7.4 Properties of Identity, Inverses, and Zero

Learning objectives

Learning objectives

• 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

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,

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,

Example 7.33

Solution

1. Recall the identity properties. Adding $0$ to a number leaves it unchanged, and multiplying a number by $1$ also leaves it unchanged.
2. Inspect part ⓐ. In $7+0=7$, the number $7$ is combined with $0$ by addition and remains $7$.
3. Name the property in part ⓐ. This equation demonstrates the identity property of addition.
4. Inspect and name part ⓑ. In $-16(1)=-16$, the number $-16$ is multiplied by $1$ and remains $-16$, so this demonstrates the identity property of multiplication.

Takeaway: Adding $0$ preserves a number by the identity property of addition, while multiplying by $1$ preserves it by the identity property of multiplication.

Use the Inverse Properties of Addition and Multiplication

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 $1?$

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

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

We call $\frac{1}{a}$ the multiplicative inverse of $a(a\ne{}0).$ 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:

Example 7.34

Solution

1. Recall the definition of an additive inverse. The additive inverse of a number is its opposite, and a number plus its opposite is always $0$.
2. Take the opposite of $13$. Changing the sign gives $-13$; checking, $13+(-13)=0$.
3. Take the opposite of $-\frac{5}{8}$. Changing the sign gives $\frac{5}{8}$; checking, $-\frac{5}{8}+\frac{5}{8}=0$.
4. Take the opposite of $0.6$. Changing the sign gives $-0.6$; checking, $0.6+(-0.6)=0$.

Takeaway: An additive inverse is the opposite of a number, so the additive inverses of $13$, $-\frac{5}{8}$, and $0.6$ are $-13$, $\frac{5}{8}$, and $-0.6$.

Example 7.35

Solution

1. Recall the definition of a multiplicative inverse. The multiplicative inverse of a nonzero number is its reciprocal, and the two values multiply to $1$.
2. Find the inverse of $9$. Since $9=\frac{9}{1}$, its reciprocal is $\frac{1}{9}$; checking, $9\cdot\frac{1}{9}=1$.
3. Find the inverse of $-\frac{1}{9}$. Flipping numerator and denominator gives $\frac{9}{-1}=-9$; checking, $-\frac{1}{9}\cdot(-9)=1$.
4. Convert $0.9$ before finding its inverse. Write $0.9=\frac{9}{10}$, so the reciprocal is $\frac{10}{9}$; checking, $\frac{9}{10}\cdot\frac{10}{9}=1$.

Takeaway: A multiplicative inverse is the reciprocal, giving $\frac{1}{9}$, $-9$, and $\frac{10}{9}$ for $9$, $-\frac{1}{9}$, and $0.9$.

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.

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 7.36

Solution

1. State the multiplication-by-zero rule. For any real number $a$, both $a\cdot0$ and $0\cdot{}a$ equal $0$.
2. Apply it to a negative integer. In $-8\cdot0$, the nonzero factor is $-8$, and the product is therefore $0$.
3. Apply it to a positive fraction. In $\frac{5}{12}\cdot0$, the nonzero factor is $\frac{5}{12}$, and the product is also $0$.
4. Apply it with zero first. In $0(2.94)$, multiplication by zero still forces the product to be $0$.

Takeaway: The product of any real number and $0$ is $0$, so $-8\cdot{}0$, $\frac{5}{12}\cdot{}0$, and $0(2.94)$ all simplify to $0$.

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.

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

Zero divided by any real number except zero is zero.

Example 7.37

Solution

1. State the division rule for zero as numerator. For any nonzero real number $a$, both $\frac{0}{a}$ and $0\div a$ equal $0$.
2. Apply it to $0\div5$. The denominator $5$ is nonzero, so the quotient is $0$.
3. Apply it to $\frac{0}{-2}$. The denominator $-2$ is nonzero, so the quotient is also $0$.
4. Apply it to $0\div\frac{7}{8}$. The divisor $\frac{7}{8}$ is nonzero, so the quotient remains $0$.

Takeaway: Zero divided by any nonzero real number is $0$, so each of $0\div5$, $\frac{0}{-2}$, and $0\div\frac{7}{8}$ equals $0$.

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 $4?$

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 $4\mathrm{÷}0,$ and so we say that division by zero is undefined.

Division by zero is undefined.

Example 7.38

Solution

1. State the rule for division by zero. For any real number $a$, both $\frac{a}{0}$ and $a\div0$ are undefined.
2. Check part ⓐ. In $7.5\div0$, the divisor is $0$, so the expression is undefined.
3. Check part ⓑ. In $\frac{-32}{0}$, the denominator is $0$, so the quotient is undefined.
4. Check part ⓒ. In $\frac{4}{9}\div0$, the divisor is again $0$, so the expression is undefined.

Takeaway: Division by zero is undefined, so $7.5\div0$, $\frac{-32}{0}$, and $\frac{4}{9}\div0$ have no defined value.

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 7.39

Solution

1. Identify the additive inverses. The terms $3x$ and $-3x$ have the same variable part and opposite coefficients, so their sum is $0$.
2. Reorder the terms. Rewrite $3x+15-3x$ as $3x+(-3x)+15$ using the commutative property of addition.
3. Cancel the opposite pair. Since $3x+(-3x)=0$, the expression becomes $0+15$.
4. Simplify with the additive identity. Adding $0$ leaves $15$, so $3x+15-3x=15$.

Takeaway: The terms $3x$ and $-3x$ are additive inverses, so they cancel to $0$ and leave the constant $15$ in $3x+15-3x$.

Example 7.40

Solution

1. Regroup the numerical factors. By the associative property, $4(0.25q)$ can be rewritten as $[4(0.25)]q$.
2. Multiply the grouped numbers. Since $4\cdot0.25=1$, the expression becomes $1q$.
3. Apply the multiplicative identity. Multiplying $q$ by $1$ leaves its coefficient unchanged, so the simplified expression is $q$.

Takeaway: Regrouping $4(0.25q)$ as $[4(0.25)]q$ gives $1.00q$, and the multiplicative identity reduces the result to $q$.

Example 7.41

Solution

1. Examine the denominator. The restriction $n\ne-5$ guarantees that $n+5\ne0$, so the denominator is never zero.
2. Use the zero-numerator rule. For a nonzero denominator $a$, $\frac{0}{a}=0$ because zero divided among $a$ equal parts is still zero.
3. Simplify. Since the denominator $n+5$ is nonzero, $\frac{0}{n+5}=0$.

Takeaway: Because $n\ne-5$, the denominator $n+5$ is nonzero, so the zero numerator makes $\frac{0}{n+5}$ simplify to $0$.

Example 7.42

Solution

1. Examine the denominator first. The denominator of $\frac{10-3p}{0}$ is the constant $0$.
2. Apply the division-by-zero rule. Because division by zero is undefined for every real numerator, the particular numerator $10-3p$ does not change that fact.
3. State the conclusion. The expression $\frac{10-3p}{0}$ has no defined value.

Takeaway: A zero denominator makes the quotient undefined regardless of its numerator, so $\frac{10-3p}{0}$ has no defined value.

Example 7.43

Solution

1. Look at the fraction factors. The factors $\frac{3}{4}$ and $\frac{4}{3}$ are reciprocals, so their product is $1$.
2. Multiply the reciprocal factors first. Since $\frac{3}{4}\cdot\frac{4}{3}=\frac{3\cdot4}{4\cdot3}=1$, the expression becomes $1(6x+12)$.
3. Apply the multiplicative identity. Multiplying by $1$ leaves the parentheses term unchanged as $6x+12$.

Takeaway: The reciprocal factors $\frac{3}{4}$ and $\frac{4}{3}$ multiply to $1$, which leaves the expression unchanged as $6x+12$.

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

Section 7.4 Exercises

Section 7.4 Exercises

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.

Use the Inverse Properties of Addition and Multiplication

In the following exercises, find the multiplicative inverse.

Use the Properties of Zero

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

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

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

Everyday Math

Writing Exercises

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?

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

ⓐ $\,23+0=23$ⓑ $\,-37(1)=-37.$

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

ⓐ $\,1\cdot{}29=29$ⓑ $\,14+0=14.$

3. Find the additive inverse: ⓐ $18$ ⓑ $\frac{7}{9}$ ⓒ $1.2$.

1. Find the additive inverse: ⓐ $47$ ⓑ $\frac{7}{13}$ ⓒ $\,8.4$.

2. Find the multiplicative inverse: ⓐ $\,5\,$ ⓑ $\,-\frac{1}{7}\,$ ⓒ $\,0.3$.

3. Find the multiplicative inverse: ⓐ $\,18$ ⓑ $\,-\frac{4}{5}$ ⓒ $\,0.6$.

1. Simplify: ⓐ $\,-14\cdot{}0\,$ ⓑ $0\cdot\frac{2}{3}\,$ ⓒ $\,(16.5)\cdot{}0.$

2. Simplify: ⓐ $\,(1.95)\cdot{}0$ ⓑ $\,0(-17)$ ⓒ $\,0\cdot\frac{5}{4}.$

3. Simplify: ⓐ $\,0\mathrm{÷}11$ ⓑ $\,\frac{0}{-6}$ ⓒ $\,0\mathrm{÷}\frac{3}{10}$.

1. Simplify: ⓐ $\,0\mathrm{÷}\frac{8}{3}$ ⓑ $\,0\mathrm{÷}(-10)$ ⓒ $\,0\mathrm{÷}12.75$.

2. Simplify: ⓐ $\,16.4\mathrm{÷}0$ ⓑ $\,\frac{-2}{0}$ ⓒ $\,\frac{1}{5}\mathrm{÷}0$.

3. Simplify: ⓐ $\,\frac{-5}{0}$ ⓑ $\,96.9\mathrm{÷}0$ ⓒ $\,\frac{4}{15}\mathrm{÷}0$

1. Simplify: $-25u-18+25u.$

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

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

1. Simplify: $\frac{18-6c}{0}.$

2. Simplify: $\frac{15-4q}{0}.$

3. Simplify: $\frac{2}{5}\cdot\frac{5}{2}(20y+50).$

1. Simplify: $\frac{3}{8}\cdot\frac{8}{3}(12z+16).$

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

3. $0+64=64$

1. $14$

2. $-19$

3. $\frac{8}{13}$

1. $-\frac{5}{12}$

2. $0.4$

3. $-0.5$

1. $\frac{6}{0}$

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

3. $92+31s-92$

1. $100(0.01p)$

2. $40(0.05n)$

3. $\frac{28-9b}{0}$, where $28-9b\ne{}0$

1. $\frac{1.75+9f}{0}$, where $1.75+9f\ne{}0$

2. $(\frac{5}{16}n-\frac{3}{7})\mathrm{÷}0$, where $\frac{5}{16}n-\frac{3}{7}\ne{}0$

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

1. $18\cdot\frac{5}{6}(15h+24)$

2. Cooking time Helen bought a $\mathrm{24-pound}$ 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.

1. ⓐ 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.

2. ⓑ Multiply $24(20\cdot\frac{1}{60}).$

3. ⓒ Which of the Properties of Real Numbers allows you to multiply $24(20\cdot\frac{1}{60})$ instead of $(24\cdot{}20)\frac{1}{60}?$

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