>>>>Please can anyone explain to me why, when any real number is
>>>>raised to the power of zero, the result is 1.

>>>It's a convenience actually since then
>>>
>>>ab·a-b = ab - b = 1.
>>
>>Yes, that's a reason for why we define it that way. But in
>>my opinion, the "real" reason is much more compelling than
>
>You mean an "additional" reason?

I must confess to being somewhat of a Platonist. From that viewpoint, I meant _the real reason_. :-)

Convenience can be a reason for doing certain things, no doubt. Given a choice of several things, their only differences being in matters of convenience, we would certainly choose the most convenient one. But if there are differences of a _fundamental_ nature between the alternatives, then convenience should not govern our choice.

Concerning leaving 00 undefined:
>>I'm not at all sure that's done most of the time _by
>>mathematicians_. But I regret to say that I agree that it is
>>done most of the time in elementary (pre-graduate school level)
>>texts.
>
>... and any level Calculus book, of course.

No. For example, just look in their sections on power series: x0 = 1, even when x = 0. (Perhaps you can find some texts which don't do that, but that is done in all the texts I've taught from.)

>>Therefore, I also briefly address the issue of 0^0 below.
>>
>>we can always think of as being present in a product,
>>whether there are any other factors or not.
>
>This is a delicate point. We are not exactly talking of the
>integers.

I'm not certain that I understand your point. Are you perhaps wishing to distinguish between the integer 0 and the real number 0 when used as exponents?

> Would you call x a factor in
>
>x×1/x = 1.

Yes, the expression on the left does have a factor of x. Indeed, it has factors of x in both numerator and denominator. (Furthermore, I'll note that those factors of x can legitimately be "cancelled", giving 1 as the result, except when x = 0.)

>(I understand what you wanted to say and accept that. The
>above is to underscore a possible ambiguity in your
>argument.)

What is that possible ambiguity?

>>x0 is the empty product, and must
>>therefore be 1, regardless of x.
>
>The argument is nice, but by the same token, 0x =
>0 regardless of x.

I don't know what you mean by "the same token". 0x is not an empty product unless x happens to be 0. We should have

0x = 0 if x > 0, 1 if x = 0, and (assuming we want to stay in the real number system) undefined if x < 0.

>>Since I've already presented an argument presumably showing
>>that 00 must be 1, perhaps I should also now, in
>>fairness, present reasons for leaving 00
>>undefined. But I can't! I am aware of not a single
>>compelling mathematical argument for leaving 00
>>undefined.
>
>I do not believe that.

Well, it's true. Note the important word "compelling".

>To me, the ambiguity created by
>"0x = 0 for all x" is sufficiently compelling to
>have second thoughts.

I'd rather that you had second thoughts about 0x = 0 for negative x! ;-)

>>So then you're wondering: Why is 00 often left
>>undefined, at least in lower level texts?
>
>With a view to Calculus and for the same reason that 0/0 is
>left undefined: there are ways and ways to approach 0.

Approach? There is no "approaching" going on, at least not in what I've been talking about. I have been talking about 00, which is an _arithmetic_ expression. The base and the exponent are _constant_. (I cannot stress that point too strongly.) But I now see what is perhaps causing your consternation, Alex.

When we consider limits in Calculus, we have certain forms which are called indeterminate. For example, knowing merely that x and y both approach 0, we cannot determine the limit of xy. Depending on _how_ x and y approach 0, many different answers are possible. For this reason, this form of limit, in which base and exponent both approach 0, is called indeterminate. Now for better or worse -- and I think the latter -- this indeterminate limit form is often denoted simply as 00. But this shorthand can easily cause confusion. It _looks_ like a mere arithmetic expression, but it's not. It would be somewhat better, I think, if this indeterminate limit form were abbreviated as "00", in the hope that the quotation marks would provide a clue to the reader that we are not literally talking about 00.

In summary:
When we refer to the limit form "00", we are talking about a situation in which the base and exponent _approach_ 0. This form is rightly called indeterminate. But in contrast, in the arithmetic expression 00, the base and exponent are _fixed_. One should not confuse the arithmetic expression and the limit form!

>>Unfortuantely, to
>>explain that gaffe would take us far beyond what's
>>appropriate for a middle school audience.
>
>This is why ... Is it not? I can't be 100% sure, but perhaps
>the sources chose to avoid giving an explanation that "would
>take the audience far beyond what's appropriate for a middle
>school audience."

Yes, I suspect that confusion between the arithmetic expression and the limit form could well contribute to the problem.

>>I'll close with a quotation from _Concrete Mathematics_ by
>>Ronald Graham, Donald Knuth, and Oren Patashnik
>>(Addison-Wesley, 2nd ed., 1994) p. 162:
>>
>>"Some textbooks leave the quantity 00
>>undefined... But this is a mistake. We must define
>>x0 = 1, for all x..."
>
>With all due respect, I understand their argument and agree
>that in Discrete Mathematics it's rather a great convenience
>to have 00 = 0.

You intended to say 1.

>But to say it's a mistake to have Calculus in mind, is a misteak.

Of course, it's no mistake to have Calculus in mind. The mistake is to think that Calculus would keep us from defining 00 as it should be! The fact that "00" is an indeterminate limit form has no bearing on whether 00 should be defined or not.

>(See B. Cipra's or E. J. Barbeau's books.)

I'd be happy to look at any particular passages you'd care to mention, if you still think they're relevant.

Regards,
David W. Cantrell