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Letting $Q$ represent a possible quotient, we get
$\frac{\text{any nonzero whole number}}{0}=Q$
Converting to the corresponding multiplication form, we have
$(\text{any nonzero whole number})=Q\times 0$
Since $Q\times 0=0$ , $(\text{any nonzero whole number})=0$ . But this is absurd. This would mean that $6=0$ , or $\text{37}=0$ . A nonzero whole number cannot equal 0! Thus,
$\frac{\text{any nonzero whole number}}{0}$ does not name a number
We are now curious about zero divided by zero $\left(\frac{0}{0}\right)$ . If we let $Q$ represent a potential quotient, we get
$\frac{0}{0}=Q$
Converting to the multiplication form,
$0=Q\times 0$
This results in
$0=0$
This is a statement that is true regardless of the number used in place of $Q$ . For example,
$\frac{0}{0}=5$ , since $0=5\times 0$ .
$\frac{0}{0}=\text{31}$ , since $0=\text{31}\times 0$ .
$\frac{0}{0}=\text{286}$ , since $0=\text{286}\times 0$ .
A unique quotient cannot be determined.
Perform, if possible, each division.
$\frac{\text{19}}{0}$ . Since division by 0 does not name a whole number, no quotient exists, and we state $\frac{\text{19}}{0}$ is undefined
$\begin{array}{c}\hfill 0\overline{)14}\end{array}$ . Since division by 0 does not name a defined number, no quotient exists, and we state $\begin{array}{c}\hfill 0\overline{)14}\end{array}$ is undefined
$\begin{array}{c}\hfill 9\overline{)0}\end{array}$ . Since division into 0 by any nonzero whole number results in 0, we have $\begin{array}{c}\hfill 0\\ \hfill 9\overline{)0}\end{array}$
$\frac{0}{7}$ . Since division into 0 by any nonzero whole number results in 0, we have $\frac{0}{7}=0$
Perform, if possible, the following divisions.
$\begin{array}{c}\hfill 0\overline{)0}\end{array}$
indeterminant
$\begin{array}{c}\hfill 0\overline{)8}\end{array}$
undefined
Divisions can also be performed using a calculator.
Divide 24 by 3.
Display Reads | ||
Type | 24 | 24 |
Press | ÷ | 24 |
Type | 3 | 3 |
Press | = | 8 |
The display now reads 8, and we conclude that $\text{24}\xf73=8$ .
Divide 0 by 7.
Display Reads | ||
Type | 0 | 0 |
Press | ÷ | 0 |
Type | 7 | 7 |
Press | = | 0 |
The display now reads 0, and we conclude that $0\xf77=0$ .
Divide 7 by 0.
Since division by zero is undefined, the calculator should register some kind of error message.
Display Reads | ||
Type | 7 | 7 |
Press | ÷ | 7 |
Type | 0 | 0 |
Press | = | Error |
The error message indicates an undefined operation was attempted, in this case, division by zero.
Use a calculator to perform each division.
$3\xf70$
An error message tells us that this operation is undefined. The particular message depends on the calculator.
$0\xf70$
An error message tells us that this operation cannot be performed. Some calculators actually set $0\xf70$ equal to 1. We know better! $0\xf70$ is indeterminant.
For the following problems, determine the quotients (if possible). You may use a calculator to check the result.
$\begin{array}{c}\hfill 7\overline{)42}\end{array}$
$\begin{array}{c}\hfill 2\overline{)14}\end{array}$
$\begin{array}{c}\hfill 1\overline{)6}\end{array}$
$\frac{\text{30}}{5}$
$\text{24}\xf78$
$\text{21}\xf77$
$\text{12}\xf74$
$\begin{array}{c}\hfill 0\overline{)0}\end{array}$
$\begin{array}{c}\hfill 6\overline{)48}\end{array}$
$\frac{\text{35}}{0}$
$\frac{0}{9}$
Write $\frac{\text{16}}{2}=8$ using three different notations.
Write $\frac{\text{27}}{9}=3$ using three different notations.
$\text{27}\xf79=3$ ; $\begin{array}{c}\hfill 9\overline{)27}\end{array}=3$ ; $\frac{\text{27}}{9}=3$
In the statement $\begin{array}{c}\hfill 4\\ \hfill 6\overline{)24}\end{array}$
6 is called the
24 is called the
4 is called the
In the statement $\text{56}\xf78=7$ ,
7 is called the
8 is called the
56 is called the
7 is quotient; 8 is divisor; 56 is dividend
( [link] ) What is the largest digit?
( [link] ) Find the sum. $\begin{array}{c}\hfill \mathrm{8,006}\\ \hfill \underline{+\mathrm{4,118}}\end{array}$
12,124
( [link] ) Find the difference. $\begin{array}{c}\hfill 631\\ \hfill \underline{-589}\end{array}$
( [link] ) Use the numbers 2, 3, and 7 to illustrate the associative property of addition.
$\begin{array}{}(2+3)+7=2+(3+7)=\text{12}\\ 5+7=2+\text{10}=\text{12}\end{array}$
( [link] ) Find the product. $\begin{array}{c}\hfill 86\\ \hfill \underline{\times 12}\end{array}$
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