# 2.2 Concepts of division of whole numbers  (Page 2/2)

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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

$\left(\text{any nonzero whole number}\right)=Q×0$

Since $Q×0=0$ , $\left(\text{any nonzero whole number}\right)=0$ . 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

## Division by zero is undefined

Division by zero does not name a number. It is, therefore, undefined.

## Division by and into zero (zero as a dividend and divisor: $\frac{0}{0}$ )

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,

$0=Q×0$

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 $0=5×0$ .

$\frac{0}{0}=\text{31}$ , since $0=\text{31}×0$ .

$\frac{0}{0}=\text{286}$ , since $0=\text{286}×0$ .

A unique quotient cannot be determined.

## Indeterminant

Since the result of the division is inconclusive, we say that $\frac{0}{0}$ is indeterminant .

## $\frac{0}{0}$ Is indeterminant

The division $\frac{0}{0}$ is indeterminant.

## Sample set b

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 $\frac{0}{7}=0$

## Practice set b

Perform, if possible, the following divisions.

$\frac{5}{0}$

undefined

$\frac{0}{4}$

0

$\begin{array}{c}\hfill 0\overline{)0}\end{array}$

indeterminant

$\begin{array}{c}\hfill 0\overline{)8}\end{array}$

undefined

$\frac{9}{0}$

undefined

$\frac{0}{1}$

0

## Calculators

Divisions can also be performed using a calculator.

## Sample set c

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 $\text{24}÷3=8$ .

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 $0÷7=0$ .

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.

## Practice set c

Use a calculator to perform each division.

$\text{35}÷7$

5

$\text{56}÷8$

7

$0÷6$

0

$3÷0$

An error message tells us that this operation is undefined. The particular message depends on the calculator.

$0÷0$

An error message tells us that this operation cannot be performed. Some calculators actually set $0÷0$ equal to 1. We know better! $0÷0$ is indeterminant.

## Exercises

For the following problems, determine the quotients (if possi­ble). You may use a calculator to check the result.

$\begin{array}{c}\hfill 4\overline{)32}\end{array}$

8

$\begin{array}{c}\hfill 7\overline{)42}\end{array}$

$\begin{array}{c}\hfill 6\overline{)18}\end{array}$

3

$\begin{array}{c}\hfill 2\overline{)14}\end{array}$

$\begin{array}{c}\hfill 3\overline{)27}\end{array}$

9

$\begin{array}{c}\hfill 1\overline{)6}\end{array}$

$\begin{array}{c}\hfill 4\overline{)28}\end{array}$

7

$\frac{\text{30}}{5}$

$\frac{\text{16}}{4}$

4

$\text{24}÷8$

$\text{10}÷2$

5

$\text{21}÷7$

$\text{21}÷3$

7

$0÷6$

$8÷0$

not defined

$\text{12}÷4$

$\begin{array}{c}\hfill 3\overline{)9}\end{array}$

3

$\begin{array}{c}\hfill 0\overline{)0}\end{array}$

$\begin{array}{c}\hfill 7\overline{)0}\end{array}$

0

$\begin{array}{c}\hfill 6\overline{)48}\end{array}$

$\frac{\text{15}}{3}$

5

$\frac{\text{35}}{0}$

$\text{56}÷7$

8

$\frac{0}{9}$

$\text{72}÷8$

9

Write $\frac{\text{16}}{2}=8$ using three different notations.

Write $\frac{\text{27}}{9}=3$ using three different notations.

$\text{27}÷9=3$ ; $\begin{array}{c}\hfill 9\overline{)27}\end{array}=3$ ; $\frac{\text{27}}{9}=3$

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 $\text{56}÷8=7$ ,

7 is called the .

8 is called the .

56 is called the .

7 is quotient; 8 is divisor; 56 is dividend

## Exercises for review

( [link] ) What is the largest digit?

( [link] ) Find the sum. $\begin{array}{c}\hfill 8,006\\ \hfill \underline{+4,118}\end{array}$

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.

$\begin{array}{}\left(2+3\right)+7=2+\left(3+7\right)=\text{12}\\ 5+7=2+\text{10}=\text{12}\end{array}$

( [link] ) Find the product. $\begin{array}{c}\hfill 86\\ \hfill \underline{×12}\end{array}$

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