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(This was asked due to the comments and downvotes on this Stackoverflow answer. I am not that good at maths, so was wondering if I had made any basic mistakes)

Ignoring limits, I would like to know if this is a valid explanation for why $\frac00$ is undefined:

$x = \frac00$
$x \cdot 0 = 0$

Hence There are an infinite number of values for $x$ as anything multiplied by $0$ is $0$.

However, it seems to have got comments, with two general themes.

Once is that you lose the values of $x$ by multiplying by $0$.

The other is that the last line is:

$x \cdot 0 = \frac00 \cdot 0$

as it involves a division by $0$.

Is there any merit to either argument? More to the point, are there any major flaws in my explanation and is there a better way of showing why $\frac00$ is undefined?

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    That 'proof' looks perfectly fine to me.2010-07-23
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    for me the explanation is ok too.2010-07-23
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    may be your question gets more good explanation if you ask for all x / 0 is undefined...2010-07-23
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    A non-negligeable number of the answers given is rather misguided...2010-07-30
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    Dear Jacob, Your explanation is pretty good, but I would make the following slight change at the end: rather than have the conclusion be that there are an infinite number of x solving the equation, interpret it as follows: there is no well-determined x that solves the equation (because any x will do!). Thus we cannot find a well-determined value for 0/0.2010-08-01
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    Dear 97832123, I think you could be more generous to the OP, and interpret the question as asking "Is this a correct explanation as to why mathematicians leave 0/0 undefined"? The answer to the question is then essentially "yes".2010-08-01
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    The key part of your argument Jacob is that $0*x=0$ for all $x$. It can be useful to identify exactly which underlying algebraic fact causes this -- it's a general phenomenon in rings. Moreover, it follows from the distributivity law. So one way to say why we don't define $0/0$ is that it would force us to give up on distributivity. While $0/0$ has no compelling definition, distributivity is a compelling idea. So that's why we don't bother trying to define $0/0$.2010-09-29
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    How many times do we get this $0/0$ question?2011-03-27
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    @Andrea Mori: I'm terribly sorry but I couldn't find any similar question; could you please post a link if you do know a similiar question?2011-03-27
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    It's true. Trying to search for "0/0" gives you a search for "00", so who can blame him/her?2011-03-27
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    @pimvdb: Dear pimvdb, I've merged your question with an older one.2011-03-27

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