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Hi CMummert.
I wish to continue the discussion here on your talk page because it is getting somewhat personal. You wrote:
Your argument, that 'you do not need to define 0^0=1 if the expression 0^0 is everywhere replaced by 1', is strange and invalid. The same argument applies to 2+2: you do not need to define 2+2=4 if you replace 2+2 with 4 every time it occurs.
You also wrote:
Yet you undid my edit in a knee-jerk fashion without comment on the talk page.
The article should be readable to the non-expert, and it is bad that the unimportant discussion about 0^0 is polluting the article. In the elementary section on integer exponents there is no disagreement that the empty product has the value 1. Please clarify your position. Bo Jacoby 10:46, 12 January 2007 (UTC).
The section Exponentiation#Zero_to_the_zero_power do not pollute the article, but the explanations of 0^0 in Exponentiation#Powers_of_zero and in Exponentiation#Powers_of_zero_2 are not clear.
The only problem with the definition 0^0=1 is that some books do not include it. The discontinuity of x^y at (0,0) is a matter of fact, no matter whether 0^0 is defined or not. So, let's stick to the definition in the article to make it readable to beginners, let the subsection Exponentiation#Zero_to_the_zero_power mention that some books leave 0^0 undefined, let's move that subsection to the section Exponentiation#Advanced_topics (which the beginner need not read) and let's include the references as we do by now.
Bo Jacoby 16:33, 12 January 2007 (UTC).
The pointers to the section on 0^0 are OK. No problem. It makes sense to say that 0^0 is defined to be 1, and then later explain that some books doesn't define it.
'Tested experimentally' is not a mathematical argument. The integer interpretation of 2+2=4 may be tested by counting pebbles, but the real number interpretation cannot be tested by adding lengths. (The teacher explaining the slide rule: Two times two is three dot nine eight. The pupil: Shouldn't it be four? Teacher: No not if it has to be quite accurate.)
Even if some mathematicians don't accept the definition, they should stop preventing other mathematicians from using it.
Just as there is only one empty set, there is also only one empty product, even if it can be expressed in several ways, say 1^0 or 0^0.
The concepts used in the subsection Exponentiation#Zero_to_the_zero_power are more advanced than the concepts required to understand basic exponentiation. That is why it should be moved to the Exponentiation#Advanced_topics section.
The elementary sections of the article should be reserved for important explanations. The synonym 'involution' is unimportant. It is never used in modern mathematics.
Bo Jacoby 00:14, 14 January 2007 (UTC).
Trovatore: Thank you. You may start by undoing CMummert's revert and explain the situation to VectorPosse.
CMummert: The Advanced topics are advanced relatively to the elementary parts of the article, not in an absolute sense.
EdC: I agree that a^b has a combinatorial interpretation that can be tested experimentally but I do not understand your example. (I get only 5*3=15 lines).
VectorPosse: I am trying to improve the article which is polluted by repeated unimportant remarks on what some books do not define.
Bo Jacoby 08:13, 14 January 2007 (UTC).
I trust that VectorPosse agrees that 0^0 is a relatively unimportant special case of n^m. It does not deserve this level of attention in the article. I am not trying to make 0^0=1 the only valid view either, but it is the view actually assumed by every user of 0^0. There is no use of an undefined expression.
EdC: n^m is the cardinality of the set of m- tuples from an n- set.
Bo Jacoby 11:49, 14 January 2007 (UTC).
Here are some running notes from my major edit this morning:
I have a few comments.
Bo Jacoby 16:53, 14 January 2007 (UTC).
1. Read the first few lines of the article Euler's constant. If you know better, then improve the article Euler's constant. If you agree, then follow the recommandation and call it Euler's number rather that Euler's constant.
2. Please note that the proof of the valid expression for negative integer x depend on . It is not sufficient that . That is why I wrote . (I did not write ). You are wellcome to rewrite the argument avoiding the sign if you have trouble understanding what it means, but please repair the damage.
3. It was never said that e2·π·i·(1/n)·1 was the only primitive root. The deleted statement was perfectly sufficient and correct. It should be reinstated.
4. This elementary article on exponentiation should not assume the reader to know an algebraic definition of π. The usual definition is geometric rather than algebraic. The deleted line contained nontrivial information and was to the point.
5. You should clean up you mess yourself, but OK, I'll do it.
6. You are wellcome.
7. OK, I'll fix the typo in Exponentiation#Exponentiation_over_sets and extend the examples. Still this interpretation is important for understanding the meaning (or one important meaning) of exponentiation, which is independent on multiplication.
8. Do what you want to do, but make it understandable to the beginner. There is no point in making branch cuts or Riemann surfaces until you understand that otherwise the function would be multivalued. One cannot solve a unrecognized problem. But go ahead and try.
9. This is great news. If I write can be, will you then (please) stop reverting my edits regarding 0^0 ?
10. Let the same rules apply to yourself as to me. My edits were reverted without comment. I treat you nicer that you treated me.
Bo Jacoby 20:22, 14 January 2007 (UTC).
The reader does not know about exponentiation. He may know that π is the ratio between circumference and diameter of a circle, but he need not understand the connection to exponentiation. Please remember to use the words Euler's Number rather that Euler's Constant, as explained under point 1. The characterization above says that the sequence 'is an increasing sequence which is bounded above', which makes sense only for real x. For non-real complex x it is nonsense. For negative integer x it does not prove that the new and old definitions of ex give the same result. The logarithm is 'typically' considered for positive arguments only, but we are dealing with nonzero complex arguments. The branch cut is not needed in this article, but the multivalued log and fractional power are needed. There is no need any more to insert that 0^0 can be defined, because it is defined by the formal definition which you did accept above (point 9). War is over. Bo Jacoby 21:30, 14 January 2007 (UTC).
"Formally, powers with positive integer exponents can be defined by the initial condition a0 = 1 and the recurrence relation an+1 = a·an " is a formal definition. How can you make yourself write that "There is no formal definition of exponentiation in this article" ? Bo Jacoby 22:19, 14 January 2007 (UTC).
Yes, I am correct. No, I am not confused. 'Can be defined' means that other definitions are possible, but here is the definition of this article. When you change your mind in the middle of the article, the reader gets infected by your serious confusion. Whether 0^0 is 'universally defined' or not does not influence the fact that the formal definition in this article is as stated. And, as I have repeatedly pointed out, you have many more changes to make in WP in order to make your point of view consistent. Many many authors, tacitly or explicitely, assume a0=1 for all a, making no pointless exception for a=0, such as you stubbornly insist to do.
Regarding point 1 and 2: Remember to fix the two errors you introduced in Exponentiation#Powers of Euler's constant.
Regarding point 7. The combinatorial interpretation is about exponentiation of nonnegative integers, not about exponentiation of sets.
Bo Jacoby 07:36, 15 January 2007 (UTC).
CMummert · talk 14:16, 15 January 2007 (UTC)
Answer to the above:
Bo Jacoby 23:25, 15 January 2007 (UTC).
I added three references for the definition of a principal nth root of unity. I am sure that I could locate dozens of sources who use the same definition. Please don't remove the definition now that it is backed up by published sources. CMummert · talk 17:38, 15 January 2007 (UTC)
The subsection Exponentiation#Powers_of_negative_real_numbers adds more to confusion than to clarification, and it obviously does not belong under the heading Exponentiation#Real_powers_of_positive_real_numbers. The problem is naturally discussed in the more general framework of Exponentiation#Complex_powers_of_complex_numbers. Bo Jacoby 09:17, 15 January 2007 (UTC).
Still a subsection called Exponentiation#Powers_of_negative_real_numbers does not belong under the heading Exponentiation#Real_powers_of_positive_real_numbers because no negative number is positive. Please place it correctly. Bo Jacoby 16:53, 15 January 2007 (UTC).
Some time ago I did a lot of work improving this article. The present high activity is good, but it also contains some backwards steps. As I do not revert edits made in good faith, I may seem impatient, and I apologize. Don't you think that the subsection on powers of negative numbers needs a concluding remark saying that the appropriate context is that of powers of complex numbers? As by now the subsection is frustrating to read, leading nowhere. Bo Jacoby 23:47, 15 January 2007 (UTC).
Some time ago I did a lot of work improving this article.
The present high activity is good, but it also contains some backwards steps.
As I do not revert edits made in good faith, ...
Also, if a couple or more of editors tell you to drop something, then drop it, especially if you are not completely sure you perfectly understand the topic at hand. Oleg Alexandrov (talk) 05:09, 17 August 2006 (UTC)
I am referring to my editing a long time before Trovatore reopened the debate by suggesting to distinguish between 00 and 00.0 , an idea which is not mainstream mathematics and which would cause trouble. It was discussed, but the positions of some editors are still unclear, as there are contradictory statements of opinion. Then the discussion changed to whether 00 = 00.0 should be defined or not. The present formal definition of the article is that a0 =1 for all a, but afterwards it is stated that this definition is not accepted by everybody, so now the reader is left in unnecessary confusion. I edited the article myself, exactly as you suggested, but CMummert reverted my edits immediately. Therefore, rather than making an edit war, I politely suggested CMummert to correct the article himself. Of course we are all in good faith and we all want a great article on exponentiation, and my comments to (and from!) CMummert prove that I do understand the topic at hand. The formula for complex exponentiation is complicated and rather useless, and there is no need to make the reader believe that it is worthwhile learning it. Even the edit called "pinpointing nonsense for CMummert to clean up" actually contained clarifications of the troublespots, not 'directions to other editors'. You offer me two pieces of advice, which alas contradict one another: Advice no 1: "If you don't like the way the article reads, Bo, please edit it yourself". Advice no 2: "Please stop right now. There are plenty of qualified editors who will fix this article right away, if only you'll let them". I did switch from advice no 1 to advice no 2, and I offer my comments on this talk page. Thank you, I did have a nice day. Don't worry, be happy, we are working towards the same goal. Bo Jacoby 20:50, 16 January 2007 (UTC).
I notice that the Durand-Kerner method is the only numerical method of solving general polynomial equations mentioned in the section "Solving polynomial equations". Does that particular method deserve special mention here? Might it not be better to point at the root-finding algorithm article? DavidCBryant 16:57, 16 January 2007 (UTC)
Most rootfinding algorithms assume the function to be real valued rather than complex valued, so the article root-finding algorithm is likely to lead the reader astray. The Durand-Kerner method utilize the fact that the function is a polynomial and finds all the complex roots. It is not the only numerical method of solving general polynomial equations, but it is simple and sufficient. The reader need not know all the methods in the history of root-finding in order to solve this simple problem. Bo Jacoby 21:04, 16 January 2007 (UTC).
It is true that I rediscovered it independently and introduced it into WP as a subsection in root-finding algorithm in september 2005. Jitze Niesen, not I, titled it Jacoby's method. Happily it turned out to be known in the litterature, otherwise you would have deleted it from WP. It is far the simplest of the methods and is easily programmed based on the explanation in the WP article. Bo Jacoby 06:15, 17 January 2007 (UTC).
Hi, I had a chance to read through the article today in its present shape, and it mostly looks okay to me modulo minor quibbles. A couple of more substantial comments:
We probably want to qualify this statement a bit more, as this terminology is far from universal. Furthermore, it is easily confused with the common notion of the "principal nth root" of an arbitrary complex number z, which usually refers to the principal value (= 1 for z=1). For example, to pull a random book off my shelf, Complex Variables for Mathematics Engineering by J. H. Mathews uses "principal nth root" in this sense, as do our articles Nth root algorithm, Nth root.
Even the MathWorld article that we cite defines "principal root of unity" as something similar to "primitive root", and only mentions the meaning above as a secondary "informal" usage. (Actually, the MathWorld article makes no sense: look at its definition for the case j=n ... I suspect it has a typo somewhere.)
The footnote says that "this terminology is especially common in the context of fast fourier transforms." Part of my research involves FFT algorithms, and I don't believe this statement is true. The only source I recall that uses this terminology is the Cormen/Leiserson/Rivest textbook (I don't doubt that there are others, it just doesn't seem "especially common"). For example, the standard Oppenheim & Schafer Discrete-Time Signal Processing textbook doesn't use the term "root of unity" at all, as far as I can tell, although it defines a symbol ; it talks in terms of the "fundamental frequency" 2π/N and its multiples. (Note that the minus sign is extremely common in the choice of primitive root for discrete Fourier transforms.) Knuth just calls it "an Nth root of unity". A widely-cited 1990 review article by Duhamel and Vetterli on FFT algorithms only uses the term "primitive root" ("[where] WN [is] the primitive Nth root of unity ..."). It is true that the quantity e2πi / n, or its conjugate, is ubiquitous in discrete Fourier transforms and related areas such as FFT algorithms, but the terminology varies.
—Steven G. Johnson 19:02, 16 January 2007 (UTC)
Lacking a widely-adopted standard terminology I used the obvious notation 11 / N for the principal N 'th root of unity (in CMummert's sense of the word principal, the N 'th root of unity with minimal positive complex argument), but StevenJ prefers e2·π·i / N, although it is not immediately obvious to the WP-reader that this expression involving trancendent numbers turns out to be algebraic, or WN although it does not reflect the fact that it is a root of unity, and although the power WNk seem to depend on k and N independently while in fact it depends only on the ratio k/N, which is obvious from the notation 1k / N . Bo Jacoby 06:46, 17 January 2007 (UTC).
Dear VectorPosse. You do not need to sound like a broken record. You have the right to keep silent. You are free to believe that the problem of 'a well-understood concept without a widely-adopted standard terminology' is unsolvable, but please grant other people the freedom to think constructively that it can somehow be solved. It was new to me that also CMummert recognized the problem. Perhaps the problem and its possible solutions will some day be described in WP to everybody's satisfaction. We are aiming at the same goal: to make a great encyclopedia. Let's cooperate to achieve that goal. Bo Jacoby 08:40, 17 January 2007 (UTC).
I think this section is too terse, and needs more examples and explanation. (Compare it to the previous section on integer powers: the difference is stark.) It should aim to be mostly accessible to, say, a high-school student or freshman who has learned the basic rules of arithmetic for complex numbers, knows trigonometry, and maybe something about polar/cartesian forms, but hasn't learned any complex analysis and has never heard of branch cuts. Euler's formula probably needs to be reiterated somewhere here (as opposed to just linked), but not rederived.
It's fine to refer the reader to the article on branch cuts and give the formal definition of the principal value, of course, but it should also give more explanation. The hypothetical reader I mentioned above will have no idea how to make sense of "a branch cut extending from the origin along the negative real axis". (Using z for both log(a) and the exponent of a in the same paragraph is also confusing.) On the other hand, the same reader should be able to grasp something like:
The section "general formula" should be expanded to give an actual derivation of this formula, from Euler's formula, by explicitly converting a+bi to polar form. (This will also simplify the formula, since it can then be written in terms of .) The section should again reiterate that a branch cut is involved in the choice of angle.
I would then suggest following it with a couple of examples, e.g. and .
I hope this is helpful; thanks for your efforts. —Steven G. Johnson 18:39, 16 January 2007 (UTC)
Bo Jacoby 21:25, 16 January 2007 (UTC).
Thank you, EdC, I'll move the graph to its proper place. You did not change a different graph, but the first one of the two graphs referred to. You are welcome to change the other one too. 06:09, 18 January 2007 (UTC).
The initial definition has been changed to "Formally, powers with positive integer exponents can be found by the initial condition a0 = 1 and the recurrence relation an+1 = a·an". (My highlighting of found as opposed to defined). The word "found" indicate to me that it is some kind of computational shortcut, but it is a definition. Also the repetition of the definition in the subsection on complex powers of complex numbers, has been removed. So now the article does not seem to contain a definition any more. It that correctly understood? Is that what the editor wants? Is it a consequence of the 00 controversy, that 'no definition' is considered better than a controversial definition? Or is it a transition between definitions? Please clarify. Bo Jacoby 16:51, 17 January 2007 (UTC).
article: "The function xy is continuous everywhere except when x and y are both 0, however." xy is also discontinuous for x=0, y<0, so the incorrect quote is removed.
article: "For this reason, it is convenient in calculus to treat 00 as an indeterminate form. " This repetition is also removed.
Bo Jacoby 05:58, 18 January 2007 (UTC).
To be honest, I think the claim is problematic as well. It makes sense in complex analysis, I suppose, as you can pick a neighborhood around x and choose a branch of the logarithm that's analytic on that neighborhood. But when we're not discussing complex numbers, negative x just doesn't work at all in any context where y may be continuously varying. With sufficient care, the claim might be rephrased to something accurate, but I don't really see as it's worth it; I think the simplest thing is to get rid of the sentence. -- Trovatore 08:38, 18 January 2007 (UTC)
To VectorPosse: It is sufficient to say that "The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity". The deleted sentence was at best superfluous, but actually misleading, because it repeated the function without repeating the domain, and then said 'everywhere'. Trovatore, for instance, was mislead. Exponentiation is a tricky business and there is no reason to hide the fact that there is a controversy. Nor is there any reason to become personal.
You say: "Sometimes it is convenient to define 0^0 = 1, and sometimes that doesn't work". Please provide an example where it doesn't work. To Trovatore: Thanks for the support. You say that "negative x just doesn't work in any context where y may be continuously varying". But negative values of x do work when y is a negative integer, except for a discontinuity at x=0.
Bo Jacoby 10:16, 18 January 2007 (UTC).
Bo Jacoby 14:06, 18 January 2007 (UTC).
I hope you forgive that I insert a headerline before your note above, because it is no longer about the discontinuity. I do not oppose your claim. The article has changed subject from mathematics to social science. This is extraordinary but probably cannot be helped.
The question no (4) above was about if we could at least be open on this controversy ? 'Revision as of 07:25, 18 January 2007' contained the subtitle 'Controversy, zero to the zero power', in order to say that the subsection is providing no information about 'zero to the zero power' but about a controversy in real-life. (The edit was undone by VectorPosse as quickly as possible). Most readers might expect an article on mathematics, and so we must be very explicite that this is not the case. Otherwise the reader gets confused and frustrated. because he does not understand what is going on. That is why the strong or weak arguments and counterarguments must enter into the article, not to convince me, but to enlighten the reader for himself to judge. Let us merely document the fact that "in real-life these arguments have not led to a consensus on the issue". Bo Jacoby 15:25, 18 January 2007 (UTC).
Thanks Steven. You was not the most pleasant editor to work with, but we had our fun and our AHA-experiences. If I had not been a little bold there would have been no article on the Durand Kerner method, and a lot of errors in WP have been corrected by me. Most of the thoughts of creative people are not original research, but has been done before, but that is in principle undecidable. The article that you had deleted on Ordinal Fraction seems to have been original. My work in the article on inferential statistics is actually also found in a paper by Karl Pearson from 1928, so that was not original research after all, even if nobody knew it, and so you had it deleted too. You may rewrite it when you have studied Pearson. You do not appreciate my work, but some other people do. I am pleased that you agree that the 0^0 controversy is unimportant, and the opposition against a notation backed up by Donald Knuth himself came to me as a surprise. Take care. Bo Jacoby 17:40, 18 January 2007 (UTC).
Oh I wish you were right, but some editors here are allergic against 0^0=1 and revert any edit using it. Thanks for the nice words. (Pearsons 1928-formula for mean value and standard deviation in inferential statistic or 'statistical induction' is no longer found i WP. Many (most?) professional statisticians do not know it). Bo Jacoby 22:51, 18 January 2007 (UTC).
I fail to see how 0^0 "arises" in the natural numbers. If I asked a typical undergraduaite freshman what they would have after taking 0 and multiplying it times itself 0 times, they would say "nothing" or "you can't do that".
Of course there is only one empty tuple, but that is not a fact about "natural numbers" it is a fact about tuples. That is, the theorem that the number of m-tuples from an n-element set is nm is only valid for n=m=0 if 0^0 has already been defined to be 1. Otherwise, the theorem would require a longer statement. CMummert · talk 14:28, 18 January 2007 (UTC)
We must do it both ways, respecting the neutral point of view, and leave the judgement to the reader:
and so on, for all the rest of mathematics. Bo Jacoby 15:46, 18 January 2007 (UTC).
I just archived this discussion page, from 12/20/06 through 12/31/06. I swear on my honor as a gentleman that I did not alter a word of it. The size of this page was reduced from 106kB to 62kB (measured with the "Edit this Page" button). That's 44kB in 12 days ≈ 3.67 kB per day, or roughly 750 words per day (31 per hour). DavidCBryant 15:32, 18 January 2007 (UTC)
I just archived this discussion page, from 1/01/07 through 1/14/07. I swear on my honor as a gentleman that I did not alter a word of it. The size of this page was reduced from 99kB to 76kB (measured with the "Edit this Page" button). That's 23kB in 14 days ≈ 1.64 kB per day, or roughly 336 words per day (14 per hour). DavidCBryant 21:39, 27 January 2007 (UTC)
I just archived this discussion page, from 1/15/07 through 1/18/07. I swear on my honor as a gentleman that I did not alter a word of it. The size of this page was reduced from 78kB to 35kB (measured with the "Edit this Page" button). That's 43kB in 4 days = 10.75 kB per day, or roughly 2,200 words per day (92 per hour). DavidCBryant 11:54, 2 February 2007 (UTC)
I tried to preserve as many of the changes made last night as I could while copyediting the article.
For real x, the power eix is the point on the unit circle, and x is the angle (1, 0,eix), measured in radian. (See Euler's formula: eix = cos(x)+i·sin(x), where cos and sin are trigonometric functions and i is the imaginary unit).
CMummert · talk 14:56, 19 January 2007 (UTC)
Bo Jacoby 15:44, 19 January 2007 (UTC).
To CMummert: The article on Trigonometric function says: "the trigonometric functions are functions of an angle". The functions cos(x) and sin(x) are trigonometric functions. So cos(x) and sin(x) are functions of an angle. Then so is cos(x) + i·sin(x), but this is equal to eix. But you said "that e^x is a function of a real number, not a function of an angle". So your point of view is at variance with the article on Trigonometric function. If you change it here, you must also change it there, but you may alternatively reconsider your point of view. SORRY! I read you as saying "that e^(ix) is a function of a real number, not a function of an angle" but you didn't. But then I do not understand your argument.
A reader of the article exponentiation can not be supposed to have any background in trigonometry which relies on power series which relies on polynomials which relies on exponentiation.
To StevenJ: The reader is probably more interested in understanding the subject than to know what is "exceedingly uncommon in mathematics". Some editor of the article, perhaps you, wrote: "Because of the periodicity explained above, the equation for has infinitely many complex solutions z". If this sentence has any meaning at all, it must be that " has infinitely many complex values z", or, eliminating z, that " has infinitely many complex values". So already here we see the first of these exceedingly uncommon cases where a multivalued function is used without selecting a particular branch cut. Later you may pick a branch cut, but the multivalued function was an inevitable intermediate step. It is often easy to express the set of solutions in terms of a particular solution, for example for logarithms:
Or for N 'th roots of unity:
Or for N 'th roots:
Bo Jacoby 01:37, 21 January 2007 (UTC).
\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) + \beta \cdot g(x) )
By the sum rule in differentiation, this is:
\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) ) + \frac{\mbox{d}}{\mbox{d} x} (\beta \cdot g(x))
By the constant factor rule in differentiation, this reduces to:
\alpha \cdot f'(x) + \beta \cdot g'(x)
Hence we have: That is correct, I was intentionally naive. The prospect of a discussion on whether an angle is different from its value called for a naive approach, (especially after the discussion about whether 0 is different from 0.0). Happily you did not insist, and the present formulation of the angle issue is quite satisfactory. Well done. You and I have equal right to edit the article. Bo Jacoby 11:02, 21 January 2007 (UTC).
To CMummert.
To DavidCBryant. The N 'th root and the logarithm are basicly defined by equations that may have more than one solution. I wrote about the set of solutions to an equation rather than about multivalued functions.
Bo Jacoby 22:08, 21 January 2007 (UTC).
The article on multivalued function refers to 'A Course of Pure Mathematics' by G. H. Hardy. There is no need to repeat it in the article on exponentiation. Bo Jacoby 12:07, 22 January 2007 (UTC).
I took the set-valued or multivalued interpretation of the logarithm away from the article and write 'solution to the equation ex=a '. The price to be paid for using principal value is that important rules, such as log(a2)=2·log(a), does not hold:
A warning to the readers would be nice. Bo Jacoby 14:22, 22 January 2007 (UTC).
Well done. Prove |eix|=1 without using Euler's formula: 1 ≤ |eix|2 = (limn(1+ix/n)n ) (limn(1−ix/n)n ) = limn(1+x2/n2)n ≤ limn(1+ε/n)n = eε for every positive ε . Just choose n>x2/ε. The only real number w satisfying 1 ≤ w ≤ eε for every positive ε is w=1. Q.E.D. Bo Jacoby 18:10, 22 January 2007 (UTC).
Thank you. Just note that a branch cut is not sufficient specification of a branch. Cutting along the negative axis splits the Riemann surface of the logarithm into branches, but each of these have the same branch cut. Bo Jacoby 23:18, 22 January 2007 (UTC).
The new warning subsection is nice. Perhaps we should introduce subsection headings: 'complex powers of positive reals' and 'real powers of unity' to deal with the useful cases az = ez·log(a) and e2πi·x where the warnings does not apply? Another subsection heading could be 'Rational powers of complex numbers'. I slightly prefer the term 'rational power' for the term 'root' because a root can be a solution to any equation like in 'root-finding method' while a rational power ar refers only to the equation xm=ar·m where m is the denominator of r such that r·m is integer. What do you think? Bo Jacoby 09:04, 23 January 2007 (UTC).
Bo Jacoby 13:39, 23 January 2007 (UTC).
"This leads to the following rules: Any number to the power 1 is itself. Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 0^0 is discussed below." As a matter of fact the argument does not lead to the 'nonzero' exception. CMummert is touchy about these things, but the definition assumed in the subsection is that an=1·a···a (n multiplications by a), and this definition applies equally well for zero as for nonzero values of a. Bo Jacoby 20:38, 25 January 2007 (UTC).
As you don't want 0^0 defined, you must change the definition "an=1·a···a, (n multiplications by a)", even if it is contemporary practice, because this definition implies 0^0=1 as a special case. One cannot accept a definition without accepting its consequences. You must find another way to explain why a0=1 for nonzero a. Bo Jacoby 00:18, 27 January 2007 (UTC).
I am talking about the subsection Exponentiation#Exponents_one_and_zero saying: "The meaning of 35 may also be viewed as 1·3·3·3·3·3 :the starting value 1 (the identity element of multiplication) is multiplied by the base as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to exponents one and zero: 31 = 1·3 = 3 and 30 = 1. This leads to the following rules: Any number to the power 1 is itself. Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 00 is discussed below." There is no justification here for restricting the second rule to nonzero numbers, and the reader, accepting the definition, is confused that the editor does not accept the conclusion. You guys who do not accept the conclusion are reponsible for making a subsection that makes sense. Bo Jacoby 07:08, 27 January 2007 (UTC).
Why such efford to conceal the fact that you are a nice person and a skilled mathematician? I regret any misunderstanding. I am not intentionally misleading; I do my best to be clear. The talk page subsection header: "edit 25-1-2007. exponent zero and one", was supposed to lead you to the article subsection: "exponent zero and one". The definition in article subsection "exponent zero and one" is exemplified by: "35=1·3·3·3·3·3", which obviously generalizes to: "an=1·a···a, (n multiplications by a)" because there is nothing special about the numbers 3 and 5, they are just examples. This leads to: "a0=1" for unrestricted a, and does not justify the restriction to only nonzero values of a. The correctness of this restriction is not my point right now. Nobody gets confused by the unrestricted definition, not even you, but students and readers do get confused by a sudden, unjustified, illogical restriction: "nonzero". This kind of stuff characterises bad math. So you have a choice to make. Either you produce another definition of the power an, one that only for nonzero values of a leads to "a0=1", or you accept both the definition, "an=1·a···a, (n multiplications by a)", and its unrestricted consequence: "a0=1". Bo Jacoby 13:09, 27 January 2007 (UTC).
Bad math is a problem. The link does not help. The definition "an=1·a···a, (n multiplications by a)" is unrestricted and must be modified if you want a restriction (as you do. I don't). Bo Jacoby 14:33, 27 January 2007 (UTC).
Two different definitions are mutually inconsistent, but no definition is inconsistent with the alternative "no definition". We do not have different definitions leading to different results, but one definition leading to one result and the alternative, "no definition", leading to no result. WP does not forces anybody to accept definitions that they do not like or do not want to use. We are all free not to understand, but don't undo edits in WP where it is explained to other people what mathematicians mean by writing x0 when x=0. Your edit (19:51, 25 January 2007) left the subsection in a state of corrupted logic. You now seem to argue that bad logic is a fact of the real world, and so this subsection should be illogical in order to reflect the real world? I don't like to argue at this level of madness. You are not more responsible for the contents of WP than anybody else. You are welcome to include the fact that some authors do not define 00, as you did, but you are not welcome to destroy the logic of other editors contributions. Bo Jacoby 12:19, 28 January 2007 (UTC).
This should make the 0^0 = DNE pretty clear:
0 = -0
0^(0) = 0^(-0) = 1/(0^0) = (1^0)/(0^0) = (1/0)^0 = DNE because 1/0 is undefined.
ARiina 14:54, 28 January 2007 (EST).
Thank you. I don't know the abbreviation DNE. 0^(0) = 1, 0^(-0) = 1, 1/(0^0) = 1/1 = 1, and (1^0)/(0^0) = 1/1 = 1, but (1/0)^0 is undefined. Division by zero must remain undefined, but zero'th power of a number is not a problem. Bo Jacoby 22:37, 28 January 2007 (UTC).
Assuming good faith, why is Trovatore reverting my edit on the definition leading to the simple rule? You cannot deny the consequence without denying the premise, as explained above. Bo Jacoby 10:31, 2 February 2007 (UTC).
DavidCBryant state that the old version was better. I had inserted section headers for clarity. Please discuss your point of view. Bo Jacoby 11:51, 2 February 2007 (UTC).
I tried to solve the following problems in the subsection on complex powers of complex numbers.
Bo Jacoby 13:28, 7 February 2007 (UTC).
Certainly, but the problems remain unsolved. Bo Jacoby 23:48, 7 February 2007 (UTC).
Quote: The function xy is continuous, however, whenever x and y are both nonnegative except when x and y are both zero. For this reason, it is convenient in calculus to treat 00 as an indeterminate form, since this allows the taking of limits (and other topological constructions) to be considered as commutative with the operation of exponentiation. (EdC's addition in italics).
The discontinuity for x=y=0 does not disappear by treating the expression 00 as an indeterminate form. The limit does not commute with exponentiation. limy→00y ≠ 00. If 00 is considered indeterminate, then commutativity implies that 0 is interminate. But 0 is not indeterminate. If 00 is considered = 1, then commutativity implies that 0 = 1, but 0 ≠ 1. In no case can continuity (meaning commutativity with lim) be saved. So actually the discontinuity is not a justification for leaving 00 undefined or indeterminate. Bo Jacoby 14:33, 8 February 2007 (UTC).
Exactly. There is no logical connection between being 'defined' in a point and being 'continuous' in a point, even if CMummert seems to believe that there is or should be such a connection. Consider the 3 cases:
So EdC is right: non sequitur. Bo Jacoby 22:32, 8 February 2007 (UTC).
Let me make the position clear for the last time. (I know Bo Jacoby already understands this and chooses to ignore it. I'll give EdC the benefit of the doubt.) EdC is right when he states that being continuous at a point and being defined at a point are not related, a priori. Fine. Nobody disputes this. So how does one extend a function to a point not in its natural domain? If the point corresponds to a removable discontinuity, then very few people object to defining the function to be the limit of the function as it approaches. If the discontinuity is not removable, as in the case of x^y as x and y approach zero, then it is usually not sensible to define it to be anything, at least from the point of view of calculus. The article as it currently stands explains this and also includes very reasonable justifications for choosing 0^0 = 1 that help other areas of mathematics work a bit more smoothly. So it is not a non sequitur to connect the ideas of continuity of a function to the definition of that function if one stipulates that in calculus it is reasonable only to define functions when they are defined by continuity in a natural way.
And now that I've written that, I'm kicking myself for going against my own advice not to engage in mathematical discussion that has already been made abundantly clear in this prolonged discussion. Please, let's not take this as carte blanche to re-debate this whole issue. I'm just trying to answer EdC since he seems more sincere and more reasonable in his attitudes than Bo Jacoby. VectorPosse 07:17, 9 February 2007 (UTC)
To VectorPosse, above:
I agree with all of the above (apart from the implication that (0, 0) is not in the natural domain of ). However, consider the situation from the standpoint of someone who is convinced that 0^0 does have a natural value, and hence is in the natural domain of . As it stands, the passage merely sets out reasons why one would not bother to define 0^0 in an analytic context. This is not a reason in itself to leave it undefined; calculus has no use for , but does not demand that that be undefined. Indeed, when learning calculus we confront pathological (i.e. typical) functions that are continuous nowhere, or continuous everywhere but differentiable nowhere, etc. So why should exponentiation be any different?
The most accessible reason I could think of is that having real exponentiation continuous everywhere it is defined allows limits to be taken through exponents. Perhaps it's not as accessible as I thought, but the article needs something there. – EdC 17:19, 9 February 2007 (UTC)
To Vectorposse. You write: "So how does one extend a function to a point not in its natural domain? If the point corresponds to a removable discontinuity, then very few people object to defining the function to be the limit of the function as it approaches. If the discontinuity is not removable, as in the case of x^y as x and y approach zero, then it is usually not sensible to define it to be anything, at least from the point of view of calculus". The statement in italics is not correct. There are indeed sensible ways in calculus to define a function value at a point of discontinuity. Consider the function f, defined for −π<x<+π by f(x)=x. It defines a fourier series, g, such that g(x)=f(x) for −π<x<+π . g is periodic: g(x+2π)=g(x). It has a nonremovable discontinuity for x=π. Nevertheless, the value g(π)=0 is a sensible way in calculus to define the function value at the point of discontinuity. This example proves your argument invalid. Bo Jacoby 19:41, 20 February 2007 (UTC).
To EdC: i can't understand why you removed my observation that 00 breaks algebraic properties of exponentiation if defines equal to 1. The exponentiation xy can be defined as the unique continuous homomorphism such that , this condition defines directly the values of for , and the function is extended to by continuity. But of course definition of 00 would break homomorphism axioms. Myrizio 01:42, 19 May 2007 (UTC)
After CMummert's latest edit:
Justifications for leaving 00 undefined include:
- The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity. [1] That is, the function xy has no continous extension including the point (0,0): along the x-axis the limit is 1, along the y-axis the limit is 0, and any intermediate limit a can be obtained using the curve y = log(a)/log(x). However, if y is an analytic function of x, or if there exists a positive constant, a, such that y < ax, then the limit is 1.
- The function zz, viewed as a function of a complex number variable z and defined as ez log z, has a logarithmic branch point at z = 0.
I don't understand how either of the two listed points have any bearing on whether 0^0 should be taken to be defined. I thought it was because continuous functions have useful properties; was I wrong? – EdC 23:57, 10 February 2007 (UTC)
*The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity. [2] That is, the function xy has no continous extension from the open first quadrant to include the point (0,0). [3]
It is uncommon in the context of elementary calculus to extend a function in a manner that makes it become discontinuous.A discontinuous extension would cause the function to lose a number of desirable properties, for example that when and , , and so would be avoided.
'Paige (the reference) does not go into great depth about why 0^0 is undefined'. Editors are not supposed to invent explanations that are not supported by the literature. If not even the reference explains why 0^0 should be left undefined, then there is no justification. Bo Jacoby 13:53, 19 February 2007 (UTC).
The article on Continuous function is inconsistent in this matter. The sentence 'The fact that a discontinuity can be removed does not make the original function continuous' is at variance with the definition, that a function is (everywhere) continuous if it is continuous at every point of the domain. Is the function x/x continuous? It is not defined for x=0, so it is not continuous on the real axis. It is continuous at every point of its domain, so it is continuous. The sign function f, defined for nonzero real values of x by f(x)=+1 for x>0 and f(x)=−1 for x<0, is continuous for every point in its domain. Yet it is usually considered discontinuous because lim an=lim bn does not imply lim f(an)=lim f(bn) in all cases, which is a usual characterization of continuous functions. So I get the point. By removing the definition of 00 you make the function f(x,y)=xy continuous at every point of the domain, which is nicer than saying that it is continuous for (x,y)≠(0,0). Right? Bo Jacoby 16:51, 24 February 2007 (UTC).
Fine! Then at last we may be able to provide a sensible explanation why some people do not want 00 to be defined: 'Some people like a function to be continuous on it's entire domain rather than on a proper subset of it's domain. So they prefer to remove the point (x,y)=(0,0) from the domain of xy rather than to admit a function xy that is defined, but not continuous, for (x,y)=(0,0)'. Are you sure that all mathematicians agree that, say, f(x) = 1/x is a continuous function? Some mathematicians might spot a discontinuity at x=0, even if f(0) is not defined. Bo Jacoby 18:06, 25 February 2007 (UTC).
Great. So I misused the word 'discontinuity' meaning 'non-removable singularity'. I assume that we agree that the singularity of xy at (x,y)=(0,0) is not removable, neither by defining, nor by undefining, 00. If you can express this with NPOV, then please do. I'm not sure I can do it myself, because the combinatorially and algebraically well-justified definition 00=1 is more elementary than the analytical concepts of singularity and discontinuity, and the advocates of undefining 00 show to be both touchy and inarticulate. Bo Jacoby 11:41, 26 February 2007 (UTC).
Last thing first: for nonnegative integers, exponentiation is easily explaned like this. "nm is the number of m-letter-words taken from an n-letter alphabet". As there is only one 0-letter-word, (namely the empty word: ""), n0 = 1, irrespective of n. Can topology be explained as easily as this?
In the article, integer exponents are explained first and noninteger exponents are explained later, being more advanced, and so the integer definition of 00 precedes the real nondefinition of 00. So, from the point of view of the logic of the article, we are "removing" (0,0) from the domain. It does not help to leave 00 undefined in the elementary section, because the definition in the elementary section implies that 00=1.
Next, the property, continuity in all points of the domain, is not taken as a rule in calculus. Functions with singularities are treated in calculus, and values in singular points may be defined, for example by fourier series, as in sawtooth wave. So the only example given in the article of one of the 'desirable properties' is invalid, and the 'number' of desirable properties lost by a discontinuous extension seem to be zero. So there is no valid justification for not defining 00=1. Your idea of reduced domain and extended domain is nice. Bo Jacoby 15:58, 2 March 2007 (UTC).
Consider the binomial coefficient. It is understood by the combinatorial interpretation, that Cn,k is the cardinality of the set of k-element subsets of a n-element set. This explanation applies for nonnegative integer values of n and k. For small values of n and k the binomial coefficient is found by elementary counting, for example
The special cases Cn,0 = | { {} } | = 1, and Cn,k = | {} | = 0 for n<k, follow from the combinatorial explanation. For bigger values of n and m counting becomes impractical, and the formula Cn,k = (n/k)·Cn−1,k−1 speeds up the calculation.
Now compare with exponentiation. It is understood by the combinatorial interpretation, that nk is the cardinality of the set of k-element tuples of a n-element set. This explanation applies for nonnegative integer values of n and k. For small values of n and k the power is found by elementary counting, for example
The special case n0 = | { () } | = 1 follow from the combinatorial explanation. For bigger values of n and m counting becomes impractical, and the formula nk = n·nk−1 speeds up the calculation. So, repeated multiplication is more advanced, and more general, than the combinatorial explanation.
The point of mentioning the sawtooth wave is not that it is has singular points, but that the value of the function at the singular points can be determined by calculus, in this case by fourier series, thus refuting the assertion that 'it is ordinarily taken as a rule in calculus that [some limit statement holds] whenever both sides of the equation are defined'. Bo Jacoby 08:09, 4 March 2007 (UTC).
I'm with you until you say that "repeated multiplication is more advanced, and more general, than the combinatorial explanation". In combinatorics, yes; but in other areas of mathematics the exponential is arrived at without invoking the cardinality of exponential sets. "Exponentiation" as a term refers to a family of operators with shared characteristics; there is no well-defined basic exponentiation operator.
For the sawtooth: ordinarily. The sawtooth is only expressible in terms of elementary functions through infinite series; everyone knows to be careful taking limits through infinite sums. If the exponential is taken undefined at (0, 0) then elementary functions preserve limits. – EdC 23:41, 5 March 2007 (UTC)
Certainly counting is more elementary than multiplication or exponentiation. That doesn't mean that taking exponential sets is more elementary than exponentiation; there are other motivations for exponentiation than exponential sets.
The sawtooth is, however, a limit; it is not elementary.
Finally, (assuming 0^0 undefined) neither of your double limits holds; since the latter is undefined at ; since the latter is undefined at . The only way to resolve the situation is to strike the axes from the definition of the exponential, in which case neither limit exists. – EdC 23:01, 6 March 2007 (UTC)
Q.E.D. Bo Jacoby 09:00, 16 March 2007 (UTC).
The editors who are supporting the idea of leaving 00 undefined claim to be mathematicians. Now let's recapitulate. There seem to be no valid justification for leaving 00 undefined. The 'justifications' in the article are not quoted from the references, but are invented by the editors, and these 'justifications' are not valid. The singularity of xy for (x,y)=(0,0) is not removed by restricting the domain of xy. The word 'nonzero' in Exponentiation#Exponents_one_and_zero is unmotivated in the context, which explains why x0=1 for all values of x. Bo Jacoby 16:22, 22 March 2007 (UTC).
An edit was made this morning with edit summary "After a long discussion the 0^0 case for integer exponents is settled." No new discussion has happened recently, and the new version has reintroduced the same absolute claims about 0^0 being 1 and being the empty product that have been discussed in great depth already. CMummert · talk 12:27, 28 March 2007 (UTC)
{{
cite journal}}
: Cite journal requires |journal=
(
help)Not only does the C99 standard not define pow(0,0)
(except to say that it may raise a domain error, and allowing implementations to define __STDC_IEC_559__
if they follow
IEC 60559), but there's nothing in the C Rationale about pow
at all, and what there is about cpow
seems unenlightening to me. I've reverted the incorrect inclusion of C in the list, but can anyone think of a reason that that reference to the Rationale should remain, or should I get rid of it too? --
Quuxplusone
22:33, 8 March 2007 (UTC)
pow (0, 0)
should return a value or raise a domain error; implementations may define __STDC_IEC_559__
if they follow
IEEE 754.[7]" What does IEEE 754 say about 0^0? –
EdC
22:41, 8 March 2007 (UTC)I was noticing that the "Powers of i" and the "Powers of e" sections are both under the heading "exponential with integer exponents." I propose that "powers of e" gets moved under "Real Powers of Positive Real Numbers", and "powers of i" goes to "Complex powers of complex numbers". Does anyone have a problem with this? -- shaile 16:03, 16 March 2007 (UTC)
I've removed the image of x^y as I thing it is missleading. I've had a bash at drawing x^y for positive x. It clearlt shows that taking limits along curves will give a limit of 0.-- Salix alba ( talk) 09:05, 3 April 2007 (UTC)
It is a nice picture. Note that on the curve y=f(x)=(log a)/(log x), the exponential xy = a. The function f is not an analytic function for x=0. If x and y approaches zero along an analytic curve, the exponential must approach 1. Bo Jacoby 21:38, 3 April 2007 (UTC).
is there a separate article on the exponential map in differential geometry? if not, should information on it be added here or should I write an article on it? SmaleDuffin 16:35, 4 April 2007 (UTC)
This article has gone from B-class to 'Good'. Well-written and explained. Attractive presentation with some kewl formulae and graphs. No math errors there I could find. Neutral and not given to controversy. Stable: um, yeah. Images: has graphs that really help the exposition. Well done and take a step up. Gifir2007 11:28, 7 April 2007 (UTC)
The third sentence of the article is Exponentiation can also be defined for exponents that are not positive integers. But it doesn't go on and say what the definition is. Obviously,
but leaving the sentence on its own can create some confusion for those who don't know basic index laws. Thanks Gizza Chat © 12:00, 7 April 2007 (UTC)
Made a new try on the lead but I guess it still needs a little improvement. Ricardo sandoval 20:54, 25 April 2007 (UTC)
The article says that 2 to the 3^4 is different from 2^3 to the 4, but it doesn't mention order of operations. Is 234 2^81 or 8^4?
The other indeterminate form is 1^infty, not 0^infty. This was fixed by an IP editor this morning, then reverted, and then I accidentally undid the reversion without leaving a useful edit summary. CMummert · talk 17:01, 9 April 2007 (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 |
Hi CMummert.
I wish to continue the discussion here on your talk page because it is getting somewhat personal. You wrote:
Your argument, that 'you do not need to define 0^0=1 if the expression 0^0 is everywhere replaced by 1', is strange and invalid. The same argument applies to 2+2: you do not need to define 2+2=4 if you replace 2+2 with 4 every time it occurs.
You also wrote:
Yet you undid my edit in a knee-jerk fashion without comment on the talk page.
The article should be readable to the non-expert, and it is bad that the unimportant discussion about 0^0 is polluting the article. In the elementary section on integer exponents there is no disagreement that the empty product has the value 1. Please clarify your position. Bo Jacoby 10:46, 12 January 2007 (UTC).
The section Exponentiation#Zero_to_the_zero_power do not pollute the article, but the explanations of 0^0 in Exponentiation#Powers_of_zero and in Exponentiation#Powers_of_zero_2 are not clear.
The only problem with the definition 0^0=1 is that some books do not include it. The discontinuity of x^y at (0,0) is a matter of fact, no matter whether 0^0 is defined or not. So, let's stick to the definition in the article to make it readable to beginners, let the subsection Exponentiation#Zero_to_the_zero_power mention that some books leave 0^0 undefined, let's move that subsection to the section Exponentiation#Advanced_topics (which the beginner need not read) and let's include the references as we do by now.
Bo Jacoby 16:33, 12 January 2007 (UTC).
The pointers to the section on 0^0 are OK. No problem. It makes sense to say that 0^0 is defined to be 1, and then later explain that some books doesn't define it.
'Tested experimentally' is not a mathematical argument. The integer interpretation of 2+2=4 may be tested by counting pebbles, but the real number interpretation cannot be tested by adding lengths. (The teacher explaining the slide rule: Two times two is three dot nine eight. The pupil: Shouldn't it be four? Teacher: No not if it has to be quite accurate.)
Even if some mathematicians don't accept the definition, they should stop preventing other mathematicians from using it.
Just as there is only one empty set, there is also only one empty product, even if it can be expressed in several ways, say 1^0 or 0^0.
The concepts used in the subsection Exponentiation#Zero_to_the_zero_power are more advanced than the concepts required to understand basic exponentiation. That is why it should be moved to the Exponentiation#Advanced_topics section.
The elementary sections of the article should be reserved for important explanations. The synonym 'involution' is unimportant. It is never used in modern mathematics.
Bo Jacoby 00:14, 14 January 2007 (UTC).
Trovatore: Thank you. You may start by undoing CMummert's revert and explain the situation to VectorPosse.
CMummert: The Advanced topics are advanced relatively to the elementary parts of the article, not in an absolute sense.
EdC: I agree that a^b has a combinatorial interpretation that can be tested experimentally but I do not understand your example. (I get only 5*3=15 lines).
VectorPosse: I am trying to improve the article which is polluted by repeated unimportant remarks on what some books do not define.
Bo Jacoby 08:13, 14 January 2007 (UTC).
I trust that VectorPosse agrees that 0^0 is a relatively unimportant special case of n^m. It does not deserve this level of attention in the article. I am not trying to make 0^0=1 the only valid view either, but it is the view actually assumed by every user of 0^0. There is no use of an undefined expression.
EdC: n^m is the cardinality of the set of m- tuples from an n- set.
Bo Jacoby 11:49, 14 January 2007 (UTC).
Here are some running notes from my major edit this morning:
I have a few comments.
Bo Jacoby 16:53, 14 January 2007 (UTC).
1. Read the first few lines of the article Euler's constant. If you know better, then improve the article Euler's constant. If you agree, then follow the recommandation and call it Euler's number rather that Euler's constant.
2. Please note that the proof of the valid expression for negative integer x depend on . It is not sufficient that . That is why I wrote . (I did not write ). You are wellcome to rewrite the argument avoiding the sign if you have trouble understanding what it means, but please repair the damage.
3. It was never said that e2·π·i·(1/n)·1 was the only primitive root. The deleted statement was perfectly sufficient and correct. It should be reinstated.
4. This elementary article on exponentiation should not assume the reader to know an algebraic definition of π. The usual definition is geometric rather than algebraic. The deleted line contained nontrivial information and was to the point.
5. You should clean up you mess yourself, but OK, I'll do it.
6. You are wellcome.
7. OK, I'll fix the typo in Exponentiation#Exponentiation_over_sets and extend the examples. Still this interpretation is important for understanding the meaning (or one important meaning) of exponentiation, which is independent on multiplication.
8. Do what you want to do, but make it understandable to the beginner. There is no point in making branch cuts or Riemann surfaces until you understand that otherwise the function would be multivalued. One cannot solve a unrecognized problem. But go ahead and try.
9. This is great news. If I write can be, will you then (please) stop reverting my edits regarding 0^0 ?
10. Let the same rules apply to yourself as to me. My edits were reverted without comment. I treat you nicer that you treated me.
Bo Jacoby 20:22, 14 January 2007 (UTC).
The reader does not know about exponentiation. He may know that π is the ratio between circumference and diameter of a circle, but he need not understand the connection to exponentiation. Please remember to use the words Euler's Number rather that Euler's Constant, as explained under point 1. The characterization above says that the sequence 'is an increasing sequence which is bounded above', which makes sense only for real x. For non-real complex x it is nonsense. For negative integer x it does not prove that the new and old definitions of ex give the same result. The logarithm is 'typically' considered for positive arguments only, but we are dealing with nonzero complex arguments. The branch cut is not needed in this article, but the multivalued log and fractional power are needed. There is no need any more to insert that 0^0 can be defined, because it is defined by the formal definition which you did accept above (point 9). War is over. Bo Jacoby 21:30, 14 January 2007 (UTC).
"Formally, powers with positive integer exponents can be defined by the initial condition a0 = 1 and the recurrence relation an+1 = a·an " is a formal definition. How can you make yourself write that "There is no formal definition of exponentiation in this article" ? Bo Jacoby 22:19, 14 January 2007 (UTC).
Yes, I am correct. No, I am not confused. 'Can be defined' means that other definitions are possible, but here is the definition of this article. When you change your mind in the middle of the article, the reader gets infected by your serious confusion. Whether 0^0 is 'universally defined' or not does not influence the fact that the formal definition in this article is as stated. And, as I have repeatedly pointed out, you have many more changes to make in WP in order to make your point of view consistent. Many many authors, tacitly or explicitely, assume a0=1 for all a, making no pointless exception for a=0, such as you stubbornly insist to do.
Regarding point 1 and 2: Remember to fix the two errors you introduced in Exponentiation#Powers of Euler's constant.
Regarding point 7. The combinatorial interpretation is about exponentiation of nonnegative integers, not about exponentiation of sets.
Bo Jacoby 07:36, 15 January 2007 (UTC).
CMummert · talk 14:16, 15 January 2007 (UTC)
Answer to the above:
Bo Jacoby 23:25, 15 January 2007 (UTC).
I added three references for the definition of a principal nth root of unity. I am sure that I could locate dozens of sources who use the same definition. Please don't remove the definition now that it is backed up by published sources. CMummert · talk 17:38, 15 January 2007 (UTC)
The subsection Exponentiation#Powers_of_negative_real_numbers adds more to confusion than to clarification, and it obviously does not belong under the heading Exponentiation#Real_powers_of_positive_real_numbers. The problem is naturally discussed in the more general framework of Exponentiation#Complex_powers_of_complex_numbers. Bo Jacoby 09:17, 15 January 2007 (UTC).
Still a subsection called Exponentiation#Powers_of_negative_real_numbers does not belong under the heading Exponentiation#Real_powers_of_positive_real_numbers because no negative number is positive. Please place it correctly. Bo Jacoby 16:53, 15 January 2007 (UTC).
Some time ago I did a lot of work improving this article. The present high activity is good, but it also contains some backwards steps. As I do not revert edits made in good faith, I may seem impatient, and I apologize. Don't you think that the subsection on powers of negative numbers needs a concluding remark saying that the appropriate context is that of powers of complex numbers? As by now the subsection is frustrating to read, leading nowhere. Bo Jacoby 23:47, 15 January 2007 (UTC).
Some time ago I did a lot of work improving this article.
The present high activity is good, but it also contains some backwards steps.
As I do not revert edits made in good faith, ...
Also, if a couple or more of editors tell you to drop something, then drop it, especially if you are not completely sure you perfectly understand the topic at hand. Oleg Alexandrov (talk) 05:09, 17 August 2006 (UTC)
I am referring to my editing a long time before Trovatore reopened the debate by suggesting to distinguish between 00 and 00.0 , an idea which is not mainstream mathematics and which would cause trouble. It was discussed, but the positions of some editors are still unclear, as there are contradictory statements of opinion. Then the discussion changed to whether 00 = 00.0 should be defined or not. The present formal definition of the article is that a0 =1 for all a, but afterwards it is stated that this definition is not accepted by everybody, so now the reader is left in unnecessary confusion. I edited the article myself, exactly as you suggested, but CMummert reverted my edits immediately. Therefore, rather than making an edit war, I politely suggested CMummert to correct the article himself. Of course we are all in good faith and we all want a great article on exponentiation, and my comments to (and from!) CMummert prove that I do understand the topic at hand. The formula for complex exponentiation is complicated and rather useless, and there is no need to make the reader believe that it is worthwhile learning it. Even the edit called "pinpointing nonsense for CMummert to clean up" actually contained clarifications of the troublespots, not 'directions to other editors'. You offer me two pieces of advice, which alas contradict one another: Advice no 1: "If you don't like the way the article reads, Bo, please edit it yourself". Advice no 2: "Please stop right now. There are plenty of qualified editors who will fix this article right away, if only you'll let them". I did switch from advice no 1 to advice no 2, and I offer my comments on this talk page. Thank you, I did have a nice day. Don't worry, be happy, we are working towards the same goal. Bo Jacoby 20:50, 16 January 2007 (UTC).
I notice that the Durand-Kerner method is the only numerical method of solving general polynomial equations mentioned in the section "Solving polynomial equations". Does that particular method deserve special mention here? Might it not be better to point at the root-finding algorithm article? DavidCBryant 16:57, 16 January 2007 (UTC)
Most rootfinding algorithms assume the function to be real valued rather than complex valued, so the article root-finding algorithm is likely to lead the reader astray. The Durand-Kerner method utilize the fact that the function is a polynomial and finds all the complex roots. It is not the only numerical method of solving general polynomial equations, but it is simple and sufficient. The reader need not know all the methods in the history of root-finding in order to solve this simple problem. Bo Jacoby 21:04, 16 January 2007 (UTC).
It is true that I rediscovered it independently and introduced it into WP as a subsection in root-finding algorithm in september 2005. Jitze Niesen, not I, titled it Jacoby's method. Happily it turned out to be known in the litterature, otherwise you would have deleted it from WP. It is far the simplest of the methods and is easily programmed based on the explanation in the WP article. Bo Jacoby 06:15, 17 January 2007 (UTC).
Hi, I had a chance to read through the article today in its present shape, and it mostly looks okay to me modulo minor quibbles. A couple of more substantial comments:
We probably want to qualify this statement a bit more, as this terminology is far from universal. Furthermore, it is easily confused with the common notion of the "principal nth root" of an arbitrary complex number z, which usually refers to the principal value (= 1 for z=1). For example, to pull a random book off my shelf, Complex Variables for Mathematics Engineering by J. H. Mathews uses "principal nth root" in this sense, as do our articles Nth root algorithm, Nth root.
Even the MathWorld article that we cite defines "principal root of unity" as something similar to "primitive root", and only mentions the meaning above as a secondary "informal" usage. (Actually, the MathWorld article makes no sense: look at its definition for the case j=n ... I suspect it has a typo somewhere.)
The footnote says that "this terminology is especially common in the context of fast fourier transforms." Part of my research involves FFT algorithms, and I don't believe this statement is true. The only source I recall that uses this terminology is the Cormen/Leiserson/Rivest textbook (I don't doubt that there are others, it just doesn't seem "especially common"). For example, the standard Oppenheim & Schafer Discrete-Time Signal Processing textbook doesn't use the term "root of unity" at all, as far as I can tell, although it defines a symbol ; it talks in terms of the "fundamental frequency" 2π/N and its multiples. (Note that the minus sign is extremely common in the choice of primitive root for discrete Fourier transforms.) Knuth just calls it "an Nth root of unity". A widely-cited 1990 review article by Duhamel and Vetterli on FFT algorithms only uses the term "primitive root" ("[where] WN [is] the primitive Nth root of unity ..."). It is true that the quantity e2πi / n, or its conjugate, is ubiquitous in discrete Fourier transforms and related areas such as FFT algorithms, but the terminology varies.
—Steven G. Johnson 19:02, 16 January 2007 (UTC)
Lacking a widely-adopted standard terminology I used the obvious notation 11 / N for the principal N 'th root of unity (in CMummert's sense of the word principal, the N 'th root of unity with minimal positive complex argument), but StevenJ prefers e2·π·i / N, although it is not immediately obvious to the WP-reader that this expression involving trancendent numbers turns out to be algebraic, or WN although it does not reflect the fact that it is a root of unity, and although the power WNk seem to depend on k and N independently while in fact it depends only on the ratio k/N, which is obvious from the notation 1k / N . Bo Jacoby 06:46, 17 January 2007 (UTC).
Dear VectorPosse. You do not need to sound like a broken record. You have the right to keep silent. You are free to believe that the problem of 'a well-understood concept without a widely-adopted standard terminology' is unsolvable, but please grant other people the freedom to think constructively that it can somehow be solved. It was new to me that also CMummert recognized the problem. Perhaps the problem and its possible solutions will some day be described in WP to everybody's satisfaction. We are aiming at the same goal: to make a great encyclopedia. Let's cooperate to achieve that goal. Bo Jacoby 08:40, 17 January 2007 (UTC).
I think this section is too terse, and needs more examples and explanation. (Compare it to the previous section on integer powers: the difference is stark.) It should aim to be mostly accessible to, say, a high-school student or freshman who has learned the basic rules of arithmetic for complex numbers, knows trigonometry, and maybe something about polar/cartesian forms, but hasn't learned any complex analysis and has never heard of branch cuts. Euler's formula probably needs to be reiterated somewhere here (as opposed to just linked), but not rederived.
It's fine to refer the reader to the article on branch cuts and give the formal definition of the principal value, of course, but it should also give more explanation. The hypothetical reader I mentioned above will have no idea how to make sense of "a branch cut extending from the origin along the negative real axis". (Using z for both log(a) and the exponent of a in the same paragraph is also confusing.) On the other hand, the same reader should be able to grasp something like:
The section "general formula" should be expanded to give an actual derivation of this formula, from Euler's formula, by explicitly converting a+bi to polar form. (This will also simplify the formula, since it can then be written in terms of .) The section should again reiterate that a branch cut is involved in the choice of angle.
I would then suggest following it with a couple of examples, e.g. and .
I hope this is helpful; thanks for your efforts. —Steven G. Johnson 18:39, 16 January 2007 (UTC)
Bo Jacoby 21:25, 16 January 2007 (UTC).
Thank you, EdC, I'll move the graph to its proper place. You did not change a different graph, but the first one of the two graphs referred to. You are welcome to change the other one too. 06:09, 18 January 2007 (UTC).
The initial definition has been changed to "Formally, powers with positive integer exponents can be found by the initial condition a0 = 1 and the recurrence relation an+1 = a·an". (My highlighting of found as opposed to defined). The word "found" indicate to me that it is some kind of computational shortcut, but it is a definition. Also the repetition of the definition in the subsection on complex powers of complex numbers, has been removed. So now the article does not seem to contain a definition any more. It that correctly understood? Is that what the editor wants? Is it a consequence of the 00 controversy, that 'no definition' is considered better than a controversial definition? Or is it a transition between definitions? Please clarify. Bo Jacoby 16:51, 17 January 2007 (UTC).
article: "The function xy is continuous everywhere except when x and y are both 0, however." xy is also discontinuous for x=0, y<0, so the incorrect quote is removed.
article: "For this reason, it is convenient in calculus to treat 00 as an indeterminate form. " This repetition is also removed.
Bo Jacoby 05:58, 18 January 2007 (UTC).
To be honest, I think the claim is problematic as well. It makes sense in complex analysis, I suppose, as you can pick a neighborhood around x and choose a branch of the logarithm that's analytic on that neighborhood. But when we're not discussing complex numbers, negative x just doesn't work at all in any context where y may be continuously varying. With sufficient care, the claim might be rephrased to something accurate, but I don't really see as it's worth it; I think the simplest thing is to get rid of the sentence. -- Trovatore 08:38, 18 January 2007 (UTC)
To VectorPosse: It is sufficient to say that "The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity". The deleted sentence was at best superfluous, but actually misleading, because it repeated the function without repeating the domain, and then said 'everywhere'. Trovatore, for instance, was mislead. Exponentiation is a tricky business and there is no reason to hide the fact that there is a controversy. Nor is there any reason to become personal.
You say: "Sometimes it is convenient to define 0^0 = 1, and sometimes that doesn't work". Please provide an example where it doesn't work. To Trovatore: Thanks for the support. You say that "negative x just doesn't work in any context where y may be continuously varying". But negative values of x do work when y is a negative integer, except for a discontinuity at x=0.
Bo Jacoby 10:16, 18 January 2007 (UTC).
Bo Jacoby 14:06, 18 January 2007 (UTC).
I hope you forgive that I insert a headerline before your note above, because it is no longer about the discontinuity. I do not oppose your claim. The article has changed subject from mathematics to social science. This is extraordinary but probably cannot be helped.
The question no (4) above was about if we could at least be open on this controversy ? 'Revision as of 07:25, 18 January 2007' contained the subtitle 'Controversy, zero to the zero power', in order to say that the subsection is providing no information about 'zero to the zero power' but about a controversy in real-life. (The edit was undone by VectorPosse as quickly as possible). Most readers might expect an article on mathematics, and so we must be very explicite that this is not the case. Otherwise the reader gets confused and frustrated. because he does not understand what is going on. That is why the strong or weak arguments and counterarguments must enter into the article, not to convince me, but to enlighten the reader for himself to judge. Let us merely document the fact that "in real-life these arguments have not led to a consensus on the issue". Bo Jacoby 15:25, 18 January 2007 (UTC).
Thanks Steven. You was not the most pleasant editor to work with, but we had our fun and our AHA-experiences. If I had not been a little bold there would have been no article on the Durand Kerner method, and a lot of errors in WP have been corrected by me. Most of the thoughts of creative people are not original research, but has been done before, but that is in principle undecidable. The article that you had deleted on Ordinal Fraction seems to have been original. My work in the article on inferential statistics is actually also found in a paper by Karl Pearson from 1928, so that was not original research after all, even if nobody knew it, and so you had it deleted too. You may rewrite it when you have studied Pearson. You do not appreciate my work, but some other people do. I am pleased that you agree that the 0^0 controversy is unimportant, and the opposition against a notation backed up by Donald Knuth himself came to me as a surprise. Take care. Bo Jacoby 17:40, 18 January 2007 (UTC).
Oh I wish you were right, but some editors here are allergic against 0^0=1 and revert any edit using it. Thanks for the nice words. (Pearsons 1928-formula for mean value and standard deviation in inferential statistic or 'statistical induction' is no longer found i WP. Many (most?) professional statisticians do not know it). Bo Jacoby 22:51, 18 January 2007 (UTC).
I fail to see how 0^0 "arises" in the natural numbers. If I asked a typical undergraduaite freshman what they would have after taking 0 and multiplying it times itself 0 times, they would say "nothing" or "you can't do that".
Of course there is only one empty tuple, but that is not a fact about "natural numbers" it is a fact about tuples. That is, the theorem that the number of m-tuples from an n-element set is nm is only valid for n=m=0 if 0^0 has already been defined to be 1. Otherwise, the theorem would require a longer statement. CMummert · talk 14:28, 18 January 2007 (UTC)
We must do it both ways, respecting the neutral point of view, and leave the judgement to the reader:
and so on, for all the rest of mathematics. Bo Jacoby 15:46, 18 January 2007 (UTC).
I just archived this discussion page, from 12/20/06 through 12/31/06. I swear on my honor as a gentleman that I did not alter a word of it. The size of this page was reduced from 106kB to 62kB (measured with the "Edit this Page" button). That's 44kB in 12 days ≈ 3.67 kB per day, or roughly 750 words per day (31 per hour). DavidCBryant 15:32, 18 January 2007 (UTC)
I just archived this discussion page, from 1/01/07 through 1/14/07. I swear on my honor as a gentleman that I did not alter a word of it. The size of this page was reduced from 99kB to 76kB (measured with the "Edit this Page" button). That's 23kB in 14 days ≈ 1.64 kB per day, or roughly 336 words per day (14 per hour). DavidCBryant 21:39, 27 January 2007 (UTC)
I just archived this discussion page, from 1/15/07 through 1/18/07. I swear on my honor as a gentleman that I did not alter a word of it. The size of this page was reduced from 78kB to 35kB (measured with the "Edit this Page" button). That's 43kB in 4 days = 10.75 kB per day, or roughly 2,200 words per day (92 per hour). DavidCBryant 11:54, 2 February 2007 (UTC)
I tried to preserve as many of the changes made last night as I could while copyediting the article.
For real x, the power eix is the point on the unit circle, and x is the angle (1, 0,eix), measured in radian. (See Euler's formula: eix = cos(x)+i·sin(x), where cos and sin are trigonometric functions and i is the imaginary unit).
CMummert · talk 14:56, 19 January 2007 (UTC)
Bo Jacoby 15:44, 19 January 2007 (UTC).
To CMummert: The article on Trigonometric function says: "the trigonometric functions are functions of an angle". The functions cos(x) and sin(x) are trigonometric functions. So cos(x) and sin(x) are functions of an angle. Then so is cos(x) + i·sin(x), but this is equal to eix. But you said "that e^x is a function of a real number, not a function of an angle". So your point of view is at variance with the article on Trigonometric function. If you change it here, you must also change it there, but you may alternatively reconsider your point of view. SORRY! I read you as saying "that e^(ix) is a function of a real number, not a function of an angle" but you didn't. But then I do not understand your argument.
A reader of the article exponentiation can not be supposed to have any background in trigonometry which relies on power series which relies on polynomials which relies on exponentiation.
To StevenJ: The reader is probably more interested in understanding the subject than to know what is "exceedingly uncommon in mathematics". Some editor of the article, perhaps you, wrote: "Because of the periodicity explained above, the equation for has infinitely many complex solutions z". If this sentence has any meaning at all, it must be that " has infinitely many complex values z", or, eliminating z, that " has infinitely many complex values". So already here we see the first of these exceedingly uncommon cases where a multivalued function is used without selecting a particular branch cut. Later you may pick a branch cut, but the multivalued function was an inevitable intermediate step. It is often easy to express the set of solutions in terms of a particular solution, for example for logarithms:
Or for N 'th roots of unity:
Or for N 'th roots:
Bo Jacoby 01:37, 21 January 2007 (UTC).
\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) + \beta \cdot g(x) )
By the sum rule in differentiation, this is:
\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) ) + \frac{\mbox{d}}{\mbox{d} x} (\beta \cdot g(x))
By the constant factor rule in differentiation, this reduces to:
\alpha \cdot f'(x) + \beta \cdot g'(x)
Hence we have: That is correct, I was intentionally naive. The prospect of a discussion on whether an angle is different from its value called for a naive approach, (especially after the discussion about whether 0 is different from 0.0). Happily you did not insist, and the present formulation of the angle issue is quite satisfactory. Well done. You and I have equal right to edit the article. Bo Jacoby 11:02, 21 January 2007 (UTC).
To CMummert.
To DavidCBryant. The N 'th root and the logarithm are basicly defined by equations that may have more than one solution. I wrote about the set of solutions to an equation rather than about multivalued functions.
Bo Jacoby 22:08, 21 January 2007 (UTC).
The article on multivalued function refers to 'A Course of Pure Mathematics' by G. H. Hardy. There is no need to repeat it in the article on exponentiation. Bo Jacoby 12:07, 22 January 2007 (UTC).
I took the set-valued or multivalued interpretation of the logarithm away from the article and write 'solution to the equation ex=a '. The price to be paid for using principal value is that important rules, such as log(a2)=2·log(a), does not hold:
A warning to the readers would be nice. Bo Jacoby 14:22, 22 January 2007 (UTC).
Well done. Prove |eix|=1 without using Euler's formula: 1 ≤ |eix|2 = (limn(1+ix/n)n ) (limn(1−ix/n)n ) = limn(1+x2/n2)n ≤ limn(1+ε/n)n = eε for every positive ε . Just choose n>x2/ε. The only real number w satisfying 1 ≤ w ≤ eε for every positive ε is w=1. Q.E.D. Bo Jacoby 18:10, 22 January 2007 (UTC).
Thank you. Just note that a branch cut is not sufficient specification of a branch. Cutting along the negative axis splits the Riemann surface of the logarithm into branches, but each of these have the same branch cut. Bo Jacoby 23:18, 22 January 2007 (UTC).
The new warning subsection is nice. Perhaps we should introduce subsection headings: 'complex powers of positive reals' and 'real powers of unity' to deal with the useful cases az = ez·log(a) and e2πi·x where the warnings does not apply? Another subsection heading could be 'Rational powers of complex numbers'. I slightly prefer the term 'rational power' for the term 'root' because a root can be a solution to any equation like in 'root-finding method' while a rational power ar refers only to the equation xm=ar·m where m is the denominator of r such that r·m is integer. What do you think? Bo Jacoby 09:04, 23 January 2007 (UTC).
Bo Jacoby 13:39, 23 January 2007 (UTC).
"This leads to the following rules: Any number to the power 1 is itself. Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 0^0 is discussed below." As a matter of fact the argument does not lead to the 'nonzero' exception. CMummert is touchy about these things, but the definition assumed in the subsection is that an=1·a···a (n multiplications by a), and this definition applies equally well for zero as for nonzero values of a. Bo Jacoby 20:38, 25 January 2007 (UTC).
As you don't want 0^0 defined, you must change the definition "an=1·a···a, (n multiplications by a)", even if it is contemporary practice, because this definition implies 0^0=1 as a special case. One cannot accept a definition without accepting its consequences. You must find another way to explain why a0=1 for nonzero a. Bo Jacoby 00:18, 27 January 2007 (UTC).
I am talking about the subsection Exponentiation#Exponents_one_and_zero saying: "The meaning of 35 may also be viewed as 1·3·3·3·3·3 :the starting value 1 (the identity element of multiplication) is multiplied by the base as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to exponents one and zero: 31 = 1·3 = 3 and 30 = 1. This leads to the following rules: Any number to the power 1 is itself. Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 00 is discussed below." There is no justification here for restricting the second rule to nonzero numbers, and the reader, accepting the definition, is confused that the editor does not accept the conclusion. You guys who do not accept the conclusion are reponsible for making a subsection that makes sense. Bo Jacoby 07:08, 27 January 2007 (UTC).
Why such efford to conceal the fact that you are a nice person and a skilled mathematician? I regret any misunderstanding. I am not intentionally misleading; I do my best to be clear. The talk page subsection header: "edit 25-1-2007. exponent zero and one", was supposed to lead you to the article subsection: "exponent zero and one". The definition in article subsection "exponent zero and one" is exemplified by: "35=1·3·3·3·3·3", which obviously generalizes to: "an=1·a···a, (n multiplications by a)" because there is nothing special about the numbers 3 and 5, they are just examples. This leads to: "a0=1" for unrestricted a, and does not justify the restriction to only nonzero values of a. The correctness of this restriction is not my point right now. Nobody gets confused by the unrestricted definition, not even you, but students and readers do get confused by a sudden, unjustified, illogical restriction: "nonzero". This kind of stuff characterises bad math. So you have a choice to make. Either you produce another definition of the power an, one that only for nonzero values of a leads to "a0=1", or you accept both the definition, "an=1·a···a, (n multiplications by a)", and its unrestricted consequence: "a0=1". Bo Jacoby 13:09, 27 January 2007 (UTC).
Bad math is a problem. The link does not help. The definition "an=1·a···a, (n multiplications by a)" is unrestricted and must be modified if you want a restriction (as you do. I don't). Bo Jacoby 14:33, 27 January 2007 (UTC).
Two different definitions are mutually inconsistent, but no definition is inconsistent with the alternative "no definition". We do not have different definitions leading to different results, but one definition leading to one result and the alternative, "no definition", leading to no result. WP does not forces anybody to accept definitions that they do not like or do not want to use. We are all free not to understand, but don't undo edits in WP where it is explained to other people what mathematicians mean by writing x0 when x=0. Your edit (19:51, 25 January 2007) left the subsection in a state of corrupted logic. You now seem to argue that bad logic is a fact of the real world, and so this subsection should be illogical in order to reflect the real world? I don't like to argue at this level of madness. You are not more responsible for the contents of WP than anybody else. You are welcome to include the fact that some authors do not define 00, as you did, but you are not welcome to destroy the logic of other editors contributions. Bo Jacoby 12:19, 28 January 2007 (UTC).
This should make the 0^0 = DNE pretty clear:
0 = -0
0^(0) = 0^(-0) = 1/(0^0) = (1^0)/(0^0) = (1/0)^0 = DNE because 1/0 is undefined.
ARiina 14:54, 28 January 2007 (EST).
Thank you. I don't know the abbreviation DNE. 0^(0) = 1, 0^(-0) = 1, 1/(0^0) = 1/1 = 1, and (1^0)/(0^0) = 1/1 = 1, but (1/0)^0 is undefined. Division by zero must remain undefined, but zero'th power of a number is not a problem. Bo Jacoby 22:37, 28 January 2007 (UTC).
Assuming good faith, why is Trovatore reverting my edit on the definition leading to the simple rule? You cannot deny the consequence without denying the premise, as explained above. Bo Jacoby 10:31, 2 February 2007 (UTC).
DavidCBryant state that the old version was better. I had inserted section headers for clarity. Please discuss your point of view. Bo Jacoby 11:51, 2 February 2007 (UTC).
I tried to solve the following problems in the subsection on complex powers of complex numbers.
Bo Jacoby 13:28, 7 February 2007 (UTC).
Certainly, but the problems remain unsolved. Bo Jacoby 23:48, 7 February 2007 (UTC).
Quote: The function xy is continuous, however, whenever x and y are both nonnegative except when x and y are both zero. For this reason, it is convenient in calculus to treat 00 as an indeterminate form, since this allows the taking of limits (and other topological constructions) to be considered as commutative with the operation of exponentiation. (EdC's addition in italics).
The discontinuity for x=y=0 does not disappear by treating the expression 00 as an indeterminate form. The limit does not commute with exponentiation. limy→00y ≠ 00. If 00 is considered indeterminate, then commutativity implies that 0 is interminate. But 0 is not indeterminate. If 00 is considered = 1, then commutativity implies that 0 = 1, but 0 ≠ 1. In no case can continuity (meaning commutativity with lim) be saved. So actually the discontinuity is not a justification for leaving 00 undefined or indeterminate. Bo Jacoby 14:33, 8 February 2007 (UTC).
Exactly. There is no logical connection between being 'defined' in a point and being 'continuous' in a point, even if CMummert seems to believe that there is or should be such a connection. Consider the 3 cases:
So EdC is right: non sequitur. Bo Jacoby 22:32, 8 February 2007 (UTC).
Let me make the position clear for the last time. (I know Bo Jacoby already understands this and chooses to ignore it. I'll give EdC the benefit of the doubt.) EdC is right when he states that being continuous at a point and being defined at a point are not related, a priori. Fine. Nobody disputes this. So how does one extend a function to a point not in its natural domain? If the point corresponds to a removable discontinuity, then very few people object to defining the function to be the limit of the function as it approaches. If the discontinuity is not removable, as in the case of x^y as x and y approach zero, then it is usually not sensible to define it to be anything, at least from the point of view of calculus. The article as it currently stands explains this and also includes very reasonable justifications for choosing 0^0 = 1 that help other areas of mathematics work a bit more smoothly. So it is not a non sequitur to connect the ideas of continuity of a function to the definition of that function if one stipulates that in calculus it is reasonable only to define functions when they are defined by continuity in a natural way.
And now that I've written that, I'm kicking myself for going against my own advice not to engage in mathematical discussion that has already been made abundantly clear in this prolonged discussion. Please, let's not take this as carte blanche to re-debate this whole issue. I'm just trying to answer EdC since he seems more sincere and more reasonable in his attitudes than Bo Jacoby. VectorPosse 07:17, 9 February 2007 (UTC)
To VectorPosse, above:
I agree with all of the above (apart from the implication that (0, 0) is not in the natural domain of ). However, consider the situation from the standpoint of someone who is convinced that 0^0 does have a natural value, and hence is in the natural domain of . As it stands, the passage merely sets out reasons why one would not bother to define 0^0 in an analytic context. This is not a reason in itself to leave it undefined; calculus has no use for , but does not demand that that be undefined. Indeed, when learning calculus we confront pathological (i.e. typical) functions that are continuous nowhere, or continuous everywhere but differentiable nowhere, etc. So why should exponentiation be any different?
The most accessible reason I could think of is that having real exponentiation continuous everywhere it is defined allows limits to be taken through exponents. Perhaps it's not as accessible as I thought, but the article needs something there. – EdC 17:19, 9 February 2007 (UTC)
To Vectorposse. You write: "So how does one extend a function to a point not in its natural domain? If the point corresponds to a removable discontinuity, then very few people object to defining the function to be the limit of the function as it approaches. If the discontinuity is not removable, as in the case of x^y as x and y approach zero, then it is usually not sensible to define it to be anything, at least from the point of view of calculus". The statement in italics is not correct. There are indeed sensible ways in calculus to define a function value at a point of discontinuity. Consider the function f, defined for −π<x<+π by f(x)=x. It defines a fourier series, g, such that g(x)=f(x) for −π<x<+π . g is periodic: g(x+2π)=g(x). It has a nonremovable discontinuity for x=π. Nevertheless, the value g(π)=0 is a sensible way in calculus to define the function value at the point of discontinuity. This example proves your argument invalid. Bo Jacoby 19:41, 20 February 2007 (UTC).
To EdC: i can't understand why you removed my observation that 00 breaks algebraic properties of exponentiation if defines equal to 1. The exponentiation xy can be defined as the unique continuous homomorphism such that , this condition defines directly the values of for , and the function is extended to by continuity. But of course definition of 00 would break homomorphism axioms. Myrizio 01:42, 19 May 2007 (UTC)
After CMummert's latest edit:
Justifications for leaving 00 undefined include:
- The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity. [1] That is, the function xy has no continous extension including the point (0,0): along the x-axis the limit is 1, along the y-axis the limit is 0, and any intermediate limit a can be obtained using the curve y = log(a)/log(x). However, if y is an analytic function of x, or if there exists a positive constant, a, such that y < ax, then the limit is 1.
- The function zz, viewed as a function of a complex number variable z and defined as ez log z, has a logarithmic branch point at z = 0.
I don't understand how either of the two listed points have any bearing on whether 0^0 should be taken to be defined. I thought it was because continuous functions have useful properties; was I wrong? – EdC 23:57, 10 February 2007 (UTC)
*The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity. [2] That is, the function xy has no continous extension from the open first quadrant to include the point (0,0). [3]
It is uncommon in the context of elementary calculus to extend a function in a manner that makes it become discontinuous.A discontinuous extension would cause the function to lose a number of desirable properties, for example that when and , , and so would be avoided.
'Paige (the reference) does not go into great depth about why 0^0 is undefined'. Editors are not supposed to invent explanations that are not supported by the literature. If not even the reference explains why 0^0 should be left undefined, then there is no justification. Bo Jacoby 13:53, 19 February 2007 (UTC).
The article on Continuous function is inconsistent in this matter. The sentence 'The fact that a discontinuity can be removed does not make the original function continuous' is at variance with the definition, that a function is (everywhere) continuous if it is continuous at every point of the domain. Is the function x/x continuous? It is not defined for x=0, so it is not continuous on the real axis. It is continuous at every point of its domain, so it is continuous. The sign function f, defined for nonzero real values of x by f(x)=+1 for x>0 and f(x)=−1 for x<0, is continuous for every point in its domain. Yet it is usually considered discontinuous because lim an=lim bn does not imply lim f(an)=lim f(bn) in all cases, which is a usual characterization of continuous functions. So I get the point. By removing the definition of 00 you make the function f(x,y)=xy continuous at every point of the domain, which is nicer than saying that it is continuous for (x,y)≠(0,0). Right? Bo Jacoby 16:51, 24 February 2007 (UTC).
Fine! Then at last we may be able to provide a sensible explanation why some people do not want 00 to be defined: 'Some people like a function to be continuous on it's entire domain rather than on a proper subset of it's domain. So they prefer to remove the point (x,y)=(0,0) from the domain of xy rather than to admit a function xy that is defined, but not continuous, for (x,y)=(0,0)'. Are you sure that all mathematicians agree that, say, f(x) = 1/x is a continuous function? Some mathematicians might spot a discontinuity at x=0, even if f(0) is not defined. Bo Jacoby 18:06, 25 February 2007 (UTC).
Great. So I misused the word 'discontinuity' meaning 'non-removable singularity'. I assume that we agree that the singularity of xy at (x,y)=(0,0) is not removable, neither by defining, nor by undefining, 00. If you can express this with NPOV, then please do. I'm not sure I can do it myself, because the combinatorially and algebraically well-justified definition 00=1 is more elementary than the analytical concepts of singularity and discontinuity, and the advocates of undefining 00 show to be both touchy and inarticulate. Bo Jacoby 11:41, 26 February 2007 (UTC).
Last thing first: for nonnegative integers, exponentiation is easily explaned like this. "nm is the number of m-letter-words taken from an n-letter alphabet". As there is only one 0-letter-word, (namely the empty word: ""), n0 = 1, irrespective of n. Can topology be explained as easily as this?
In the article, integer exponents are explained first and noninteger exponents are explained later, being more advanced, and so the integer definition of 00 precedes the real nondefinition of 00. So, from the point of view of the logic of the article, we are "removing" (0,0) from the domain. It does not help to leave 00 undefined in the elementary section, because the definition in the elementary section implies that 00=1.
Next, the property, continuity in all points of the domain, is not taken as a rule in calculus. Functions with singularities are treated in calculus, and values in singular points may be defined, for example by fourier series, as in sawtooth wave. So the only example given in the article of one of the 'desirable properties' is invalid, and the 'number' of desirable properties lost by a discontinuous extension seem to be zero. So there is no valid justification for not defining 00=1. Your idea of reduced domain and extended domain is nice. Bo Jacoby 15:58, 2 March 2007 (UTC).
Consider the binomial coefficient. It is understood by the combinatorial interpretation, that Cn,k is the cardinality of the set of k-element subsets of a n-element set. This explanation applies for nonnegative integer values of n and k. For small values of n and k the binomial coefficient is found by elementary counting, for example
The special cases Cn,0 = | { {} } | = 1, and Cn,k = | {} | = 0 for n<k, follow from the combinatorial explanation. For bigger values of n and m counting becomes impractical, and the formula Cn,k = (n/k)·Cn−1,k−1 speeds up the calculation.
Now compare with exponentiation. It is understood by the combinatorial interpretation, that nk is the cardinality of the set of k-element tuples of a n-element set. This explanation applies for nonnegative integer values of n and k. For small values of n and k the power is found by elementary counting, for example
The special case n0 = | { () } | = 1 follow from the combinatorial explanation. For bigger values of n and m counting becomes impractical, and the formula nk = n·nk−1 speeds up the calculation. So, repeated multiplication is more advanced, and more general, than the combinatorial explanation.
The point of mentioning the sawtooth wave is not that it is has singular points, but that the value of the function at the singular points can be determined by calculus, in this case by fourier series, thus refuting the assertion that 'it is ordinarily taken as a rule in calculus that [some limit statement holds] whenever both sides of the equation are defined'. Bo Jacoby 08:09, 4 March 2007 (UTC).
I'm with you until you say that "repeated multiplication is more advanced, and more general, than the combinatorial explanation". In combinatorics, yes; but in other areas of mathematics the exponential is arrived at without invoking the cardinality of exponential sets. "Exponentiation" as a term refers to a family of operators with shared characteristics; there is no well-defined basic exponentiation operator.
For the sawtooth: ordinarily. The sawtooth is only expressible in terms of elementary functions through infinite series; everyone knows to be careful taking limits through infinite sums. If the exponential is taken undefined at (0, 0) then elementary functions preserve limits. – EdC 23:41, 5 March 2007 (UTC)
Certainly counting is more elementary than multiplication or exponentiation. That doesn't mean that taking exponential sets is more elementary than exponentiation; there are other motivations for exponentiation than exponential sets.
The sawtooth is, however, a limit; it is not elementary.
Finally, (assuming 0^0 undefined) neither of your double limits holds; since the latter is undefined at ; since the latter is undefined at . The only way to resolve the situation is to strike the axes from the definition of the exponential, in which case neither limit exists. – EdC 23:01, 6 March 2007 (UTC)
Q.E.D. Bo Jacoby 09:00, 16 March 2007 (UTC).
The editors who are supporting the idea of leaving 00 undefined claim to be mathematicians. Now let's recapitulate. There seem to be no valid justification for leaving 00 undefined. The 'justifications' in the article are not quoted from the references, but are invented by the editors, and these 'justifications' are not valid. The singularity of xy for (x,y)=(0,0) is not removed by restricting the domain of xy. The word 'nonzero' in Exponentiation#Exponents_one_and_zero is unmotivated in the context, which explains why x0=1 for all values of x. Bo Jacoby 16:22, 22 March 2007 (UTC).
An edit was made this morning with edit summary "After a long discussion the 0^0 case for integer exponents is settled." No new discussion has happened recently, and the new version has reintroduced the same absolute claims about 0^0 being 1 and being the empty product that have been discussed in great depth already. CMummert · talk 12:27, 28 March 2007 (UTC)
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(
help)Not only does the C99 standard not define pow(0,0)
(except to say that it may raise a domain error, and allowing implementations to define __STDC_IEC_559__
if they follow
IEC 60559), but there's nothing in the C Rationale about pow
at all, and what there is about cpow
seems unenlightening to me. I've reverted the incorrect inclusion of C in the list, but can anyone think of a reason that that reference to the Rationale should remain, or should I get rid of it too? --
Quuxplusone
22:33, 8 March 2007 (UTC)
pow (0, 0)
should return a value or raise a domain error; implementations may define __STDC_IEC_559__
if they follow
IEEE 754.[7]" What does IEEE 754 say about 0^0? –
EdC
22:41, 8 March 2007 (UTC)I was noticing that the "Powers of i" and the "Powers of e" sections are both under the heading "exponential with integer exponents." I propose that "powers of e" gets moved under "Real Powers of Positive Real Numbers", and "powers of i" goes to "Complex powers of complex numbers". Does anyone have a problem with this? -- shaile 16:03, 16 March 2007 (UTC)
I've removed the image of x^y as I thing it is missleading. I've had a bash at drawing x^y for positive x. It clearlt shows that taking limits along curves will give a limit of 0.-- Salix alba ( talk) 09:05, 3 April 2007 (UTC)
It is a nice picture. Note that on the curve y=f(x)=(log a)/(log x), the exponential xy = a. The function f is not an analytic function for x=0. If x and y approaches zero along an analytic curve, the exponential must approach 1. Bo Jacoby 21:38, 3 April 2007 (UTC).
is there a separate article on the exponential map in differential geometry? if not, should information on it be added here or should I write an article on it? SmaleDuffin 16:35, 4 April 2007 (UTC)
This article has gone from B-class to 'Good'. Well-written and explained. Attractive presentation with some kewl formulae and graphs. No math errors there I could find. Neutral and not given to controversy. Stable: um, yeah. Images: has graphs that really help the exposition. Well done and take a step up. Gifir2007 11:28, 7 April 2007 (UTC)
The third sentence of the article is Exponentiation can also be defined for exponents that are not positive integers. But it doesn't go on and say what the definition is. Obviously,
but leaving the sentence on its own can create some confusion for those who don't know basic index laws. Thanks Gizza Chat © 12:00, 7 April 2007 (UTC)
Made a new try on the lead but I guess it still needs a little improvement. Ricardo sandoval 20:54, 25 April 2007 (UTC)
The article says that 2 to the 3^4 is different from 2^3 to the 4, but it doesn't mention order of operations. Is 234 2^81 or 8^4?
The other indeterminate form is 1^infty, not 0^infty. This was fixed by an IP editor this morning, then reverted, and then I accidentally undid the reversion without leaving a useful edit summary. CMummert · talk 17:01, 9 April 2007 (UTC)