Since the square root of 0 is 0, using the general rule that a number
divided by its square root equals its square root, is 0 (the number)
divided by 0 (its square root) defined as 0 (its square root again)?

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Request for Question Clarification by justaskscott-ga on 05 Nov 2003 18:33 PST

This is the type of question to drive even a mathematician insane. 
You can cite the rule about square roots -- but someone else can
invoke the rule that "every number divided by itself equals one", and
yet another person can say, "any number divided by zero equals
infinity".  The best rule I have seen is "never divide by zero".

A good quote on why the answer is indetermine is:

"The number D is a divisor of a number M if there is an integer K for
which M=KxD.  K is called the quotient of M divided by D.  Since 0=K×0
for all integers K, zero is a divisor of zero.  Indeed, any number may
be substituted for K, so that the choice of quotient for zero divided
by zero is arbitrary.  The quotient is said to be indeterminate."

"Zero Divided by Zero", by E. E. McDonnell
The ACM Portal
http://delivery.acm.org/10.1145/810000/803689/p295-mcdonnell.pdf?key1=803689&key2=1545808601&coll=GUIDE&dl=ACM&CFID=13804836&CFTOKEN=72687185

If this answer (with additional citations) would be sufficient, I
would be happy to post it.

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Request for Question Clarification by justaskscott-ga on 05 Nov 2003 18:35 PST

I see that you have posted a second question on this topic.  You might
want to cancel the second question.  However, I should note that
another Researcher is working on that question as I write this.  So
you may want to wait to see whether that Researcher posts another
proposed or actual answer.

Hello victorperpetua,

Zero divided by zero has meaning only in the context of slopes and
their ratios. Outside of this context zero divided by zero is known as
an ?indeterminate form?. An indeterminate form may have any value
whatsoever according to the functions concerned.
Your attempt to find a value for zero divided by zero fails when you
divide by zero. In this context division by zero is not allowed!
If you want to increase your understanding of the matter I recommend
the book ?THE NOTHING THAT IS ? a natural history of zero? by Robert
Kaplan.

Hi vperpetua,
              referring to your question.. I think that "the general
rule that a number divided by its square root equals its square root "
is only applicable for natural numbers.. and I think 0 is excluded
from that. .

In response to kweldude's comment, the natural numbers are the set of
all integers greater than 0.  In fact, all complex numbers excluding 0
fulfill this square root rule.  For instance, (-4)^.5 = 2i, and (2i)^2
= -4.  This rule even holds true for irrational numbers: (2^.5)^.5 =
2^.25, and (2^.25)^2 = 2^.5  It is only zero that comes up with such
infuriating problems.  0/0 is a problem that will never have a
satisfactory answer.