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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 $a,a\ne 0,a\xb7\frac{1}{a}=1.$
We’ll formally state the inverse properties here:
Find the additive inverse of ⓐ $\frac{5}{8}$ ⓑ $0.6$ ⓒ $\mathrm{-8}$ ⓓ $-\phantom{\rule{0.2em}{0ex}}\frac{4}{3}.$
To find the additive inverse, we find the opposite.
Find the additive inverse of: ⓐ $\frac{7}{9}$ ⓑ $1.2$ ⓒ $\mathrm{-14}$ ⓓ $-\phantom{\rule{0.2em}{0ex}}\frac{9}{4}.$
ⓐ $-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}$ ⓑ $\mathrm{-1.2}$ ⓒ $14$ ⓓ $\frac{9}{4}$
Find the additive inverse of: ⓐ $\frac{7}{13}$ ⓑ $8.4$ ⓒ $\mathrm{-46}$ ⓓ $-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}.$
ⓐ $-\phantom{\rule{0.2em}{0ex}}\frac{7}{13}$ ⓑ $\mathrm{-8.4}$ ⓒ $46$ ⓓ $\frac{5}{2}$
Find the multiplicative inverse of ⓐ $9$ ⓑ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{9}$ ⓒ $0.9.$
To find the multiplicative inverse, we find the reciprocal.
Find the multiplicative inverse of ⓐ $4$ ⓑ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{7}$ ⓒ $0.3$
ⓐ $\frac{1}{4}$ ⓑ $\mathrm{-7}$ ⓒ $\frac{10}{3}$
Find the multiplicative inverse of ⓐ $18$ ⓑ $-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}$ ⓒ $0.6.$
ⓐ $\frac{1}{18}$ ⓑ $-\phantom{\rule{0.2em}{0ex}}\frac{5}{4}$ ⓒ $\frac{5}{3}$
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.
For any real number a .
The product of any real number and 0 is 0.
What about division involving zero? What is $0\xf73?$ 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,
We can check division with the related multiplication fact.
So we know $0\xf73=0$ because $0\xb73=0.$
For any real number a , except $0,$ $\frac{0}{a}=0$ and $0\xf7a=0.$
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: $4\xf70=?$ means $?\xb70=4.$ 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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