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From my abstract algebra knowledge of a group, there is no operation called division. Why is it shown here? The proper way to write the inverse is just x^-1 never 1/x, which is not meaningful in an algebraic group. Perhaps this should better be explained in the page, and remove references to 1/x. In general math, yes, 1/x is certainly a valid way to write x^-1, but not in abstract algebra. Conceptually we all handle it the same way, convert 1/x to x^-1 but 1/x is not the operation 1 divided by x. It is the multiplicative inverse of x, that's all. In the group Z3, under multiplication, there is no element 1/2, since there are only elements 0,1,2. However, 1/2 means 2^-1 which is defined, and ultimately 2^-1 = 2. This is easier to explain than the nonsense that results by using the operation division, which would be 1/2 = 2... 24.34.198.111 00:27, 27 April 2007 (UTC)
Sorry, I had to correct my question, I have made several mistakes when I wrote it before.
What I meant was:
In the computation of power for two complex numbers a and b as:
ab
it's used that:
ab=eb log(a)
But then, in the section Failure of power and logarithm identities, it is said that the identity log(ab)=b log(a) do not hold in general, and it's prooven with an example.
How is it possible then to equal ab=eb log(a) , since you need to use the property log(ab)=b log(a) to write it in that way? What makes that true in this case if a and b are any complex numbers?
Kaexar 01:50, 19 July 2007 (UTC)
But you made me notice that I wrote my question wrong. Thanks again.
Kaexar 01:50, 19 July 2007 (UTC)
Makholm and CBM. Note that the power series of e^x is not used in this elementary article at all. Nor is the differential equation or the continued fraction. All that stuff is taken care of in the article exponential function. This article on exponentiation is aimed at readers who do not know about exponentiation already. Superfluous material is likely to confuse the reader and make him stop reading. Holes in the argumentation likewise. The power series is a hindrance for reading and understanding. The limit without explanation is a hindrance for reading and understanding. Not everything that is true should be written everywhere. It should serve a purpose. Bo Jacoby 17:16, 8 August 2007 (UTC).
The limit definition of ex is used to show definition compatibility. The power series is important, but not here. To many readers a formula is a hindrance, and an unnecessary formula is an unnecessary hindrance. If ex is defined by the power series the reader will think:
If, on the other hand, ex is defined by the limit the reader will think:
To the service of the reader we should stick to the limit definition. You are welcome to explan |n|→ ∞.
Bo Jacoby 14:15, 9 August 2007 (UTC).
The limit definition is not simpler than the power series definition, but the definition compatibility is much easier to see using the limit definition than using the power series definition. Do you understand the argument above? Of course people reading mathematics should read formulas, but the formulas should serve a purpose, and the power series formula serves no purpose in this context. Why do you want to make these changes, Carl? Bo Jacoby 22:43, 10 August 2007 (UTC).
By now the article does not even assert compatibility between the two definitions. The main article on exponential function does not explain the compatibility either. The reader is left clueless. The number e is not explicitely defined, and the two formulas for ex are neither shown to be equal to one another nor to be equal to the old definition of ex for integer x. So the subsection is useless to the beginner and comprehensible only to the reader to whom it is superfluous. The important formula ex = limn(1−x/n)−n, which before was included as ex = lim|n|(1+x/n)n, has now been omitted, and it is not found in exponential function. The proof for definition compatibility does not really require advanced real (or complex) analysis, because if convergence fails no alternative definition is offered and so no incompatibility issue occurs. You are welcome to include convergence arguments in exponential function, but exponentiation should be kept elementary. Your present edit is a backwards step. I do not question your good faith, but you should take more care. Bo Jacoby 07:59, 13 August 2007 (UTC).
The name of the article exponentiation is attractive to beginners, (unlike for example Quantum gravity which is more scary), and the article links to the main article, exponential function for readers who want details. The purpose of introducing the exponential function ex in this article is to prepare the way for the exponentiation ax. Bo Jacoby 11:30, 13 August 2007 (UTC).
In the context of complex analysis the limit definition ez = limn(1+z/n)n, is equally important, and the geometrical interpretation of ei·x = limn(1+i·x/n)n is more straightforward than the geometrical interpretation of ei·x = Σn(i·x)n/n! Bo Jacoby 19:24, 13 August 2007 (UTC). Note also that the subsection you changed, Exponentiation#Powers_of_e, is part of the section Exponentiation#Exponentiation_with_integer_exponents, where it used to belong but no longer belongs. Please clean it up. Bo Jacoby 06:17, 15 August 2007 (UTC).
In " If a is a real number, and n is a positive integer, then the unique real solution with the same sign as a to the equation \ x^n = a is called the principal n^th root of a, and is denoted \sqrt[n]{a}."
When you say "the unique solution" is it not implicit that there always is one? (not the case!!) And why use the "the same sign"? (what about a=0??) Wouldn't it be more clear/didactic to explain what happens dividing in cases, the n's into even or odd and the a's into positive, 0, and negative?
Why use the term "principal n^th root"? It only makes real sense after someone studied complex numbers, which still takes a few years for most students. Ricardo sandoval 04:28, 24 August 2007 (UTC)
I was pointing to the absence of principal even root of negative numbers as in n-th root. So the term "the unique" doesn't apply. Someone could define the principal nth root for these number but they would not be real anyway.
I see the text is an improvement over the previous one (to which you referred) and it is even better now. But there is still the problem above. When remarking about the nth root of zero, I was pointing to the more trivial fact that zero has no sign, in the usual sense ( I am not considering the sgn function as a definition of sign). Although someone should expect the principal nth root of someone "without sign" (0) should also be "without sign" (0), this is not stated explicitly. Ricardo sandoval 03:43, 25 August 2007 (UTC)
In the section Rational powers of positive real numbers it is implicit that the result is also positive, i.e that the negative root should not be taken, or the principal value should be used. I think that should be made explicit. —Preceding unsigned comment added by TerryM--re ( talk • contribs) 00:11, 4 November 2007 (UTC)
This article has been reviewed as part of Wikipedia:WikiProject Good articles/Project quality task force. In reviewing the article against the Good article criteria, I have found there are some issues that need to be addressed. The article is under-referenced. Also, it's not specific enough for stating references in general. I am giving seven days for improvements to be made. If issues are addressed, the article will remain listed as a Good article. Otherwise, it will be delisted. If improved after it has been delisted, it may be nominated at WP:GAC. Feel free to drop a message on my talk page if you have any questions. Regards, OhanaUnited Talk page 21:32, 8 September 2007 (UTC)
Whoever can take care of the images that display equations, check out the first "Justification" under the heading "Zero to the zero power." It claims X^X -> 1, where it should be X^0 NinjaSkitch 06:06, 1 October 2007 (UTC)
I realize that "a to the power n" is correct terminology, but there is another correct notation that could be included there, "a to the exponent n". My math teacher tells us that that is the correct form, and that "a to the power n" is incorrect. However, I don't believe her, and I think both are correct. Is my math teacher wrong, or is she right? ZtObOr 01:42, 9 October 2007 (UTC)
EDIT: I forgot to include this bit. My math teacher's reasoning is that since the "n" in "a to the power n" is called the exponent, that it should really be called "a to the exponent n", since the word "exponent" refers to the exponent "n" in the first place, even if the word "power" is in its place.. —Preceding unsigned comment added by Ztobor ( talk • contribs) 01:45, 9 October 2007 (UTC) (Edited 4/10/2008)
Perhaps it is too pedantic to say that "a to the power n" is not a notation; it is a verbal description, or terminology. The notation is . As to the answer to your question: is the exponent, as your teacher rightly says. It is also the power. The exponent is the symbol that we use to designate a power. Therefore I prefer to say "a to the power n" as it doesn't confuse the thing with the symbol that we use to describe it. This is rather like the distinction between a number and a numeral. Of course you may choose to finesse the problem by saying "a to the n". TerryM--re 00:01, 4 November 2007 (UTC) I should add that the term index is also almost synonymous with exponent. In fact now that I think about it, I'm not sure that the terms are well defined; they are certainly used almost interchangeably by many people. TerryM--re 00:31, 4 November 2007 (UTC)
Is there any article to link to, to mention the cis function? This function can make working with Euler and De Moivre's formulas a lot easier. It is clearer to write on paper and leads to less mistakes, not to mention faster. An online example of cis, although not specific: http://oakroadsystems.com/twt/sumdiff.htm
Also, a link to an article describing implementing complex exponention in software: http://www.efg2.com/Lab/Mathematics/Complex/Numbers.htm The code is in Pascal/Delphi. JWhiteheadcc 06:05, 2 December 2007 (UTC)
This section, when I came upon it, claimed that Cantor's theorem implies that in the context of calculus. This is nonsense. If is interpreted as the limit of 2g(x) as x goes to infinity (where g(x) is a real valued function which goes to infinity), then this is just a limit of real numbers and not some kind of cardinal which is "larger" than another infinite limit of real numbers. Not larger in the sense of Cantor's theorem, anyway. This is not a cardinal.
So I fixed up the section so that it at least says correct things instead of nonsense, but I don't think it's a great section. I wouldn't mind if someone just deleted it, but then someone else would probably repost the same misconceptions later on.
By the way, I don't know where on earth I could find citations for this kind of thing. It's clear to any mathematician familiar with the relevant definitions. —Preceding unsigned comment added by 70.245.113.97 ( talk) 01:43, 11 December 2007 (UTC)
Am I right in thinking that tends to when n>1, remains at 1 when n=1, tends to 0 when abs(n)<1, and tends to when n<-1? Note: This being my first contribution, I don't want to make changes directly, in case of misinterpretation of the intent of the article! MessyBlob ( talk) 18:48, 5 April 2008 (UTC)
In this section, it is heavily implied that the Microsoft Calculator is a programming language, which is obviously wrong. Maybe it should be kept for simplicity or perhaps a rewording of the statement is needed. —Preceding unsigned comment added by 88.108.69.114 ( talk) 19:37, 17 May 2008 (UTC)
the only way i see this = 1 is the ^0 is subtracting 1 less so 0^0 = 0-(-1)= 1 —Preceding unsigned comment added by 208.91.185.12 ( talk) 21:31, 2 October 2008 (UTC)
I'm surprised the 00 section does not mention all that Knuth has said in his paper Two Notes on Notation. Can someone add it? The paper: TeX source, doi: 10.2307/2325085, arXiv: math/9205211.
Just to follow up on this: The article now cites it (more precisely, it cites the published version instead of the preprint). -- FactSpewer ( talk) 01:58, 18 October 2008 (UTC)
I'm not a math person, but I think I see an error in section 1.5 Identities and properties and would like someone help me check this out.
In the last line of the section the statement is a^b^c = a^(b^c), but the example preceding this seems to show just the opposite.
Then there is an inequality a^b^c ≠ (a^b)^c. Should this be an equality not an inequality?
And, finally, the article states "Without parentheses to modify the order of calculation, the order is usually understood to be from right to left". Should this be "from left to right"?
```` —Preceding unsigned comment added by Lou27182 ( talk • contribs) 16:14, 27 July 2008 (UTC)
The arguments for defining to be 1 are overwhelming even in analysis, and I feel that the article should be rewritten to reflect this. For instance, in addition to the analysis reasons already given (the power series for and the derivative of ), there is the case in the geometric series formula
itself a special case of the binomial formula for general exponents.
I feel that the arguments for leaving undefined stem from a misguided insistence that functions be continuous or differentiable on their domains. Here are rebuttals to the arguments currently in the article:
The best solution is to define , and to accept that the function will be discontinuous at and that no holomorphic branch of will exist in a neighborhood of 0. Of course, defining is not the same as saying that whenever and are tending to 0; the article should remain clear on this point.
BjornPoonen ( talk) 05:31, 11 October 2008 (UTC)
I agree with BjornPoonen. Your objections are to the point. The quote: "it may be best to treat 00 as an ill-defined quantity" is unjustified and without references. Bo Jacoby ( talk) 10:52, 11 October 2008 (UTC).
Thank you, Salix and Dmcq, for your comments, and in particular for pointing out the archive (even if it does not contain any arguments resembling my analogies with and ). Your arguments that the article should continue to mention that some authors leave undefined are well taken, and I agree with you on this point. What I feel is unjustified is the implication that the definition is useful only in the discrete side of mathematics. I would suggest removing "There are two principal treatments in practice, one from discrete mathematics and the other from analysis." I would also suggest removing the blanket statement "In general, mathematical analysis treats as undefined...". Citing some textbooks that leave it undefined does not mean that this is the general approach in analysis, when other textbooks define . BjornPoonen ( talk) 17:26, 11 October 2008 (UTC)
The changes made have improved the article and I wish to thank the editors involved. I dislike, however, the following sentence: "The rule in calculus that limx→a f(x)g(x) = (limx→a f(x))limx→a g(x) whenever both sides of the equation are defined would fail if 00 were defined", (my italics), because the italiced assertion is not a general rule of calculus, and because it refers to 00 being defined which we have just allowed, and because an equation (defined number)=(indeterminate form) is as false as (defined number)=(another defined number) and (defined number)=(undefined number). I hope we can agree to omit the sentence. Bo Jacoby ( talk) 12:30, 18 October 2008 (UTC).
After looking it over, I too think that the "rule of calculus" sentence should be removed or at least qualified, but for a different reason than Bo Jacoby: according to the definitions in the article,
Should we interpret this as saying that (-2)1 should be left undefined? (I hope not! That would be a disaster!) -- FactSpewer ( talk) 15:55, 18 October 2008 (UTC)
I agree that the limit of a function f(x) as x approaches a can be defined without f(a) itself being defined (I gather that this is what you meant by your remark about limit points), but the point I was trying to make was this:
In summary, I think this sentence about the "rule of calculus" sentence really needs to be removed. -- FactSpewer ( talk) 06:44, 20 October 2008 (UTC)
Dear Dmcq, I appreciate your effort to explain the "rule of calculus" sentence to me, but I am still unclear as to what it is trying to say. I have no objection to keeping it if it can be made precise and a reference for it (in the form you are stating it) is given. Let me suggest a couple possible readings of the sentence (expanding it in more detail, just to make sure we are on the same page), and ask you which one you mean.
I believe it is trying to say the following:
My question is what is meant by "exponentiation is continuous". Which of the following is intended as a hypothesis for the application of the rule?
(Admittedly, some of these would render the "rule" false or useless.) Of course, I'm not suggesting that we write all this in the article.
Finally, in what textbook do you find this "rule of calculus"? For the time being, I'll flag the sentence as needing a citation.
Thank you, -- FactSpewer ( talk) 02:47, 22 October 2008 (UTC)
I know it isn't a primary source but perhaps you an agree the statement at the start of Indeterminate form is correct:
It then goes on to say 00 is an indeterminate form and if you look in any textbook they'll say the same thing. Now if the algebraic operation after replacing the subexpression gives a defined number then the substitution does give enough information to determine the original limit. This is quite different from the case of Pow(x,y)=xy where (x,y)≠0 and Pow(0,0)=1. This does not lead to an indeterminate form. One cannot say that limit Pow(x,y) = Pow(limit x, limit y) if both sides are defined. Pow is not an algebraic operation in the terms of the definition.
Of course it would be possible to say the usual rules for indeterminate forms only applies for addition, subtraction, multiplication and division but don't apply for exponentiation. But that is not what is done. What is done is that 00 is treated as an indeterminate form.
As to the question about continuity the answer is number 3. Think of the same question as applied to the indeterminate form 0/0. If it had a defined value then it wouldn't be an indeterminate form. And division isn't continuous at 0,0. If divide(x,y)=x/y except divide(0,0)=1 was defined then it couldn't be used as an indeterminate form where one could move the limits inside the brackets. Without the defined value divide(0,0)=1 we could use it as an indeterminate form with a bit extra about moving limits in this particular function. Indeterminate form is just a special case of a function with a discontinuity that has no defined value at the discontinuity and is used because people didn't want to have a slew of special rules for dealing with straightforward expressions. Dmcq ( talk) 09:35, 22 October 2008 (UTC)
To keep the discussion constructive, I'd like this new section to be devoted exclusively to feedback on the proposed draft. Absolute statements that 0^0 is 1 or that 0^0 is undefined can go elsewhere.
No one over the past few days has had any objection to the proposed corrections outside the 0^0 section, so I will now restore those corrections, while leaving the 0^0 section untouched for the time being.
As for the proposed 0^0 section, a few people on the Talk:Exponentiation page have come out generally in support of it while also saying that it will need some minor fixing that can be done later. I believe the only person so far who has explicitly said that it is not an improvement is Steven G. Johnson. Steven: I would like to understand better what specifically in the new draft you are objecting to. I think we all agree with you that several references state explicitly that 0^0 is undefined, but I do not understand why this is incompatible with the proposed draft. Indeed, the proposed draft cites some such references, while also citing references for the opposite point of view, that 0^0 is 1. -- FactSpewer ( talk) 18:10, 16 October 2008 (UTC)
I just updated the proposed draft. Now it contains only the 0^0 section (the corrections in other sections having been implemented already). I also attempted to incorporate the suggestions made by those of you who commented on it. In particular, the opening sentences were rewritten so that it starts less abruptly, and so that it begins the discussion in a balanced way. Feedback on this draft is welcome, and should go here, in this section of Talk:Exponentiation. -- FactSpewer ( talk) 15:53, 17 October 2008 (UTC)
The sentence
looks mathematically incorrect to me.
Consider the identity log ab = b log a, for instance, for complex numbers a and b with a ≠ 0.
If we are treating log as a multivalued function, and A denotes one possible value of log a, then log a is the set of numbers of the form A + 2πin, where n ranges over integers. So b log a is the set of numbers of the form b(A + 2πin).
On the other hand, ab is defined as exp(b log a), which is the set of numbers of the form exp(b(A + 2πin)), so log ab is the set of numbers of the form b(A + 2πin) + 2πim, where m and n range over integers.
So the set log ab contains the set b log a, and for most values of b, it is strictly larger, meaning that the identity fails. -- FactSpewer ( talk) 01:30, 17 October 2008 (UTC)
I am unhappy with some of the very recent edits by Dmcq:
-- FactSpewer ( talk) 06:19, 20 October 2008 (UTC)
I've had a go at the multi-valued bits at the start of the complex exponentiation section and for Clausen's paradox and tried to make them a bit less off-putting. Dmcq ( talk) 08:00, 21 October 2008 (UTC)
The present structure of the article is that powers of real numbers are treated before powers of complex numbers. I suggest that powers of positive real numbers are finalized before powers of negative real numbers and powers of complex numbers, because the discontinuous rational powers of negative numbers are relatively unimportant and confusing. Comments, please. Bo Jacoby ( talk) 11:46, 20 October 2008 (UTC).
1 Exponentiation with integer exponents
2 Powers of positive real numbers
3 Powers of negative real numbers
4 Complex powers of real numbers
5 Powers of complex numbers
QUOTE:
UNQUOTE
In the chain of equalities
the weak links are equality sign number two and five, and not number three and four, no matter how is defined.
Bo Jacoby ( talk) 12:24, 20 October 2008 (UTC).
In the chain
the weak links are equality signs number three and six, and not number four and five, no matter how or if is defined. The point is whether is continuous or not.
If you are allowed to write then you are also allowed to write and if you are allowed to write then you are also allowed to write Bo Jacoby ( talk) 15:37, 20 October 2008 (UTC).
The "limit rule" is invalid independently on whether 0/0 is defined or not. A definition of 0/0 does not invalidate any legitimite calculation. The weak link is "lim(2x/x)=(lim 2x)/(lim x)", not "(lim 2x)/(lim x)=0/0" which is true because lim 2x=0 and lim x=0. (Her "lim" means "limx→0"). Similarily, defining 00 is not harmful in any context, and 'undefining' 00 solves no problem at all, and the 'justifications' for leaving 00 undefined are logically invalid. Bo Jacoby ( talk) 10:40, 21 October 2008 (UTC).
Defining 0/0 solves no problem, but defining 00 does solve a problem, and creates no problem. It is OK to report that some authors do not define 00, but it is not OK to report logically invalid justifications which seem to be original research. See WP:OR. Bo Jacoby ( talk) 14:21, 21 October 2008 (UTC).
The rule in calculus that lim x→a f(x)^g(x) = (lim x→a f(x))^(lim x→a g(x)) whenever both sides of the equation are defined and exponentiation is continuous is not a general rule of calculus but was coined for the purpose. "If 0^0 was defined as 1 then both sides of the rule would be defined giving the false conclusion that 1/e = 1" is not true, because exponentiation is not continuous if 0^0 is defined. Bo Jacoby ( talk) 16:08, 21 October 2008 (UTC).
Isn't the section on Exponentiation#Branches of the complex logarithm a bit over the top? I don't think this article needs anything quite like that if it can be dealt with better in the branch cut or complex logarithm articles. I certainly was quite reluctant to put in even the bit about Riemann surface but I though a quick reference and a picture were called for. Dmcq ( talk)
Dear Dmcq, Thank you for the new sentence in the 0^0 section. It's better than what was there before. We might still consider removing the sentences
since they are redundant with what came before and with being a counterexample to Moebius's statement. Anyway, I leave it to others to decide this.
-- FactSpewer ( talk) 00:01, 25 October 2008 (UTC)
See the example above in "redundant equations", by Factspewer. This example does not show that 0^0 is undefined in the algebra of limits for real numbers, it is much more basic than that! Suitably altered, it shows that exponentiation cannot be, in general, defined on the real numbers. In particular, there is no sensible way to define the binary function of exponentiation for the input pair of real numbers (0,0).
This is where it helps to know the (a) construction of integers in terms of numbers (0,1,2,3,4,5,... for which there can be no conceptual reduction - you can only draw a tally in the sand and wave your arms around until people understand what you mean), rationals in terms of integers, and reals in terms of rationals. More precisely for example, we have a definition of integers in terms of equivalence classes of number-number pairs, rationals in terms of equivalence classes of integer-integer pairs, reals in (for example) terms of equivalence classes of Cauchy sequences of rationals. Knowing about this sort of thing can clarify things. Some may perjoratively call this stuff "logic", but it isn't really, it's just very nit-picking mathematics. But it is this sort of contemplation that led to the theory of p-adic numbers, so it is not entirely stupid.
Anyway, the confusion around "0^0=1" (see next paragraph for why the inverted commas), present even in professional mathematicians, results from a conflation of integers and reals and their power functions. These should technically not be conflated, it technically makes no sense (not wrong, nonsensical) to say the real number 5 equals the number 5. Of course we identify numbers with the corresponding real numbers via the embedding of N in R. Addition, multiplication and order are preserved, but exponentiation is not quite. This embedding does not allow one to talk about x^y, where x and y are real number variables ranging over R. So for the purposes of resolving talk about 0^0 where 0 signifies the real number 0, one has to abandon this habitual (usually harmless) identification of 0 and 0', 1 and 1', 2 and 2', etc.
Note that any phrase "0^0=..." makes no literal sense until you have clarified that you mean the real number 0. The binary power (to stop confusion with exp) function on numbers (0,1,2,3,4,5,...) can very well be defined at the number pair (0,0), by defining 0^0=1. This causes no problems and makes perfect sense. Likewise for integers. For reals it is not so simple, in fact there is no way to define exponentiation at (0,0).
Let us use the Cauchy sequences definition of reals. Then in the above example write 2 instead of e and write 1/n instead of x, to get a convergent (and hence Cauchy) sequence. Then insist that power:RxR->R is such that 0^0=1. Bang! You have destroyed the Cauchy definition of the real number system.
Okay, then we reject power:RxR->R being such that 0^0=1. What about the stuff raised in the article as a reason why 0^0=1 might be good to have, for example infinite series for e^x at x=0, geometric series at x=0, binomial theorem at x=0, power rule for derivatives at n=1 and x=0, etc. We want to get that sorted. And we can, because nothing I said above indicates that we cannot have a function power:RxN->R with all the usual properties. R is a field, and this is just the usual field exponentiation applied to R. Naturally we can have power:RxZ->R also.
In power:RxZ->R, power(0,0) is defined and the usual interpretation of power(0,0)=1 is true. In power:RxR->R, power(0,0) is undefined. We cannot have a binary operation on R with the properties that we want a power function to satisfy, that is defined everywhere. Se la vi, there isn't always a function that does what you want. Bozo9 19:59, 25 October 2008
The assertion is false even if 00 is undefined. That is just an elementary observation made by a reader who need not write a book about it. It is not a reasonable job to write a false statement in the article. Bo Jacoby ( talk) 14:25, 26 October 2008 (UTC).
IEEE evaluates the boolean expression (e^(-1))=NaN to false, and (e^(-1))=1 evaluates to false, so, also according to IEEE, is false irrespectively on whether 00 is NaN or 1. Thank you for this referenced authority in support of my point of view. What I feel has nothing to do with it. Bo Jacoby ( talk) 16:04, 26 October 2008 (UTC).
That's how much discussion has been spent on the 0^0 question in this Talk page to date. Realize that you're not going to convince someone who has been arguing with essentially all other editors on this subject for two years now.
To quote CMummert from last year:
Need we continue now? —Steven G. Johnson ( talk) 19:17, 28 October 2008 (UTC)
The derivation that e^x = e^k for integer x=k was removed by Dmcq. Why? Bo Jacoby ( talk) 00:05, 30 October 2008 (UTC).
What was the problematic step? Bo Jacoby ( talk) 00:17, 30 October 2008 (UTC).
I think that, correct or incorrect, the derivation isn't needed. In an article at this level, the typical reader isn't going to be interested in or aided by proofs. Moreover, the notation in the derivation that was there,
involves things like "|n| → ∞" that are not typically encountered in a first calculus course. Also, e itself is defined as
higher in the article (without the absolute value). I don't see that the appearance of the absolute value would be at all clear to a naive reader. But I also don't think it would be worth writing a long explanation for this derivation, which is already a tangential point in the article. — Carl ( CBM · talk) 01:03, 30 October 2008 (UTC)
To Dmcq. Thank you for the apology. I hope you get better. To Carl. When k<0 and n>0 then m=nk<0 , and one needs the limit The exponential function is central to the generalization from integer to noninteger exponents. Without a hint about the new definition of e^x being a generalization of the old definition, the notation e^x is unmotivated. The text can be improved but I don't think it should be omitted. Bo Jacoby ( talk) 08:54, 30 October 2008 (UTC).
There is an article on exponential function, and an article on e, so I think that this subsection of the article on exponentiation should not contain more detail than what is used in the later subsections. That's why I removed a sentence. I do not think that my change made the text neither inaccessible nor aimed at a special audience. (Remember the rule: improve rather than revert). I made the change for a reason. It is important to note that not only must the exponent be large, but also the deviation from unity be small in order that Euler's indeterminate form be finite. (Happily we are not discussing indeterminate forms here). So the words "while the number goes to one" were included, as I still think they ought to be. An explanation why (e)k=ek for negative integer k is still missing, but an explanation why ex+y = ex·ey would suffice, I think. Then we could also omit the proof that (e)k = ek for positive integer k. I don't consider the number 1.0011000 to be an 'inline proof'. Bo Jacoby ( talk) 14:34, 31 October 2008 (UTC).
Let's take one step at a time. Do you object against "and is defined as the limit as the power goes to infinity" being replaced with "and is defined as the limit as the power goes to infinity while the deviation from one goes to zero" ? Step two. Do you think that the reader cannot identify 1.0011000 and (1+1/1000)1000 ? 15:58, 31 October 2008 (UTC).
The function exp(x) is important because it equals its own derivative, and hence arises in any application where something grows or decays in proportion to its current value. The number e is important not because it is the limit of (1+1/n)^n, but because it is the number such that e^x = exp(x) and hence gives a convenient notation for the exponential function that reminds the user of its multiplicative property.
With this in mind, one possibility might be to define exp(x) first (as the function equaling its derivative with exp(0)=1, and/or by the power series, and/or by the limit - probably it is best to mention all three, with the first one being primary), and then to state that there is a number e such that e^x (as defined earlier in the article) equals exp(x). Then there is no confusion between the notations, and as a bonus the formula that exp(k) equals e.e....e is an automatic consequence. -- FactSpewer ( talk) 01:44, 1 November 2008 (UTC)
I think that the article on exponentiation should explain ax where a is a positive real number and x is an arbitrary complex number. The exponential function ex is a convenient intermediate step. (This is to answer FactSpewer's question of the purpose of the section on ex). There are two common definitions of ex , namely the limit lim(1+1/n)n and the Taylor series Σxn/n! . The power an where n is an integer has just been explained, but some readers may at this stage not be comfortable with the theory of derivatives, which is more advanced than the theory of exponentiation. That's why I preferred the limit definition to the Taylor series definition in this context. Perhaps we should have a very short big exponent subsection , telling that limn→∞an is zero for |a| < 1 and infinite for |a| > 1 and equal to unity for |a| < 1 and undefined elsewhere, e.g. for a = −1 . This motivates investigating a big power of a number close to one. It is interesting that (1+x/n)n is virtually independent on n for big values of n. (1% of interest in 10 years almost equals 0.5% of interest in 20 years). I don't know an equally elementary motivation for studying the Taylor series. I think that the equally easy formulas lim(1+(x+y)/n)n = lim(1+x/n)n·lim(1+y/n)n and Σ(x+y)n/n! = (Σxn/n!)·(Σyn/n!) should be found in the article on exponential function. Bo Jacoby ( talk) 06:30, 2 November 2008 (UTC).
The two reasons that Dmcq gave are the same as the ones I would have given, except that it is not needed for real exponentiation (which has already been defined via the limit process!) Given that these are the only two goals, and that most of the material here fits better in either the e article or the exponential function article, I would suggest eliminating this section and placing the following sentence in the section where exponentiation of positive real numbers is defined.
And then of course, the function e^z would be mentioned later on when defining complex exponentiation. Is there anything else that needs to be said here that is not redundant with what is (or should be) in the e and exponential function articles?-- FactSpewer ( talk) 08:07, 3 November 2008 (UTC)
it seems as if Dmcq and I agree that ex should be defined here, as it is, in order to prepare for defining ax, but that FactSpewer disagrees. Bo Jacoby ( talk) 22:03, 3 November 2008 (UTC).
The latest reordering of the subsections is not satisfactory. The value of (1+n−1)n for big integer values of n belongs naturally in the subsection big exponents of the section on integer exponents. The definition of ex belongs naturally under the heading powers of positive real numbers, and equally naturally immediately after the definition of e, which was last in the previous section. The characterizations of a1/n = e(ln a)/n as the positive solution of the algebraic equation xn = a should be included, but not confuse the chain of logic. Bo Jacoby ( talk) 22:53, 5 November 2008 (UTC).
For big values of n, (a+bn−1+cn−2+···)n goes to infinity if a > 1 and towards zero if |a| < 1 , and towards a positive value, (eb), if a = 1. These facts belong to big exponents. Bo Jacoby ( talk) 00:00, 6 November 2008 (UTC).
How then make the logical connection between the integer exponents and the exponential function? Bo Jacoby ( talk) 07:56, 6 November 2008 (UTC).
If a section is called powers of e, then e should be defined or explained already. Otherwise the reader cannot understand the header. The reader should be able to read exponentiation without having to study exponential function. The dictum of repetition is fine for oral presentations, but not necessarily for an encyclopedia, where the reader is free to reread several times. I would hate to sacrifice logic on the altar of smalltalk. Bo Jacoby ( talk) 12:13, 6 November 2008 (UTC).
I don't think we disagree as much as it seems, nor that I remove explanations in the beginning. I am the one who want to define e before it is used. :-) Bo Jacoby ( talk) 13:30, 6 November 2008 (UTC).
I know what I did and I know what you did: you introduced a big gap between big exponents and powers of e which are closely connected logically. The algebraic definition of fractional exponents does not lead to complex exponents, but you gave it priority in favour of the powers of e section that is the road towards generalization of exponentiation. Yes, here we differ. But we do not differ in our attempts to write a good encyclopedia. You are not right in assuming that I want to remove explanations. Bo Jacoby ( talk) 23:30, 6 November 2008 (UTC).
I'd like to have
Both serial and random access reading of the article should be possible. Bo Jacoby ( talk) 09:08, 8 November 2008 (UTC).
I have raised mention of this at Wikipedia_talk:WikiProject_Mathematics#Exponentiation wars Dmcq ( talk) 13:36, 8 November 2008 (UTC)
Bo Jacoby ( talk) 23:11, 8 November 2008 (UTC).
It occurs to me that I don't have a standard name for the basic identity ab+c=abac. Is there one? Dmcq ( talk) 09:55, 1 November 2008 (UTC)
There may be names for it, but I don't think they are universally used. When a is e, I think some people call it the addition formula for the exponential function, or the functional equation for the exponential function. I'd suggest leaving it unnamed.-- FactSpewer ( talk) 03:43, 2 November 2008 (UTC)
The notation in big exponents is fairly obvious, but not the usual notation used elsewhere that I know of. Is this a standard way of writing limits anywhere? I found I had problems translating it into English, the 'for' in the statement I translated as 'as'. Dmcq ( talk) 16:07, 5 November 2008 (UTC)
About Loadmaster's edit: Does Wikipedia have a standard for hyphenation in words like nonzero? If not, is there a notable reference that recommends inserting a hyphen? Table 6.1 of the Chicago Manual of Style lists non as a prefix forming closed compounds --- nonzero, nonnegative, etc. WardenWalk ( talk) 21:36, 8 November 2008 (UTC)
I'm wondering if the rule for negative reals with odd integer powers holds when a non-integer power is split into a product having an integer exponent and a fractional exponent. Given a non-integer a and non-zero x, then for some integer n such that a = n+f,
Does it then follow that if x is a negative real and n is an odd integer, that the result has a negative real component? For example, is it true that
Compare this to an equivalent equation that uses an even integer n:
It appears that the two results differ in sign, which seems to be inconsistent. Another example:
Is this correct, or did I miss something? — Loadmaster ( talk) 20:27, 11 November 2008 (UTC)
I like this section very much. However, in the discussion of the limit of f(x)g(x) when f(x) and g(x) tend to 0, these functions are required to be "positive-valued". To me, this makes perfect sense for f. But it seems pointless for g, unless we want to allow f to take the value 0, which is of little benefit.
I would suggest we require f to be strictly positive, and remove any restriction on the sign of g. That way, the possible limits can be anything in [0,+∞], instead of [0,1]. The part about [0,1] gives one the misleading impression that 00 is really trying to be a number between 0 and 1, if one doesn't pay close attention to the restriction imposed on g. 67.150.253.241 ( talk) 14:39, 15 November 2008 (UTC)
I looked back and there used to be a section 'Powers with infinity'. I propose bringing something similar back to be a real or complex number equivalent to Large exponents the main purpose of which would be to talk about real and complex limits, and I'd put it just after the 0^0 section. I note a bit has just been added to the 0^0 section which would fit nicely as a basis for the new section. The new section would duplicate a bit in the current Large exponent section but that would be unsuitable for adding to as it is right at the top dealing with integers. Dmcq ( talk) 12:28, 19 November 2008 (UTC)
About a month ago, I tried rewriting the complex logarithm material in exponentiation, but the consensus here was that the material belonged in the complex logarithm article instead. An expanded version of this material has since been put there, so I will now trim the complex logarithm material in exponentiation, replacing it with links where appropriate.
The section on powers of the imaginary unit now seems out of place. The main dichotomy in the definitions of exponentiation is between integer exponents and real/complex exponents. Integer powers of i belong in the Exponentiation with Integer Exponents section (it matters very little whether the base is real or not). -- FactSpewer ( talk) 05:37, 26 November 2008 (UTC)
I don't feel strongly about the positioning of the "powers of i" section, so if you do, then let's just leave it where it is. By the way, could you explain what you meant by "general principles"? Also, why is the definition of i^n more complicated than the definition of r^s for real numbers r and s? I would have said that the opposite is true (look at how much space is devoted to the definitions). I'm not sure I understood the points you were making here.
I agree that having the "powers of negative real numbers" section refer to the complex number section is reasonable.
I like the spiral picture for logs too! It hasn't been deleted from Wikipedia, but just moved to the complex logarithm page. Same for the color map; these items are primarily relevant to the complex logarithm.
The removal of the sentence "The computation of complex powers is facilitated..." was unintentional; sorry about that; I'll try to put it back. On the other hand, probably that section that it refers to could use some trimming too. Do we think that exponentiation formulas like
are worth remembering? -- FactSpewer ( talk) 01:01, 27 November 2008 (UTC)
Regarding "same picture in different articles": agreed. I just didn't feel that these pictures, though, fit the context very well here. So I feel that it's better not to include those pictures here.
Moving Complex power of a complex number after Roots of arbitrary complex numbers is an interesting idea, but I think it is better to keep it as it is, given that this is an article about exponentiation. My feeling is that the section on roots is there just to explain how complex exponentiation (after it is defined) relates to the concept of roots.
I'll try adding your idea to have the -1 section refer forward to the i section. -- FactSpewer ( talk) 20:04, 29 November 2008 (UTC)
What about the speed of calculation is it polynomial timme? -- Melab±1 ☎ 01:12, 30 December 2008 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 |
From my abstract algebra knowledge of a group, there is no operation called division. Why is it shown here? The proper way to write the inverse is just x^-1 never 1/x, which is not meaningful in an algebraic group. Perhaps this should better be explained in the page, and remove references to 1/x. In general math, yes, 1/x is certainly a valid way to write x^-1, but not in abstract algebra. Conceptually we all handle it the same way, convert 1/x to x^-1 but 1/x is not the operation 1 divided by x. It is the multiplicative inverse of x, that's all. In the group Z3, under multiplication, there is no element 1/2, since there are only elements 0,1,2. However, 1/2 means 2^-1 which is defined, and ultimately 2^-1 = 2. This is easier to explain than the nonsense that results by using the operation division, which would be 1/2 = 2... 24.34.198.111 00:27, 27 April 2007 (UTC)
Sorry, I had to correct my question, I have made several mistakes when I wrote it before.
What I meant was:
In the computation of power for two complex numbers a and b as:
ab
it's used that:
ab=eb log(a)
But then, in the section Failure of power and logarithm identities, it is said that the identity log(ab)=b log(a) do not hold in general, and it's prooven with an example.
How is it possible then to equal ab=eb log(a) , since you need to use the property log(ab)=b log(a) to write it in that way? What makes that true in this case if a and b are any complex numbers?
Kaexar 01:50, 19 July 2007 (UTC)
But you made me notice that I wrote my question wrong. Thanks again.
Kaexar 01:50, 19 July 2007 (UTC)
Makholm and CBM. Note that the power series of e^x is not used in this elementary article at all. Nor is the differential equation or the continued fraction. All that stuff is taken care of in the article exponential function. This article on exponentiation is aimed at readers who do not know about exponentiation already. Superfluous material is likely to confuse the reader and make him stop reading. Holes in the argumentation likewise. The power series is a hindrance for reading and understanding. The limit without explanation is a hindrance for reading and understanding. Not everything that is true should be written everywhere. It should serve a purpose. Bo Jacoby 17:16, 8 August 2007 (UTC).
The limit definition of ex is used to show definition compatibility. The power series is important, but not here. To many readers a formula is a hindrance, and an unnecessary formula is an unnecessary hindrance. If ex is defined by the power series the reader will think:
If, on the other hand, ex is defined by the limit the reader will think:
To the service of the reader we should stick to the limit definition. You are welcome to explan |n|→ ∞.
Bo Jacoby 14:15, 9 August 2007 (UTC).
The limit definition is not simpler than the power series definition, but the definition compatibility is much easier to see using the limit definition than using the power series definition. Do you understand the argument above? Of course people reading mathematics should read formulas, but the formulas should serve a purpose, and the power series formula serves no purpose in this context. Why do you want to make these changes, Carl? Bo Jacoby 22:43, 10 August 2007 (UTC).
By now the article does not even assert compatibility between the two definitions. The main article on exponential function does not explain the compatibility either. The reader is left clueless. The number e is not explicitely defined, and the two formulas for ex are neither shown to be equal to one another nor to be equal to the old definition of ex for integer x. So the subsection is useless to the beginner and comprehensible only to the reader to whom it is superfluous. The important formula ex = limn(1−x/n)−n, which before was included as ex = lim|n|(1+x/n)n, has now been omitted, and it is not found in exponential function. The proof for definition compatibility does not really require advanced real (or complex) analysis, because if convergence fails no alternative definition is offered and so no incompatibility issue occurs. You are welcome to include convergence arguments in exponential function, but exponentiation should be kept elementary. Your present edit is a backwards step. I do not question your good faith, but you should take more care. Bo Jacoby 07:59, 13 August 2007 (UTC).
The name of the article exponentiation is attractive to beginners, (unlike for example Quantum gravity which is more scary), and the article links to the main article, exponential function for readers who want details. The purpose of introducing the exponential function ex in this article is to prepare the way for the exponentiation ax. Bo Jacoby 11:30, 13 August 2007 (UTC).
In the context of complex analysis the limit definition ez = limn(1+z/n)n, is equally important, and the geometrical interpretation of ei·x = limn(1+i·x/n)n is more straightforward than the geometrical interpretation of ei·x = Σn(i·x)n/n! Bo Jacoby 19:24, 13 August 2007 (UTC). Note also that the subsection you changed, Exponentiation#Powers_of_e, is part of the section Exponentiation#Exponentiation_with_integer_exponents, where it used to belong but no longer belongs. Please clean it up. Bo Jacoby 06:17, 15 August 2007 (UTC).
In " If a is a real number, and n is a positive integer, then the unique real solution with the same sign as a to the equation \ x^n = a is called the principal n^th root of a, and is denoted \sqrt[n]{a}."
When you say "the unique solution" is it not implicit that there always is one? (not the case!!) And why use the "the same sign"? (what about a=0??) Wouldn't it be more clear/didactic to explain what happens dividing in cases, the n's into even or odd and the a's into positive, 0, and negative?
Why use the term "principal n^th root"? It only makes real sense after someone studied complex numbers, which still takes a few years for most students. Ricardo sandoval 04:28, 24 August 2007 (UTC)
I was pointing to the absence of principal even root of negative numbers as in n-th root. So the term "the unique" doesn't apply. Someone could define the principal nth root for these number but they would not be real anyway.
I see the text is an improvement over the previous one (to which you referred) and it is even better now. But there is still the problem above. When remarking about the nth root of zero, I was pointing to the more trivial fact that zero has no sign, in the usual sense ( I am not considering the sgn function as a definition of sign). Although someone should expect the principal nth root of someone "without sign" (0) should also be "without sign" (0), this is not stated explicitly. Ricardo sandoval 03:43, 25 August 2007 (UTC)
In the section Rational powers of positive real numbers it is implicit that the result is also positive, i.e that the negative root should not be taken, or the principal value should be used. I think that should be made explicit. —Preceding unsigned comment added by TerryM--re ( talk • contribs) 00:11, 4 November 2007 (UTC)
This article has been reviewed as part of Wikipedia:WikiProject Good articles/Project quality task force. In reviewing the article against the Good article criteria, I have found there are some issues that need to be addressed. The article is under-referenced. Also, it's not specific enough for stating references in general. I am giving seven days for improvements to be made. If issues are addressed, the article will remain listed as a Good article. Otherwise, it will be delisted. If improved after it has been delisted, it may be nominated at WP:GAC. Feel free to drop a message on my talk page if you have any questions. Regards, OhanaUnited Talk page 21:32, 8 September 2007 (UTC)
Whoever can take care of the images that display equations, check out the first "Justification" under the heading "Zero to the zero power." It claims X^X -> 1, where it should be X^0 NinjaSkitch 06:06, 1 October 2007 (UTC)
I realize that "a to the power n" is correct terminology, but there is another correct notation that could be included there, "a to the exponent n". My math teacher tells us that that is the correct form, and that "a to the power n" is incorrect. However, I don't believe her, and I think both are correct. Is my math teacher wrong, or is she right? ZtObOr 01:42, 9 October 2007 (UTC)
EDIT: I forgot to include this bit. My math teacher's reasoning is that since the "n" in "a to the power n" is called the exponent, that it should really be called "a to the exponent n", since the word "exponent" refers to the exponent "n" in the first place, even if the word "power" is in its place.. —Preceding unsigned comment added by Ztobor ( talk • contribs) 01:45, 9 October 2007 (UTC) (Edited 4/10/2008)
Perhaps it is too pedantic to say that "a to the power n" is not a notation; it is a verbal description, or terminology. The notation is . As to the answer to your question: is the exponent, as your teacher rightly says. It is also the power. The exponent is the symbol that we use to designate a power. Therefore I prefer to say "a to the power n" as it doesn't confuse the thing with the symbol that we use to describe it. This is rather like the distinction between a number and a numeral. Of course you may choose to finesse the problem by saying "a to the n". TerryM--re 00:01, 4 November 2007 (UTC) I should add that the term index is also almost synonymous with exponent. In fact now that I think about it, I'm not sure that the terms are well defined; they are certainly used almost interchangeably by many people. TerryM--re 00:31, 4 November 2007 (UTC)
Is there any article to link to, to mention the cis function? This function can make working with Euler and De Moivre's formulas a lot easier. It is clearer to write on paper and leads to less mistakes, not to mention faster. An online example of cis, although not specific: http://oakroadsystems.com/twt/sumdiff.htm
Also, a link to an article describing implementing complex exponention in software: http://www.efg2.com/Lab/Mathematics/Complex/Numbers.htm The code is in Pascal/Delphi. JWhiteheadcc 06:05, 2 December 2007 (UTC)
This section, when I came upon it, claimed that Cantor's theorem implies that in the context of calculus. This is nonsense. If is interpreted as the limit of 2g(x) as x goes to infinity (where g(x) is a real valued function which goes to infinity), then this is just a limit of real numbers and not some kind of cardinal which is "larger" than another infinite limit of real numbers. Not larger in the sense of Cantor's theorem, anyway. This is not a cardinal.
So I fixed up the section so that it at least says correct things instead of nonsense, but I don't think it's a great section. I wouldn't mind if someone just deleted it, but then someone else would probably repost the same misconceptions later on.
By the way, I don't know where on earth I could find citations for this kind of thing. It's clear to any mathematician familiar with the relevant definitions. —Preceding unsigned comment added by 70.245.113.97 ( talk) 01:43, 11 December 2007 (UTC)
Am I right in thinking that tends to when n>1, remains at 1 when n=1, tends to 0 when abs(n)<1, and tends to when n<-1? Note: This being my first contribution, I don't want to make changes directly, in case of misinterpretation of the intent of the article! MessyBlob ( talk) 18:48, 5 April 2008 (UTC)
In this section, it is heavily implied that the Microsoft Calculator is a programming language, which is obviously wrong. Maybe it should be kept for simplicity or perhaps a rewording of the statement is needed. —Preceding unsigned comment added by 88.108.69.114 ( talk) 19:37, 17 May 2008 (UTC)
the only way i see this = 1 is the ^0 is subtracting 1 less so 0^0 = 0-(-1)= 1 —Preceding unsigned comment added by 208.91.185.12 ( talk) 21:31, 2 October 2008 (UTC)
I'm surprised the 00 section does not mention all that Knuth has said in his paper Two Notes on Notation. Can someone add it? The paper: TeX source, doi: 10.2307/2325085, arXiv: math/9205211.
Just to follow up on this: The article now cites it (more precisely, it cites the published version instead of the preprint). -- FactSpewer ( talk) 01:58, 18 October 2008 (UTC)
I'm not a math person, but I think I see an error in section 1.5 Identities and properties and would like someone help me check this out.
In the last line of the section the statement is a^b^c = a^(b^c), but the example preceding this seems to show just the opposite.
Then there is an inequality a^b^c ≠ (a^b)^c. Should this be an equality not an inequality?
And, finally, the article states "Without parentheses to modify the order of calculation, the order is usually understood to be from right to left". Should this be "from left to right"?
```` —Preceding unsigned comment added by Lou27182 ( talk • contribs) 16:14, 27 July 2008 (UTC)
The arguments for defining to be 1 are overwhelming even in analysis, and I feel that the article should be rewritten to reflect this. For instance, in addition to the analysis reasons already given (the power series for and the derivative of ), there is the case in the geometric series formula
itself a special case of the binomial formula for general exponents.
I feel that the arguments for leaving undefined stem from a misguided insistence that functions be continuous or differentiable on their domains. Here are rebuttals to the arguments currently in the article:
The best solution is to define , and to accept that the function will be discontinuous at and that no holomorphic branch of will exist in a neighborhood of 0. Of course, defining is not the same as saying that whenever and are tending to 0; the article should remain clear on this point.
BjornPoonen ( talk) 05:31, 11 October 2008 (UTC)
I agree with BjornPoonen. Your objections are to the point. The quote: "it may be best to treat 00 as an ill-defined quantity" is unjustified and without references. Bo Jacoby ( talk) 10:52, 11 October 2008 (UTC).
Thank you, Salix and Dmcq, for your comments, and in particular for pointing out the archive (even if it does not contain any arguments resembling my analogies with and ). Your arguments that the article should continue to mention that some authors leave undefined are well taken, and I agree with you on this point. What I feel is unjustified is the implication that the definition is useful only in the discrete side of mathematics. I would suggest removing "There are two principal treatments in practice, one from discrete mathematics and the other from analysis." I would also suggest removing the blanket statement "In general, mathematical analysis treats as undefined...". Citing some textbooks that leave it undefined does not mean that this is the general approach in analysis, when other textbooks define . BjornPoonen ( talk) 17:26, 11 October 2008 (UTC)
The changes made have improved the article and I wish to thank the editors involved. I dislike, however, the following sentence: "The rule in calculus that limx→a f(x)g(x) = (limx→a f(x))limx→a g(x) whenever both sides of the equation are defined would fail if 00 were defined", (my italics), because the italiced assertion is not a general rule of calculus, and because it refers to 00 being defined which we have just allowed, and because an equation (defined number)=(indeterminate form) is as false as (defined number)=(another defined number) and (defined number)=(undefined number). I hope we can agree to omit the sentence. Bo Jacoby ( talk) 12:30, 18 October 2008 (UTC).
After looking it over, I too think that the "rule of calculus" sentence should be removed or at least qualified, but for a different reason than Bo Jacoby: according to the definitions in the article,
Should we interpret this as saying that (-2)1 should be left undefined? (I hope not! That would be a disaster!) -- FactSpewer ( talk) 15:55, 18 October 2008 (UTC)
I agree that the limit of a function f(x) as x approaches a can be defined without f(a) itself being defined (I gather that this is what you meant by your remark about limit points), but the point I was trying to make was this:
In summary, I think this sentence about the "rule of calculus" sentence really needs to be removed. -- FactSpewer ( talk) 06:44, 20 October 2008 (UTC)
Dear Dmcq, I appreciate your effort to explain the "rule of calculus" sentence to me, but I am still unclear as to what it is trying to say. I have no objection to keeping it if it can be made precise and a reference for it (in the form you are stating it) is given. Let me suggest a couple possible readings of the sentence (expanding it in more detail, just to make sure we are on the same page), and ask you which one you mean.
I believe it is trying to say the following:
My question is what is meant by "exponentiation is continuous". Which of the following is intended as a hypothesis for the application of the rule?
(Admittedly, some of these would render the "rule" false or useless.) Of course, I'm not suggesting that we write all this in the article.
Finally, in what textbook do you find this "rule of calculus"? For the time being, I'll flag the sentence as needing a citation.
Thank you, -- FactSpewer ( talk) 02:47, 22 October 2008 (UTC)
I know it isn't a primary source but perhaps you an agree the statement at the start of Indeterminate form is correct:
It then goes on to say 00 is an indeterminate form and if you look in any textbook they'll say the same thing. Now if the algebraic operation after replacing the subexpression gives a defined number then the substitution does give enough information to determine the original limit. This is quite different from the case of Pow(x,y)=xy where (x,y)≠0 and Pow(0,0)=1. This does not lead to an indeterminate form. One cannot say that limit Pow(x,y) = Pow(limit x, limit y) if both sides are defined. Pow is not an algebraic operation in the terms of the definition.
Of course it would be possible to say the usual rules for indeterminate forms only applies for addition, subtraction, multiplication and division but don't apply for exponentiation. But that is not what is done. What is done is that 00 is treated as an indeterminate form.
As to the question about continuity the answer is number 3. Think of the same question as applied to the indeterminate form 0/0. If it had a defined value then it wouldn't be an indeterminate form. And division isn't continuous at 0,0. If divide(x,y)=x/y except divide(0,0)=1 was defined then it couldn't be used as an indeterminate form where one could move the limits inside the brackets. Without the defined value divide(0,0)=1 we could use it as an indeterminate form with a bit extra about moving limits in this particular function. Indeterminate form is just a special case of a function with a discontinuity that has no defined value at the discontinuity and is used because people didn't want to have a slew of special rules for dealing with straightforward expressions. Dmcq ( talk) 09:35, 22 October 2008 (UTC)
To keep the discussion constructive, I'd like this new section to be devoted exclusively to feedback on the proposed draft. Absolute statements that 0^0 is 1 or that 0^0 is undefined can go elsewhere.
No one over the past few days has had any objection to the proposed corrections outside the 0^0 section, so I will now restore those corrections, while leaving the 0^0 section untouched for the time being.
As for the proposed 0^0 section, a few people on the Talk:Exponentiation page have come out generally in support of it while also saying that it will need some minor fixing that can be done later. I believe the only person so far who has explicitly said that it is not an improvement is Steven G. Johnson. Steven: I would like to understand better what specifically in the new draft you are objecting to. I think we all agree with you that several references state explicitly that 0^0 is undefined, but I do not understand why this is incompatible with the proposed draft. Indeed, the proposed draft cites some such references, while also citing references for the opposite point of view, that 0^0 is 1. -- FactSpewer ( talk) 18:10, 16 October 2008 (UTC)
I just updated the proposed draft. Now it contains only the 0^0 section (the corrections in other sections having been implemented already). I also attempted to incorporate the suggestions made by those of you who commented on it. In particular, the opening sentences were rewritten so that it starts less abruptly, and so that it begins the discussion in a balanced way. Feedback on this draft is welcome, and should go here, in this section of Talk:Exponentiation. -- FactSpewer ( talk) 15:53, 17 October 2008 (UTC)
The sentence
looks mathematically incorrect to me.
Consider the identity log ab = b log a, for instance, for complex numbers a and b with a ≠ 0.
If we are treating log as a multivalued function, and A denotes one possible value of log a, then log a is the set of numbers of the form A + 2πin, where n ranges over integers. So b log a is the set of numbers of the form b(A + 2πin).
On the other hand, ab is defined as exp(b log a), which is the set of numbers of the form exp(b(A + 2πin)), so log ab is the set of numbers of the form b(A + 2πin) + 2πim, where m and n range over integers.
So the set log ab contains the set b log a, and for most values of b, it is strictly larger, meaning that the identity fails. -- FactSpewer ( talk) 01:30, 17 October 2008 (UTC)
I am unhappy with some of the very recent edits by Dmcq:
-- FactSpewer ( talk) 06:19, 20 October 2008 (UTC)
I've had a go at the multi-valued bits at the start of the complex exponentiation section and for Clausen's paradox and tried to make them a bit less off-putting. Dmcq ( talk) 08:00, 21 October 2008 (UTC)
The present structure of the article is that powers of real numbers are treated before powers of complex numbers. I suggest that powers of positive real numbers are finalized before powers of negative real numbers and powers of complex numbers, because the discontinuous rational powers of negative numbers are relatively unimportant and confusing. Comments, please. Bo Jacoby ( talk) 11:46, 20 October 2008 (UTC).
1 Exponentiation with integer exponents
2 Powers of positive real numbers
3 Powers of negative real numbers
4 Complex powers of real numbers
5 Powers of complex numbers
QUOTE:
UNQUOTE
In the chain of equalities
the weak links are equality sign number two and five, and not number three and four, no matter how is defined.
Bo Jacoby ( talk) 12:24, 20 October 2008 (UTC).
In the chain
the weak links are equality signs number three and six, and not number four and five, no matter how or if is defined. The point is whether is continuous or not.
If you are allowed to write then you are also allowed to write and if you are allowed to write then you are also allowed to write Bo Jacoby ( talk) 15:37, 20 October 2008 (UTC).
The "limit rule" is invalid independently on whether 0/0 is defined or not. A definition of 0/0 does not invalidate any legitimite calculation. The weak link is "lim(2x/x)=(lim 2x)/(lim x)", not "(lim 2x)/(lim x)=0/0" which is true because lim 2x=0 and lim x=0. (Her "lim" means "limx→0"). Similarily, defining 00 is not harmful in any context, and 'undefining' 00 solves no problem at all, and the 'justifications' for leaving 00 undefined are logically invalid. Bo Jacoby ( talk) 10:40, 21 October 2008 (UTC).
Defining 0/0 solves no problem, but defining 00 does solve a problem, and creates no problem. It is OK to report that some authors do not define 00, but it is not OK to report logically invalid justifications which seem to be original research. See WP:OR. Bo Jacoby ( talk) 14:21, 21 October 2008 (UTC).
The rule in calculus that lim x→a f(x)^g(x) = (lim x→a f(x))^(lim x→a g(x)) whenever both sides of the equation are defined and exponentiation is continuous is not a general rule of calculus but was coined for the purpose. "If 0^0 was defined as 1 then both sides of the rule would be defined giving the false conclusion that 1/e = 1" is not true, because exponentiation is not continuous if 0^0 is defined. Bo Jacoby ( talk) 16:08, 21 October 2008 (UTC).
Isn't the section on Exponentiation#Branches of the complex logarithm a bit over the top? I don't think this article needs anything quite like that if it can be dealt with better in the branch cut or complex logarithm articles. I certainly was quite reluctant to put in even the bit about Riemann surface but I though a quick reference and a picture were called for. Dmcq ( talk)
Dear Dmcq, Thank you for the new sentence in the 0^0 section. It's better than what was there before. We might still consider removing the sentences
since they are redundant with what came before and with being a counterexample to Moebius's statement. Anyway, I leave it to others to decide this.
-- FactSpewer ( talk) 00:01, 25 October 2008 (UTC)
See the example above in "redundant equations", by Factspewer. This example does not show that 0^0 is undefined in the algebra of limits for real numbers, it is much more basic than that! Suitably altered, it shows that exponentiation cannot be, in general, defined on the real numbers. In particular, there is no sensible way to define the binary function of exponentiation for the input pair of real numbers (0,0).
This is where it helps to know the (a) construction of integers in terms of numbers (0,1,2,3,4,5,... for which there can be no conceptual reduction - you can only draw a tally in the sand and wave your arms around until people understand what you mean), rationals in terms of integers, and reals in terms of rationals. More precisely for example, we have a definition of integers in terms of equivalence classes of number-number pairs, rationals in terms of equivalence classes of integer-integer pairs, reals in (for example) terms of equivalence classes of Cauchy sequences of rationals. Knowing about this sort of thing can clarify things. Some may perjoratively call this stuff "logic", but it isn't really, it's just very nit-picking mathematics. But it is this sort of contemplation that led to the theory of p-adic numbers, so it is not entirely stupid.
Anyway, the confusion around "0^0=1" (see next paragraph for why the inverted commas), present even in professional mathematicians, results from a conflation of integers and reals and their power functions. These should technically not be conflated, it technically makes no sense (not wrong, nonsensical) to say the real number 5 equals the number 5. Of course we identify numbers with the corresponding real numbers via the embedding of N in R. Addition, multiplication and order are preserved, but exponentiation is not quite. This embedding does not allow one to talk about x^y, where x and y are real number variables ranging over R. So for the purposes of resolving talk about 0^0 where 0 signifies the real number 0, one has to abandon this habitual (usually harmless) identification of 0 and 0', 1 and 1', 2 and 2', etc.
Note that any phrase "0^0=..." makes no literal sense until you have clarified that you mean the real number 0. The binary power (to stop confusion with exp) function on numbers (0,1,2,3,4,5,...) can very well be defined at the number pair (0,0), by defining 0^0=1. This causes no problems and makes perfect sense. Likewise for integers. For reals it is not so simple, in fact there is no way to define exponentiation at (0,0).
Let us use the Cauchy sequences definition of reals. Then in the above example write 2 instead of e and write 1/n instead of x, to get a convergent (and hence Cauchy) sequence. Then insist that power:RxR->R is such that 0^0=1. Bang! You have destroyed the Cauchy definition of the real number system.
Okay, then we reject power:RxR->R being such that 0^0=1. What about the stuff raised in the article as a reason why 0^0=1 might be good to have, for example infinite series for e^x at x=0, geometric series at x=0, binomial theorem at x=0, power rule for derivatives at n=1 and x=0, etc. We want to get that sorted. And we can, because nothing I said above indicates that we cannot have a function power:RxN->R with all the usual properties. R is a field, and this is just the usual field exponentiation applied to R. Naturally we can have power:RxZ->R also.
In power:RxZ->R, power(0,0) is defined and the usual interpretation of power(0,0)=1 is true. In power:RxR->R, power(0,0) is undefined. We cannot have a binary operation on R with the properties that we want a power function to satisfy, that is defined everywhere. Se la vi, there isn't always a function that does what you want. Bozo9 19:59, 25 October 2008
The assertion is false even if 00 is undefined. That is just an elementary observation made by a reader who need not write a book about it. It is not a reasonable job to write a false statement in the article. Bo Jacoby ( talk) 14:25, 26 October 2008 (UTC).
IEEE evaluates the boolean expression (e^(-1))=NaN to false, and (e^(-1))=1 evaluates to false, so, also according to IEEE, is false irrespectively on whether 00 is NaN or 1. Thank you for this referenced authority in support of my point of view. What I feel has nothing to do with it. Bo Jacoby ( talk) 16:04, 26 October 2008 (UTC).
That's how much discussion has been spent on the 0^0 question in this Talk page to date. Realize that you're not going to convince someone who has been arguing with essentially all other editors on this subject for two years now.
To quote CMummert from last year:
Need we continue now? —Steven G. Johnson ( talk) 19:17, 28 October 2008 (UTC)
The derivation that e^x = e^k for integer x=k was removed by Dmcq. Why? Bo Jacoby ( talk) 00:05, 30 October 2008 (UTC).
What was the problematic step? Bo Jacoby ( talk) 00:17, 30 October 2008 (UTC).
I think that, correct or incorrect, the derivation isn't needed. In an article at this level, the typical reader isn't going to be interested in or aided by proofs. Moreover, the notation in the derivation that was there,
involves things like "|n| → ∞" that are not typically encountered in a first calculus course. Also, e itself is defined as
higher in the article (without the absolute value). I don't see that the appearance of the absolute value would be at all clear to a naive reader. But I also don't think it would be worth writing a long explanation for this derivation, which is already a tangential point in the article. — Carl ( CBM · talk) 01:03, 30 October 2008 (UTC)
To Dmcq. Thank you for the apology. I hope you get better. To Carl. When k<0 and n>0 then m=nk<0 , and one needs the limit The exponential function is central to the generalization from integer to noninteger exponents. Without a hint about the new definition of e^x being a generalization of the old definition, the notation e^x is unmotivated. The text can be improved but I don't think it should be omitted. Bo Jacoby ( talk) 08:54, 30 October 2008 (UTC).
There is an article on exponential function, and an article on e, so I think that this subsection of the article on exponentiation should not contain more detail than what is used in the later subsections. That's why I removed a sentence. I do not think that my change made the text neither inaccessible nor aimed at a special audience. (Remember the rule: improve rather than revert). I made the change for a reason. It is important to note that not only must the exponent be large, but also the deviation from unity be small in order that Euler's indeterminate form be finite. (Happily we are not discussing indeterminate forms here). So the words "while the number goes to one" were included, as I still think they ought to be. An explanation why (e)k=ek for negative integer k is still missing, but an explanation why ex+y = ex·ey would suffice, I think. Then we could also omit the proof that (e)k = ek for positive integer k. I don't consider the number 1.0011000 to be an 'inline proof'. Bo Jacoby ( talk) 14:34, 31 October 2008 (UTC).
Let's take one step at a time. Do you object against "and is defined as the limit as the power goes to infinity" being replaced with "and is defined as the limit as the power goes to infinity while the deviation from one goes to zero" ? Step two. Do you think that the reader cannot identify 1.0011000 and (1+1/1000)1000 ? 15:58, 31 October 2008 (UTC).
The function exp(x) is important because it equals its own derivative, and hence arises in any application where something grows or decays in proportion to its current value. The number e is important not because it is the limit of (1+1/n)^n, but because it is the number such that e^x = exp(x) and hence gives a convenient notation for the exponential function that reminds the user of its multiplicative property.
With this in mind, one possibility might be to define exp(x) first (as the function equaling its derivative with exp(0)=1, and/or by the power series, and/or by the limit - probably it is best to mention all three, with the first one being primary), and then to state that there is a number e such that e^x (as defined earlier in the article) equals exp(x). Then there is no confusion between the notations, and as a bonus the formula that exp(k) equals e.e....e is an automatic consequence. -- FactSpewer ( talk) 01:44, 1 November 2008 (UTC)
I think that the article on exponentiation should explain ax where a is a positive real number and x is an arbitrary complex number. The exponential function ex is a convenient intermediate step. (This is to answer FactSpewer's question of the purpose of the section on ex). There are two common definitions of ex , namely the limit lim(1+1/n)n and the Taylor series Σxn/n! . The power an where n is an integer has just been explained, but some readers may at this stage not be comfortable with the theory of derivatives, which is more advanced than the theory of exponentiation. That's why I preferred the limit definition to the Taylor series definition in this context. Perhaps we should have a very short big exponent subsection , telling that limn→∞an is zero for |a| < 1 and infinite for |a| > 1 and equal to unity for |a| < 1 and undefined elsewhere, e.g. for a = −1 . This motivates investigating a big power of a number close to one. It is interesting that (1+x/n)n is virtually independent on n for big values of n. (1% of interest in 10 years almost equals 0.5% of interest in 20 years). I don't know an equally elementary motivation for studying the Taylor series. I think that the equally easy formulas lim(1+(x+y)/n)n = lim(1+x/n)n·lim(1+y/n)n and Σ(x+y)n/n! = (Σxn/n!)·(Σyn/n!) should be found in the article on exponential function. Bo Jacoby ( talk) 06:30, 2 November 2008 (UTC).
The two reasons that Dmcq gave are the same as the ones I would have given, except that it is not needed for real exponentiation (which has already been defined via the limit process!) Given that these are the only two goals, and that most of the material here fits better in either the e article or the exponential function article, I would suggest eliminating this section and placing the following sentence in the section where exponentiation of positive real numbers is defined.
And then of course, the function e^z would be mentioned later on when defining complex exponentiation. Is there anything else that needs to be said here that is not redundant with what is (or should be) in the e and exponential function articles?-- FactSpewer ( talk) 08:07, 3 November 2008 (UTC)
it seems as if Dmcq and I agree that ex should be defined here, as it is, in order to prepare for defining ax, but that FactSpewer disagrees. Bo Jacoby ( talk) 22:03, 3 November 2008 (UTC).
The latest reordering of the subsections is not satisfactory. The value of (1+n−1)n for big integer values of n belongs naturally in the subsection big exponents of the section on integer exponents. The definition of ex belongs naturally under the heading powers of positive real numbers, and equally naturally immediately after the definition of e, which was last in the previous section. The characterizations of a1/n = e(ln a)/n as the positive solution of the algebraic equation xn = a should be included, but not confuse the chain of logic. Bo Jacoby ( talk) 22:53, 5 November 2008 (UTC).
For big values of n, (a+bn−1+cn−2+···)n goes to infinity if a > 1 and towards zero if |a| < 1 , and towards a positive value, (eb), if a = 1. These facts belong to big exponents. Bo Jacoby ( talk) 00:00, 6 November 2008 (UTC).
How then make the logical connection between the integer exponents and the exponential function? Bo Jacoby ( talk) 07:56, 6 November 2008 (UTC).
If a section is called powers of e, then e should be defined or explained already. Otherwise the reader cannot understand the header. The reader should be able to read exponentiation without having to study exponential function. The dictum of repetition is fine for oral presentations, but not necessarily for an encyclopedia, where the reader is free to reread several times. I would hate to sacrifice logic on the altar of smalltalk. Bo Jacoby ( talk) 12:13, 6 November 2008 (UTC).
I don't think we disagree as much as it seems, nor that I remove explanations in the beginning. I am the one who want to define e before it is used. :-) Bo Jacoby ( talk) 13:30, 6 November 2008 (UTC).
I know what I did and I know what you did: you introduced a big gap between big exponents and powers of e which are closely connected logically. The algebraic definition of fractional exponents does not lead to complex exponents, but you gave it priority in favour of the powers of e section that is the road towards generalization of exponentiation. Yes, here we differ. But we do not differ in our attempts to write a good encyclopedia. You are not right in assuming that I want to remove explanations. Bo Jacoby ( talk) 23:30, 6 November 2008 (UTC).
I'd like to have
Both serial and random access reading of the article should be possible. Bo Jacoby ( talk) 09:08, 8 November 2008 (UTC).
I have raised mention of this at Wikipedia_talk:WikiProject_Mathematics#Exponentiation wars Dmcq ( talk) 13:36, 8 November 2008 (UTC)
Bo Jacoby ( talk) 23:11, 8 November 2008 (UTC).
It occurs to me that I don't have a standard name for the basic identity ab+c=abac. Is there one? Dmcq ( talk) 09:55, 1 November 2008 (UTC)
There may be names for it, but I don't think they are universally used. When a is e, I think some people call it the addition formula for the exponential function, or the functional equation for the exponential function. I'd suggest leaving it unnamed.-- FactSpewer ( talk) 03:43, 2 November 2008 (UTC)
The notation in big exponents is fairly obvious, but not the usual notation used elsewhere that I know of. Is this a standard way of writing limits anywhere? I found I had problems translating it into English, the 'for' in the statement I translated as 'as'. Dmcq ( talk) 16:07, 5 November 2008 (UTC)
About Loadmaster's edit: Does Wikipedia have a standard for hyphenation in words like nonzero? If not, is there a notable reference that recommends inserting a hyphen? Table 6.1 of the Chicago Manual of Style lists non as a prefix forming closed compounds --- nonzero, nonnegative, etc. WardenWalk ( talk) 21:36, 8 November 2008 (UTC)
I'm wondering if the rule for negative reals with odd integer powers holds when a non-integer power is split into a product having an integer exponent and a fractional exponent. Given a non-integer a and non-zero x, then for some integer n such that a = n+f,
Does it then follow that if x is a negative real and n is an odd integer, that the result has a negative real component? For example, is it true that
Compare this to an equivalent equation that uses an even integer n:
It appears that the two results differ in sign, which seems to be inconsistent. Another example:
Is this correct, or did I miss something? — Loadmaster ( talk) 20:27, 11 November 2008 (UTC)
I like this section very much. However, in the discussion of the limit of f(x)g(x) when f(x) and g(x) tend to 0, these functions are required to be "positive-valued". To me, this makes perfect sense for f. But it seems pointless for g, unless we want to allow f to take the value 0, which is of little benefit.
I would suggest we require f to be strictly positive, and remove any restriction on the sign of g. That way, the possible limits can be anything in [0,+∞], instead of [0,1]. The part about [0,1] gives one the misleading impression that 00 is really trying to be a number between 0 and 1, if one doesn't pay close attention to the restriction imposed on g. 67.150.253.241 ( talk) 14:39, 15 November 2008 (UTC)
I looked back and there used to be a section 'Powers with infinity'. I propose bringing something similar back to be a real or complex number equivalent to Large exponents the main purpose of which would be to talk about real and complex limits, and I'd put it just after the 0^0 section. I note a bit has just been added to the 0^0 section which would fit nicely as a basis for the new section. The new section would duplicate a bit in the current Large exponent section but that would be unsuitable for adding to as it is right at the top dealing with integers. Dmcq ( talk) 12:28, 19 November 2008 (UTC)
About a month ago, I tried rewriting the complex logarithm material in exponentiation, but the consensus here was that the material belonged in the complex logarithm article instead. An expanded version of this material has since been put there, so I will now trim the complex logarithm material in exponentiation, replacing it with links where appropriate.
The section on powers of the imaginary unit now seems out of place. The main dichotomy in the definitions of exponentiation is between integer exponents and real/complex exponents. Integer powers of i belong in the Exponentiation with Integer Exponents section (it matters very little whether the base is real or not). -- FactSpewer ( talk) 05:37, 26 November 2008 (UTC)
I don't feel strongly about the positioning of the "powers of i" section, so if you do, then let's just leave it where it is. By the way, could you explain what you meant by "general principles"? Also, why is the definition of i^n more complicated than the definition of r^s for real numbers r and s? I would have said that the opposite is true (look at how much space is devoted to the definitions). I'm not sure I understood the points you were making here.
I agree that having the "powers of negative real numbers" section refer to the complex number section is reasonable.
I like the spiral picture for logs too! It hasn't been deleted from Wikipedia, but just moved to the complex logarithm page. Same for the color map; these items are primarily relevant to the complex logarithm.
The removal of the sentence "The computation of complex powers is facilitated..." was unintentional; sorry about that; I'll try to put it back. On the other hand, probably that section that it refers to could use some trimming too. Do we think that exponentiation formulas like
are worth remembering? -- FactSpewer ( talk) 01:01, 27 November 2008 (UTC)
Regarding "same picture in different articles": agreed. I just didn't feel that these pictures, though, fit the context very well here. So I feel that it's better not to include those pictures here.
Moving Complex power of a complex number after Roots of arbitrary complex numbers is an interesting idea, but I think it is better to keep it as it is, given that this is an article about exponentiation. My feeling is that the section on roots is there just to explain how complex exponentiation (after it is defined) relates to the concept of roots.
I'll try adding your idea to have the -1 section refer forward to the i section. -- FactSpewer ( talk) 20:04, 29 November 2008 (UTC)
What about the speed of calculation is it polynomial timme? -- Melab±1 ☎ 01:12, 30 December 2008 (UTC)