1.9 Properties of real numbers (Page 3/10)

Elementary algebra Page 3 / 10

We have the statement that 2 times a blank space equals 1. Then it is stated that “We know 2 times 1/2 equals 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 . The reciprocal of $a$ number is its multiplicative inverse. A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number $a, a \neq 0, a \cdot \frac{1}{a} = 1 .$

We’ll formally state the inverse properties here:

Inverse property

\begin{matrix} of addition & For any real number a, & a + (− a) = 0 \\ − a . is the additive inverse of a . \\ A number and its opposite add to zero. \\ 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 . \\ A number and its reciprocal multiply to one. \end{matrix}

Find the additive inverse of ⓐ $\frac{5}{8}$ ⓑ $0.6$ ⓒ $−8$ ⓓ $- \frac{4}{3} .$

Solution

To find the additive inverse, we find the opposite.

ⓐ The additive inverse of $\frac{5}{8}$ is the opposite of $\frac{5}{8} .$ The additive inverse of $\frac{5}{8}$ is $- \frac{5}{8} .$
ⓑ The additive inverse of 0.6 is the opposite of 0.6. The additive inverse of 0.6 is $−0.6 .$
ⓒ The additive inverse of $−8$ is the opposite of $−8 .$ We write the opposite of $−8$ as $− (−8),$ and then simplify it to 8. Therefore, the additive inverse of $−8$ is 8.
ⓓ The additive inverse of $- \frac{4}{3}$ is the opposite of $- \frac{4}{3} .$ We write this as $− (- \frac{4}{3}),$ and then simplify to $\frac{4}{3} .$ Thus, the additive inverse of $- \frac{4}{3}$ is $\frac{4}{3} .$

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Find the additive inverse of: ⓐ $\frac{7}{9}$ ⓑ $1.2$ ⓒ $−14$ ⓓ $- \frac{9}{4} .$

ⓐ $- \frac{7}{9}$ ⓑ $−1.2$ ⓒ $14$ ⓓ $\frac{9}{4}$

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Find the additive inverse of: ⓐ $\frac{7}{13}$ ⓑ $8.4$ ⓒ $−46$ ⓓ $- \frac{5}{2} .$

ⓐ $- \frac{7}{13}$ ⓑ $−8.4$ ⓒ $46$ ⓓ $\frac{5}{2}$

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

Solution

To find the multiplicative inverse, we find the reciprocal.

ⓐ The multiplicative inverse of 9 is the reciprocal of 9, which is $\frac{1}{9} .$ Therefore, the multiplicative inverse of 9 is $\frac{1}{9} .$
ⓑ The multiplicative inverse of $- \frac{1}{9}$ is the reciprocal of $- \frac{1}{9},$ which is $−9 .$ Thus, the multiplicative inverse of $- \frac{1}{9}$ is $−9 .$
ⓒ To find the multiplicative inverse of 0.9, we first convert 0.9 to a fraction, $\frac{9}{10} .$ Then we find the reciprocal of the fraction. The reciprocal of $\frac{9}{10}$ is $\frac{10}{9} .$ So the multiplicative inverse of 0.9 is $\frac{10}{9} .$

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Find the multiplicative inverse of ⓐ $4$ ⓑ $- \frac{1}{7}$ ⓒ $0.3$

ⓐ $\frac{1}{4}$ ⓑ $−7$ ⓒ $\frac{10}{3}$

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

ⓐ $\frac{1}{18}$ ⓑ $- \frac{5}{4}$ ⓒ $\frac{5}{3}$

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Use the properties of zero

The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.

Multiplication by zero

For any real number a .

a \cdot 0 = 0 0 \cdot a = 0

The product of any real number and 0 is 0.

What about division involving zero? What is $0 \div 3 ?$ Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,

0 \div 3 = 0

We can check division with the related multiplication fact.

12 \div 6 = 2 because 2 \cdot 6 = 12 .

So we know $0 \div 3 = 0$ because $0 \cdot 3 = 0 .$

Division of zero

For any real number a , except $0,$ $\frac{0}{a} = 0$ and $0 \div a = 0 .$

Zero divided by any real number except zero is zero.

Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact: $4 \div 0 = ?$ means $? \cdot 0 = 4 .$ Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4.

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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