Associative Property
If a, b, and c are real numbers then… (a+b)+c=a+(b+c)(a+b)+c=a+(b+c) (a·b)·c=a·(b·c)(a·b)·c=a·(b·c)
Identity Property 00 is the additive identity 11 is the multiplicative identity
For any real number a, a+0=a0+a=aa+0=a0+a=a a·1=a1·a=aa·1=a1·a=a
Inverse Property aais the additive inverse of aa a,a0a,a0
1/a1/a is the multiplicative inverse of aa
For any real number a, a+(a)=0a+(a)=0 a·1a=1a·1a=1
Distributive Property
If a,b,ca,b,c are real numbers, then a(b+c)=ab+aca(b+c)=ab+ac
Properties of Zero
For any real number a,
a0=00a=0a0=00a=0
For any real number a,a0a,a0 0a=00a=0
a0a0 is undefined
Table 7.1 Properties of Real Numbers

Media

ACCESS ADDITIONAL ONLINE RESOURCES

Section 7.4 Exercises

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.

158.

101 + 0 = 101 101 + 0 = 101

159.

3 5 ( 1 ) = 3 5 3 5 ( 1 ) = 3 5

160.

−9 · 1 = −9 −9 · 1 = −9

161.

0 + 64 = 64 0 + 64 = 64

Use the Inverse Properties of Addition and Multiplication

In the following exercises, find the multiplicative inverse.

162.

8 8

163.

14 14

164.

−17 −17

165.

−19 −19

166.

7 12 7 12

167.

8 13 8 13

168.

3 10 3 10

169.

5 12 5 12

170.

0.8 0.8

171.

0.4 0.4

172.

−0.2 −0.2

173.

−0.5 −0.5

Use the Properties of Zero

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

174.

48 · 0 48 · 0

175.

0 6 0 6

176.

3 0 3 0

177.

22 · 0 22 · 0

178.

0 ÷ 11 12 0 ÷ 11 12

179.

6 0 6 0

180.

0 3 0 3

181.

0 ÷ 7 15 0 ÷ 7 15

182.

0 · 8 15 0 · 8 15

183.

( −3.14 ) ( 0 ) ( −3.14 ) ( 0 )

184.

5.72 ÷ 0 5.72 ÷ 0

185.

1 10 0 1 10 0

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

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

186.

19 a + 44 19 a 19 a + 44 19 a

187.

27 c + 16 27 c 27 c + 16 27 c

188.

38 + 11 r 38 38 + 11 r 38

189.

92 + 31 s 92 92 + 31 s 92

190.

10 ( 0.1 d ) 10 ( 0.1 d )

191.

100 ( 0.01 p ) 100 ( 0.01 p )

192.

5 ( 0.6 q ) 5 ( 0.6 q )

193.

40 ( 0.05 n ) 40 ( 0.05 n )

194.

0r+200r+20, where r−20r−20

195.

0s+130s+13, where s−13s−13

196.

0u4.990u4.99, where u4.99u4.99

197.

0v65.10v65.1, where v65.1v65.1

198.

0÷(x12)0÷(x12), where x12x12

199.

0÷(y16)0÷(y16), where y16y16

200.

325a0325a0, where 325a0325a0

201.

289b0289b0, where 289b0289b0

202.

2.1+0.4c02.1+0.4c0, where 2.1+0.4c02.1+0.4c0

203.

1.75+9f01.75+9f0, where 1.75+9f01.75+9f0

204.

(34+910m)÷0(34+910m)÷0, where 34+910m034+910m0

205.

(516n37)÷0(516n37)÷0, where 516n370516n370

206.

9 10 · 10 9 ( 18 p 21 ) 9 10 · 10 9 ( 18 p 21 )

207.

5 7 · 7 5 ( 20 q 35 ) 5 7 · 7 5 ( 20 q 35 )

208.

15 · 3 5 ( 4 d + 10 ) 15 · 3 5 ( 4 d + 10 )

209.

18 · 5 6 ( 15 h + 24 ) 18 · 5 6 ( 15 h + 24 )

Everyday Math

210.

Insurance copayment Carrie had to have 55 fillings done. Each filling cost $80.$80. Her dental insurance required her to pay 20%20% of the cost. Calculate Carrie’s cost

  1. by finding her copay for each filling, then finding her total cost for 55 fillings, and

  2. by multiplying 5(0.20)(80).5(0.20)(80).

  3. Which of the Properties of Real Numbers did you use for part (b)?

211.

Cooking time Helen bought a 24-pound24-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 2020 minutes per pound cooking time.

  1. Calculate the length of time needed to roast the turkey by multiplying 24·2024·20 to find the number of minutes and then multiplying the product by 160160 to convert minutes into hours.

  2. Multiply 24(20·160).24(20·160).

  3. Which of the Properties of Real Numbers allows you to multiply 24(20·160)24(20·160) instead of (24·20)160?(24·20)160?

Writing Exercises

212.

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

213.

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

Self Check

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

.

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?

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction
Citation information

© Jul 24, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.