This post at squareCircleZ (a very nice enrichment and support website for students and teachers of mathematics) raises the conundrum of how to define 0^0 if all positive x give x^0=1 and 0^x=0.
Actually, many mathematicians would consider 0^0 to be undefined because of the possibility of getting different values as limits of x^y with x and y both approaching zero, but it is indeed a useful exercise for students to consider why zac would say that 0^0 “is most commonly regarded as having value 1”, as well as whether or not the possibility of alternative choices represents a breakdown of consistency in mathematics.
The argument that 0^3=0,0^2=0,0^1=0 (etc) imply 0^0=0 does, at first, seem just as convincing as the other one, but actually it is a bit “one-sided”, since one could just as well say 0^(-3) is undefined (or “infinite” if you like to put it that way) and ditto for 0^(-2), 0^(-1), etc, so why shouldn’t the same be true for 0^(-0) (which is of course the same as 0^0)?
On the other hand, x^0=1 is true not just for positive x but also for negative ones (and in fact for all nonzero complex numbers). So to extend the definition by defining 0^0=1 has the advantage of creating a continuous extension of x^0 to all real (and complex) numbers.
So x^0=1 can be usefully extended in a continuous way to all numbers but 0^x=0 cannot. This may be one reason why some mathematicians find it useful to adopt the 0^0=1 definition, but it is not the only one.
Other reasons include:
– the convenience of being able to consider power series in x valid at x=0 without having to separate out the constant term, and
– the fact that it makes for the completion of Pascal’s Triangle with a 1 at the top.
But perhaps best of all is the fact that in set theory the power n^p has a natural definition as the number of ways of mapping a set of p elements into a set of n elements, and if n=p=0 this would be the number of ways of mapping the empty set into itself (which can be done in one way if you agree that nothing going to nothing should be counted as a mapping).
Nonetheless, it remains just a definition chosen by us – as is each step in the process of extension from the more primitive cases of b^p where b and p are both positive integers and where the power is defined by repeated multiplication.
In fact, it often happens that a concept or operation defined in a restricted context has more than one reasonable extension to a broader context and so we can’t use both without giving them different names. Often people are careless and use the same name for the extended operation and it is usually not a problem because everyone makes the same choice. But if different people make different choices and both fail to change the name then this does lead to an inconsistency. But of course in such cases it is just the use of language that is inconsistent. The basic facts that the language is describing remain (so far as we know) without contradiction. So, to the extent that different choices are made, perhaps we should consider it as an inconsistency of mathematicians rather than of mathematics.
(In the case of 0^0 though, I doubt that any mathematician prefers the 0^0=0 definition. So in this case the difference, if any, is between defining it as 1 or not at all – which I think is a less serious confusion than actually having two candidate definitions. A situation where that more problematic kind of difference does occur is in the case of fractional powers of negative numbers, but even there the fact that we can make different choices about how to name things does not really mean that the facts themselves are actually inconsistent.)