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Find the additive inverse: ⓐ $18$ ⓑ $\frac{7}{9}$ ⓒ $1.2$ .
Find the additive inverse: ⓐ $47$ ⓑ $\frac{7}{13}$ ⓒ $\phantom{\rule{0.2em}{0ex}}8.4$ .
Find the multiplicative inverse: ⓐ $9\phantom{\rule{0.2em}{0ex}}$ ⓑ $-\frac{1}{9}\phantom{\rule{0.2em}{0ex}}$ ⓒ $0.9$ .
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, $\mathrm{-9}.$
ⓒ 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: ⓐ $\phantom{\rule{0.2em}{0ex}}5\phantom{\rule{0.2em}{0ex}}$ ⓑ $\phantom{\rule{0.2em}{0ex}}-\frac{1}{7}\phantom{\rule{0.2em}{0ex}}$ ⓒ $\phantom{\rule{0.2em}{0ex}}0.3$ .
Find the multiplicative inverse: ⓐ $\phantom{\rule{0.2em}{0ex}}18$ ⓑ $\phantom{\rule{0.2em}{0ex}}-\frac{4}{5}$ ⓒ $\phantom{\rule{0.2em}{0ex}}0.6$ .
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 $0.$
For any real number $a,$
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}\mathrm{-8}\xb70$ ⓑ $\phantom{\rule{0.2em}{0ex}}\frac{5}{12}\xb70$ ⓒ $\phantom{\rule{0.2em}{0ex}}0(2.94)$ .
ⓐ | |
$\mathrm{-8}\cdot 0$ | |
The product of any real number and 0 is 0. | $0$ |
ⓑ | |
$\frac{5}{12}\xb70$ | |
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: ⓐ $\phantom{\rule{0.2em}{0ex}}\mathrm{-14}\xb70\phantom{\rule{0.2em}{0ex}}$ ⓑ $0\xb7\frac{2}{3}\phantom{\rule{0.2em}{0ex}}$ ⓒ $\phantom{\rule{0.2em}{0ex}}(16.5)\xb70.$
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}(1.95)\xb70$ ⓑ $\phantom{\rule{0.2em}{0ex}}0(\mathrm{-17})$ ⓒ $\phantom{\rule{0.2em}{0ex}}0\xb7\frac{5}{4}.$
What about dividing with $0?$ Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are $0$ cookies to share, so each person gets $0$ cookies.
Remember that we can always check division with the related multiplication fact. So, we know that
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.
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}0\xf75\phantom{\rule{0.2em}{0ex}}$ ⓑ $\frac{0}{\mathrm{-2}}$ ⓒ $\phantom{\rule{0.2em}{0ex}}0\xf7\frac{7}{8}$ .
ⓐ | |
$0\xf75$ | |
Zero divided by any real number, except 0, is zero. | $0$ |
ⓑ | |
$\frac{0}{\mathrm{-2}}$ | |
Zero divided by any real number, except 0, is zero. | $0$ |
ⓒ | |
$0\xf7\frac{7}{8}$ | |
Zero divided by any real number, except 0, is zero. | $0$ |
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}0\xf711$ ⓑ $\phantom{\rule{0.2em}{0ex}}\frac{0}{\mathrm{-6}}$ ⓒ $\phantom{\rule{0.2em}{0ex}}0\xf7\frac{3}{10}$ .
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}0\xf7\frac{8}{3}$ ⓑ $\phantom{\rule{0.2em}{0ex}}0\xf7(\mathrm{-10})$ ⓒ $\phantom{\rule{0.2em}{0ex}}0\xf712.75$ .
Now let’s think about dividing a number by zero. What is the result of dividing $4$ by $0?$ Think about the related multiplication fact. Is there a number that multiplied by $0$ gives $4?$
Since any real number multiplied by $0$ equals $0,$ there is no real number that can be multiplied by $0$ to obtain $4.$ We can conclude that there is no answer to $4\xf70,$ and so we say that division by zero is undefined.
For any real number $a,\phantom{\rule{0.2em}{0ex}}\frac{a}{0},$ and $a\xf70$ are undefined.
Division by zero is undefined.
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}7.5\xf70$ ⓑ $\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-32}}{0}$ ⓒ $\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\xf70$ .
ⓐ | |
$7.5\xf70$ | |
Division by zero is undefined. | undefined |
ⓑ | |
$\frac{\mathrm{-32}}{0}$ | |
Division by zero is undefined. | undefined |
ⓒ | |
$\frac{4}{9}\xf70$ | |
Division by zero is undefined. | undefined |
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}16.4\xf70$ ⓑ $\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-2}}{0}$ ⓒ $\phantom{\rule{0.2em}{0ex}}\frac{1}{5}\xf70$ .
Simplify: ⓐ $\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-5}}{0}$ ⓑ $\phantom{\rule{0.2em}{0ex}}96.9\xf70$ ⓒ $\phantom{\rule{0.2em}{0ex}}\frac{4}{15}\xf70$
We summarize the properties of zero.
Multiplication by Zero: For any real number $a,$
$\phantom{\rule{4em}{0ex}}\begin{array}{c}a\xb70=0\phantom{\rule{2em}{0ex}}0\xb7a=0\phantom{\rule{2em}{0ex}}\text{The product of any number and 0 is 0.}\hfill \end{array}$
Division by Zero: For any real number $a,\phantom{\rule{0.2em}{0ex}}a\ne 0$
$\phantom{\rule{4em}{0ex}}\begin{array}{c}\frac{0}{a}=0\phantom{\rule{0.5em}{0ex}}\text{Zero divided by any real number, except itself, is zero.}\hfill \end{array}$
$\phantom{\rule{4em}{0ex}}\begin{array}{c}\frac{a}{0}\phantom{\rule{0.2em}{0ex}}\text{is undefined. Division by zero is undefined.}\hfill \end{array}$
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