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in arabic sites which is interested in maths , i find many topics like ,here is a proof that 0=2 .

and we answer that the proof is wrong as we can't divide by zero .

but i really wonder , why can't we divide by zero ?

i think the reason that mathematicians refused dividing by zero is that make us into a contradictions like $1=2=3=...$ and things like this , but those are facts about the physical world , why should mathematicians obey the outside world ?

i also have read a news in BBC about a new theorem which find special cases where we can divide by zero , but not details was mentioned i think , have any one had any idea about this ?

Andrés E. Caicedo
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FNH
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    In math you are free to invent a system where division by $0$ is allowed, but it wouldn't be very interesting since all numbers would be equal. – Cheerful Parsnip Jun 24 '13 at 01:30
  • @GrumpyParsnip , yes , this is the same which i thought , this will make $1=2=3=...$. – FNH Jun 24 '13 at 01:35
  • You lost me at "why should mathematicians obey the outside world?". Can you explain this statement a bit more thoroughly? Because without context it's rather perplexing... – rurouniwallace Jun 24 '13 at 01:39
  • @ZettaSuro , i mean that people who say that we refuse dividing by zero because if this is the case then 0=2 but we not this is not true. so i would ask , why should we agree with the outside- physical-world ? i think that amWhy's answer answer this question . – FNH Jun 24 '13 at 01:50
  • In your mind, what would be $5/0$? – Ink Jun 24 '13 at 01:57
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    @Ink , the answer is , there is no thing in my mind = $5/0$ , also in my mind there is nothing equal to the square root of $-1$ but we use the $i$ in every day mathematics ! . – FNH Jun 24 '13 at 02:02
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    @ZettaSuro: This may or may not be what Maths Lover had in mind, but I'd like to add that mathematics is an abstraction. In math, we can create concepts and structures at will -- not because we necessarily want to use these concepts to describe the real world, but just because we want to. – Jesse Madnick Jun 24 '13 at 02:04

3 Answers3

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By definition, for any $x,y$ in a field, $\frac{x}{y}$ is the unique field element $z$ (if it exists) such that $zy=x$. If $y=0$ and $x\ne 0,$ then no such $z$ exists. If $x=0$ and $y=0,$ then we don't have uniqueness.

That's why we don't/can't define division by $0$.

Now, if we expand the definition of a field to allow $0=1$, then it is proved easily that $0$ is the only element of the field--making such a field a supremely uninteresting structure.

Cameron Buie
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    if $x=0$ and $y=0$, then we don't have uniqueness unless $0$ is the only element in the field, in which division by $0$ makes perfect sense. Never forget the trivial cases. Note that in such a field $1=0$, and the definitions of the 'integers' $n \overset{def}= 1 + 1 + \dots + 1$ (n times) also make sense because they give $n = 0$. – Patrick Da Silva Jun 24 '13 at 01:49
  • True, but I was operating under the "$0\ne 1$" field convention. – Cameron Buie Jun 24 '13 at 01:52
  • @PatrickDaSilva I thought most people didn't take ${0}$ to be a field. – Ink Jun 24 '13 at 02:13
  • @Ink : Most people don't, but it's not non-sense to think of it as a field. But it is certainly a ring in which the equation $zx = y$ always have a solution $z$ for given $x$ and $y$, thus you could call $0 = 0/0$ in some sense. – Patrick Da Silva Jun 24 '13 at 05:37
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You are correct that division by zero results in statements such as $1 = 2 = 3 =\cdots$, but that is not just a statement about facts in the real world (or "physical world"): mathematics as we know it would fall apart if this is allowed. What, after all, can be said, mathematically, if we have that $1 = 2 = 3 = \cdots$?

Just consider the implications, as they are vast, if we division by zero defined, and hence "meaningful"...

How would you define division if you allow division by zero? See, e.g., this answer regarding division by zero, with respect to how we define division, and division as we know it would fail if we were permit division by zero.

Indeed, how would we define $0^{-1}$ so that $0\cdot 0^{-1} = 1$?

These questions are simple prompts to suggest that to allow/define division by zero would entail having to redefine all axioms of arithmetic, and the field axioms, and much more. I.e., any successful redefining and reconstruction of a consistent system which is also consistent with allowing division by zero would yield, as first suggested in the comments, a system which would be exceedingly uninteresting, even perhaps meaningless.

Community
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amWhy
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  • so the hall of the problem is that this will not interesting but it's not that we can't ? . i think i agree with that . – FNH Jun 24 '13 at 01:49
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    Yes...it could be done, but we wouldn't recognize the same mathematics and axiomatic systems as we know it, "all things being equal": very uninteresting indeed! – amWhy Jun 24 '13 at 01:50
  • Is my question is so stupid to this degree to get -5 ?! haha ! in arabic sites , such questions is very popular ! – FNH Jun 24 '13 at 02:03
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    I don't think so (the -5)...many here don't know your sincerity. I know you well enough to take you seriously. There are popular sites in English that attract such questions, but one has to learn to filter out or gauge which sites have merit and authority, and which sites are for "popular consumption"/entertainment only. Many people think about math as being just a collection of rules, and think breaking one rule won't matter (i.e., defining division by zero). But in truth, most of math "hangs together" by axioms and chains of implication", so altering one axiom has a domino effect... – amWhy Jun 24 '13 at 02:08
  • yes , this is true ,we have to make this distinguish , i remember that one time , there was a long conversation between me and someone who see that mathematics is like Magic ! we shouldn't trust it ! and his argument was that mathematically 0=2 and this is wrong , i tried to make him understand what does it mean to have axiomatic system and how does modern maths work? but he was not convinced , so i said , i have to make sure that my understanding is right . Regards. – FNH Jun 24 '13 at 02:14
  • @BabakS. , there is a site called , Google ejabat which is created by google and is similar to yahoo answers " Q&A site " there is a section of the site for mathematics , such questions is asked many times , also , I've seen such questions in a sites called , الرياضيات رمز ، الرياضيات العربيه which means Arabian Mathematics , Mathematics symbol . also those question always asked on facebook group in arabic which is concerned with mathematics and physics :) . – FNH Jul 12 '13 at 10:03
  • @BabakS. , Which new question ? I didn't understand your point . may you clarify ? :) – FNH Jul 12 '13 at 10:17
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    @MathsLover: I mean don't you have any questions here for challenging? That's it. :) – Mikasa Jul 12 '13 at 10:18
  • @BabakS. , I have many good question which will be asked in next few days :) the last two weeks , I couldn't study so there was no questions :) – FNH Jul 12 '13 at 11:12
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Please note that as you imply, mathematics in general is actually quite a free field in which where able to invent systems in which we can do pretty much anything we want so you could come up with a system in which you can divide by zero. However, usually these systems all have their own axioms and generally for good reason and the reason we can't normally divide by zero is, in a sense because of the axioms we use for normal arithmetic.

Really, division is just the opposite of multiplication. When I divide 6 by 2, I'm really just looking for the number that I multiply by 2 to get 6 (which is 3). However, say I wanted to divide 6 by 0. What number can I multiply by 0 to get 6???

I'd also like to take you back to how you would have first learnt about division (presumably) which is "how many lots of one number can I get into another?". When I ask what is 6 divided by 2, I'm asking how many 2's go into 6 and since $6=2+2+2$, the answer is 3. However, how many 0's go into 6? Even if I keep adding 0 to itself infinitely many times, I'm never going to get to 6, so the answer is undefined.

In short, yes we are free to make new systems with new rules that allow us to divide by zero, but in our current system it is just not sensible.

john
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