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Hướng dẫn học tập > Prealgebra

Using the Identity and Inverse Properties of Addition and Subtraction

Learning Outcomes

  • Identify the identity properties of multiplication and addition
  • Use the inverse property of addition and multiplication to simplify expressions

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 00 the additive identity. For example, 13+0−14+00+(−3x)13−14−3x\begin{array}{ccccc}\hfill 13+0\hfill & & \hfill -14+0\hfill & & \hfill 0+\left(-3x\right)\hfill \\ \hfill 13\hfill & & \hfill -14\hfill & & \hfill -3x\hfill \end{array} What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call 11 the multiplicative identity. For example, 43⋅1−27⋅11⋅6y543−276y5\begin{array}{ccccc}\hfill 43\cdot 1\hfill & & \hfill -27\cdot 1\hfill & & \hfill 1\cdot \frac{6y}{5}\hfill \\ \hfill 43\hfill & & \hfill -27\hfill & & \hfill \frac{6y}{5}\hfill \end{array}

Identity Properties

The identity property of addition: for any real number aa, a+0=a(0)+a=a0 is called the additive identity\begin{array}{}\\ \hfill a+0=a(0)+a=a\hfill \\ \hfill \text{0 is called the}\mathbf{\text{ additive identity}}\hfill \end{array} The identity property of multiplication: for any real number aa a⋅1=a(1)⋅a=a1 is called the multiplicative identity\begin{array}{c}\hfill a\cdot 1=a(1)\cdot a=a\hfill \\ \hfill \text{1 is called the}\mathbf{\text{ multiplicative identity}}\hfill \end{array}

example

Identify whether each equation demonstrates the identity property of addition or multiplication. 1. 7+0=77+0=7 2. −16(1)=−16-16\left(1\right)=-16 Solution:
1.
7+0=77+0=7
We are adding 0. We are using the identity property of addition.
2.
−16(1)=−16-16\left(1\right)=-16
We are multiplying by 1. We are using the identity property of multiplication.

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[ohm_question]146481[/ohm_question]

 Use the Inverse Properties of Addition and Multiplication

What number added to 5 gives the additive identity, 0?
5+=05 + =0 We know 5+(−−5)=05+(\color {red}{--5})=0
What number added to −6 gives the additive identity, 0?
−6+=0-6 + =0 We know −−6+6=0--6+\color {red}{6}=0
Notice that in each case, the missing number was the opposite of the number. We call −a-a the additive inverse of aa. The opposite of a number is its additive inverse. A number and its opposite add to 00, which is the additive identity. What number multiplied by 23\frac{2}{3} gives the multiplicative identity, 1?1? In other words, two-thirds times what results in 1?1?
23⋅=1\frac{2}{3}\cdot =1 We know 23⋅32=1\frac{2}{3}\cdot\color{red}{\frac{3}{2}}=1
What number multiplied by 22 gives the multiplicative identity, 1?1? In other words two times what results in 1?1?
2⋅=12\cdot =1 We know 2⋅12=12\cdot\color{red}{\frac{1}{2}}=1
Notice that in each case, the missing number was the reciprocal of the number. We call 1a\frac{1}{a} the multiplicative inverse of a(a≠0).a\left(a\ne 0\right)\text{.} The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to 11, which is the multiplicative identity. We’ll formally state the Inverse Properties here:

Inverse Properties

Inverse Property of Addition for any real number aa, a+(−a)=0−a is the additive inverse of a.\begin{array}{}\\ \hfill a+\left(-a\right)=0\hfill \\ \hfill -a\text{ is the}\mathbf{\text{ additive inverse }}\text{of }a.\hfill \end{array} Inverse Property of Multiplication for any real number a≠0a\ne 0, a⋅1a=11ais the multiplicative inverse of a.\begin{array}{}\\ \\ \hfill a\cdot \frac{1}{a}=1\hfill \\ \hfill \frac{1}{a}\text{is the}\mathbf{\text{ multiplicative inverse }}\text{of }a.\hfill \end{array}

example

Find the additive inverse of each expression: 1. 1313 2. −58-\frac{5}{8} 3. 0.60.6

Answer: Solution: To find the additive inverse, we find the opposite. 1. The additive inverse of 1313 is its opposite, −13-13. 2. The additive inverse of −58-\frac{5}{8} is its opposite, 58\frac{5}{8}. 3. The additive inverse of 0.60.6 is its opposite, −0.6-0.6.

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[ohm_question]146482[/ohm_question]

example

Find the multiplicative inverse: 1. 99 2. −19-\frac{1}{9} 3. 0.90.9

Answer: Solution: To find the multiplicative inverse, we find the reciprocal. 1. The multiplicative inverse of 99 is its reciprocal, 19\frac{1}{9}. 2. The multiplicative inverse of −19-\frac{1}{9} is its reciprocal, −9-9. 3. To find the multiplicative inverse of 0.90.9, we first convert 0.90.9 to a fraction, 910\frac{9}{10}. Then we find the reciprocal, 109\frac{10}{9}.

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[ohm_question]146483[/ohm_question] [ohm_question]146519[/ohm_question] [ohm_question]146520[/ohm_question]

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