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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.

  1. ⓐ \(\ 7+0=7\)

  2. ⓑ \(\ -16(1)=-16\)

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.

  1. ⓐ The additive inverse of \(13\) is its opposite, \(-13.\)

  2. ⓑ The additive inverse of \(-\frac{5}{8}\) is its opposite, \(\frac{5}{8}.\)

  3. ⓒ 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.

  1. ⓐ The multiplicative inverse of \(9\) is its reciprocal, \(\frac{1}{9}.\)

  2. ⓑ The multiplicative inverse of \(-\frac{1}{9}\) is its reciprocal, \(-9.\)

  3. ⓒ 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 .

PropertyOf AdditionOf 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.

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Практика (40)

Сначала попробуйте решить каждую на бумаге. Откройте ответ, чтобы проверить себя; проверенные задачи можно открыть в решателе со всеми шагами.

  1. Find the opposite of \(-4.\)

    Показать ответ

    \(4\)

  2. Find the reciprocal of \(\frac{5}{2}.\)

    Показать ответ

    \(\frac{2}{5}\)

  3. Multiply: \(\frac{3a}{5}\cdot \frac{9}{2a}.\)

    Показать ответ

    \(\frac{27}{10}\)

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

    1. ⓐ \(\ 7+0=7\)

    2. ⓑ \(\ -16(1)=-16\)

    Показать ответ
    ⓐ
    \(7+0=7\)
    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.

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

    ⓐ \(\ 23+0=23\)ⓑ \(\ -37(1)=-37.\)

    Показать ответ

    1. ⓐ identity property of addition
    2. ⓑ identity property of multiplication

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

    ⓐ \(\ 1\cdot 29=29\)ⓑ \(\ 14+0=14.\)

    Показать ответ

    1. ⓐ identity property of multiplication
    2. ⓑ identity property of addition

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

    Показать ответ

    To find the additive inverse, we find the opposite.

    1. ⓐ The additive inverse of \(13\) is its opposite, \(-13.\)

    2. ⓑ The additive inverse of \(-\frac{5}{8}\) is its opposite, \(\frac{5}{8}.\)

    3. ⓒ The additive inverse of \(0.6\) is its opposite, \(-0.6.\)

  8. Find the additive inverse: ⓐ \(18\) ⓑ \(\frac{7}{9}\) ⓒ \(1.2\).

    Показать ответ

    1. ⓐ \(\ -18\)
    2. ⓑ \(\ -\frac{7}{9}\)
    3. ⓒ \(\ -1.2\)

  9. Find the additive inverse: ⓐ \(47\) ⓑ \(\frac{7}{13}\) ⓒ \(\ 8.4\).

    Показать ответ
    1. ⓐ \(\ -47\)
    2. ⓑ \(\ -\frac{7}{13}\)
    3. ⓒ \(\ -8.4\)

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

    Показать ответ

    To find the multiplicative inverse, we find the reciprocal.

    1. ⓐ The multiplicative inverse of \(9\) is its reciprocal, \(\frac{1}{9}.\)

    2. ⓑ The multiplicative inverse of \(-\frac{1}{9}\) is its reciprocal, \(-9.\)

    3. ⓒ To find the multiplicative inverse of \(0.9,\) we first convert \(0.9\) to a fraction, \(\frac{9}{10}.\) Then we find the reciprocal, \(\frac{10}{9}.\)

  11. Find the multiplicative inverse: ⓐ \(\ 5\) ⓑ \(\ -\frac{1}{7}\) ⓒ \(\ 0.3\).

    Показать ответ
    1. ⓐ \(\ \frac{1}{5}\)
    2. ⓑ \(\ -7\)
    3. ⓒ \(\ \frac{10}{3}\)

  12. Find the multiplicative inverse: ⓐ \(\ 18\) ⓑ \(\ -\frac{4}{5}\) ⓒ \(\ 0.6\).

    Показать ответ

    1. ⓐ \(\ \frac{1}{18}\)
    2. ⓑ \(\ -\frac{5}{4}\)
    3. ⓒ \(\ \frac{5}{3}\)

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

    Показать ответ
    ⓐ
    \(-8⋅0\)
    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.\(0\)

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

    Показать ответ

    1. ⓐ 0
    2. ⓑ 0
    3. ⓒ 0

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

    Показать ответ

    1. ⓐ 0
    2. ⓑ 0
    3. ⓒ 0

  16. Simplify: ⓐ \(\ 0\div 5\) ⓑ \(\frac{0}{-2}\) ⓒ \(\ 0\div \frac{7}{8}\).

    Показать ответ
    ⓐ
    \(0\div 5\)
    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.\(0\)

  17. Simplify: ⓐ \(\ 0\div 11\) ⓑ \(\ \frac{0}{-6}\) ⓒ \(\ 0\div \frac{3}{10}\).

    Показать ответ

    1. ⓐ 0
    2. ⓑ 0
    3. ⓒ 0

  18. Simplify: ⓐ \(\ 0\div \frac{8}{3}\) ⓑ \(\ 0\div (-10)\) ⓒ \(\ 0\div 12.75\).

    Показать ответ

    1. ⓐ 0
    2. ⓑ 0
    3. ⓒ 0

  19. Simplify: ⓐ \(\ 7.5\div 0\) ⓑ \(\ \frac{-32}{0}\) ⓒ \(\ \frac{4}{9}\div 0\).

    Показать ответ
    ⓐ
    \(7.5\div 0\)
    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

  20. Simplify: ⓐ \(\ 16.4\div 0\) ⓑ \(\ \frac{-2}{0}\) ⓒ \(\ \frac{1}{5}\div 0\).

    Показать ответ

    1. ⓐ undefined
    2. ⓑ undefined
    3. ⓒ undefined

  21. Simplify: ⓐ \(\ \frac{-5}{0}\) ⓑ \(\ 96.9\div 0\) ⓒ \(\ \frac{4}{15}\div 0\)

    Показать ответ

    1. ⓐ undefined
    2. ⓑ undefined
    3. ⓒ undefined

  22. Simplify: \(3x+15-3x.\)

    Показать ответ
    \(3x+15-3x\)
    Notice the additive inverses, \(3x\) and \(-3x\).\(0+15\)
    Add.\(15\)

  23. Simplify: \(-12z+9+12z.\)

    Показать ответ

    9

  24. Simplify: \(-25u-18+25u.\)

    Показать ответ

    −18

  25. Simplify: \(4(0.25q).\)

    Показать ответ
    \(4(0.25q)\)
    Regroup, using the associative property.\([4(0.25)]q\)
    Multiply.\(1.00q\)
    Simplify; 1 is the multiplicative identity.\(q\)

  26. Simplify: \(2(0.5p).\)

    Показать ответ

    p

  27. Simplify: \(25(0.04r).\)

    Показать ответ

    r

  28. Simplify: \(\frac{0}{n+5}\), where \(n\ne -5\).

    Показать ответ
    \(\frac{0}{n+5}\)
    Zero divided by any real number except itself is zero.\(0\)

  29. Simplify: \(\frac{0}{m+7}\), where \(m\ne -7\).

    Показать ответ

    0

  30. Simplify: \(\frac{0}{d-4}\), where \(d\ne 4\).

    Показать ответ

    0

  31. Simplify: \(\frac{10-3p}{0}.\)

    Показать ответ
    \(\frac{10-3p}{0}\)
    Division by zero is undefined.undefined

  32. Simplify: \(\frac{18-6c}{0}.\)

    Показать ответ

    undefined

  33. Simplify: \(\frac{15-4q}{0}.\)

    Показать ответ

    undefined

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

    Показать ответ

    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.\(6x+12\)

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

    Показать ответ

    20y + 50

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

    Показать ответ

    12z + 16

  37. \(101+0=101\)

  38. \(\frac{3}{5}(1)=\frac{3}{5}\)

    Показать ответ

    identity property of multiplication

  39. \(-9\cdot 1=-9\)

  40. \(\frac{7}{12}\)

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Как: Properties of Identity, Inverses, and Zero

  1. Recognize the identity properties of addition and multiplication
  2. Use the inverse properties of addition and multiplication
  3. Use the properties of zero
  4. Simplify expressions using the properties of identities, inverses, and zero

Вопросы, которые люди задают

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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