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Prealgebra

7.4 Properties of Identity, Inverses, and Zero

Prealgebra7.4 Properties of Identity, Inverses, and Zero
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

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 7.4

Before you get started, take this readiness quiz.

  1. Find the opposite of −4.−4.
    If you missed this problem, review Example 3.3.
  2. Find the reciprocal of 52.52.
    If you missed this problem, review Example 4.29.
  3. 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=2914+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=101101+0=101

159.

35(1)=3535(1)=35

160.

−9·1=−9−9·1=−9

161.

0+64=640+64=64

Use the Inverse Properties of Addition and Multiplication

In the following exercises, find the multiplicative inverse.

162.

88

163.

1414

164.

−17−17

165.

−19−19

166.

712712

167.

813813

168.

310310

169.

512512

170.

0.80.8

171.

0.40.4

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.

48·048·0

175.

0606

176.

3030

177.

22·022·0

178.

0÷11120÷1112

179.

6060

180.

0303

181.

0÷7150÷715

182.

0·8150·815

183.

(−3.14)(0)(−3.14)(0)

184.

5.72÷05.72÷0

185.

11001100

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

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

186.

19a+4419a19a+4419a

187.

27c+1627c27c+1627c

188.

38+11r3838+11r38

189.

92+31s9292+31s92

190.

10(0.1d)10(0.1d)

191.

100(0.01p)100(0.01p)

192.

5(0.6q)5(0.6q)

193.

40(0.05n)40(0.05n)

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.

910·109(18p21)910·109(18p21)

207.

57·75(20q35)57·75(20q35)

208.

15·35(4d+10)15·35(4d+10)

209.

18·56(15h+24)18·56(15h+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?

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