7.5: Properties of Identity, Inverses, and Zero
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Skills to Develop
 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!
Before you get started, take this readiness quiz.
 Find the opposite of −4. If you missed this problem, review Example 3.1.3.
 Find the reciprocal of \(\dfrac{5}{2}\). If you missed this problem, review Example 4.4.11.
 Multiply: \(\dfrac{3a}{5} \cdot \dfrac{9}{2a}\). If you missed this problem, review Example 4.3.9.
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,
\[\begin{split} 13 + 0 \qquad 14 &+ 0 \qquad 0 + (3x) \\ 13 \qquad \qquad &14 \qquad \; 3x \end{split}\]
Definition: Identity Properties
The identity property of addition: for any real number a,
\[a + 0 = a \qquad 0 + a = a\]
0 is called the additive identity
The identity property of multiplication: for any real number a
\[a \cdot 1 = a \qquad 1 \cdot a = a\]
1 is called the multiplicative identity
Example \(\PageIndex{1}\):
Identify whether each equation demonstrates the identity property of addition or multiplication. (a) 7 + 0 = 7 (b) −16(1) = −16
Solution
(a) 7 + 0 = 7
We are adding 0.  We are using the identity property of addition. 
(b) −16(1) = −16
We are multiplying by 1.  We are using the identity property of multiplication. 
Exercise \(\PageIndex{1}\):
Identify whether each equation demonstrates the identity property of addition or multiplication: (a) 23 + 0 = 23 (b) −37(1) = −37.
 Answer a

identity property of addition
 Answer b

identity property of multiplication
Exercise \(\PageIndex{2}\):
Identify whether each equation demonstrates the identity property of addition or multiplication: (a) 1 • 29 = 29 (b) 14 + 0 = 14.
 Answer a

identity property of multiplication
 Answer b

additive identity
Use the Inverse Properties of Addition and Multiplication
What number added to 5 gives the additive identity, 0?
5 + _____ = 0  We know \(5 + (\textcolor{red}{5}) = 0\) 
What number added to −6 gives the additive identity, 0?
6 + _____ = 0  We know \(6 + (\textcolor{red}{6}) = 0\) 
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 \(\dfrac{2}{3}\) gives the multiplicative identity, 1? In other words, twothirds times what results in 1?
\(\dfrac{2}{3} \cdot\) _____ = 1  We know \(\dfrac{2}{3} \cdot \textcolor{red}{\dfrac{3}{2}} = 1\) 
What number multiplied by 2 gives the multiplicative identity, 1? In other words two times what results in 1?
2 • _____ = 1  We know \(2 \cdot \textcolor{red}{\dfrac{1}{2}} = 1\) 
Notice that in each case, the missing number was the reciprocal of the number.
We call \(\dfrac{1}{a}\) the multiplicative inverse of a (a ≠ 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:
Definition: 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 ≠ 0,
\[a \cdot \dfrac{1}{a} = 1\]
\(\dfrac{1}{a}\) is the multiplicative inverse of a.
Example \(\PageIndex{2}\):
Find the additive inverse of each expression: (a) 13 (b) \(− \dfrac{5}{8}\) (c) 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 \(− \dfrac{5}{8}\) is its opposite, \(\dfrac{5}{8}\).
 The additive inverse of 0.6 is its opposite, −0.6.
Exercise \(\PageIndex{3}\):
Find the additive inverse of each expression: (a) 18 (b) \(\dfrac{7}{9}\) (c) 1.2.
 Answer a

\(18\)
 Answer b

\(\frac{7}{9}\)
 Answer c

\(1.2\)
Exercise \(\PageIndex{4}\):
Find the additive inverse of each expression: (a) 47(b) \(\dfrac{7}{13}\) (c) 8.4.
 Answer a

\(47\)
 Answer b

\(\frac{7}{13}\)
 Answer c

\(8.4\)
Example \(\PageIndex{3}\):
Find the multiplicative inverse: (a) 9 (b) \(− \dfrac{1}{9}\) (c) 0.9.
Solution
To find the additive inverse, we find the opposite.
 The multiplicative inverse of 9 is its reciprocal, \(\dfrac{1}{9}\).
 The multiplicative inverse of \(− \dfrac{1}{9}\) is its reciprocal, 9.
 To find the multiplicative inverse of 0.9, we first convert 0.9 to a fraction, \(\dfrac{9}{10}\). Then we find the reciprocal, \(\dfrac{10}{9}\).
Exercise \(\PageIndex{5}\):
Find the multiplicative inverse: (a) 5 (b) \(− \dfrac{1}{7}\) (c) 0.3.
 Answer a

\(\frac{1}{5}\)
 Answer b

\(7\)
 Answer c

\(\frac{10}{3}\)
Exercise \(\PageIndex{6}\):
Find the multiplicative inverse: (a) 18 (b) \(− \dfrac{4}{5}\) (c) 0.6.
 Answer a

\(\frac{1}{18}\)
 Answer b

\(\frac{5}{4}\)
 Answer c

\(\frac{5}{3}\)
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.
Definition: Multiplication by Zero
For any real number a,
\[a \cdot 0 = 0 \qquad 0 \cdot a = 0\]
Example \(\PageIndex{4}\):
Simplify: (a) −8 • 0 (b) \(\dfrac{5}{12} \cdot 0\) (c) 0(2.94).
Solution
(a) −8 • 0
The product of any real number and 0 is 0.  0 
(b) \(\dfrac{5}{12} \cdot 0\)
The product of any real number and 0 is 0.  0 
(c) 0(2.94)
The product of any real number and 0 is 0.  0 
Exercise \(\PageIndex{7}\):
Simplify: (a) −14 • 0 (b) \(0 \cdot \dfrac{2}{3}\) (c) (16.5) • 0.
 Answer a

\(0\)
 Answer b

\(0\)
 Answer c

\(0\)
Exercise \(\PageIndex{8}\):
Simplify: (a) (1.95) • 0 (b) 0(17) (c) \(0 \cdot \dfrac{5}{4}\).
 Answer a

\(0\)
 Answer b

\(0\)
 Answer c

\(0\)
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
\[0 \div 3 = 0\; because\; 0 \cdot 3 = 0 \ldotp\]
Definition: Division of Zero
For any real number a, except 0, \(\dfrac{0}{a}\) = 0 and 0 ÷ a = 0.
Zero divided by any real number except zero is zero.
Example \(\PageIndex{5}\):
Simplify: (a) 0 ÷ 5 (b) \(\dfrac{0}{−2}\) (c) 0 ÷ \(\dfrac{7}{8}\).
Solution
(a) 0 ÷ 5
Zero divided by any real number, except 0, is zero.  0 
(b) \(\dfrac{0}{−2}\)
Zero divided by any real number, except 0, is zero.  0 
(c) 0 ÷ \(\dfrac{7}{8}\)
Zero divided by any real number, except 0, is zero.  0 
Exercise \(\PageIndex{9}\):
Simplify: (a) 0 ÷ 11 (b) \(\dfrac{0}{−6}\) (c) 0 ÷ \(\dfrac{3}{10}\).
 Answer a

\(0\)
 Answer b

\(0\)
 Answer c

\(0\)
Exercise \(\PageIndex{10}\):
Simplify: (a) 0 ÷ \(\dfrac{8}{3}\) (b) 0 ÷ (10) (c) 0 ÷ 12.75.
 Answer a

\(0\)
 Answer b

\(0\)
 Answer c

\(0\)
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 = ___ means ___ • 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.
Definition: Division by Zero
For any real number a, \(\dfrac{a}{0}\), and a ÷ 0 are undefined.
Division by zero is undefined.
Example \(\PageIndex{6}\):
Simplify: (a) 7.5 ÷ 0 (b) \(\dfrac{−32}{0}\) (c) \(\dfrac{4}{9}\) ÷ 0.
Solution
(a) 7.5 ÷ 0
Division by zero is undefined.  undefined 
(b) \(\dfrac{−32}{0}\)
Division by zero is undefined.  undefined 
(c) \(\dfrac{4}{9}\) ÷ 0
Division by zero is undefined.  undefined 
Exercise \(\PageIndex{11}\):
Simplify: (a) 16.4 ÷ 0 (b) \(\dfrac{−2}{0}\) (c) \(\dfrac{1}{5}\) ÷ 0.
 Answer a

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 Answer b

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 Answer c

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Exercise \(\PageIndex{12}\):
Simplify: (a) \(\dfrac{−5}{0}\) (b) 96.9 ÷ 0 (c) \(\dfrac{4}{15}\) ÷ 0.
 Answer a

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 Answer b

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 Answer c

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We summarize the properties of zero.
Definition: properties of zero
Multiplication by Zero: For any real number a,
\[a \cdot 0 = 0 \qquad 0 \cdot a = 0\]
The product of any number and 0 is 0.
Division by Zero: For any real number a, a ≠ 0
\[\dfrac{0}{a} = 0\]
Zero divided by any real number, except itself, is zero.
\[\dfrac{a}{0}\; is\; undefined \ldotp\]
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 \(\PageIndex{7}\):
Simplify: 3x + 15 − 3x.
Solution
Notice the additive inverses, 3x and −3x.  0 + 15 
Add.  15 
Exercise \(\PageIndex{13}\):
Simplify: −12z + 9 + 12z.
 Answer

9
Exercise \(\PageIndex{14}\):
Simplify: −25u − 18 + 25u.
 Answer

18
Example \(\PageIndex{8}\):
Simplify: 4(0.25q).
Solution
Regroup, using the associative property.  [4(0.25)]q 
Multiply.  1.00q 
Simplify; 1 is the multiplicative identity.  q 
Exercise \(\PageIndex{15}\):
Simplify: 2(0.5p).
 Answer

p
Exercise \(\PageIndex{16}\):
Simplify: 25(0.04r).
 Answer

r
Example \(\PageIndex{9}\):
Simplify: \(\dfrac{0}{n + 5}\), where n ≠ −5.
Solution
Zero divided by any real number except itself is zero.  0 
Exercise \(\PageIndex{17}\):
Simplify: \(\dfrac{0}{m + 7}\), where m ≠ −7.
 Answer

0
Exercise \(\PageIndex{18}\):
Simplify: \(\dfrac{0}{d  4}\), where d ≠ 4.
 Answer

0
Example \(\PageIndex{10}\):
Simplify: \(\dfrac{10 − 3p}{0}\).
Solution
Division by zero is undefined.  undefined 
Exercise \(\PageIndex{19}\):
Simplify: \(\dfrac{18 − 6c}{0}\).
 Answer

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Exercise \(\PageIndex{20}\):
Simplify: \(\dfrac{15 − 4q}{0}\).
 Answer

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Example \(\PageIndex{11}\):
Simplify: \(\dfrac{3}{4} \cdot \dfrac{4}{3}\)(6x + 12).
Solution
We cannot combine the terms in parentheses, so we multiply the two fractions first.
Multiply; the product of reciprocals is 1.  1(6x + 12) 
Simplify by recognizing the multiplicative identity.  6x + 12 
Exercise \(\PageIndex{21}\):
Simplify: \(\dfrac{2}{5} \cdot \dfrac{5}{2}\)(20y+ 50).
 Answer

20y + 50
Exercise \(\PageIndex{22}\):
Simplify: \(\dfrac{3}{8} \cdot \dfrac{8}{3}\)(12z+ 16).
 Answer

12z + 16
All the properties of real numbers we have used in this chapter are summarized in Table \(\PageIndex{1}\).
Property  Of Addition  Of Multiplication 

Commutative Property  
If a and b are real numbers then…  a + b = b + a  a • b = b • a 
Associative Property  
If a, b, and c are real numbers then…  (a + b) + c = a + (b + c)  (a • b) • c = (b • 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 • 1 = a 1 • a = a 
Inverse Property  −a is the additive inverse of a 
a, a ≠ 0 1/a is the multiplicative inverse of a 
For any real number a,  a + (−a) = 0  a • 1/a = 1 
Distributive Property  
If a, b, c are real numbers, then a(b + c) = ab + ac  
Properties of Zero  
For any real number a, 
a • 0 = 0 0 • a = 0 

For any real number a where a ≠ 0 
\(\dfrac{0}{a} = 0\) \(\dfrac{a}{0}\) is undefined 
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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
 \(\dfrac{3}{5}(1) = \dfrac{3}{5}\)
 −9 • 1 = −9
 0 + 64 = 64
Use the Inverse Properties of Addition and Multiplication
In the following exercises, find the multiplicative inverse.
 8
 14
 −17
 −19
 \(\dfrac{7}{12}\)
 \(\dfrac{8}{13}\)
 \(− \dfrac{3}{10}\)
 \(− \dfrac{5}{12}\)
 0.8
 0.4
 −0.2
 −0.5
Use the Properties of Zero
In the following exercises, simplify using the properties of zero.
 48 • 0
 \(\dfrac{0}{6}\)
 \(\dfrac{3}{0}\)
 22 • 0
 0 ÷ \(\dfrac{11}{12}\)
 \(\dfrac{6}{0}\)
 \(\dfrac{0}{3}\)
 0 ÷ \(\dfrac{7}{15}\)
 0 • \(\dfrac{8}{15}\)
 (−3.14)(0)
 5.72 ÷ 0
 \(\dfrac{\dfrac{1}{10}}{0}\)
Simplify Expressions using the Properties of Identities, Inverses, and Zero
In the following exercises, simplify using the properties of identities, inverses, and zero.
 19a + 44 − 19a
 27c + 16 − 27c
 38 + 11r − 38
 92 + 31s − 92
 10(0.1d)
 100(0.01p)
 5(0.6q)
 40(0.05n)
 \(\dfrac{0}{r + 20}\), where r ≠ −20
 \(\dfrac{0}{s + 13}\), where s ≠ −13
 \(\dfrac{0}{u − 4.99}\), where u ≠ 4.99
 \(\dfrac{0}{v − 65.1}\), where v ≠ 65.1
 0 ÷ \(\left(x − \dfrac{1}{2}\right)\), where x ≠ \(\dfrac{1}{2}\)
 0 ÷ \(\left(y − \dfrac{1}{6}\right)\), where y ≠ \(\dfrac{1}{6}\)
 \(\dfrac{32 − 5a}{0}\), where 32 − 5a ≠ 0
 \(\dfrac{28 − 9b}{0}\), where 28 − 9b ≠ 0
 \(\dfrac{2.1 + 0.4c}{0}\), where 2.1 + 0.4c ≠ 0
 \(\dfrac{1.75 + 9 f}{0}\), where 1.75 + 9 f ≠ 0
 \(\left\dfrac{3}{4} + \dfrac{9}{10}m \right) \div 0\), where \(\dfrac{3}{4} + \dfrac{9}{10}m \neq 0\)
 \(\left(\dfrac{5}{16}n − \dfrac{3}{7}\right) \div 0\), where \(\dfrac{5}{16}n − \dfrac{3}{7} \neq 0\)
 \(\dfrac{9}{10} \cdot \dfrac{10}{9}\)(18p − 21)
 \(\dfrac{5}{7} \cdot \dfrac{7}{5}\)(20q − 35)
 15 \(\cdot \dfrac{3}{5}\)(4d + 10)
 18 \(\cdot \dfrac{5}{6}\)(15h + 24)
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(0.20)(80).
 Which of the Properties of Real Numbers did you use for part (b)?
 Cooking time Helen bought a 24pound 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 · 20 to find the number of minutes and then multiplying the product by 1 60 to convert minutes into hours.
 Multiply 24\(\left(20 \cdot \dfrac{1}{60}\right)\).
 Which of the Properties of Real Numbers allows you to multiply 24\(\left(20 \cdot \dfrac{1}{60}\right)\) instead of (24 • 20)\(\dfrac{1}{60}\)?
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?
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) 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?
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne AnthonySmith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1fa2...49835c3c@5.191."