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}

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

ⓐ $7 + 0 = 7$
ⓑ $−16 (1) = −16$

Solution

This table illustrates the identity property of addition, showing an example where adding zero to a number results in the same number.
ⓐ
	$7 + 0 = 7$
We are adding 0.	We are using the identity property of addition.

This table illustrates the identity property of multiplication, showing that multiplying any number by one results in the original number.
ⓑ
	$−16 (1) = −16$
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?
$5 +_____= 0$
What number added to −6 gives the additive identity, 0?
$−6 +_____= 0$

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 $\frac{2}{3}$ 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 $a (a \neq 0) .$ 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:

Find the additive inverse of each expression: ⓐ $13$ ⓑ $- \frac{5}{8}$ ⓒ $0.6$ .

Solution

To find the additive inverse, we find the opposite.

ⓐ The additive inverse of $13$ is its opposite, $−13 .$
ⓑ The additive inverse of $- \frac{5}{8}$ is its opposite, $\frac{5}{8} .$
ⓒ The additive inverse of $0.6$ is its opposite, $−0.6 .$

Find the multiplicative inverse: ⓐ $9$ ⓑ $- \frac{1}{9}$ ⓒ $0.9$ .

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

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 $0 .$

Simplify: ⓐ $−8 \cdot 0$ ⓑ $\frac{5}{12} \cdot 0$ ⓒ $0 (2.94)$ .

Solution

This table illustrates the mathematical property that the product of any real number and zero is zero, accompanied by an example.
ⓐ
	$−8 \cdot 0$
The product of any real number and 0 is 0.	$0$

An example illustrating the multiplication property of zero: any number multiplied by zero equals zero.
ⓑ
	$\frac{5}{12} \cdot 0$
The product of any real number and 0 is 0.	$0$

Table demonstrating the Zero Property of Multiplication with its statement and an example.
ⓒ
	$0 (2.94)$
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.

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

Solution

Illustrates the rule that zero divided by any non-zero real number results in zero, with a mathematical example.
ⓐ
	$0 \div 5$
Zero divided by any real number, except 0, is zero.	$0$

Mathematical rule: Zero divided by any non-zero real number equals zero, illustrated with an example.
ⓑ
	$\frac{0}{−2}$
Zero divided by any real number, except 0, is zero.	$0$

Demonstrates the rule: zero divided by any non-zero real number equals zero, with an example.
ⓒ
	$0 \div \frac{7}{8}$
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 $4 \div 0,$ and so we say that division by zero is undefined.

Division by zero is undefined.

Simplify: ⓐ $7.5 \div 0$ ⓑ $\frac{−32}{0}$ ⓒ $\frac{4}{9} \div 0$ .

Solution

This table demonstrates the undefined outcome when performing division by zero with a mathematical expression.
ⓐ
	$7.5 \div 0$
Division by zero is undefined.	undefined

This table illustrates that division by zero is undefined, presenting an example expression and its corresponding status.
ⓑ
	$\frac{−32}{0}$
Division by zero is undefined.	undefined

Demonstrates that division by zero is undefined, with a mathematical example (4/9 0) and the general rule.
ⓒ
	$\frac{4}{9} \div 0$
Division by zero is undefined.	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.

Simplify: $3 x + 15 - 3 x .$

Solution

Simplification of the algebraic expression 3x + 15 - 3x, demonstrating the use of additive inverses.
	$3 x + 15 - 3 x$
Notice the additive inverses, $3 x$ and $−3 x$ .	$0 + 15$
Add.	$15$

Simplify: $4 (0.25 q) .$

Solution

This table illustrates the step-by-step simplification of the expression 4(0.25q) to q, applying mathematical properties.
	$4 (0.25 q)$
Regroup, using the associative property.	$[4 (0.25)] q$
Multiply.	$1.00 q$
Simplify; 1 is the multiplicative identity.	$q$

Simplify: $\frac{0}{n + 5}$ , where $n \neq −5$ .

Solution

This table illustrates the mathematical property: zero divided by any non-zero real number is zero.
	$\frac{0}{n + 5}$
Zero divided by any real number except itself is zero.	$0$

Simplify: $\frac{10 - 3 p}{0} .$

Solution

This table explains that division by zero is undefined, showing a mathematical example and its corresponding status.
	$\frac{10 - 3 p}{0}$
Division by zero is undefined.	undefined

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

Solution

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

Illustrates simplifying an algebraic expression step-by-step using reciprocals and the multiplicative identity.
	$\frac{3}{4} \cdot \frac{4}{3} (6 x + 12)$
Multiply; the product of reciprocals is 1.	$1 (6 x + 12)$
Simplify by recognizing the multiplicative identity.	$6 x + 12$

All the properties of real numbers we have used in this chapter are summarized in Table 20.

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 $a$	$a, a \neq 0$ $1 / a$ 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 $a, a \neq 0$	$\frac{0}{a} = 0$ $\frac{a}{0}$ 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 $a \div 0$ 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.

$101 + 0 = 101$

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

Solution

identity property of multiplication

$−9 \cdot 1 = −9$

$0 + 64 = 64$

Solution

identity property of addition

Use the Inverse Properties of Addition and Multiplication

In the following exercises, find the multiplicative inverse.

$8$

$14$

Solution

$\frac{1}{14}$

$−17$

$−19$

Solution

$- \frac{1}{19}$

$\frac{7}{12}$

$\frac{8}{13}$

Solution

$\frac{13}{8}$

$- \frac{3}{10}$

$- \frac{5}{12}$

Solution

$- \frac{12}{5}$

$0.8$

$0.4$

Solution

$\frac{5}{2}$

$−0.2$

$−0.5$

Solution

−2

Use the Properties of Zero

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

$48 \cdot 0$

$\frac{0}{6}$

Solution

0

$\frac{3}{0}$

$22 \cdot 0$

Solution

0

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

$\frac{6}{0}$

Solution

undefined

$\frac{0}{3}$

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

Solution

0

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

$(−3.14) (0)$

Solution

0

$5.72 \div 0$

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

Solution

undefined

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

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

$19 a + 44 - 19 a$

$27 c + 16 - 27 c$

Solution

16

$38 + 11 r - 38$

$92 + 31 s - 92$

Solution

31s

$10 (0.1 d)$

$100 (0.01 p)$

Solution

p

$5 (0.6 q)$

$40 (0.05 n)$

Solution

2n

$\frac{0}{r + 20}$ , where $r \neq −20$

$\frac{0}{s + 13}$ , where $s \neq −13$

Solution

0

$\frac{0}{u - 4.99}$ , where $u \neq 4.99$

$\frac{0}{v - 65.1}$ , where $v \neq 65.1$

Solution

0

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

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

Solution

0

$\frac{32 - 5 a}{0}$ , where $32 - 5 a \neq 0$

$\frac{28 - 9 b}{0}$ , where $28 - 9 b \neq 0$

Solution

undefined

$\frac{2.1 + 0.4 c}{0}$ , where $2.1 + 0.4 c \neq 0$

$\frac{1.75 + 9 f}{0}$ , where $1.75 + 9 f \neq 0$

Solution

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 $\frac{5}{16} n - \frac{3}{7} \neq 0$

Solution

undefined

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

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

Solution

20q − 35

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

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

Solution

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 $5 (0.20) (80) .$

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 $24 \cdot 20$ 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 $(24 \cdot 20) \frac{1}{60} ?$

Solution

ⓐ 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?

Solution

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!'

ⓑ 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?