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Letting Q size 12{Q} {} represent a possible quotient, we get

any nonzero whole number 0 = Q size 12{ { {"any nonzero whole number"} over {0} } =Q} {}

Converting to the corresponding multiplication form, we have

( any nonzero whole number ) = Q × 0 size 12{ \( "any nonzero whole number" \) =Q times 0} {}

Since Q × 0 = 0 size 12{Q times 0=0} {} , ( any nonzero whole number ) = 0 size 12{ \( "any nonzero whole number" \) =0} {} . But this is absurd. This would mean that 6 = 0 size 12{6=0} {} , or 37 = 0 size 12{"37"=0} {} . A nonzero whole number cannot equal 0! Thus,

any nonzero whole number 0 size 12{ { {"any nonzero whole number"} over {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: 0 0 )

We are now curious about zero divided by zero 0 0 size 12{ left ( { {0} over {0} } right )} {} . If we let Q size 12{Q} {} represent a potential quotient, we get

0 0 = Q size 12{ { {0} over {0} } =Q} {}

Converting to the multiplication form,

0 = Q × 0 size 12{0=Q times 0} {}

This results in

0 = 0 size 12{0=0} {}

This is a statement that is true regardless of the number used in place of Q size 12{Q} {} . For example,

0 0 = 5 size 12{ { {0} over {0} } =5} {} , since 0 = 5 × 0 size 12{0=5 times 0} {} .

0 0 = 31 size 12{ { {0} over {0} } ="31"} {} , since 0 = 31 × 0 size 12{0="31" times 0} {} .

0 0 = 286 size 12{ { {0} over {0} } ="286"} {} , since 0 = 286 × 0 size 12{0="286" times 0} {} .

A unique quotient cannot be determined.

Indeterminant

Since the result of the division is inconclusive, we say that 0 0 size 12{ { {0} over {0} } } {} is indeterminant .

0 0 size 12{ { {0} over {0} } } {} Is indeterminant

The division 0 0 size 12{ { {0} over {0} } } {} is indeterminant.

Sample set b

Perform, if possible, each division.

19 0 size 12{ { {"19"} over {0} } } {} . Since division by 0 does not name a whole number, no quotient exists, and we state 19 0 size 12{ { {"19"} over {0} } } {} is undefined

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0 14 . Since division by 0 does not name a defined number, no quotient exists, and we state 0 14 is undefined

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9 0 . Since division into 0 by any nonzero whole number results in 0, we have 0 9 0

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0 7 size 12{ { {0} over {7} } } {} . Since division into 0 by any nonzero whole number results in 0, we have 0 7 = 0 size 12{ { {0} over {7} } =0} {}

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Practice set b

Perform, if possible, the following divisions.

5 0 size 12{ { {5} over {0} } } {}

undefined

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0 4 size 12{ { {0} over {4} } } {}

0

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9 0 size 12{ { {9} over {0} } } {}

undefined

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0 1 size 12{ { {0} over {1} } } {}

0

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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 24 ÷ 3 = 8 size 12{"24" div 3=8} {} .

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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 size 12{0 div 7=0} {} .

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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.

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Practice set c

Use a calculator to perform each division.

35 ÷ 7 size 12{"35" div 7} {}

5

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56 ÷ 8 size 12{"56" div 8} {}

7

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0 ÷ 6 size 12{0 div 6} {}

0

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3 ÷ 0 size 12{3 div 0} {}

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

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0 ÷ 0 size 12{0 div 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.

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Exercises

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

30 5 size 12{ { {"30"} over {5} } } {}

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16 4 size 12{ { {"16"} over {4} } } {}

4

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24 ÷ 8 size 12{"24" div 8} {}

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10 ÷ 2 size 12{"10" div 2} {}

5

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21 ÷ 7 size 12{"21" div 7} {}

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21 ÷ 3 size 12{"21" div 3} {}

7

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0 ÷ 6 size 12{0 div 6} {}

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8 ÷ 0 size 12{8 div 0} {}

not defined

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12 ÷ 4 size 12{"12" div 4} {}

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15 3 size 12{ { {"15"} over {3} } } {}

5

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35 0 size 12{ { {"35"} over {0} } } {}

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56 ÷ 7 size 12{"56" div 7} {}

8

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0 9 size 12{ { {0} over {9} } } {}

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72 ÷ 8 size 12{"72" div 8} {}

9

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Write 16 2 = 8 size 12{ { {"16"} over {2} } =8} {} using three different notations.

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Write 27 9 = 3 size 12{ { {"27"} over {9} } =3} {} using three different notations.

27 ÷ 9 = 3 size 12{"27" div 9=3} {} ; 9 27 = 3 ; 27 9 = 3 size 12{ { {"27"} over {9} } =3} {}

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In the statement 4 6 24

6 is called the .

24 is called the .

4 is called the .

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In the statement 56 ÷ 8 = 7 size 12{"56" div 8=7} {} ,

7 is called the .

8 is called the .

56 is called the .

7 is quotient; 8 is divisor; 56 is dividend

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Exercises for review

( [link] ) What is the largest digit?

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( [link] ) Find the sum. 8,006 + 4,118 ̲

12,124

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( [link] ) Find the difference. 631 - 589 ̲

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( [link] ) Use the numbers 2, 3, and 7 to illustrate the associative property of addition.

( 2 + 3 ) + 7 = 2 + ( 3 + 7 ) = 12 5 + 7 = 2 + 10 = 12 alignl { stack { size 12{ \( 2+3 \) +7=2+ \( 3+7 \) ="12"} {} #size 12{5+7=2+"10"="12"} {} } } {}

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( [link] ) Find the product. 86 × 12 ̲

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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