I believe Emil is right about the usual convention being that 0^0 = 1, on the grounds that this makes x^0 = 1 for all x.
However, in some particular context, it is possible that an author might
create a local, temporary convention that 0^0 = 0, e.g. if he wanted it to be true for all a >= 0 that the function x |--> a^x is continuous on the non-negative real axis.
It's probably wise to check in each case - I don't think one can completely take it for granted that 0^0 = 1, in the same way that one can take it for granted that 0 + x = x, and 0*x = 0, for all x.
As for 0/0: here it is much less a matter of convention. Some very general conventions do enter into the explanation, but so does rational argument. One need not quake in fear of the remembered authority of the school mathematics teacher in order to see why there is no assignment of a value to 0/0 that "makes sense". It's something you can see for yourself, as follows.
In any case where one mathematical object, y, can be "divided" by another, x, I think it is true (if only by a very well-established convention) that what the operation of dividing by x means
is that it is the inverse
of the operation of multiplication by x.
"Inverse"? "Multiplication"? One thing at a time.
An inverse of a function f: A --> B is a function g: B --> A such that g(f(a)) = a for every a in A, and f(g(b)) = b for every b in B.
In words, if you think of a function as an operation converting some sort of mathematical values into some other sort of mathematical values (possibly the same sort, it doesn't matter), then an inverse
of the function is an operation which "undoes" it, whichever order you perform the two operations in. (A function can have only one inverse.)
(I hope that's clear! If not, this article isn't going to be much help.)
What "multiplication" means depends on the context (this is the price of modern mathematical abstraction, but it brings benefits), but I think it will always be true that the set of mathematical objects in question form a commutative group
under multiplication - so long as 0 is excluded.
It doesn't matter very much what this means in detail; it's just a quick way of saying that some of the usual laws of arithmetic are satisfied.
The element 0 has to be excluded, because it is a consequence of the well-established conventions governing the way this symbol is used that 0*x = x*0 = 0, for every x belonging to the set of objects in question.
(Quick proof of this fact: 0*x + 0*x = (0 + 0)*x = 0*x, therefore 0*x = 0, by cancellation. This can be justified strictly by reasoning from axioms.)
Therefore, the operation of multiplying by 0 has no inverse; because it takes every value to 0, there is no possible function which can "undo" its effect. (That is, unless 0 itself is the only value being considered, in which case all arithmetic reduces to the uninteresting 0 + 0 = 0, 0*0 = 0!)
In symbols, suppose the function x |--> x*0 (i.e. the function which converts any given number x to x*0 = 0) had an inverse, to be denoted (temporarily) by y |--> y/0 (i.e. it converts any given number y to some number y/0 - we don't need to specify what this is).
By the definition of inverse
, we must have (x*0)/0 = x for all x, and (y/0)*0 = y for all y. The second of these identities implies y = 0 for all y.
The first identity has already been shown to be absurd, in a previous post. It implies x = 0/0 = z, for all x and all z, so all the numbers being considered must all be equal to one another, and therefore all must be equal to 0.
I'm just giving another side of the same argument.
There is another use of the word "inverse" in mathematics, which is related to this one, but might cause confusion in the present context, because it could also be used to give an alternative proof: that there is no number which can be an "inverse" of 0 (in this second sense).
This alternative sense of the word is as follows (you will find it if you look up the term "commutative group",which I used earlier):
With respect to a given operation of multiplication, for which there is an "identity" element, 1, an "inverse" of an element x is an element y such that x*y = y*x = 1. ("Identity" here means that 1*x = x*1 = x, for all x.)
You might care to try giving a proof that 0 can have no "inverse" in this sense. (It's not hard! What's a little harder is to see how this argument relates to the other one. I mention it only in case someone comes across the other definition and gets confused.)