A thorough explanation on why division by zero is undefined? [duplicate] Announcing the...

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A thorough explanation on why division by zero is undefined? [duplicate]



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30 pm US/Eastern)why division by zero is not possibleDivision by $0$Division by zeroInterpreting divisionLong division: 24158 divided 6Why is there no “remainder” in multiplicationMental ArithmeticIs it possible to follow another way to perform this calculation steps?How is division comparable to subtraction?Q: Quadratic Division - How to divide two quadratics?Real and Non-real Numbers; Value of Zero?I want to learn math from basics the Indian way and am looking for a book to guide me and some workbooks to practice. Any recommendations?What is a/b/c/d? That is: What is the correct order for multiple consecutive division operations?












3












$begingroup$



This question already has an answer here:




  • why division by zero is not possible [duplicate]

    3 answers




A Quick Note



I know there is a slough of related questions on stack exchange, but none of them really seem to answer my question. This post is the closest in relationship to my question, but the answer simply expresses a high level mathematical explanation, and not an example I can teach my kids. Growing up, my school always taught that division by zero was undefined or not allowed, but never really explained why, or how this was true.



The proposed duplicate has a very good answer, that I understand, however I'm not so sure my kids would understand that answer. The accepted answer will have to be understood by children under age 10 with a minimal working knowledge of multiplication and division.





Getting Started



The other day, I was working on a project at home in which I performed division by zero with a double precision floating point number in my code. This isn't always undefined in the computer world and can sometimes result in $infty$. The reason for this is clearly explained in IEEE 754 and quite thoroughly in this Stackoverflow post:




Division by zero (an operation on finite operands gives an exact infinite result, e.g., $frac{1}{0}$ or $log{0}$) (returns ±$infty$ by default).




Now, this got me thinking about basic arithmetic and how to prove each operation, and I created a mental inconsistency between multiplication and division.





Multiplication



As this is an important part of the thought process that lead me down this mental rabbit hole, I am including the elementary explanation of multiplication.




  • If I place $10$ marbles on my desk, $3$ times, I have placed $30$ marbles on my desk.


    • This is expressed as $10 cdot 3 = 30$ and is true.



  • If I place $10$ marbles on my desk, $0$ times, I have placed $0$ marbles on my desk.


    • This is expressed as $10 cdot 0 = 0$ and is true.




These two scenarios are true no matter what numbers are used.





Division



This is where things took an unexpected turn in my mind.



Let's say that I am a wandering saint and I have 50 apples. I want to help the hungry people of the world so I give my apples away freely. Now, let's handle two similar scenarios.




  • I come across $10$ people, and I want to give them all of my apples, I also want to ensure that each person receives the same number of apples. With $50$ apples to disperse across $10$ people, this means each person receives $5$ apples.


    • This is expressed as $frac{50}{10} = 5$ and is true.




However, let's say I have the same $50$ apples, and I come across a town where no one is hungry, and no one wants my apples. Well, I have $50$ apples, and I have $0$ people to give them to, so I still have $50$ apples. I didn't disperse my apples evenly across any number of people, so it's still the same bag of $50$ apples.



I believe this may be my mind's way of bending the facts here, and that I've convinced myself that I'm dividing $50$ zero times, but in fact I may have divided $50$ one time (by me). But it has me thinking, if I divide a pizza into zero equal slices, well then I essentially didn't slice the pizza and thus still just have an entire pizza.





My Question



How can it be proved thoroughly, not just with math, but with an example explanation (understandable by children) that division by zero is truly undefined?










share|cite|improve this question











$endgroup$



marked as duplicate by Dietrich Burde, Somos, José Carlos Santos, Javi, BlueRaja - Danny Pflughoeft 8 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 2




    $begingroup$
    The explanations at this duplicate (linked to the above) give a thorough explanation, right? It has so many examples in the answers, too. This problem really has been explained so many times, with so many good answers.
    $endgroup$
    – Dietrich Burde
    12 hours ago








  • 2




    $begingroup$
    In your apple scenario, you are actually dividing 50 apples by 1 (giving them all to yourself). Here are some thoughts I typed up, which may be helpful: ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf You may like section F best , which talks intuitively about a photocopy machine shrinking/enlarging and the concept of invertible/non-invertible operations.
    $endgroup$
    – Michael
    12 hours ago








  • 1




    $begingroup$
    You have tricked yourself with your apples/pizza. If you were divvying up the $50$ apples to $10$ people, then you would have no apples left over. Similarly, if you came across $25$ people, you'd have none left over (and each other person would have $2$). The whole scenario ignores you, and assumes that you have none left over. If you take a fair share instead, then you are dividing it by a number one greater. To model division by zero in that case, even you yourself would disappear, and so would the remaining apples. Where did they go?
    $endgroup$
    – Theo Bendit
    12 hours ago








  • 1




    $begingroup$
    @Somos : Based on reading her post, I believe she wants the answer of $1/0$ to be 1, and $50/0$ to be 50, based on her pizza scenario, but she also knows that is not quite right. PerpetualJ : Before getting intuition about dividing 50 pizzas by 0 (which is not defined) you might try intuition on dividing 50 pizzas by $1/2$ (which is defined).
    $endgroup$
    – Michael
    10 hours ago








  • 1




    $begingroup$
    Furthermore, the conditions you require for a satisfactory answer are, dare I say, impossibly high. Today, we seem to think little children can understand anything only if the right explanation is made to them, but that's so confused -- there are some things that are naturally beyond people of different intellectual capacities. Be sure that later on your child would understand why $1/0$ makes no sense, once they are ready for it. No need to hurry.
    $endgroup$
    – Allawonder
    9 hours ago
















3












$begingroup$



This question already has an answer here:




  • why division by zero is not possible [duplicate]

    3 answers




A Quick Note



I know there is a slough of related questions on stack exchange, but none of them really seem to answer my question. This post is the closest in relationship to my question, but the answer simply expresses a high level mathematical explanation, and not an example I can teach my kids. Growing up, my school always taught that division by zero was undefined or not allowed, but never really explained why, or how this was true.



The proposed duplicate has a very good answer, that I understand, however I'm not so sure my kids would understand that answer. The accepted answer will have to be understood by children under age 10 with a minimal working knowledge of multiplication and division.





Getting Started



The other day, I was working on a project at home in which I performed division by zero with a double precision floating point number in my code. This isn't always undefined in the computer world and can sometimes result in $infty$. The reason for this is clearly explained in IEEE 754 and quite thoroughly in this Stackoverflow post:




Division by zero (an operation on finite operands gives an exact infinite result, e.g., $frac{1}{0}$ or $log{0}$) (returns ±$infty$ by default).




Now, this got me thinking about basic arithmetic and how to prove each operation, and I created a mental inconsistency between multiplication and division.





Multiplication



As this is an important part of the thought process that lead me down this mental rabbit hole, I am including the elementary explanation of multiplication.




  • If I place $10$ marbles on my desk, $3$ times, I have placed $30$ marbles on my desk.


    • This is expressed as $10 cdot 3 = 30$ and is true.



  • If I place $10$ marbles on my desk, $0$ times, I have placed $0$ marbles on my desk.


    • This is expressed as $10 cdot 0 = 0$ and is true.




These two scenarios are true no matter what numbers are used.





Division



This is where things took an unexpected turn in my mind.



Let's say that I am a wandering saint and I have 50 apples. I want to help the hungry people of the world so I give my apples away freely. Now, let's handle two similar scenarios.




  • I come across $10$ people, and I want to give them all of my apples, I also want to ensure that each person receives the same number of apples. With $50$ apples to disperse across $10$ people, this means each person receives $5$ apples.


    • This is expressed as $frac{50}{10} = 5$ and is true.




However, let's say I have the same $50$ apples, and I come across a town where no one is hungry, and no one wants my apples. Well, I have $50$ apples, and I have $0$ people to give them to, so I still have $50$ apples. I didn't disperse my apples evenly across any number of people, so it's still the same bag of $50$ apples.



I believe this may be my mind's way of bending the facts here, and that I've convinced myself that I'm dividing $50$ zero times, but in fact I may have divided $50$ one time (by me). But it has me thinking, if I divide a pizza into zero equal slices, well then I essentially didn't slice the pizza and thus still just have an entire pizza.





My Question



How can it be proved thoroughly, not just with math, but with an example explanation (understandable by children) that division by zero is truly undefined?










share|cite|improve this question











$endgroup$



marked as duplicate by Dietrich Burde, Somos, José Carlos Santos, Javi, BlueRaja - Danny Pflughoeft 8 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 2




    $begingroup$
    The explanations at this duplicate (linked to the above) give a thorough explanation, right? It has so many examples in the answers, too. This problem really has been explained so many times, with so many good answers.
    $endgroup$
    – Dietrich Burde
    12 hours ago








  • 2




    $begingroup$
    In your apple scenario, you are actually dividing 50 apples by 1 (giving them all to yourself). Here are some thoughts I typed up, which may be helpful: ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf You may like section F best , which talks intuitively about a photocopy machine shrinking/enlarging and the concept of invertible/non-invertible operations.
    $endgroup$
    – Michael
    12 hours ago








  • 1




    $begingroup$
    You have tricked yourself with your apples/pizza. If you were divvying up the $50$ apples to $10$ people, then you would have no apples left over. Similarly, if you came across $25$ people, you'd have none left over (and each other person would have $2$). The whole scenario ignores you, and assumes that you have none left over. If you take a fair share instead, then you are dividing it by a number one greater. To model division by zero in that case, even you yourself would disappear, and so would the remaining apples. Where did they go?
    $endgroup$
    – Theo Bendit
    12 hours ago








  • 1




    $begingroup$
    @Somos : Based on reading her post, I believe she wants the answer of $1/0$ to be 1, and $50/0$ to be 50, based on her pizza scenario, but she also knows that is not quite right. PerpetualJ : Before getting intuition about dividing 50 pizzas by 0 (which is not defined) you might try intuition on dividing 50 pizzas by $1/2$ (which is defined).
    $endgroup$
    – Michael
    10 hours ago








  • 1




    $begingroup$
    Furthermore, the conditions you require for a satisfactory answer are, dare I say, impossibly high. Today, we seem to think little children can understand anything only if the right explanation is made to them, but that's so confused -- there are some things that are naturally beyond people of different intellectual capacities. Be sure that later on your child would understand why $1/0$ makes no sense, once they are ready for it. No need to hurry.
    $endgroup$
    – Allawonder
    9 hours ago














3












3








3


2



$begingroup$



This question already has an answer here:




  • why division by zero is not possible [duplicate]

    3 answers




A Quick Note



I know there is a slough of related questions on stack exchange, but none of them really seem to answer my question. This post is the closest in relationship to my question, but the answer simply expresses a high level mathematical explanation, and not an example I can teach my kids. Growing up, my school always taught that division by zero was undefined or not allowed, but never really explained why, or how this was true.



The proposed duplicate has a very good answer, that I understand, however I'm not so sure my kids would understand that answer. The accepted answer will have to be understood by children under age 10 with a minimal working knowledge of multiplication and division.





Getting Started



The other day, I was working on a project at home in which I performed division by zero with a double precision floating point number in my code. This isn't always undefined in the computer world and can sometimes result in $infty$. The reason for this is clearly explained in IEEE 754 and quite thoroughly in this Stackoverflow post:




Division by zero (an operation on finite operands gives an exact infinite result, e.g., $frac{1}{0}$ or $log{0}$) (returns ±$infty$ by default).




Now, this got me thinking about basic arithmetic and how to prove each operation, and I created a mental inconsistency between multiplication and division.





Multiplication



As this is an important part of the thought process that lead me down this mental rabbit hole, I am including the elementary explanation of multiplication.




  • If I place $10$ marbles on my desk, $3$ times, I have placed $30$ marbles on my desk.


    • This is expressed as $10 cdot 3 = 30$ and is true.



  • If I place $10$ marbles on my desk, $0$ times, I have placed $0$ marbles on my desk.


    • This is expressed as $10 cdot 0 = 0$ and is true.




These two scenarios are true no matter what numbers are used.





Division



This is where things took an unexpected turn in my mind.



Let's say that I am a wandering saint and I have 50 apples. I want to help the hungry people of the world so I give my apples away freely. Now, let's handle two similar scenarios.




  • I come across $10$ people, and I want to give them all of my apples, I also want to ensure that each person receives the same number of apples. With $50$ apples to disperse across $10$ people, this means each person receives $5$ apples.


    • This is expressed as $frac{50}{10} = 5$ and is true.




However, let's say I have the same $50$ apples, and I come across a town where no one is hungry, and no one wants my apples. Well, I have $50$ apples, and I have $0$ people to give them to, so I still have $50$ apples. I didn't disperse my apples evenly across any number of people, so it's still the same bag of $50$ apples.



I believe this may be my mind's way of bending the facts here, and that I've convinced myself that I'm dividing $50$ zero times, but in fact I may have divided $50$ one time (by me). But it has me thinking, if I divide a pizza into zero equal slices, well then I essentially didn't slice the pizza and thus still just have an entire pizza.





My Question



How can it be proved thoroughly, not just with math, but with an example explanation (understandable by children) that division by zero is truly undefined?










share|cite|improve this question











$endgroup$





This question already has an answer here:




  • why division by zero is not possible [duplicate]

    3 answers




A Quick Note



I know there is a slough of related questions on stack exchange, but none of them really seem to answer my question. This post is the closest in relationship to my question, but the answer simply expresses a high level mathematical explanation, and not an example I can teach my kids. Growing up, my school always taught that division by zero was undefined or not allowed, but never really explained why, or how this was true.



The proposed duplicate has a very good answer, that I understand, however I'm not so sure my kids would understand that answer. The accepted answer will have to be understood by children under age 10 with a minimal working knowledge of multiplication and division.





Getting Started



The other day, I was working on a project at home in which I performed division by zero with a double precision floating point number in my code. This isn't always undefined in the computer world and can sometimes result in $infty$. The reason for this is clearly explained in IEEE 754 and quite thoroughly in this Stackoverflow post:




Division by zero (an operation on finite operands gives an exact infinite result, e.g., $frac{1}{0}$ or $log{0}$) (returns ±$infty$ by default).




Now, this got me thinking about basic arithmetic and how to prove each operation, and I created a mental inconsistency between multiplication and division.





Multiplication



As this is an important part of the thought process that lead me down this mental rabbit hole, I am including the elementary explanation of multiplication.




  • If I place $10$ marbles on my desk, $3$ times, I have placed $30$ marbles on my desk.


    • This is expressed as $10 cdot 3 = 30$ and is true.



  • If I place $10$ marbles on my desk, $0$ times, I have placed $0$ marbles on my desk.


    • This is expressed as $10 cdot 0 = 0$ and is true.




These two scenarios are true no matter what numbers are used.





Division



This is where things took an unexpected turn in my mind.



Let's say that I am a wandering saint and I have 50 apples. I want to help the hungry people of the world so I give my apples away freely. Now, let's handle two similar scenarios.




  • I come across $10$ people, and I want to give them all of my apples, I also want to ensure that each person receives the same number of apples. With $50$ apples to disperse across $10$ people, this means each person receives $5$ apples.


    • This is expressed as $frac{50}{10} = 5$ and is true.




However, let's say I have the same $50$ apples, and I come across a town where no one is hungry, and no one wants my apples. Well, I have $50$ apples, and I have $0$ people to give them to, so I still have $50$ apples. I didn't disperse my apples evenly across any number of people, so it's still the same bag of $50$ apples.



I believe this may be my mind's way of bending the facts here, and that I've convinced myself that I'm dividing $50$ zero times, but in fact I may have divided $50$ one time (by me). But it has me thinking, if I divide a pizza into zero equal slices, well then I essentially didn't slice the pizza and thus still just have an entire pizza.





My Question



How can it be proved thoroughly, not just with math, but with an example explanation (understandable by children) that division by zero is truly undefined?





This question already has an answer here:




  • why division by zero is not possible [duplicate]

    3 answers








arithmetic






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 12 hours ago







PerpetualJ

















asked 12 hours ago









PerpetualJPerpetualJ

20417




20417




marked as duplicate by Dietrich Burde, Somos, José Carlos Santos, Javi, BlueRaja - Danny Pflughoeft 8 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Dietrich Burde, Somos, José Carlos Santos, Javi, BlueRaja - Danny Pflughoeft 8 hours ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 2




    $begingroup$
    The explanations at this duplicate (linked to the above) give a thorough explanation, right? It has so many examples in the answers, too. This problem really has been explained so many times, with so many good answers.
    $endgroup$
    – Dietrich Burde
    12 hours ago








  • 2




    $begingroup$
    In your apple scenario, you are actually dividing 50 apples by 1 (giving them all to yourself). Here are some thoughts I typed up, which may be helpful: ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf You may like section F best , which talks intuitively about a photocopy machine shrinking/enlarging and the concept of invertible/non-invertible operations.
    $endgroup$
    – Michael
    12 hours ago








  • 1




    $begingroup$
    You have tricked yourself with your apples/pizza. If you were divvying up the $50$ apples to $10$ people, then you would have no apples left over. Similarly, if you came across $25$ people, you'd have none left over (and each other person would have $2$). The whole scenario ignores you, and assumes that you have none left over. If you take a fair share instead, then you are dividing it by a number one greater. To model division by zero in that case, even you yourself would disappear, and so would the remaining apples. Where did they go?
    $endgroup$
    – Theo Bendit
    12 hours ago








  • 1




    $begingroup$
    @Somos : Based on reading her post, I believe she wants the answer of $1/0$ to be 1, and $50/0$ to be 50, based on her pizza scenario, but she also knows that is not quite right. PerpetualJ : Before getting intuition about dividing 50 pizzas by 0 (which is not defined) you might try intuition on dividing 50 pizzas by $1/2$ (which is defined).
    $endgroup$
    – Michael
    10 hours ago








  • 1




    $begingroup$
    Furthermore, the conditions you require for a satisfactory answer are, dare I say, impossibly high. Today, we seem to think little children can understand anything only if the right explanation is made to them, but that's so confused -- there are some things that are naturally beyond people of different intellectual capacities. Be sure that later on your child would understand why $1/0$ makes no sense, once they are ready for it. No need to hurry.
    $endgroup$
    – Allawonder
    9 hours ago














  • 2




    $begingroup$
    The explanations at this duplicate (linked to the above) give a thorough explanation, right? It has so many examples in the answers, too. This problem really has been explained so many times, with so many good answers.
    $endgroup$
    – Dietrich Burde
    12 hours ago








  • 2




    $begingroup$
    In your apple scenario, you are actually dividing 50 apples by 1 (giving them all to yourself). Here are some thoughts I typed up, which may be helpful: ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf You may like section F best , which talks intuitively about a photocopy machine shrinking/enlarging and the concept of invertible/non-invertible operations.
    $endgroup$
    – Michael
    12 hours ago








  • 1




    $begingroup$
    You have tricked yourself with your apples/pizza. If you were divvying up the $50$ apples to $10$ people, then you would have no apples left over. Similarly, if you came across $25$ people, you'd have none left over (and each other person would have $2$). The whole scenario ignores you, and assumes that you have none left over. If you take a fair share instead, then you are dividing it by a number one greater. To model division by zero in that case, even you yourself would disappear, and so would the remaining apples. Where did they go?
    $endgroup$
    – Theo Bendit
    12 hours ago








  • 1




    $begingroup$
    @Somos : Based on reading her post, I believe she wants the answer of $1/0$ to be 1, and $50/0$ to be 50, based on her pizza scenario, but she also knows that is not quite right. PerpetualJ : Before getting intuition about dividing 50 pizzas by 0 (which is not defined) you might try intuition on dividing 50 pizzas by $1/2$ (which is defined).
    $endgroup$
    – Michael
    10 hours ago








  • 1




    $begingroup$
    Furthermore, the conditions you require for a satisfactory answer are, dare I say, impossibly high. Today, we seem to think little children can understand anything only if the right explanation is made to them, but that's so confused -- there are some things that are naturally beyond people of different intellectual capacities. Be sure that later on your child would understand why $1/0$ makes no sense, once they are ready for it. No need to hurry.
    $endgroup$
    – Allawonder
    9 hours ago








2




2




$begingroup$
The explanations at this duplicate (linked to the above) give a thorough explanation, right? It has so many examples in the answers, too. This problem really has been explained so many times, with so many good answers.
$endgroup$
– Dietrich Burde
12 hours ago






$begingroup$
The explanations at this duplicate (linked to the above) give a thorough explanation, right? It has so many examples in the answers, too. This problem really has been explained so many times, with so many good answers.
$endgroup$
– Dietrich Burde
12 hours ago






2




2




$begingroup$
In your apple scenario, you are actually dividing 50 apples by 1 (giving them all to yourself). Here are some thoughts I typed up, which may be helpful: ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf You may like section F best , which talks intuitively about a photocopy machine shrinking/enlarging and the concept of invertible/non-invertible operations.
$endgroup$
– Michael
12 hours ago






$begingroup$
In your apple scenario, you are actually dividing 50 apples by 1 (giving them all to yourself). Here are some thoughts I typed up, which may be helpful: ee.usc.edu/stochastic-nets/docs/divide-by-zero.pdf You may like section F best , which talks intuitively about a photocopy machine shrinking/enlarging and the concept of invertible/non-invertible operations.
$endgroup$
– Michael
12 hours ago






1




1




$begingroup$
You have tricked yourself with your apples/pizza. If you were divvying up the $50$ apples to $10$ people, then you would have no apples left over. Similarly, if you came across $25$ people, you'd have none left over (and each other person would have $2$). The whole scenario ignores you, and assumes that you have none left over. If you take a fair share instead, then you are dividing it by a number one greater. To model division by zero in that case, even you yourself would disappear, and so would the remaining apples. Where did they go?
$endgroup$
– Theo Bendit
12 hours ago






$begingroup$
You have tricked yourself with your apples/pizza. If you were divvying up the $50$ apples to $10$ people, then you would have no apples left over. Similarly, if you came across $25$ people, you'd have none left over (and each other person would have $2$). The whole scenario ignores you, and assumes that you have none left over. If you take a fair share instead, then you are dividing it by a number one greater. To model division by zero in that case, even you yourself would disappear, and so would the remaining apples. Where did they go?
$endgroup$
– Theo Bendit
12 hours ago






1




1




$begingroup$
@Somos : Based on reading her post, I believe she wants the answer of $1/0$ to be 1, and $50/0$ to be 50, based on her pizza scenario, but she also knows that is not quite right. PerpetualJ : Before getting intuition about dividing 50 pizzas by 0 (which is not defined) you might try intuition on dividing 50 pizzas by $1/2$ (which is defined).
$endgroup$
– Michael
10 hours ago






$begingroup$
@Somos : Based on reading her post, I believe she wants the answer of $1/0$ to be 1, and $50/0$ to be 50, based on her pizza scenario, but she also knows that is not quite right. PerpetualJ : Before getting intuition about dividing 50 pizzas by 0 (which is not defined) you might try intuition on dividing 50 pizzas by $1/2$ (which is defined).
$endgroup$
– Michael
10 hours ago






1




1




$begingroup$
Furthermore, the conditions you require for a satisfactory answer are, dare I say, impossibly high. Today, we seem to think little children can understand anything only if the right explanation is made to them, but that's so confused -- there are some things that are naturally beyond people of different intellectual capacities. Be sure that later on your child would understand why $1/0$ makes no sense, once they are ready for it. No need to hurry.
$endgroup$
– Allawonder
9 hours ago




$begingroup$
Furthermore, the conditions you require for a satisfactory answer are, dare I say, impossibly high. Today, we seem to think little children can understand anything only if the right explanation is made to them, but that's so confused -- there are some things that are naturally beyond people of different intellectual capacities. Be sure that later on your child would understand why $1/0$ makes no sense, once they are ready for it. No need to hurry.
$endgroup$
– Allawonder
9 hours ago










6 Answers
6






active

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That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports?" If you have a definition of "division", then you can ask whether that definition can be applied to zero. For instance, if you define division such that $xdiv y$ means "Give the number $z$ such that $y cdot z =x$", there is no such number in the standard real number system for $y=0$. If we're required to have that $(xdiv y) cdot y=x$, then that doesn't work when $y$ is equal to zero. In computer languages where x/0 returns an object for which multiplication is defined, you do not have that (x)*0 == x. So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.



Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1
    $endgroup$
    – PerpetualJ
    9 hours ago










  • $begingroup$
    @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples.
    $endgroup$
    – D. Patrick
    8 hours ago





















5












$begingroup$

Consider a problem where you have to divide a finite number like $5$ by zero. $5div0$ is essentially a request for some number which when multiplied by zero gives $5$:



$$5div0=Nimplies 0cdot N=5.$$



Is there a number that when multiplied by zero gives you $5$? The answer is clearly no because any number times zero always gives you zero. Therefore, $5div0$ is left undefined. "Undefined" here basically means that we can't explain what $5div0$ really means.



What about the case $0div0$?
$$0div0=Nimplies 0cdot N=0.$$



We know that any number times zero is zero. This means $N$ can be any number. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0div0$ is also left undefined.





Here's another very simple example for good measure. You have $7$ pizzas and you want to divide them among zero people. How much pizza will each person get? Well, you have no people to give the pizzas to. You can pose that question and even write it mathematically as $7div0$, but what could possibly be the answer to this question? Practically speaking, this is unanswerable. In other words, it's not clear what the statement $7div0$ in this context means. In math-speak, we would say that this is undefined.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    +1 : I agree that being undefined is also quite undignified. =)
    $endgroup$
    – Michael
    10 hours ago





















3












$begingroup$

My understanding of division by zero goes back to the definition of rings.
Let $R$ be a commutative ring and $a,bin R$ with $b$ a unit in $R$.
Then define the fraction $a/b$ as follows:
$$frac{a}{b} = acdot b^{-1}$$
i.e., division by $b$ is defined by multiplication with the inverse of $b$.



Since the zero element $0$ in a ring is absorbing (i.e., $acdot 0 = 0 = 0cdot a$) and thus not a unit, division by $0$ is not defined.






share|cite|improve this answer









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  • 5




    $begingroup$
    The question clearly states that the explanation is targeted towards children.
    $endgroup$
    – user1952500
    12 hours ago






  • 3




    $begingroup$
    This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..."
    $endgroup$
    – Michael
    12 hours ago



















3












$begingroup$

Numberphile shows how if you define division by 0 as infinity (a specific kind of infinity) you get a 1=2 proof. Division, is basically a loop subtraction with a counter variable.(jives well with multiplication as repeated addition.) Division by 0 then, makes an infinite loop. And the limits have 2 values.






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    1












    $begingroup$

    I explain it like this, ignoring the definition gap on purpose:



    Have a look at the graph of $x/x$: it's a straight line at $y=1$. In this graph, we clearly see that $0/0=1$.



    Then, look at $5x/x$: it's a straight line at $y=5$. We clearly see that $5*0/0=5$. Now this could be interpreted as $(5*0)/0 = 0/0 = 1$ (using the result of the graph before) or as $5*(0/0)=5$ (also using the result from before).



    You can repeat this with other numbers as well, so the children can see that the result is arbitrary.



    This should make it pretty clear that if we allow division by zero, other laws cannot hold. So it's better undefined.






    share|cite|improve this answer









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      1












      $begingroup$

      I think you've pretty much got it and others have gone into more detail on the various bits of maths related to this.



      The simplest way to describe this for kids I can think of:




      1. Anything multiplied by zero is zero. Easy.

      2. Multiplying anything with two non-zero values gives a non-zero value. Shouldn't be a problem.

      3. Division of the result by one of those values gives you the other. We did the opposite of 2.


      So 3 X 4 = 12, 12 / 4 gives you 3. 3 was the number you multiplied by 4 to get 12.
      We have a result 12, we ask "12 / 0 what was the other value multiplied by 0 to get 12"?



      There is no such number because of statement 1 - hence divide by zero is undefined.






      share|cite|improve this answer








      New contributor




      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      $endgroup$













      • $begingroup$
        Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function.
        $endgroup$
        – cyborg
        8 hours ago


















      6 Answers
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      6 Answers
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      9












      $begingroup$

      That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports?" If you have a definition of "division", then you can ask whether that definition can be applied to zero. For instance, if you define division such that $xdiv y$ means "Give the number $z$ such that $y cdot z =x$", there is no such number in the standard real number system for $y=0$. If we're required to have that $(xdiv y) cdot y=x$, then that doesn't work when $y$ is equal to zero. In computer languages where x/0 returns an object for which multiplication is defined, you do not have that (x)*0 == x. So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.



      Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times.






      share|cite|improve this answer









      $endgroup$









      • 1




        $begingroup$
        The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1
        $endgroup$
        – PerpetualJ
        9 hours ago










      • $begingroup$
        @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples.
        $endgroup$
        – D. Patrick
        8 hours ago


















      9












      $begingroup$

      That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports?" If you have a definition of "division", then you can ask whether that definition can be applied to zero. For instance, if you define division such that $xdiv y$ means "Give the number $z$ such that $y cdot z =x$", there is no such number in the standard real number system for $y=0$. If we're required to have that $(xdiv y) cdot y=x$, then that doesn't work when $y$ is equal to zero. In computer languages where x/0 returns an object for which multiplication is defined, you do not have that (x)*0 == x. So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.



      Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times.






      share|cite|improve this answer









      $endgroup$









      • 1




        $begingroup$
        The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1
        $endgroup$
        – PerpetualJ
        9 hours ago










      • $begingroup$
        @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples.
        $endgroup$
        – D. Patrick
        8 hours ago
















      9












      9








      9





      $begingroup$

      That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports?" If you have a definition of "division", then you can ask whether that definition can be applied to zero. For instance, if you define division such that $xdiv y$ means "Give the number $z$ such that $y cdot z =x$", there is no such number in the standard real number system for $y=0$. If we're required to have that $(xdiv y) cdot y=x$, then that doesn't work when $y$ is equal to zero. In computer languages where x/0 returns an object for which multiplication is defined, you do not have that (x)*0 == x. So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.



      Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times.






      share|cite|improve this answer









      $endgroup$



      That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports?" If you have a definition of "division", then you can ask whether that definition can be applied to zero. For instance, if you define division such that $xdiv y$ means "Give the number $z$ such that $y cdot z =x$", there is no such number in the standard real number system for $y=0$. If we're required to have that $(xdiv y) cdot y=x$, then that doesn't work when $y$ is equal to zero. In computer languages where x/0 returns an object for which multiplication is defined, you do not have that (x)*0 == x. So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.



      Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered 9 hours ago









      AcccumulationAcccumulation

      7,4342619




      7,4342619








      • 1




        $begingroup$
        The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1
        $endgroup$
        – PerpetualJ
        9 hours ago










      • $begingroup$
        @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples.
        $endgroup$
        – D. Patrick
        8 hours ago
















      • 1




        $begingroup$
        The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1
        $endgroup$
        – PerpetualJ
        9 hours ago










      • $begingroup$
        @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples.
        $endgroup$
        – D. Patrick
        8 hours ago










      1




      1




      $begingroup$
      The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1
      $endgroup$
      – PerpetualJ
      9 hours ago




      $begingroup$
      The example following glorified subtraction as is given with glorified addition for multiplication was perfect! Being able to explain why division by zero is undefined requires breaking the process down into terms they can understand. I can see their counter argument coming back as “well, you still have 50 apples” though. +1
      $endgroup$
      – PerpetualJ
      9 hours ago












      $begingroup$
      @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples.
      $endgroup$
      – D. Patrick
      8 hours ago






      $begingroup$
      @PerpetualJ, I left a similar comment before I realized there was already an answer to this effect. :( In any case, I think you're hung up on the people and possession. If you give 10 people 50 apples, the apples still exist. Maybe think about baskets instead? "If I have 50 apples, I can put 1 apple into 10 baskets 5 times." I still have 50 apples; they're just in baskets now. "If I have 50 apples, I can put 1 apple into 0 baskets infinite times." Either way, I still have 50 apples.
      $endgroup$
      – D. Patrick
      8 hours ago













      5












      $begingroup$

      Consider a problem where you have to divide a finite number like $5$ by zero. $5div0$ is essentially a request for some number which when multiplied by zero gives $5$:



      $$5div0=Nimplies 0cdot N=5.$$



      Is there a number that when multiplied by zero gives you $5$? The answer is clearly no because any number times zero always gives you zero. Therefore, $5div0$ is left undefined. "Undefined" here basically means that we can't explain what $5div0$ really means.



      What about the case $0div0$?
      $$0div0=Nimplies 0cdot N=0.$$



      We know that any number times zero is zero. This means $N$ can be any number. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0div0$ is also left undefined.





      Here's another very simple example for good measure. You have $7$ pizzas and you want to divide them among zero people. How much pizza will each person get? Well, you have no people to give the pizzas to. You can pose that question and even write it mathematically as $7div0$, but what could possibly be the answer to this question? Practically speaking, this is unanswerable. In other words, it's not clear what the statement $7div0$ in this context means. In math-speak, we would say that this is undefined.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        +1 : I agree that being undefined is also quite undignified. =)
        $endgroup$
        – Michael
        10 hours ago


















      5












      $begingroup$

      Consider a problem where you have to divide a finite number like $5$ by zero. $5div0$ is essentially a request for some number which when multiplied by zero gives $5$:



      $$5div0=Nimplies 0cdot N=5.$$



      Is there a number that when multiplied by zero gives you $5$? The answer is clearly no because any number times zero always gives you zero. Therefore, $5div0$ is left undefined. "Undefined" here basically means that we can't explain what $5div0$ really means.



      What about the case $0div0$?
      $$0div0=Nimplies 0cdot N=0.$$



      We know that any number times zero is zero. This means $N$ can be any number. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0div0$ is also left undefined.





      Here's another very simple example for good measure. You have $7$ pizzas and you want to divide them among zero people. How much pizza will each person get? Well, you have no people to give the pizzas to. You can pose that question and even write it mathematically as $7div0$, but what could possibly be the answer to this question? Practically speaking, this is unanswerable. In other words, it's not clear what the statement $7div0$ in this context means. In math-speak, we would say that this is undefined.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        +1 : I agree that being undefined is also quite undignified. =)
        $endgroup$
        – Michael
        10 hours ago
















      5












      5








      5





      $begingroup$

      Consider a problem where you have to divide a finite number like $5$ by zero. $5div0$ is essentially a request for some number which when multiplied by zero gives $5$:



      $$5div0=Nimplies 0cdot N=5.$$



      Is there a number that when multiplied by zero gives you $5$? The answer is clearly no because any number times zero always gives you zero. Therefore, $5div0$ is left undefined. "Undefined" here basically means that we can't explain what $5div0$ really means.



      What about the case $0div0$?
      $$0div0=Nimplies 0cdot N=0.$$



      We know that any number times zero is zero. This means $N$ can be any number. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0div0$ is also left undefined.





      Here's another very simple example for good measure. You have $7$ pizzas and you want to divide them among zero people. How much pizza will each person get? Well, you have no people to give the pizzas to. You can pose that question and even write it mathematically as $7div0$, but what could possibly be the answer to this question? Practically speaking, this is unanswerable. In other words, it's not clear what the statement $7div0$ in this context means. In math-speak, we would say that this is undefined.






      share|cite|improve this answer











      $endgroup$



      Consider a problem where you have to divide a finite number like $5$ by zero. $5div0$ is essentially a request for some number which when multiplied by zero gives $5$:



      $$5div0=Nimplies 0cdot N=5.$$



      Is there a number that when multiplied by zero gives you $5$? The answer is clearly no because any number times zero always gives you zero. Therefore, $5div0$ is left undefined. "Undefined" here basically means that we can't explain what $5div0$ really means.



      What about the case $0div0$?
      $$0div0=Nimplies 0cdot N=0.$$



      We know that any number times zero is zero. This means $N$ can be any number. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0div0$ is also left undefined.





      Here's another very simple example for good measure. You have $7$ pizzas and you want to divide them among zero people. How much pizza will each person get? Well, you have no people to give the pizzas to. You can pose that question and even write it mathematically as $7div0$, but what could possibly be the answer to this question? Practically speaking, this is unanswerable. In other words, it's not clear what the statement $7div0$ in this context means. In math-speak, we would say that this is undefined.







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited 54 mins ago

























      answered 11 hours ago









      Michael RybkinMichael Rybkin

      4,664522




      4,664522












      • $begingroup$
        +1 : I agree that being undefined is also quite undignified. =)
        $endgroup$
        – Michael
        10 hours ago




















      • $begingroup$
        +1 : I agree that being undefined is also quite undignified. =)
        $endgroup$
        – Michael
        10 hours ago


















      $begingroup$
      +1 : I agree that being undefined is also quite undignified. =)
      $endgroup$
      – Michael
      10 hours ago






      $begingroup$
      +1 : I agree that being undefined is also quite undignified. =)
      $endgroup$
      – Michael
      10 hours ago













      3












      $begingroup$

      My understanding of division by zero goes back to the definition of rings.
      Let $R$ be a commutative ring and $a,bin R$ with $b$ a unit in $R$.
      Then define the fraction $a/b$ as follows:
      $$frac{a}{b} = acdot b^{-1}$$
      i.e., division by $b$ is defined by multiplication with the inverse of $b$.



      Since the zero element $0$ in a ring is absorbing (i.e., $acdot 0 = 0 = 0cdot a$) and thus not a unit, division by $0$ is not defined.






      share|cite|improve this answer









      $endgroup$









      • 5




        $begingroup$
        The question clearly states that the explanation is targeted towards children.
        $endgroup$
        – user1952500
        12 hours ago






      • 3




        $begingroup$
        This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..."
        $endgroup$
        – Michael
        12 hours ago
















      3












      $begingroup$

      My understanding of division by zero goes back to the definition of rings.
      Let $R$ be a commutative ring and $a,bin R$ with $b$ a unit in $R$.
      Then define the fraction $a/b$ as follows:
      $$frac{a}{b} = acdot b^{-1}$$
      i.e., division by $b$ is defined by multiplication with the inverse of $b$.



      Since the zero element $0$ in a ring is absorbing (i.e., $acdot 0 = 0 = 0cdot a$) and thus not a unit, division by $0$ is not defined.






      share|cite|improve this answer









      $endgroup$









      • 5




        $begingroup$
        The question clearly states that the explanation is targeted towards children.
        $endgroup$
        – user1952500
        12 hours ago






      • 3




        $begingroup$
        This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..."
        $endgroup$
        – Michael
        12 hours ago














      3












      3








      3





      $begingroup$

      My understanding of division by zero goes back to the definition of rings.
      Let $R$ be a commutative ring and $a,bin R$ with $b$ a unit in $R$.
      Then define the fraction $a/b$ as follows:
      $$frac{a}{b} = acdot b^{-1}$$
      i.e., division by $b$ is defined by multiplication with the inverse of $b$.



      Since the zero element $0$ in a ring is absorbing (i.e., $acdot 0 = 0 = 0cdot a$) and thus not a unit, division by $0$ is not defined.






      share|cite|improve this answer









      $endgroup$



      My understanding of division by zero goes back to the definition of rings.
      Let $R$ be a commutative ring and $a,bin R$ with $b$ a unit in $R$.
      Then define the fraction $a/b$ as follows:
      $$frac{a}{b} = acdot b^{-1}$$
      i.e., division by $b$ is defined by multiplication with the inverse of $b$.



      Since the zero element $0$ in a ring is absorbing (i.e., $acdot 0 = 0 = 0cdot a$) and thus not a unit, division by $0$ is not defined.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered 12 hours ago









      WuestenfuxWuestenfux

      5,6211513




      5,6211513








      • 5




        $begingroup$
        The question clearly states that the explanation is targeted towards children.
        $endgroup$
        – user1952500
        12 hours ago






      • 3




        $begingroup$
        This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..."
        $endgroup$
        – Michael
        12 hours ago














      • 5




        $begingroup$
        The question clearly states that the explanation is targeted towards children.
        $endgroup$
        – user1952500
        12 hours ago






      • 3




        $begingroup$
        This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..."
        $endgroup$
        – Michael
        12 hours ago








      5




      5




      $begingroup$
      The question clearly states that the explanation is targeted towards children.
      $endgroup$
      – user1952500
      12 hours ago




      $begingroup$
      The question clearly states that the explanation is targeted towards children.
      $endgroup$
      – user1952500
      12 hours ago




      3




      3




      $begingroup$
      This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..."
      $endgroup$
      – Michael
      12 hours ago




      $begingroup$
      This is the first time I have seen someone explain division by zero by starting out "Let $R$ be a commutative ring..."
      $endgroup$
      – Michael
      12 hours ago











      3












      $begingroup$

      Numberphile shows how if you define division by 0 as infinity (a specific kind of infinity) you get a 1=2 proof. Division, is basically a loop subtraction with a counter variable.(jives well with multiplication as repeated addition.) Division by 0 then, makes an infinite loop. And the limits have 2 values.






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        Numberphile shows how if you define division by 0 as infinity (a specific kind of infinity) you get a 1=2 proof. Division, is basically a loop subtraction with a counter variable.(jives well with multiplication as repeated addition.) Division by 0 then, makes an infinite loop. And the limits have 2 values.






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          Numberphile shows how if you define division by 0 as infinity (a specific kind of infinity) you get a 1=2 proof. Division, is basically a loop subtraction with a counter variable.(jives well with multiplication as repeated addition.) Division by 0 then, makes an infinite loop. And the limits have 2 values.






          share|cite|improve this answer









          $endgroup$



          Numberphile shows how if you define division by 0 as infinity (a specific kind of infinity) you get a 1=2 proof. Division, is basically a loop subtraction with a counter variable.(jives well with multiplication as repeated addition.) Division by 0 then, makes an infinite loop. And the limits have 2 values.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 12 hours ago









          Roddy MacPheeRoddy MacPhee

          1,049118




          1,049118























              1












              $begingroup$

              I explain it like this, ignoring the definition gap on purpose:



              Have a look at the graph of $x/x$: it's a straight line at $y=1$. In this graph, we clearly see that $0/0=1$.



              Then, look at $5x/x$: it's a straight line at $y=5$. We clearly see that $5*0/0=5$. Now this could be interpreted as $(5*0)/0 = 0/0 = 1$ (using the result of the graph before) or as $5*(0/0)=5$ (also using the result from before).



              You can repeat this with other numbers as well, so the children can see that the result is arbitrary.



              This should make it pretty clear that if we allow division by zero, other laws cannot hold. So it's better undefined.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                I explain it like this, ignoring the definition gap on purpose:



                Have a look at the graph of $x/x$: it's a straight line at $y=1$. In this graph, we clearly see that $0/0=1$.



                Then, look at $5x/x$: it's a straight line at $y=5$. We clearly see that $5*0/0=5$. Now this could be interpreted as $(5*0)/0 = 0/0 = 1$ (using the result of the graph before) or as $5*(0/0)=5$ (also using the result from before).



                You can repeat this with other numbers as well, so the children can see that the result is arbitrary.



                This should make it pretty clear that if we allow division by zero, other laws cannot hold. So it's better undefined.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  I explain it like this, ignoring the definition gap on purpose:



                  Have a look at the graph of $x/x$: it's a straight line at $y=1$. In this graph, we clearly see that $0/0=1$.



                  Then, look at $5x/x$: it's a straight line at $y=5$. We clearly see that $5*0/0=5$. Now this could be interpreted as $(5*0)/0 = 0/0 = 1$ (using the result of the graph before) or as $5*(0/0)=5$ (also using the result from before).



                  You can repeat this with other numbers as well, so the children can see that the result is arbitrary.



                  This should make it pretty clear that if we allow division by zero, other laws cannot hold. So it's better undefined.






                  share|cite|improve this answer









                  $endgroup$



                  I explain it like this, ignoring the definition gap on purpose:



                  Have a look at the graph of $x/x$: it's a straight line at $y=1$. In this graph, we clearly see that $0/0=1$.



                  Then, look at $5x/x$: it's a straight line at $y=5$. We clearly see that $5*0/0=5$. Now this could be interpreted as $(5*0)/0 = 0/0 = 1$ (using the result of the graph before) or as $5*(0/0)=5$ (also using the result from before).



                  You can repeat this with other numbers as well, so the children can see that the result is arbitrary.



                  This should make it pretty clear that if we allow division by zero, other laws cannot hold. So it's better undefined.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 10 hours ago









                  Thomas WellerThomas Weller

                  256212




                  256212























                      1












                      $begingroup$

                      I think you've pretty much got it and others have gone into more detail on the various bits of maths related to this.



                      The simplest way to describe this for kids I can think of:




                      1. Anything multiplied by zero is zero. Easy.

                      2. Multiplying anything with two non-zero values gives a non-zero value. Shouldn't be a problem.

                      3. Division of the result by one of those values gives you the other. We did the opposite of 2.


                      So 3 X 4 = 12, 12 / 4 gives you 3. 3 was the number you multiplied by 4 to get 12.
                      We have a result 12, we ask "12 / 0 what was the other value multiplied by 0 to get 12"?



                      There is no such number because of statement 1 - hence divide by zero is undefined.






                      share|cite|improve this answer








                      New contributor




                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.






                      $endgroup$













                      • $begingroup$
                        Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function.
                        $endgroup$
                        – cyborg
                        8 hours ago
















                      1












                      $begingroup$

                      I think you've pretty much got it and others have gone into more detail on the various bits of maths related to this.



                      The simplest way to describe this for kids I can think of:




                      1. Anything multiplied by zero is zero. Easy.

                      2. Multiplying anything with two non-zero values gives a non-zero value. Shouldn't be a problem.

                      3. Division of the result by one of those values gives you the other. We did the opposite of 2.


                      So 3 X 4 = 12, 12 / 4 gives you 3. 3 was the number you multiplied by 4 to get 12.
                      We have a result 12, we ask "12 / 0 what was the other value multiplied by 0 to get 12"?



                      There is no such number because of statement 1 - hence divide by zero is undefined.






                      share|cite|improve this answer








                      New contributor




                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.






                      $endgroup$













                      • $begingroup$
                        Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function.
                        $endgroup$
                        – cyborg
                        8 hours ago














                      1












                      1








                      1





                      $begingroup$

                      I think you've pretty much got it and others have gone into more detail on the various bits of maths related to this.



                      The simplest way to describe this for kids I can think of:




                      1. Anything multiplied by zero is zero. Easy.

                      2. Multiplying anything with two non-zero values gives a non-zero value. Shouldn't be a problem.

                      3. Division of the result by one of those values gives you the other. We did the opposite of 2.


                      So 3 X 4 = 12, 12 / 4 gives you 3. 3 was the number you multiplied by 4 to get 12.
                      We have a result 12, we ask "12 / 0 what was the other value multiplied by 0 to get 12"?



                      There is no such number because of statement 1 - hence divide by zero is undefined.






                      share|cite|improve this answer








                      New contributor




                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.






                      $endgroup$



                      I think you've pretty much got it and others have gone into more detail on the various bits of maths related to this.



                      The simplest way to describe this for kids I can think of:




                      1. Anything multiplied by zero is zero. Easy.

                      2. Multiplying anything with two non-zero values gives a non-zero value. Shouldn't be a problem.

                      3. Division of the result by one of those values gives you the other. We did the opposite of 2.


                      So 3 X 4 = 12, 12 / 4 gives you 3. 3 was the number you multiplied by 4 to get 12.
                      We have a result 12, we ask "12 / 0 what was the other value multiplied by 0 to get 12"?



                      There is no such number because of statement 1 - hence divide by zero is undefined.







                      share|cite|improve this answer








                      New contributor




                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.









                      share|cite|improve this answer



                      share|cite|improve this answer






                      New contributor




                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.









                      answered 9 hours ago









                      cyborgcyborg

                      1111




                      1111




                      New contributor




                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.





                      New contributor





                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.






                      cyborg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                      Check out our Code of Conduct.












                      • $begingroup$
                        Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function.
                        $endgroup$
                        – cyborg
                        8 hours ago


















                      • $begingroup$
                        Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function.
                        $endgroup$
                        – cyborg
                        8 hours ago
















                      $begingroup$
                      Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function.
                      $endgroup$
                      – cyborg
                      8 hours ago




                      $begingroup$
                      Addendum: the basic mistake in the question is that how you get the answer - adding, subtracting or anything else - the process or algorithm of it - does not define the result. As different algorithms give different answers that shows the point. From a purely functional perspective where there are no algorithms this is the most sensible mapping for the inverse of the multiplication function.
                      $endgroup$
                      – cyborg
                      8 hours ago



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