Okay Marin let me take a crack at this:
Marin said:
Dividing by zero appears to be incorrect, but where does the prohibition of this operation lie? I mean, we're all taught at school that division over 0 is not defined, but in calculus you define this operation as infinity
**(at least, my teacher taught me that division of a number by zero gives infinity and division of a number by infinity gives zero, as well as division of zero over infinity is zero)
I see it as an inconsistency in the definition of the operations - every filed that includes these operations is not closed with respect to division by zero, is it? Or generalised (correct me if I'm wrong) in every field (vector space) the interference of the inverse element of the multiplication with the neutral element of the addition collapses, or in other words the neutral element of the addition has no inverse with respect to the multiplication. Why is that? Is this for every vector space the case?
So, first off, if your teacher taught you that division of a number by zero gives infinity and division of a number by infinity gives zero, *your teacher was wrong*. However I don't think that was what happened. I think probably what happened is your teacher did something sneaky and didn't clearly explain it. I think that what your teacher probably said was that:
lim n->0 ( x/n = infinity )
lim n->infinity ( x/n = 0 )
...for all x.
I.E., *IN THE LIMIT*, any number divided by zero becomes infinity and any number divided by infinity becomes zero. Something being true in the limit is quite different from it being true in general! This distinction is important because in the limit the rules are different, in the limit things like "n/0" and "infinity" have a well defined meaning, normally "n/0" and "infinity" are undefined concepts.
So that aside, as for "why" you can't define a field where you can divide by zero:
So if you look at the field axioms, they simply conspicuously fail to describe what happens when you divide by zero. Looking at the wikipedia version of the field axioms axiom #5 is:
# Additive and multiplicative inverses: For every a belonging to F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any nonzero a, i.e. for any a ≠ 0, there exists an element a^−1 in F, such that a · a^-1 = 1.
This does not say what happens to 0. It just declines to specifically say anything about division by zero at all. (The vector space axioms, on the other hand, decline to say anything about whether ANY item in the vector space has a multiplicative inverse, period!)
But, although this axiom doesn't specifically say what happens when you divide by zero, it is possible to *derive* from this axiom what happens when you divide by zero.
The section on fields in my copy of "Introductory Modern Algebra" by Saul Stahl contains a proof that begins like this:
Set x = a * 0. By the distributivity of addition and multiplication,
x = a * 0 = a * (0 + 0) = a * 0 + a * 0 = x + x
Consequently
a * 0 = x = x + (x + (-x)) = (x + x) + (-x) = x + (-x) = 0
If we stop the proof there, we've just proven something interesting: a * 0 = 0, regardless of a.
This makes it very easy to prove by contradiction that b / 0 does not exist for any b:
Set y = b / 0 , for some nonzero b
We can rewrite that as y * 0 = b
But a * 0 = 0 for all a, therefore b = 0
?!? contradiction
Now, here's the trick: This proof depends on the field axioms, so you can get around this if you decide to declare that your "b/0" element is exempt from some of the field axioms, but once you start doing this it would no longer be a field (and I'm sure, but am not going to try to prove, that it would not be a vector space or a module or a ring either... you may want to check the axioms for a vector space and see whether the proof above applies to vector spaces too). People *DO* define sloppier structures where "b/0" is well defined or where something called "infinity" is in the member set, but you can't do that and still keep all the nice consistent properties that make people want to use fields and vector spaces and such.
