johnhemming wrote:Gengulphus wrote:In particular, the function x^x is 'privileged' with respect to the question "What is the limit of x^x as x tends to zero?", since it appears in the question, but it is not 'privileged' with respect to the question "What is ZERO to the power of ZERO?".
Having driven to Stirchley and back considering this issue I see what you say and have some sympathy with it. However, considering the nature of the function I think basically the power of zero is something that has a defined result because of the way the function operates which is that it ignores the thing that it is the power of.
Basically, the defined result of any function in mathematics is what mathematicians define it to be, and they then explore the consequences of the definition they've chosen. Some definitions lead to more interesting consequences than others, and/or to consequences that seem to reflect what happens in the real world better, and those are the definitions that get studied more by other mathematicians and used more in science. And when particular definitions become very widely studied, pretty much to the exclusion of all others, they and their consequences tend to get named, and the name
is that set of definitions and consequences - not as a matter of naturally-occurring fact, but basically by convention among mathematicians. So for example, what mathematicians call the "real numbers" have a specific definition (or to be a bit more precise, a number of different definitions, all of which are provably equivalent): the numbers resulting from that definition have interesting properties and are widely applicable to the real world - so much so that outside of mathematics, they are generally just thought of as "numbers". But they are just a mathematical construct, not a naturally-occurring fact, and there do seem to be imperfections in their description of the real world, e.g. due to quantum effects (for instance, the real number 10^(-1000000) exists as a mathematical construct, but whether a distance of 10^(-1000000) metres or a time period of 10^(-1000000) seconds exists in any meaningful sense in the real world seems highly doubtful).
In this case, if a mathematician wants to define x^y in the usual way if (x,y) is not (0,0) and augment it with (for instance) 0^0 = pi, they're entirely free to do so. However, if they want to communicate effectively with other mathematicians about their results, they need to call out that difference from the usual mathematical conventions (and it seems highly unlikely that their results will be of any particular interest, so they're unlikely to get the usual mathematical conventions changed). Furthermore, in this case, the usual mathematical conventions depend on which area of mathematics they're in. In combinatorics and a number of other areas involving integers, the usual mathematical convention is to define 0^0 = 1, basically because continuity doesn't apply to functions of integers, and defining 0^0 = 1 so makes a lot of formulae involving integers produce correct results (the reason why it does that lies in the combinatoric definition of x^y that I mentioned in an earlier post, as the number of ways one can choose an ordered sequence of y objects from x different types of object, with duplicate choices allowed). In areas involving real numbers, continuity does apply to functions of real numbers, and the fact that
no definition of 0^0 can make x^y continuous at (x,y) = (0,0) means that it often makes most sense just to leave 0^0 undefined.
Though I will add that in areas involving real numbers, if one
does want to define 0^0 for some reason, defining 0^0 = 1 usually makes most sense. The reason is that to make x^y approach a value other than 1 as (x,y) approaches (0,0), one needs to make x approach 0 far more quickly than y approaches 0. For instance, I can make x^y approach 0.1 (and in fact be 0.1 at every point along the approach) as (x,y) approaches (0,0) by going along the path (x,y) = (0.1,1), (0.01,0.5), (0.001,0.333...), (0.0001,0.25), (0.00001,0.2), etc, with the nth step along that path being at (x,y) = (0.1^n,1/n). In that, x approaches 0 very quickly and y approaches 0 pretty slowly - for example, by the tenth step, x is down to 0.0000000001 and y has only just got down to 0.1.
So in some loose sense, the exercise done in the video of gathering numerical evidence about how x^y behaves will find that it approaches 1 for almost all choices of how one makes (x,y) approach (0,0), and so choosing to define 0^0 = 1 makes it more nearly continuous at (0,0) than any other choice. I.e. if a mathematician chooses to define 0^0 = 1, they're less likely to go wrong if they inadvertently assume that x^y is continuous at (x,y) = (0,0) than for any other choice - but that assumption will still be a flaw in any proofs they come up with using it.
Finally, I should perhaps mention that this example of the appropriate mathematical definitions depending on the area of mathematics involved in a way that non-mathematicians usually won't know about isn't by any means unique. As another example, there's the question "What is infinity plus 1?", to which the 'popular' answer is "Infinity". Mathematically, though, the answer is:
In real numbers: "There's no such real number as infinity"
In formulae involving limits: "Infinity is a symbol indicating a particular type of limit, not a number that you can do arithmetic on"
In cardinal numbers: "Infinity plus 1 and 1 plus infinity are both infinity, for 'infinity' being any infinite cardinal number"
In ordinal numbers: "Infinity plus 1 is distinct from infinity, but 1 plus infinity is equal to infinity, for 'infinity' being any infinite ordinal number"
In surreal numbers: "Infinity plus 1 is equal to 1 plus infinity, but distinct from infinity, for 'infinity' being any infinite surreal number"
and quite possibly some others I haven't thought of!
Gengulphus