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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
7hours 36 min - 4hours 50 min
Tanis 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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