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Properties of Identity, Inverses, and Zero
Recognize the identity properties of addition and multiplication
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,
\[\begin{array}{lllll}13+0 & \ & -14+0 & \ & 0+(-3x) \\ 13 & \ & -14 & \ & -3x\end{array}\]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,
\[\begin{array}{lllll}43\cdot 1 & \ & -27\cdot 1 & \ & 1\cdot \frac{6y}{5} \\ 43 & \ & -27 & \ & \frac{6y}{5}\end{array}\]Example
Try it.
Identify whether each equation demonstrates the identity property of addition or multiplication.
-
ⓐ \(\ 7+0=7\)
-
ⓑ \(\ -16(1)=-16\)
Solution
| ⓐ | |
| \(7+0=7\) | |
| We are adding 0. | We are using the identity property of addition. |
| ⓑ | |
| \(-16(1)=-16\) | |
| 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? | |
| \(5+\underline{\hspace{1.5em}}=0\) | |
| What number added to −6 gives the additive identity, 0? | |
| \(-6+\underline{\hspace{1.5em}}=0\) |
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 \(1?\)
| \(\frac{2}{3}\cdot \underline{\hspace{1.5em}}=1\) |
What number multiplied by \(2\) gives the multiplicative identity, \(1?\) In other words two times what results in \(1?\)
| \(2\cdot \underline{\hspace{1.5em}}=1\) |
Notice that in each case, the missing number was the reciprocal of the number.
We call \(\frac{1}{a}\) the multiplicative inverse of \(a(a\ne 0)\text{.}\) 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
Try it.
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, \(-0.6.\)
Example
Try it.
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.
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
Try it.
Simplify: ⓐ \(\ -8\cdot 0\) ⓑ \(\ \frac{5}{12}\cdot 0\) ⓒ \(\ 0(2.94)\).
Solution
| ⓐ | |
| \(-8⋅0\) | |
| The product of any real number and 0 is 0. | \(0\) |
| ⓑ | |
| \(\frac{5}{12}\cdot 0\) | |
| The product of any real number and 0 is 0. | \(0\) |
| ⓒ | |
| \(0(2.94)\) | |
| The product of any real number and 0 is 0. | \(0\) |
Condensed: the full section is in OpenStax Prealgebra 2e.
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
Try it.
Simplify: \(3x+15-3x.\)
Solution
| \(3x+15-3x\) | |
| Notice the additive inverses, \(3x\) and \(-3x\). | \(0+15\) |
| Add. | \(15\) |
Example
Try it.
Simplify: \(4(0.25q).\)
Solution
| \(4(0.25q)\) | |
| Regroup, using the associative property. | \([4(0.25)]q\) |
| Multiply. | \(1.00q\) |
| Simplify; 1 is the multiplicative identity. | \(q\) |
Example
Try it.
Simplify: \(\frac{0}{n+5}\), where \(n\ne -5\).
Solution
| \(\frac{0}{n+5}\) | |
| Zero divided by any real number except itself is zero. | \(0\) |
Example
Try it.
Simplify: \(\frac{10-3p}{0}.\)
Solution
| \(\frac{10-3p}{0}\) | |
| Division by zero is undefined. | undefined |
Example
Try it.
Simplify: \(\frac{3}{4}\cdot \frac{4}{3}(6x+12).\)
Solution
We cannot combine the terms in parentheses, so we multiply the two fractions first.
| \(\frac{3}{4}\cdot \frac{4}{3}(6x+12)\) | |
| Multiply; the product of reciprocals is 1. | \(1(6x+12)\) |
| Simplify by recognizing the multiplicative identity. | \(6x+12\) |
All the properties of real numbers we have used in this chapter are summarized in .
| 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 | \(-\text{a}\)is the additive inverse of \(a\) | \(a,a\ne 0\) \(1/\text{a}\) is the multiplicative inverse of \(a\) |
| For any real number a, | \(a+\text{(}\text{-}\text{a}\text{)}\ =0\) | \(a\cdot \frac{1}{a}=1\) |
| Distributive Property \(\)If \(a,b,c\) are real numbers, then \(a(b+c)=ab+ac\) | ||
| Properties of Zero | ||
| For any real number a, | \(\begin{array}{l}a⋅0=0 \\ 0⋅a=0\end{array}\) | |
| For any real number \(a,a\ne 0\) | \(\frac{0}{a}=0\) \(\)\(\frac{a}{0}\) 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: \(a⋅1=a\ 1⋅a=a\) 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: \((a\ne 0)\ a⋅\frac{1}{a}=1\ \frac{1}{a}\) is the multiplicative inverse of a
- Properties of Zero
- Multiplication by Zero: For any real number a, \(\begin{array}{lllllll}a⋅0=0 & & & 0⋅a=0 & & & \text{The product of any number and 0 is 0.}\end{array}\)
- Division of Zero: For any real number a, \(\begin{array}{lllllll}\frac{0}{a}=0 & & & & & & \text{Zero divided by any real number, except itself, is zero.}\end{array}\)
- Division by Zero: For any real number a, \(\frac{a}{0}\) is undefined and \(a\div 0\) is undefined. Division by zero is undefined.
Properties of Identity, Inverses, and Zero
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.
Try it.
\(101+0=101\)
Try it.
\(\frac{3}{5}(1)=\frac{3}{5}\)
Solution
identity property of multiplication
Try it.
\(-9\cdot 1=-9\)
Use the Inverse Properties of Addition and Multiplication
In the following exercises, find the multiplicative inverse.
Try it.
\(\frac{7}{12}\)
Try it.
\(\frac{8}{13}\)
Solution
\(\frac{13}{8}\)
Try it.
\(-\frac{3}{10}\)
Try it.
\(-\frac{5}{12}\)
Solution
\(-\frac{12}{5}\)
Use the Properties of Zero
In the following exercises, simplify using the properties of zero.
Try it.
\(48\cdot 0\)
Try it.
\(\frac{0}{6}\)
Solution
0
Try it.
\(\frac{3}{0}\)
Try it.
\(22\cdot 0\)
Solution
0
Try it.
\(0\div \frac{11}{12}\)
Try it.
\(\frac{6}{0}\)
Solution
undefined
Try it.
\(\frac{0}{3}\)
Try it.
\(0\div \frac{7}{15}\)
Solution
0
Try it.
\(0\cdot \frac{8}{15}\)
Try it.
\((-3.14)(0)\)
Solution
0
Try it.
\(5.72\div 0\)
Try it.
\(\frac{\frac{1}{10}}{0}\)
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.
Try it.
\(19a+44-19a\)
Try it.
\(27c+16-27c\)
Solution
16
Try it.
\(38+11r-38\)
Try it.
\(92+31s-92\)
Solution
31s
Try it.
\(10(0.1d)\)
Try it.
\(100(0.01p)\)
Solution
p
Try it.
\(40(0.05n)\)
Solution
2n
Try it.
\(\frac{0}{r+20}\), where \(r\ne -20\)
Try it.
\(\frac{0}{s+13}\), where \(s\ne -13\)
Solution
0
Try it.
\(\frac{0}{u-4.99}\), where \(u\ne 4.99\)
Try it.
\(\frac{0}{v-65.1}\), where \(v\ne 65.1\)
Solution
0
Try it.
\(0\div (x-\frac{1}{2})\), where \(x\ne \frac{1}{2}\)
Try it.
\(0\div (y-\frac{1}{6})\), where \(y\ne \frac{1}{6}\)
Solution
0
Try it.
\(\frac{32-5a}{0}\), where \(32-5a\ne 0\)
Try it.
\(\frac{28-9b}{0}\), where \(28-9b\ne 0\)
Solution
undefined
Try it.
\(\frac{2.1+0.4c}{0}\), where \(2.1+0.4c\ne 0\)
Try it.
\(\frac{1.75+9f}{0}\), where \(1.75+9f\ne 0\)
Solution
undefined
Try it.
\((\frac{3}{4}+\frac{9}{10}m)\div 0\), where \(\frac{3}{4}+\frac{9}{10}m\ne 0\)
Try it.
\((\frac{5}{16}n-\frac{3}{7})\div 0\), where \(\frac{5}{16}n-\frac{3}{7}\ne 0\)
Solution
undefined
Try it.
\(\frac{9}{10}\cdot \frac{10}{9}(18p-21)\)
Try it.
\(\frac{5}{7}\cdot \frac{7}{5}(20q-35)\)
Solution
20q − 35
Try it.
\(15\cdot \frac{3}{5}(4d+10)\)
Try it.
\(18\cdot \frac{5}{6}(15h+24)\)
Solution
225h + 360
Try it.
In your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.
Try it.
How can the use of the properties of real numbers make it easier to simplify expressions?
Solution
Answers will vary.
Condensed: the full section is in OpenStax Prealgebra 2e.
이제 너 계산기는 이것을 해결하지 않지만, 그 조각은 계산 가능합니다. 아래의 하나를 시도하거나 자신의 것을 입력하십시오.
연습 (40)
먼저 종이에 각각을 시도해 보세요. 확인하기 위해 답을 드러내세요. 확인된 답은 각 단계의 솔버에서 열 수 있습니다.
-
Find the opposite of \(-4.\)
답을 드러내세요
\(4\)
-
Find the reciprocal of \(\frac{5}{2}.\)
답을 드러내세요
\(\frac{2}{5}\)
-
Multiply: \(\frac{3a}{5}\cdot \frac{9}{2a}.\)
답을 드러내세요
\(\frac{27}{10}\)
-
Identify whether each equation demonstrates the identity property of addition or multiplication.
-
ⓐ \(\ 7+0=7\)
-
ⓑ \(\ -16(1)=-16\)
답을 드러내세요
ⓐ \(7+0=7\) We are adding 0. We are using the identity property of addition. ⓑ \(-16(1)=-16\) We are multiplying by 1. We are using the identity property of multiplication. -
-
Identify whether each equation demonstrates the identity property of addition or multiplication:
ⓐ \(\ 23+0=23\)ⓑ \(\ -37(1)=-37.\)
답을 드러내세요
- ⓐ identity property of addition
- ⓑ identity property of multiplication
-
Identify whether each equation demonstrates the identity property of addition or multiplication:
ⓐ \(\ 1\cdot 29=29\)ⓑ \(\ 14+0=14.\)
답을 드러내세요
- ⓐ identity property of multiplication
- ⓑ identity property of addition
-
Find the additive inverse of each expression: ⓐ \(13\) ⓑ \(-\frac{5}{8}\) ⓒ \(\ 0.6\).
답을 드러내세요
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 additive inverse: ⓐ \(18\) ⓑ \(\frac{7}{9}\) ⓒ \(1.2\).
답을 드러내세요
- ⓐ \(\ -18\)
- ⓑ \(\ -\frac{7}{9}\)
- ⓒ \(\ -1.2\)
-
Find the additive inverse: ⓐ \(47\) ⓑ \(\frac{7}{13}\) ⓒ \(\ 8.4\).
답을 드러내세요
- ⓐ \(\ -47\)
- ⓑ \(\ -\frac{7}{13}\)
- ⓒ \(\ -8.4\)
-
Find the multiplicative inverse: ⓐ \(9\) ⓑ \(-\frac{1}{9}\) ⓒ \(0.9\).
답을 드러내세요
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, \(\frac{10}{9}.\)
-
-
Find the multiplicative inverse: ⓐ \(\ 5\) ⓑ \(\ -\frac{1}{7}\) ⓒ \(\ 0.3\).
답을 드러내세요
- ⓐ \(\ \frac{1}{5}\)
- ⓑ \(\ -7\)
- ⓒ \(\ \frac{10}{3}\)
-
Find the multiplicative inverse: ⓐ \(\ 18\) ⓑ \(\ -\frac{4}{5}\) ⓒ \(\ 0.6\).
답을 드러내세요
- ⓐ \(\ \frac{1}{18}\)
- ⓑ \(\ -\frac{5}{4}\)
- ⓒ \(\ \frac{5}{3}\)
-
Simplify: ⓐ \(\ -8\cdot 0\) ⓑ \(\ \frac{5}{12}\cdot 0\) ⓒ \(\ 0(2.94)\).
답을 드러내세요
ⓐ \(-8⋅0\) The product of any real number and 0 is 0. \(0\) ⓑ \(\frac{5}{12}\cdot 0\) The product of any real number and 0 is 0. \(0\) ⓒ \(0(2.94)\) The product of any real number and 0 is 0. \(0\) -
Simplify: ⓐ \(\ -14\cdot 0\) ⓑ \(0\cdot \frac{2}{3}\) ⓒ \(\ (16.5)\cdot 0.\)
답을 드러내세요
- ⓐ 0
- ⓑ 0
- ⓒ 0
-
Simplify: ⓐ \(\ (1.95)\cdot 0\) ⓑ \(\ 0(-17)\) ⓒ \(\ 0\cdot \frac{5}{4}.\)
답을 드러내세요
- ⓐ 0
- ⓑ 0
- ⓒ 0
-
Simplify: ⓐ \(\ 0\div 5\) ⓑ \(\frac{0}{-2}\) ⓒ \(\ 0\div \frac{7}{8}\).
답을 드러내세요
ⓐ \(0\div 5\) Zero divided by any real number, except 0, is zero. \(0\) ⓑ \(\frac{0}{-2}\) Zero divided by any real number, except 0, is zero. \(0\) ⓒ \(0\div \frac{7}{8}\) Zero divided by any real number, except 0, is zero. \(0\) -
Simplify: ⓐ \(\ 0\div 11\) ⓑ \(\ \frac{0}{-6}\) ⓒ \(\ 0\div \frac{3}{10}\).
답을 드러내세요
- ⓐ 0
- ⓑ 0
- ⓒ 0
-
Simplify: ⓐ \(\ 0\div \frac{8}{3}\) ⓑ \(\ 0\div (-10)\) ⓒ \(\ 0\div 12.75\).
답을 드러내세요
- ⓐ 0
- ⓑ 0
- ⓒ 0
-
Simplify: ⓐ \(\ 7.5\div 0\) ⓑ \(\ \frac{-32}{0}\) ⓒ \(\ \frac{4}{9}\div 0\).
답을 드러내세요
ⓐ \(7.5\div 0\) Division by zero is undefined. undefined ⓑ \(\frac{-32}{0}\) Division by zero is undefined. undefined ⓒ \(\frac{4}{9}\div 0\) Division by zero is undefined. undefined -
Simplify: ⓐ \(\ 16.4\div 0\) ⓑ \(\ \frac{-2}{0}\) ⓒ \(\ \frac{1}{5}\div 0\).
답을 드러내세요
- ⓐ undefined
- ⓑ undefined
- ⓒ undefined
-
Simplify: ⓐ \(\ \frac{-5}{0}\) ⓑ \(\ 96.9\div 0\) ⓒ \(\ \frac{4}{15}\div 0\)
답을 드러내세요
- ⓐ undefined
- ⓑ undefined
- ⓒ undefined
-
Simplify: \(3x+15-3x.\)
답을 드러내세요
\(3x+15-3x\) Notice the additive inverses, \(3x\) and \(-3x\). \(0+15\) Add. \(15\) -
Simplify: \(-12z+9+12z.\)
답을 드러내세요
9
-
Simplify: \(-25u-18+25u.\)
답을 드러내세요
−18
-
Simplify: \(4(0.25q).\)
답을 드러내세요
\(4(0.25q)\) Regroup, using the associative property. \([4(0.25)]q\) Multiply. \(1.00q\) Simplify; 1 is the multiplicative identity. \(q\) -
Simplify: \(2(0.5p).\)
답을 드러내세요
p
-
Simplify: \(25(0.04r).\)
답을 드러내세요
r
-
Simplify: \(\frac{0}{n+5}\), where \(n\ne -5\).
답을 드러내세요
\(\frac{0}{n+5}\) Zero divided by any real number except itself is zero. \(0\) -
Simplify: \(\frac{0}{m+7}\), where \(m\ne -7\).
답을 드러내세요
0
-
Simplify: \(\frac{0}{d-4}\), where \(d\ne 4\).
답을 드러내세요
0
-
Simplify: \(\frac{10-3p}{0}.\)
답을 드러내세요
\(\frac{10-3p}{0}\) Division by zero is undefined. undefined -
Simplify: \(\frac{18-6c}{0}.\)
답을 드러내세요
undefined
-
Simplify: \(\frac{15-4q}{0}.\)
답을 드러내세요
undefined
-
Simplify: \(\frac{3}{4}\cdot \frac{4}{3}(6x+12).\)
답을 드러내세요
We cannot combine the terms in parentheses, so we multiply the two fractions first.
\(\frac{3}{4}\cdot \frac{4}{3}(6x+12)\) Multiply; the product of reciprocals is 1. \(1(6x+12)\) Simplify by recognizing the multiplicative identity. \(6x+12\) -
Simplify: \(\frac{2}{5}\cdot \frac{5}{2}(20y+50).\)
답을 드러내세요
20y + 50
-
Simplify: \(\frac{3}{8}\cdot \frac{8}{3}(12z+16).\)
답을 드러내세요
12z + 16
-
\(101+0=101\)
-
\(\frac{3}{5}(1)=\frac{3}{5}\)
답을 드러내세요
identity property of multiplication
-
\(-9\cdot 1=-9\)
-
\(\frac{7}{12}\)
여기서 사용된 기호
기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.
어떻게: Properties of Identity, Inverses, and Zero
- 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
사람들이 묻는 질문
Why does the order of operations matter?
Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.
How do I check an arithmetic answer?
Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.
Why are fractions harder than decimals?
They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.
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