# 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. $\begin{array}{c}\hfill 86\\ \hfill \underline{×12}\end{array}$

#### Questions & Answers

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Rafiq
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Damian
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LITNING Reply
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LITNING
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Bob
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brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
7hours 36 min - 4hours 50 min
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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