>>>... 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.)
>
>It was a while that I last taught Calculus (or, for that
>matter, anything else). If the memory serves, they did
>universally use the shorthand Sakxk for the
>infinite sum a0 + a1x + ..., but I
>have no recollection of any instance where the proper point
>has been made or the matter of x0 = 1 at all
>taken up.

Sorry for my delay in responding, Alex.

When talking about empty products, it's particularly nice to consider Taylor's series. The kth term involves

f(k)(a), k!, and (x - a)k.

When k happens to be 0, _all three_ of these quantities are empty "products". As mentioned earlier in this thread, the latter two (which are products, literally) are 1, the identity element for multiplication. But what about f(k)(a) when k happens to be 0? In other words, what must the 0th derivative of our function be? Well, instead of writing the kth derivative of f as f(k), it's nicer here to write it using the alternative notation

Dkf.

In this form, it's clear that the operator D must be applied k times. (For example, D2, the "product" of D and D, is an operator causing us to differentiate twice.) Then D0 is the empty "product" of the operator D. Therefore, it must be the _identity_ for operators, and that identity operator applied to our function f gives just f itself: D0f = f.

Of course, I suppose that most people's intuition tells them that, differentiating 0 times, the function f is unchanged. But I think it's also nice to see, via the empty product principle, that D0 must be the identity operator.

>>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!
>
>Again, I see your point and enjoy your argument but I can't
>accept the value judgement concealed behind "... as it
>should be."

If it would make you any happier, feel free to replace "should" by "must", above and below.

Consider the statement "23 = 8 is as it should be." Do you really think that there is some unacceptable value judgment concealed therein? I don't. Similarly, I don't hesitate to say that 00 = 1 is as it should be.

>Look how far we (the humanity, I mean) got
>(developing mathematics) without this point having been
>settled.

Yes, I see your point. But...

This morning, while pulling up a vine in my garden, I got a thorn in my finger. The thorn didn't hurt much at all. I could have continued with my gardening, without any noticeable detriment, without ever having removed the thorn. But why not remove it at the first opportunity?

To me, having 00 undefined is just such a thorn. Sure, we could live with it and do well enough. But why not remove the thorn?

>>>(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.
>
>What's relevant here is an inordinate amount of ways the
>kids get confused in. The "empty sum" and "empty product",
>however natural they are to your (or mine) mind's
>organization are bound to confuse some. I can't substantiate
>my opinion in any way but I think that they may be enjoyable
>more than helpful. In other words, they provide an
>additional angle to what one has already accepted or they
>bring more confusion to one who is struggling anyway.

You raise pedagogical concerns. Clearly, such concerns are valid. But of course we know that what is _pedagogically_ nice cannot dictate what is _mathematically_ correct. What is mathematically correct is determined first; only subsequently may we consider how that can best be taught.

But in fact, I consider having x0 = 1 for _all_ x to be not only a mathematical necessity, but also pedagogically easier to handle than leaving 00 undefined. After all, if we explain well why _any_ 0th power must be 1, then we are done. Period. (And then, much later, when the student is studying limits, he will be cautioned that the fact that the limit form called "00" is indeterminate has nothing to do with 00 = 1.) But if we were to have x0 = 1 for all nonzero x, leaving 00 undefined, we would have to explain to students not only why x0 should be 1 for all nonzero x, but also explain why 00 should be left undefined. I could not, in honesty, do the latter. As I said before, I know of no compelling mathematical reason for leaving 00 undefined. But if you know of any such reason, Alex (or anyone else), please give it!

Regards,
David Cantrell