Prove that $0/0 \ne 0$

Question

I asked this earlier today and received lots of confusion and misunderstanding. Here are some clarifications:

I am not asking for intuitive reasoning that division by zero is impossible or nonsensical. I am asking for a mathematical proof that $0/0$ is specifically not equal to zero. Also, the argument that $a/b$ is the unique solution to $a=bc$ does not apply in this scenario because $b=0$ therefore $b/b \ne 1 $*$ $ and you can't move it to the other side of the equation by multiplying the equation by $b$.

Finally, please don't mark this as a duplicate of my previous question. Said question was marked as a duplicate of something that didn't answer the question. Nobody responded to the question post with an answer to the question.

Sorry for previous confusion. Once again, please prove that $0/0 \ne 0$. And once again, please try to avoid an exclusively intuitive proof.

*There exist multiple proofs for this, and one (of which I know) for the more broad statement: $0/0 \ne c \ :\ c/c=1$. In order to avoid confusion, I won't attempt to transcribe said proof for the broad statement.

Answer 1

I'll try to sum up some of the comments and answers here.

Your question is either :

Prove that $\frac 00 \neq 0$.

or

Prove that $\frac 00$ can not be defined as the number $0$.

First, you must understand that those two questions are not the same question at all.

• In the first, $\frac 00$ seems to be defined as some number, and you ask why this number is not $0$. This would be the same idea as asking :

  Prove that $\sqrt{2} \neq 1.41$

  This is something we can prove, because we know the definition of $\sqrt{2}$, and thus it's fairly easy to show it's not $1.41$.

  Unfortunately here, there is no definition of $\frac 00$, so this question can not be answered. To simplify a bit, your question is exactly the same as asking :

  Prove that $\bowtie \neq 0$

  Well nobody can, because nobody knows what $\bowtie$ is. You must provide a definition of $\bowtie$ so that the question can be answered. Similarly, you must provide a definition of $\frac 00$ so that we can answer your question... which leads to the second question.

• The second question is a bit more interesting. Indeed, anybody can come up and say :

  I proclaim that from now one, $\frac 00 = 0$

  and similarly anybody can come up and say that $\frac 00 = 1$, or $\frac 00 = e$, or even $\frac 00 = \texttt{a hot banana}$.

  Indeed, since the symbol $\frac 00$ has no definition, you can pretty much decide to use it to represent what you like. So in a sense, if you want to, you can use $\frac 00$ as another way to write $0$, as much as I can use $\texttt{a hot banana}$ as my way to say $0$.

  Now why shouldn't you do that ? Well because writing $\frac 00$ implies the concept of dividing $0$ by $0$, since the communauty of mathematicians have agreed that $\frac ab$ is the way to talk about the division of $a$ by $b$. And the problem that then arises is that the division $0$ by $0$ makes no sense - as pointing out by @vrugtehagel in his answer, $0 = 0 \cdot c$ does not have a unique solution.

From there, you have two options : either you agree with the definition of division, which you really should, either you don't. If you don't, you must come up with another defintion of what is meant by dividing two numbers. Once you've done that, we will be able to tell if, within your definition, $\frac 00 = 0$.

Answer 2

Division is defined by the following (no matter what you think about it, if you have a different definition you should state it):

$a/b$ is the unique solution $c$ to $a=bc$

So what is $0/0$? It should be the unique solution to $0=0\cdot c$, but that equation doesn't have a unique solution (sure $0$ is one, but so is $23$). Hence we say $0/0$ "doesn't exist" because it doesn't represent a unique number so we can't really work with it.

And yes, this could also mean we say $0/0\neq 0$, but I don't even agree with that; in fact, I'd say saying "prove $0/0\neq 0$" isn't even a valid question, because $0/0$ simply doesn't exist (uniquely). It would be similar if I asked you "prove that I am not a blooperyfoobar". It doesn't make sense, because a blooperyfoobar doesn't exist, so you can't really say whether or not I am one.

Answer 3

I notice that in your original question, your edits responded to the question I gave in my comments (thanks). So I felt I should try to give a more detailed discussion here, this will be in line with the above comment by @Arthur on "properties" (and possibly in line with other comments).

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I assume that you can reach self-satisfying contradictions if you assume $1/0$ exists as a real number. So let us try to reduce the problem to that case.

Let's suppose we allow $0/0$ to be a real number $c$ (possibly $c=0$). Now we must also specify what algebraic manipulations of $\cdot/0$ we allow. For example if we allow associativity of products we get: $$ c=0/0 = (0\cdot 1)/0 = 0 \cdot (1/0) $$ The issue here is that this implicitly assumes $1/0$ is a quantity that exists, which we already know is a faulty assumption. So we either reach a contradiction, or we must disallow such manipulations. One might arrive at using $1/0$ another way: The definition of dividing by zero is multiplying by $0^{-1}$, which is the same as $1/0$. So now we have to "disallow" this natural definition. Finally we might consider the distributive property: $$ c = 0/0 = (1+-1)/0 = 1/0 + -1/0 $$ and this is bad since it assumes both $1/0$ and $-1/0$ are defined. So we must "disallow" that manipulation too.

You can object to my argument/s above by saying that you are fine with such "disallowing." You will define $0/0$ as a real number, but then you will significantly restrict the algebra manipulations (and $0^{-1}$ definitions) that we are allowed to use for $\cdot/0$. In that case, nobody knows what manipulations/definitions are allowed (since you haven't told us), and such restrictions seem to basically say the same thing as "if you ever see $0/0$ you are not allowed to manipulate it in any of the usual ways." But, something that cannot be manipulated is not very useful. Most people would prefer to follow the convention $\cdot/0$ is not defined.