Re: 1^Infinity
- To: mathgroup at smc.vnet.net
- Subject: [mg38322] Re: 1^Infinity
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Thu, 12 Dec 2002 01:34:54 -0500 (EST)
- Organization: NewsReader.Com Subscriber
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Selwyn Hollis <selwynh at earthlink.net> wrote: > It is astonishing that debates like this keep coming up. To anyone who > disagrees with the notion that 1^Infinity is indeterminate, I suggest > that you write something called "a new kind of calculus." But you might > want to learn the old kind first. I taught calculus for many years. Unless I've misunderstood Ted, his question has nothing to do with calculus. (If I have misunderstood you, Ted, please let me know!) What you're thinking about, Selwyn, cannot be debated -- at least not by reasonable people: Certain _limit_ forms, such as "1^oo" and "0^0", are indeterminate. That is simply a fact. It cannot be altered (unless we were to change the power function x^y substantially, which surely we don't want to do). But, when we say that a certain limit form is indeterminate, it is important to know not only what that _does_ mean, but also what it does _not_ mean! Let's take the limit form "0^0" as an example. Saying that it is indeterminate means that, as x and y approach 0, x^y may approach any of many different possible values (or the limit may not exist). [The word "indeterminate" is appropriate in that, merely knowing that both base and power approach 0, we do not have enough information to be able to determine the limit, if it exists.] This indeterminacy is due to the fact that f(x,y) = x^y has an essential singularity at (0,0). But saying that the limit form "0^0" is indeterminate does _not_ mean that the simple arithmetic expression 0^0 need be undefined. In the arithmetic expression 0^0, both base and power _are_ 0, they are constant. No limits are involved. Whether the arithmetic expression 0^0 should be defined as 1, as many prominent mathematicians (including Euler, Knuth, Graham, and Kahan) have suggested, or should be undefined, is apparently still open to debate by reasonable people. Now the OP had asked about 1^Infinity. If he was asking about the limit form "1^Infinity", then, just as for "0^0", there can be no reasonable debate. That limit form is indeterminate, period. But, since he mentioned no limits, I had naturally assumed that he had in mind the simple arithmetic expression 1^Infinity, in which the base does not merely approach 1, but rather _is_ 1, and the power does not merely approach Infinity, by rather _is_ Infinity. Both base and power are constants. (In case you're balking at the notion of the power being the constant Infinity: Of course, there is no such constant in the real number system. But such a constant does exist in an extension of the reals.) Just as for the arithmetic expression 0^0, whether the arithmetic expression 1^Infinity should be defined as 1, or be undefined, is still open to debate by reasonable people. Note: Of course, defining the arithmetic expressions 0^0 and 1^Infinity to be 1 would in no way alter the fact that f(x,y) = x^y has essential singularities at (0,0) and (1,Infinity), and so would in no way alter the fact that the limit forms "0^0 and "1^Infinity" are indeterminate. David Cantrell -- -------------------- http://NewsReader.Com/ -------------------- Usenet Newsgroup Service New Rate! $9.95/Month 50GB
- References:
- Re: 1^Infinity
- From: "David W. Cantrell" <DWCantrell@sigmaxi.org>
- Re: 1^Infinity