Talk:Exponentiation/Archive 4

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Exponentiation, 0^0

Moved from User Talk:CMummert

Hi CMummert.

I wish to continue the discussion here on your talk page because it is getting somewhat personal. You wrote:

The series for e^x is just one power series; it isn't any different from the general power series in that it can be written e^x = 1 + \sum_{n=1}^\infty x^n/n! or \sum_{n=0}^\infty x^n/n!.

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:

As always, feel free to edit the article; I do not remove things in a knee-jerk fashion, and several of your previous edits have improved the exposition. CMummert 20:55, 19 December 2006 (UTC)

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).

I did leave a comment in the the edit summary. The formula you added contradicted the text above it by once again claiming that 0^0 = 1. And even if the formula was correct, it was redundant to the text above it.
I don't believe the section on 0^0 is "polluting" the article; I think it is clarifying the article, which otherwise dealt with 0^0 in a haphazard fashion. The 0^0 section was started when information that didn't belong at Empty product was moved here. Is there some part of the 0^0 section that you think is not readable by beginners? It seems pretty clear to me. CMummert · talk 14:07, 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.

  1. It is confusing that an expression is defined and undefined at the same time. I do not mind it being mentioned that some books leave 0^0 undefined, but as the only proposed definition is: 0^0=1, that definition should be stated as the definition; anybody not defining the expression 0^0 are not using the expression 0^0 either, and so no harm is done by tolerating people to use the definition 0^0=1, and the beginner need not to know that some books do not define 0^0.
  2. It is not clear how 0^0 can be defined for integers but not defined for reals, (or defined in discrete mathematics but not defined in continuous mathematic). In all other cases of mathematics the reals are treated as a generalization of the integers, so that a correct whole number formula is also correct as a real number formula when the numbers happen to be whole. (2+2=4 is true for integer 2 and 4 as well as for real numbers 2 and 4). It is very confusing to the beginner as well as to me that this rule seem to be violated.

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 necessary because of the way that the article is structured; it makes sense to cover 0^0 only once, rather than five times, and so in the other places a pointer to the section where it is covered is helpful to the reader. Otherwise the reader might wonder why 0^0 is not covered. It certainly doesn't make sense to say that 0^0 is defined to be 1, and then later explain why it isn't actually defined to be 1.
0^0 isn't defined and undefined at the same time - some authors define it, and some don't. There is no universal consensus in mathematics that it is defined at all. It's not just that some books don't include it - not all mathematicians would accept that 0^0 is defined to be 1 in the same way the 2+2 is defined to be 4. Unlike 2+2 = 4, which can be tested experimentally, 0^0 has no real-world significance and cannot be tested. I don't believe your claim that a beginner would be unable to read the section on 0^0 and see what is going on.
Also, please don't move things to "advanced topics" section. The title of that section is already troubling - how do we decide what is an "advanced" topic? - and the material there is should be spread around. In particular, the synonym involution should be in the lead paragraph, because that is where synonyms go in all other articles on WP. CMummert · talk 16:56, 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).

Who exactly is preventing anyone from using such a definition? Come on, out with it, give us their names. We'll take care of it. --Trovatore 00:32, 14 January 2007 (UTC)
In case I wasn't explicit enough, before: the entire section on "advanced topics" needs to be sorted and integrated with the rest of the article. There is nothing "advanced" in this article. The fact that the value of 0^0 cannot be empirically tested is quite relevant to its status as a convention rather than a fact. CMummert · talk 00:58, 14 January 2007 (UTC)
Huh? Of course 0^0=1 can be empirically tested. Draw two circles on the ground, and put 3 stones in the first circle and 5 in the second. Then 5^3 is the number of ways you can draw lines, a single line from each of the stones in the first circle to one of the stones in the second circle; that gives 125 ways. (Try it!) Now remove all the stones from both circles; there is exactly one way to draw lines as specified, which is to leave the ground blank. In other words, you have just empirically demonstrated that 0^0=1. –EdC 01:43, 14 January 2007 (UTC)
Yet again, we are wandering into debating the math, and that is beside the point. Yes, it should be debated once. When that is done and consensus is not reached, then we write the article to reflect all points of view neutrally. The article should mention both points of view and it does. Aren't we finished here, Bo Jacoby? Your side of the argument has plenty of room in the article, so why do you insist on edging out ours (or continuing to insert your own in more and more places scattered around the article)? You have raised no new point in weeks. Give it a rest. VectorPosse 01:51, 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).

You are not the arbiter of what is important. (Neither am I. That's the point here. I'm not trying to make my view the only valid one.) And using words like "vandalism" and "polluted" is not helping your case. VectorPosse 08:21, 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.

1^4 = | { (1,1,1,1) } | = 1. There is one 4-tuple from a 1-set.
2^3 = | { (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) } | = 8. There are 8 3-tupples from a 2-set.
3^2 = | { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } | = 9. There are 9 2-tuples from a 3-set.
4^1 = | { (1) (2) (3) (4) } | = 4. There are 4 1-tuples from a 4-set.
5^0 = | { () } | = 1. There is 1 0-tuple.

Bo Jacoby 11:49, 14 January 2007 (UTC).

Yep. For example, (2, 1, 2) corresponds to drawing a line from rock A to rock 2, from rock B to rock 1, and from rock C to rock 2. The "lines between rocks" model serves to demonstrate that a purely empirical interpretation is possible. I agree that the tuple model is a lot more concise in written text, though. –EdC 13:35, 14 January 2007 (UTC)
Re the "interpretation" section: please stop re-adding things redundant things, especially ones that seem to argue that 0^0 = 1. My goal here is to have the article neutrally describe the situation in the real world, where 0^0 is sometimes defined as a convenience and someltimes left udefined.
I don't know which "real world" you're living in, but in the real world I know, 0^0 is defined to equal 1 because that's what its value is. Convenience has nothing to do with it, any more than 1+1 is defined to equal 2 as a convenience. –EdC 13:52, 14 January 2007 (UTC)
The information on "set exponentiation" is already buried down in the advanced topics. Moreover nm can denote the cardinality of the set of functions from a set of size m to a set of size n, but it can denote other things as well. The word "is" is inappropriate when linking notation with the meaning of the notation. This mistake has appeared in this article before with claims that 0^0 "is" an empty product. A symbolic expression can represent a concept, but cannot be equal to it.
True, but in that case I hope you will amend the claim that exponentiation is repeated multiplication. –EdC 13:51, 14 January 2007 (UTC)
Yes. There are lots of settings where it doesn't represent repeated multiplication. CMummert · talk 14:53, 14 January 2007 (UTC)
I'm going to refactor the advanced topics this morning, which will take care of the set exponentiation at the same time. CMummert · talk 13:25, 14 January 2007 (UTC)

Refactoring article

Here are some running notes from my major edit this morning:

  • Synonyms such as involution belong in the lead.
  • The expression \lim_{x \to \pm\infty} is not standard.
  • For the purpose of defining power functions, it doesn't matter how the logarithm is defined
  • The stuff that was there about complex logarithms was a mishmash of erroneous and nonstandard concepts. For example, the expression e2·π·i·(1/n)·1 on its own cannot define a primitive root of unity, and the logarithm is not usually defined as a multivalued function. I cleaned it up some. Please discuss additions before reinserting them in the article. —The preceding unsigned comment was added by CMummert (talk • contribs) 14:48, 14 January 2007 (UTC).

I have a few comments.

  1. Euler's constant has two meanings, e or γ. You erased the correct link.
  2. The expression \lim_{n \to \pm\infty} may not be used a lot, but it was explained, and its removal has now made the calculation invalid for negative x-values.
  3. The expression e2·π·i·(1/n)·1 is a primitive root of unity. It may not be the only one, but what was written was correct, and there was a link to root of unity for details.
  4. Now π is undefined. You removed the definition.
  5. The definition of ex extends to complex x, not only to real x, as is now written. The logic of the article of going from integer to complex exponents has now been destroyed.
  6. I corrected some obvious typos. Try Show preview before Save page.
  7. The interpretation subsection gave clarity which the discussion page has proved necessary.
  8. In complex analysis the logarithm must be considered a multivalued function.
  9. I wonder when you find this: "Formally, powers with positive integer exponents can be defined by the initial condition a0 = 1 and the recurrence relation an+1 = a·an ". When you find it you must destroy it because it leads to a conclusion which you dislike. And you must modify a lot of other formulas from WP as well. Enjoy.
  10. Why not discuss before editing?

Bo Jacoby 16:53, 14 January 2007 (UTC).

Thanks for fixing several typos in the article. Here are some responses to your points.
  1. The link to E (mathematical constant) is still there. It is not inaccurate to call this number Euler's constant.
  2. The expression \lim_{n \to \pm\infty} is never used, so far as I can tell, in the context of calculus or complex analysis. Can you provide any reference where it appears? The definition e^x = \lim_{n \to \infty} (1+x/n)^n is valid regardless of whether x is negative or positive.
  3. There is more than one primitive nth root of unity for any n > 2. Your expression only covered one of them.
  4. This is an encyclopedia article, not an axiomatic treatment. There is no need to define π.
  5. Feel free to rephrase this.
  6. Thanks.
  7. The interpretation section, as I explained above, gave one interpretation as if it is the only interpretation. That section is still present under "powers of sets".
  8. Not only is it false that the logarithm "must" be multivalued, in the standard presentation of complex analysis it is not multivalued - for each branch cut there is a single-valued logarithm function defined on the complement of that branch cut. Even if you switch to a Riemann surface, the log function is still single valued - the point of a Riemann surface is to eliminate the branch cut by changing the topology.
  9. I did read that phrase, and I decided that "can be" defined is accurate. "Is always defined" would not be accurate. Similarly, the definition of exponentiation as a Z action on an group can be used to define 0^0 in the integers, but is not always done.
  10. Nobody is required to discuss something before editing. If, after editing, the consensus of editors is that the edit is appropriate, then it will remain, and if the consensus is that it isn't useful it will be changed. It is a bad idea to repeatedly reintroduce things that a consensus of editors has deemed inappropriate, however; this is why I continue to ask you to discuss before reintroducing the claim that 0^0 = 1 in all settintgs. I have backed up this request with specifc references where 0^0 is not defined, including one that calls 0^0 = 1 a "myth".
I asked at the math project talk page for other editors to watch this page and comment on it. Hopefully some of them will do so. CMummert · talk 18:42, 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 e^x = \lim_{n \to \infty} (1+x/n)^n for negative integer x depend on e = \lim_{n \to -\infty} (1+1/n)^n. It is not sufficient that e = \lim_{n \to +\infty} (1+1/n)^n. That is why I wrote e = \lim_{n \to \pm \infty} (1+1/n)^n. (I did not write \lim_{x \to \pm\infty} ). You are wellcome to rewrite the argument avoiding the \pm 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).

You are always free to edit the article, as is everyone else. I don't understand your reference to some hypothetical "beginner"; there is no need to write the article for a 5 year old reader. Everyone with a trivial amount of education knows what π is. Euler's constant is a fine way to describe the base of the natural logarithm.
Here is an argument for why e^x can be defined as the appropriate limit. Note that the limit is taken as n goes to infinity, not +- infinity.
Your continued support for multivalued logarithms is self-consistent, but not the way that the logarithm is typically considered. The article on branch cuts is the place to explain their motivation, not this article.
There is no need to insert "0^0 can be defined to be 1" in multiple places in the article; the reasons for this possible definition are throughly covered once in the section on 0^0. Just because I left one indirect reference doesn't mean I support adding more of them.
CMummert · talk 20:52, 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).

There is no formal definition of exponentiation in this article, which is why I removed the claim "according to the definition." Several other editors have also pointed out that 0^0 is not universally defined to be 1, and this is the list time I will comment on that matter here. Your comments on the talk page are just repeating the same arguments, so I will stop responding, again, until something new comes up. CMummert · talk 21:44, 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).

Perhaps you are correct that I should have deleted that sentence the first time through. You are confused about the difference between "can be" and "is".
I edited the section on roots of unity as you suggested. CMummert · talk 22:24, 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).

2007-1-15

  • I didn't delete the reference to a "multivalued logarithm", because it isn't completely inaccurate, but the article needs to point out that the actual logarithm function used to define the complex exponential must be single valued; that's why you need a branch cut. Moving to a Riemann surface does give continuity, but then you are no longer talking about points on the complex plane, you are talking about points on a Riemann surface.
  • I would be glad to rewrite section on e^x if you feel it is erroneous. By removing the derivation that is there, I could make that section more clear and complete. The article on exponential function is the right place for derivations.
  • Notation such as {a1/n} is not correct - the set notation needs to include a desription of what values n ranges over. Also, by convention, a1/2 denotes the principal value of the square root, not the set of possible values. I will not be drawn into discussing that matter here.
  • The combinatorial interpretation doesn't belong in the section on integer exponents. I moved it near the other application section about polynomial equations. Let me point out here, as well as my edit comments, that the term "m-set" is not standard in any way.
  • Comments in the article such as "A complicated and rather useless formula is known for the complex powers of complex numbers." don't seem appropriate to me.

CMummert · talk 14:16, 15 January 2007 (UTC)

Answer to the above:

  1. No, the logarithm function used to define the complex exponential need not be single valued; that's why a branch cut is not needed.
  2. The new definition for ex should give the same function values as the old definition for integer values of x. The explanation in the article is now insufficient. An elementary article should refer to other articles only for further reading, not for understanding the basics.
  3. Do you suggest a better notation to distinguish between the set of values {a1/n} and a single value a1/n ?
  4. The subsection on combinatorial interpretation is the most elementary part of the entire article as it does not even use multiplication. It should be moved to the very beginning and serve as the definition of nm for nonnegative values of n and m. Then the present definition would become a theorem.
  5. Right. The appropriate approach to a useless formula is to remove it entirely.

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)

This is nice. The definition of principal value in WP is that the principal value of the log of a positive number is real, and so the principal value of 11/N is e2·π·i·0/N=1, which is quite unimportant, rather than e2·π·i·1/N which is the useful and interesting choice, which I prefer, but which other WP editors have banned. You might like (or hate) to discuss this matter with User:Stevenj who opposed me in this discussion. Bo Jacoby 18:13, 15 January 2007 (UTC).
The power function az is single valued, like any other ordinary function. Although you may think it is more convenient to define it as a multiply valued function, this is not the way it is actually commonly done. To make it single valued, a branch cut is needed.
Your comments about using a combinatorial lemma to define exponentiation differ from your usual claim that the most pedestrian definitions need to be given. Nobody defines exponentiation of integers using that combinatorial lemma (except indirectly in set theory, but that is covered already). Doing so here would be bizarre. The basic definition of exponentiation is in terms of repeated multiplication, as the article reflects.
This article does not need to stand on its own. It is fine to leave things undefined and link to their respective articles for definitions, or to give brief discussion with a pointer to the main article. This is not an axiomatic treatment or a textbook.
I fixed the notation to be correct already. The key is to use English rather than symbols if the symbols aren't correct. CMummert · talk 23:47, 15 January 2007 (UTC)

Powers of negative real numbers

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).

I agree, generally speaking. I read through this yesterday, and decided to quit editing the article at exactly this point ... the bit about odd/even powers of negative real numbers and limits is probably too complicated for the audience. I've dropped the second paragraph, and added a little more verbiage to the first one. Does it look better now? DavidCBryant 11:39, 15 January 2007 (UTC)
A common question among students is whether you can raise a negative real number to an irrational power when working with real numbers only. The motivation for this section is to address that question. CMummert · talk 14:18, 15 January 2007 (UTC)
OK, I'll cook up a short paragraph that explains the fact that a limit has to be unique, or else it doesn't exist. And I'll try to make it clearer than the previous explanation was (it was clear enough to me, but then analysis is sort of my bag – I was thinking of other readers). But it will have to wait for a little while ... I need to go offline and do something utilitarian right now.  ;^> DavidCBryant 16:20, 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).

Give me a break, will you, Bo? I've made the changes both of you requested. You were in here hollering about it just 33 minutes after I said I had some business in real life. I needed a couple of hours. Anyway, the section has been expanded, so hopefully it's clearer now. And it's been separated from "positive real numbers". Oh -- I also added a sentence to the part about ex, because I couldn't find the fact that ex is positive definite for real x anywhere else in the article. DavidCBryant 18:56, 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).

You know, Bo, your remarks above would almost be funny if they weren't so patently false.

Some time ago I did a lot of work improving this article.

Yesterday I spent roughly five hours patching up some little things (not substantive stuff, but fixing the style of in-line expressions, and correcting minor grammatical errors). You tore a lot of that out right away. So then I spent three hours digging through the page history, to identify the saboteur. You're not "improving" this article, Bo. You're ruining it. Consistently, methodically, and almost continuously ruining it.

The present high activity is good, but it also contains some backwards steps.

I agree that it contains some backwards steps. You are responsible for most of them.

As I do not revert edits made in good faith, ...

No, you don't revert them ... you just yank them out by the roots, and replace them with your own opinions. Please don't try to explain what "good faith" is. By your own actions, clearly documented below, you have recently demonstrated that you are not acting in good faith yourself.
  • At 15:01 14 Jan 2007 CMummert saved a reasonably good version of this article. It did contain some typos, but the math was good.
  • At 22:11 14 Jan 2007 Bo Jacoby made this edit. He inserted editorial comments directly into the article itself, in clear violation of established procedure. (If you don't like the way the article reads, Bo, please edit it yourself. If you want to discuss the changes, please write something on the talk page. Under no circumstances whatsoever ought you place directions to other editors in the body of the article. I think CMummert deserves an apology.)
  • At 10:24 15 Jan 2007 Bo Jacoby made this edit. He took a neutral sentence – "A formula is known for the complex powers of complex numbers", moved it to a new section and inserted a heading, so it would be quite conspicuous, and reinserted these words: "A complicated and rather useless formula is known for the complex powers of complex numbers." That's in clear violation of the NPOV standard.
I don't really have a lot else to say to you, Bo Jacoby, except this. Please stop. Please stop right now. There are plenty of qualified editors who will fix this article right away, if only you'll let them. Oh -- it's not as if this pattern of behavior is new, or unusual. Here's a quote from your talk page.

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)

Have a great day, Bo! DavidCBryant 13:01, 16 January 2007 (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).

Durand-Kerner method

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)

Sounds fine to me. There are lots of root-finding algorithms, as you point out. CMummert · talk 17:35, 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).

Most of the algorithms in the section "Finding roots of polynomials" of the root-finding article work for complex as well as real roots (e.g. Jenkins-Traub, Laguerre, Bairstow, Aberth, splitting circles, and companion matrices all find complex roots). It makes the most sense to link to a general overview when there are so many applicable methods.
Of course, your preference for the Durand-Kerner article is understandable, given that you originally titled it Jacoby's method.
—Steven G. Johnson 22:43, 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).

Some comments

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:

Principal nth root of unity

article: The number e2πi (1/n) is the principal nth root of unity

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 WN = exp( − 2πi / N); 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)

I'm not real big on algebra, but I do have an indistinct memory of the "principal primitive nth root of unity" in discussions of cyclotomic fields and such, so named because gcd(1, n) = 1 for every positive natural number n. Is that what someone else (CMummert? EdC?) was driving at here? DavidCBryant 19:41, 16 January 2007 (UTC)
I edited the article to make the definition more qualified, and removed the link to Fourier transforms. Unfortunately, this is another example of a well-understood concept without a widely-adopted standard terminology. CMummert · talk 19:57, 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).

It is not "obvious notation". I hate to sound like a broken record, but I have to reiterate: we are not here to debate the merits of your system, Bo. Even if we agreed that 11/N was a good substitute (given that a standard terminology is lacking), it would not be proper to introduce it here. You really don't need to keep defending it. Your point is moot. VectorPosse 07:25, 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).

It is improper to suggest that I keep silent on the matter. Watch your words, please. And no, I do not grant you the "freedom" to use Wikipedia to do your constructive thinking on the matter. Do that on your own time and follow well-established Wikipedia guidelines while you are here. You must have consensus and the article must reflect published conventions. When such conventions are lacking, we may explain that to the reader and use one or more of the more commonly available conventions.
I'm happy to cooperate. It is the opinion of several editors here that you are the one being generally uncooperative. VectorPosse 18:17, 17 January 2007 (UTC)

Complex Powers of Complex Numbers

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:

Because of the periodicity explained above, the equation ex = a for x = loga has infinitely many complex solutions x: for any solution x, the number x+2πi is also a solution. So, the complex logarithm could be defined as a multivalued function. In order to assign a specific value to az = ezloga, however, we must choose a particular value of loga. The most common choice, known as the principal value of the logarithm, is the complex number loga whose imaginary part lies in the interval ( − π, + π]. Such a choice is known as a branch cut, and the principal value corresponds to a branch cut extending from the origin along the negative real axis. Although the complex logarithm is commonly used in the formal definition of complex exponentiation az, for computation one often first converts a to polar form, as described in more detail below.

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 r = \sqrt{a^2 + b^2}.) 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. ( − 2)3 + i and ii.

I hope this is helpful; thanks for your efforts. —Steven G. Johnson 18:39, 16 January 2007 (UTC)

Thanks for taking the time to comment on the article. I incorporated the paragraph above and rephrased the rest of that section. I agree it was much more terse than the rest of the article. I agree that "general formula" would be better replaced with an explanation of how to use polar form to compute complex powers. CMummert · talk 20:09, 16 January 2007 (UTC)

Suggestions for improvement of the graphs

  1. The graph: "Exponentiation with various bases: red has base e, green has base 10, and blue has base 1.7 " would be more instructive like this: "Exponentiation with various bases: red has base e, green has base 10, black has base 1, blue has base 1/e and magenta has base 0.1. ". And the graph belongs in the subsection Exponentiation#Real_powers_of_positive_real_numbers.
  2. The graph: "From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8 " would be more instructive like this: "From top to bottom: x−8, x−4, x−2, x−1, x−1/2, x−1/4,x−1/8, x0, x1/8, x1/4, x1/2, x1, x2, x4, x8 "

Bo Jacoby 21:25, 16 January 2007 (UTC).

Also, using colour (where avoidable) to convey information discriminates against people with colourblindness. I'm sure there's a WP: guideline about this somewhere. –EdC 21:44, 16 January 2007 (UTC)
Yup: WP:MOS#Color coding and WP:WAI#Color. –EdC 21:46, 16 January 2007 (UTC)
The best thing is almost always to put the labels directly on the graph. i.e. don't use a legend, just put text labels next to/over the curves. Since the exponent is written in a small font as a superscript, it might be good to label the curves with the exponenent alone (i.e. add text labels "8", "7", ... to the curves, and then say in the caption that the curves are x^b where b is given by the label). Direct labelling avoids an indirection, and also circumvents problems for color-blind readers (or black-and-white printed copies). —Steven G. Johnson 23:54, 16 January 2007 (UTC)
Done, thanks. –EdC 21:42, 17 January 2007 (UTC)
Thanks Ed, but you changed a different graph. I think we were referring to Image:Root_graphs.png. —Steven G. Johnson 01:38, 18 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).

Have all definitions disappeared?

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).

In practice, there isn't a "true" definition of exponentiation; there are lots of different definitions that mostly agree with each other. This article is not an axiomatic treatment of exponentiation, so there is no need to choose one definition here at the expense of all the other ones. The article as it stands seems like a good survey of the various interpretations of exponentiation in different contexts.
Moreover, I suspect from your posts higher on this page that your goal in getting a "formal definition" introduced is so that you can reintroduce your claims that 0^0 = 1, backed by the "formal definition" in the article. The article currently states "In many settings, 00 is defined to be 1." That seems accurate to me; I don't see what more you could hope for while still acknowledging that there are settings in which 0^0 is not defined to be 1. CMummert · talk 17:24, 17 January 2007 (UTC)
Yes, the article has been changed from a short explanation to the common reader into a general survey of interest to the authors only. I do not approve. I acknowledge that there are settings where 0^0 is not used, and therefore not defined, but where 0^0 is used it invariably means one, so there is no value in not defining it, and the fact that some authors do not define it is of marginal interest to the reader who wants to understand exponentiation. Your willingness to sacrifice clarity and definitions in order not to define 0^0 by accident is amazing. (The initial condition a0=1 is a little simpler that a1=a, because it contains the variable a only once). Bo Jacoby 07:08, 18 January 2007 (UTC).
a0=1 may indeed by simpler than a1=a, but the section is explicitly discussing positive integer exponents. Zero is not a positive integer. --Trovatore 08:25, 18 January 2007 (UTC)
Yes. But the section could be changed to discuss nonnegative integer exponents. Benefit: More generality, simpler definition, connection to combinatorial interpretation. Drawback: The 00 controversy. Bo Jacoby 10:57, 18 January 2007 (UTC)
Well, I just put the words "are defined" back into the sentence "Formally, ..." Then I realized I had made a mistake, but Trovatore fixed it before I could do it myself. This is exciting! DavidCBryant 18:35, 17 January 2007 (UTC)

discontinuities

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).

If something is incorrect but fixable, then fix it. Do not remove it. (And I trusted you too quickly by changing it to reflect the above comment. At the very least, it should be for x = 0 and y less than or equal to zero.) And there is nothing controversial about it. Sometimes it is convenient to define 0^0 = 1, and sometimes that doesn't work (10). Your arguments to the contrary are specious. I tire of hearing your twisted logic about not mentioning the undefined case, even though that's the one that every one of my calculus students need to learn about(1). VectorPosse 08:21, 18 January 2007 (UTC)
Turns out I trusted you WAY too much. The paragraph clearly says "nonnegative x and y" at the beginning, so it was fine as it was. I reverted myself and learned an important lesson about trying to give even the slightest credence to Bo Jabocy. VectorPosse 08:35, 18 January 2007 (UTC)
Oh, and one more thing. I don't know if you did it on purpose or not, but try not to make edits to this section first, and then make a whole bunch more edits. Are you doing this because you think it will harder to revert? The undo feature actually makes it quite easy. I'll give you the benefit of the doubt this time, but it would help if it didn't seem like you were trying to "sneak" it in. VectorPosse 08:25, 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)

Oh, hadn't seen the thing about nonnegative x and y. I still sort of think the sentence is marginal. The natural domain for the function (x,y) |-> xy, in the real C0 category, is x>0. I don't see the value in straining to include x=0, y>0. --Trovatore 08:59, 18 January 2007 (UTC)
Of course, restricting x > 0, y>0 still leaves xy with no continuous extension to 00, while it does extend continuously to points where just one of x or y is zero. CMummert · talk 12:20, 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).

The difference here is that Trovatore gave a reasonable mathematical justification and pinpointed the potential source of confusion (2). You just deleted it, calling it inaccurate. I reiterate my previous point: fix it instead of deleting it (3).
As for the controversy, I also reiterate: there isn't one (4). Two conventions happily coexist in the article (5). CMummert did a great job writing it up in a neutral way. Why draw attention to it as if one point of view or the other might be "false" or "questionable". They aren't. I register my strong objection to using the word "controversy" to draw unnecessary attention to a rather unimportant example of exponentiation.
About "becoming personal": please spare me. You have no room to talk about getting personal. (Remember when you called my revert "vandalism" (6). Other examples abound.) My comment is a statement of fact based on your behaviour. I cannot trust that your actions are always motivated by a desire to improve the article. At least some of them only serve to push your point of view (7). I am not claiming that you don't believe you are improving the article and I'm not claiming that nothing you do improves the article (8). I'm simply saying that I cannot allow myself to trust that any given edit you make is correctly motivated. Therefore I cannot afford to give the "slightest credence" to your claims, even if some of them turn out to be correct or well-motivated (9).
And finally, about your call for my examples: You've been given many and I will not get sucked in to your pointless debates (10). VectorPosse 10:51, 18 January 2007 (UTC)
  1. I did not understand what you say that your calculus students need to learn about, is it 'twisted logic' or is it 'not mentioning' or is it 'the undefined case' ? And why do they need to learn about that?
  2. I too did pinpoint the source of confusion : "xy is also discontinuous for x=0, y<0".
  3. Trovatore and I agree that it should be deleted rather than fixed, because it contains nothing new.
  4. I refer to the controversy amongst mathematicians, not amongst WP editors. My edit was perfectly neutral: I merely told the reader that a controversy exists as documented in the reference section where different authors express different opinions on the matter. I agree that the unimportant example should not draw attention.
  5. The coexistence between the two conventions is a happy one only for some editors, not for the readers. It is devastating for the article that definitions are unclear, and it is unclear what is the reason for introducing the confusion.
  6. If your revert, which to me looked like vandalism, was in fact made in good faith in accordance with WP policies, then I do apologize for my mistake in calling it vandalism.
  7. Pushing my point of view is improving the article, in my point of view. You have got another point of view. That is no reason to get upset.
  8. Now, if you believe that I believe that I am improving the article, why not trust that my actions are motivated by a desire to improve the article? Your logic is less than perfect here.
  9. Here is an exercise in probability theory for you. If i out of n claims turn out to be correct or well-motivated, what is the mean value and the standard deviation of the credence that a rational person should give to such claims? (Hint: the credence is beta distributed). And what is the probability that the next claim will be correct or well-motivated?
  10. You owe to the readers to share your hesitations against the definition. I have seen no examples where the definition doesn't work, (which does not mean that the definition makes a discontinuous function continuous. It doesn't). None of the Justifications for leaving 00 undefined are logically valid, as they don't have the consequence that one should leave 00 undefined.

Bo Jacoby 14:06, 18 January 2007 (UTC).

Controversy

It doesn't matter whether the arguments in favor of the definition are strong or weak - the fact is that they are not universally accepted, and this lack of acceptance has been documented with reliable sources. The WP policies are clear that if a consensus on an issue does not exist in the real world it is not appropriate to claim in a WP article that one side or the other is "correct" - the article should neutrally describe the situation. To repeat: it makes no difference how mathematically compelling you feel the arguments in favor of the definition are, because in real-life these arguments have not led to a consensus on the issue. The article currently says "In many settings, 0^0 is defined to be 1." This seems perfectly accurate to me. CMummert · talk 14:19, 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).

A lack of uniform notation is not the same thing as a "controversy", and this article is no different from other articles on mathematical subjects in Wikipedia in neutrally reporting the various notations in common use. (Nor is there anything wrong with "social science", and indeed articles on mathematical topics should include well-sourced historical background, etc., as well as the bare mathematical theorems and definitions. Wikipedia is an encyclopedia, not a mathematics textbook.) There are many areas of mathematics, like this, where notation is not completely uniform across subfields, but professionals rarely get excited or actively argue about such things. (Dubbing the big yawn of 00 inconsistencies, in particular, a "controversy" is grossly overstating the attention that anyone gives that matter in modern times.) —Steven G. Johnson 15:58, 18 January 2007 (UTC)
(Not that anyone has ever convinced Bo that Wikipedia is not the place to promote his notational reforms, from declaring a uniform convention for 00 to introducing the convention \sqrt{1} = -1 to a new definition for the DFT to a new function notation to a new fraction notation, despite dozens of editors telling him the same things over and over again in numerous contexts. Nor do I expect anyone to convince him now; only weight of numbers keeps him restrained. Thanks for wasting everyone's time, Bo.) —Steven G. Johnson 15:58, 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).

No one here is "opposing" the 0^0=1 convention (and I suspect that most of us prefer it); you are chasing shadows. We are opposing the false implication that this convention is universal, or that the longstanding alternative convention of 0^0 as indeterminate is considered "incorrect" in all subfields of mathematics. (And by the way, I didn't touch your inferential statistics article; other editors, including at least one professional statistician, apparently felt it was misleading and that any worthwhile content in it should be merged into other articles.) You are clearly capable of making positive contributions; it's unfortunate that so much of your energy is spent waging pointless notational battles against multitudes of other editors. —Steven G. Johnson 20:24, 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).

"Arises in the natural numbers"

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:

  1. The number of n-tuples taken from a set containing m elements equals mn for all nonnegative integers n and m.
  2. The number of n-tuples taken from a set containing m elements equals mn for all nonnegative integers n and m except for m=n=0 where it equals 1.

and so on, for all the rest of mathematics. Bo Jacoby 15:46, 18 January 2007 (UTC).

I agree with CMummert; the fact about tuples is already mentioned, and this is not the same thing as a fact that inevitably "arises" about the natural numbers. Natural numbers, including zero, were doubtless in use for centuries before 0^0 = 1 "arose". (Bo, the article already points out that 0^0=1 removes special cases from combinatorics formulas; there is nothing more that needs to be added here.) —Steven G. Johnson 16:09, 18 January 2007 (UTC)
There you go again, confusing the motivation for exponentiation in repeated multiplication with the operation itself. The point that needs to be made is that 0^0 = 1 is not just a convenience, it is the inevitable result of any foundational treatment of the naturals, and so is necessary if we are to have numbers that can be treated consistently in set theory and logic.
Also, I don't get what the emphasis here on tuples is. The foundational operation for exponentiation is the function operation \langle X, Y\rangle \mapsto \{f: Y \to X\}, just as the foundational operation for multiplication is the Cartesian product and the foundational operation for addition is the disjoint union. –EdC 17:48, 18 January 2007 (UTC)
There are two problems with that analysis. The first is the assumption that the definitions used by regular mathematicians, especially by analysts, agree with those used set theorists. I don't mind that you rephrased the article to remove the word "is" near the beginning, but for most people exponentiation with positive integer exponents is defined as repeated multiplication, not in terms of the cardinality of a set of functions. Similarly, few regular mathematicians would say that multiplication of integers is defined by considering cardinalities of Cartesian products. As S. Johnson points out, people were using these arithmetic operations for centuries before the concept of "Cartesian product" or "function space" had been invented.
The second problem is that you assume that the only foundational treatment of the natural numbers is as finite cardinals. It is perfectly possible in ZFC to define the real numbers as the unique complete ordered Archimedean field, define the complex numbers from the reals, define complex exponentiation, and finally define the natural numbers as the positive elements of the prime subring of the reals. (In that scheme, 0^0 would remain undefined. CMummert · talk 18:08, 18 January 2007 (UTC)
Perhaps change the article to say "This definition arises in set-theoretic developments of the natural numbers, ..." or some such thing? —Steven G. Johnson 20:34, 18 January 2007 (UTC)
Mathematicians used exponentiation centuries before ZFC. Bo Jacoby 23:00, 18 January 2007 (UTC).
Yes. They used natural numbers and exponentiation without either running into the problem of 0^0 or the issue of set theory, function spaces, etc. CMummert · talk 23:13, 18 January 2007 (UTC)
More than a century before ZFC, Cauchy and Libri discussed 0^0. Bo Jacoby 10:32, 19 January 2007 (UTC).
Ah, forgot about that. Actually, I'm not sure I was ever shown that construction; sure we did the construction of the reals as a complete ordered field, but there was always the naturals hovering in the background. I guess the issue then is that discrete mathematicians (including computer scientists) tend to confuse the naturals with the finite cardinals (esp. the ZFC cardinals).
I've added "This definition arises in foundational treatments of the natural numbers as finite cardinals..."; hope this can be made OK. I wonder whether we might be able to extract something useful from this discussion, regarding how the issue is driven by questions of ontological priority among the number systems. –EdC 00:16, 19 January 2007 (UTC)
Your recent edit "arises ... finite cardinals" is great. That is exactly the context where the empty product, empty function, etc. are important.
I may have been too hasty in my second claim above; it is certainly possible to define the complex numbers, and then the naturals as their prime subring. But in order to define the complex exponential without first (at least implicitly) defining integral powers of complex numbers, and thus without developing power series, is not as obviously easy as I thought it was this afternoon. It seems like I have to prove things about analytic continuation without using power series. So I apologize for that comment.
I originally tried to use "discrete mathematics" and "continuous mathematics" to describe something like the priority of the number systems, but I didn't see any tenable way to do it and so I gave up. CMummert · talk 00:35, 19 January 2007 (UTC)

Archiving

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)

Editing 2007-1-19

I tried to preserve as many of the changes made last night as I could while copyediting the article.

  • I kept the unit circle interpretation. The most problematic phrase was the one below this bullet. In addition to the grammar problems, it had the problem of confusing a real number with an ordered triple. There is no need to define π, i, sin, cos, etc. because this is not a "formal" treatment of exponentiation. We can assume the reader is familiar with polar coordinates, as well, so we don't need to talk about the orientation of the angle here.

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).

  • In an informal sense the "complex logarithm" is multivalued, but whenever a function log(x) is used it is a single-valued function.
  • The second root is indeed called the 2nd root. Of course it isn't called the 2th root, but that is just a fact about English grammar. I also moved the square, cube root stuff to the section on general roots just below, because it is not unique to roots of unity.
  • The reference for principal root of unity got moved to the wrong sentence. It is worth reading the footnote to see what it is actually citing before breaking sentences in half. Also, footnotes follow punctuation.

CMummert · talk 14:56, 19 January 2007 (UTC)

  1. It was common practice since antiquity that angle B in triangle ABC is called angle ABC, preferably with the words 'triangle' and 'angle' replaced by a symbols. But why not simply omit the controversial point and write: "For real x, the power eix is a point on the unit circle, and x is the angle". There is not much doubt about which angle we are talking about.
  2. The reason why I explain π, i, sin, cos is that it makes the text comprehensible to a wider audience. The reader which I have in mind do not know about exponentiation, and far less about trigonometry, and absolutely nothing about polar coordinates. It is not because I want a formal treatment. And no harm is done by telling the reader something that he already knows, it may make him/her feel comfortable.
  3. The symbol log a sometimes means the set of solutions, and sometimes any solution, and sometimes a particular solution to the equation ex=a. Let's be open about it.
  4. Really! I never heard of the 2nd root, only the square root. Perhaps some reader has also heard about the square root and will like to be ensured that we are talking about the same thing.
  5. Thanks.

Bo Jacoby 15:44, 19 January 2007 (UTC).

The problem with (1) is that a real number is not the same thing as an angle in the ABC sense, and e^x is a function of a real number, not a function of an angle. The current wording is essentially what you suggest but is semantically correct. When it is useful you can wikilink to terms rather than defining them. But in general we don't wikilink or define every term; the reader is assumed to have some reasonable background before reading the article. If you can produce a reference, in print, that uses notation like A = log(z), where A is a set and z is a complex number, then I will be glad to add this to the article. It is exceedingly uncommon in mathematics to use multivalued functions. The terms square root and cube root are still in the article. CMummert · talk 16:09, 19 January 2007 (UTC)
It would be less confusing to say, "it is exceedingly uncommon in mathematics to use multivalued functions without selecting a particular branch cut." Although multivalued functions can be represented as a map to a set of values, their explicit description in such terms is rare in my experience. In common informal usage the term "multivalued function" refers to any function that has several possible values (as opposed to having a set of simultaneous values), of which a single value is usually selected. As in "the square root of a positive real is multivalued, in that we could choose to define it as either positive or negative, but by convention we usually choose the positive result." (For example, the Complex Variables book by Mathews, and the Mathematics books by Aleksandrov, Kolmogorov, and Lavrent'ev, talk informally about multivalued functions as having "several possible values", one of which is selected via a branch cut, without ever formally defining them as maps to sets.) —Steven G. Johnson 18:17, 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 ez = a for z = loga has infinitely many complex solutions z". If this sentence has any meaning at all, it must be that "z = loga has infinitely many complex values z", or, eliminating z, that "loga 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:

( the set of values of log(x) ) = { (any value of log(x) ) + 2πi·k | k in Z }.

Or for N 'th roots of unity:

( the set of values of 11 / N ) = { (any primitive value of 11 / N )k | 1 ≤ k ≤ N }.

Or for N 'th roots:

( the set of values of a1 / N ) = (any value of a1 / N ) · ( the set of values of 11 / N ) = { (any value of a1 / N ) · (any primitive value of 11 / N )k | 1 ≤ k ≤ N }.

Bo Jacoby 01:37, 21 January 2007 (UTC).

I think that you are intentionally being naive about the trigonometric functions. The situation with complex logarithms has already been explained above. Please stop repeating the same arguments; I will not comment on these issues again, but reserve the right to edit the article. CMummert · talk 03:25, 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).

Edit 2007-1-21

I reverted your recent changes. In addition to making the article less clear, the part about log was incorrect. I did move your comments about the real logarithm to the appropriate section of the article, and expanded them there where they fit in.
Whenever the notation 'log(z)' is used, it is used to denote a single-valued function, not a multivalued function. The use of Log(z) with a capital log is not common. The common notation is to use lowercase log for what you called Log and to use no symbolic name at all for a multivalued logarithm function. Please find a reference for your notation before introducting it to the article. CMummert · talk 13:44, 21 January 2007 (UTC)
Just to add to the confusion, I have seen some references to Log(z) as the principal branch of the natural logarithm in the complex plane. I don't think it's standard notation, though. As far as I can tell, treatment of the natural logarithm as a "multi-valued function" varies a lot from one author to another. Formally correct authors take care to insist that a "function" is always single-valued. Books about applied math, or engineering math, are not nearly so careful. Modern presentations of mathematical ideas range all over the lot, from the hopelessly naive to the maddeningly formal. Oh – FWIW, I think the math articles in Wikipedia should adopt the convention that a function is single-valued, and let "relationships" be multi-valued. DavidCBryant 14:23, 21 January 2007 (UTC)
PS I almost forgot ... Whittaker and Watson define Log as the real-valued natural logarithm of a positive real number (this is the real principal branch), and define log(z) = Log(|z|) + iarg(z). Therefore this particular argument is really about the argument. Cute!  ;^> DavidCBryant 14:51, 21 January 2007 (UTC)
I have seen many places where an author use the word "multivalued" in the context of logarithms. The article as it stands says "the logarithm could be defined as a multivalued function" to reflect this reality. What I have never seen is notation like
\{ 2 + 2k\pi : k \in \mathbb{Z} \} = \log(e^2)
Have you ever seen a book that uses the symbolic log function as a multivalued function in that way? CMummert · talk 14:37, 21 January 2007 (UTC)
No. Never. If I ran across that in a book, I'd stop reading right there. And I actually read all the way through a book once, a long time ago, that purportedly "proved" that π = 22/7. DavidCBryant 14:51, 21 January 2007 (UTC)

To CMummert.

  1. You are even faster to revert than to improve. Ask questions first and give other editors the benefit of the doubt for a while. It gives such a nice impression of politeness, and we all serve as role models for fellow editors.
  2. The notation Log is used in the WP article Principal value. Why not read the reference before deleting from the article?
  3. The branch cut is a cut between branches, not a choice of a branch. Several branches have the same branch cut. So your version of the article is incorrect in this respect and an improvement is definitely called for.
  4. The reference to Euler's formula here is no good because Euler's formula assumes more knowledge on the part of the reader than should be expected at this stage, as I have explained before. Shall we include the proof that |eix| = 1 for real x ?
  5. I did not write \{ 2 + 2k\pi : k \in \mathbb{Z} \} = \log(e^2) or even \{ 2 + 2k\pi i : k \in \mathbb{Z} \} = \log(e^2) , but I explained with English words, as you have earlier suggested, that '( the set of values of log(z) ) = { (any value of log(z) ) + 2πi·k : k in Z }'. I was carefully explicite about the set of possible values. That would be in your example: '( the set of values of log(e2) ) = { 2 + 2πi·k : k in Z }'. (I don't mind using {:} for {|}).
  6. Your example is a little tricky because ex is usually the well defined complex exponential function rather than merely the special case a=e of the non-uniquely defined exponentiation ax. Some authors write exp(z) for the exponential function to avoid this ambiguity, but this is probably unnecessary, and the notation ex is almost universally used in applications, including eix for trigonometry.
  7. You should tell the readers that complex exponentiation ax for irrational b is not really useful except for a=e.
  8. You should include a comment on exponential growth and exponential decay in the subsection on real exponentiation.

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).

I have adressed many of those points in earlier comments, and I will not be drawn into discussing the same issues again and again. In particular, I have already pointed out that no wikipedia article can serve as an authoritative source for another article, and that providing links is the usually the correct way to aid readers who don't have the appropriate background. If you can't provide any printed reference to back up your ideas about multivalued logarithms, then you are not in a position to argue they should be included. I can easily provide references to the standard conception of the complex logarithm as a single-valued function defined using a branch cut. So my request for printed sources about multivalued logarithms still stands. CMummert · talk 22:52, 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'm not asking for a vague pointer to a reference that uses the term "multivalued function". I am asking for a reference that treats the symbolic expression log(z) as a set-valued function in the context of complex analysis. And I am looking for a direct quote, with bibliographic details and page numbers, showing that the book uses log(z) to denote a set-valued function. CMummert · talk 13:10, 22 January 2007 (UTC)

Edit 2007-1-22

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:

iπ = log(−1) = log((−i)2) ≠ 2·log(−i) = 2·(−iπ/2) = −iπ.

A warning to the readers would be nice. Bo Jacoby 14:22, 22 January 2007 (UTC).

I was about to put two other examples here, so the previous comment is fortunately timed. I agree that another section should be added to the section on complex exponentials point out that familiar rules for real powers and logarithms fail to hold in the complex setting.
Two more examples:
(-1)^{1/2}(-1)^{1/2} = i\cdot i = -1
(-1)^{1/2}(-1)^{1/2} \not = (-1\cdot -1)^{1/2} = 1
and
i = (-1)^{1/2} = \left (\frac{1}{-1}\right )^{1/2} \not = \frac{1^{1/2}}{(-1)^{1/2}} = \frac{1}{i} = -i
(I inserted i three times in the identity by Bo Jacoby because log(i) = iπ not π.) CMummert · talk 14:35, 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).

There is no reason for this article to prove in any way that |e^ix| = 1. This is not a textbook nor an axiomatic development. The changes you made this morning were great; I tried to leave the structure alone but made some prose and notation changes. The only problem is the description of the branches of the logarithm, which isn't quite correct, but I left it to think of the minimal change that would make it correct. A branch is essentially a maximal consistent cut of the Riemann surface; you have to specify the shape of the cut to completely determine the branch. CMummert · talk 18:31, 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).

Real powers of unity are not interesting, and not worth talking about here. Is it true that the identities all hold for complex powers of positive reals?
I don't think that rational powers of complex numbers need a separate section; they are not defined separately in practice, there is already a short discussion of them, and there isn't much more to say. CMummert · talk 12:18, 23 January 2007 (UTC)
  1. The real exponents powers of unity, (not the real powers of unity, you are right), e2πi·x, are used in fourier transform and circular motion, but they may not be recognized as real powers of unity because of the ban against the worksaving notation 1x. Just as the primitive N 'th root of unity e2πi / N is not written 11/N but is still called an N 'th roots of unity.
  2. Yes I think so. When the base is positive, then the log is real, and then there is no way the rules will lead to nonreal logarithms of the base. az+w = e(z+w)·log(a) = ez·log(a)+w·log(a) = ez·log(a)·ew·log(a) = az·aw . (az)w = (ez·log(a))w = ez·log(a)·w = az·w . [P.S. Oh no, I am too hasty on the last one. −1 = eπi = e(2πi)·(1/2) ≠ (e2πi)1/2 = 11/2 = 1. Bo Jacoby 12:15, 24 January 2007 (UTC)]
  3. I basicly agree that there isn't much more to say, but I suggest subsection-headers to structure what is said.

Bo Jacoby 13:39, 23 January 2007 (UTC).

edit 25-1-2007. exponent zero and one.

"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).

You are still trying to use logic, but the issue of 0^0 is one of contemporary pactice rather than of logic. The section on 0^0 already describes the reasoning you inserted today, and there is a clear link in the material quoted above in question to the section on 0^0; if a reader is confused about why zero is excluded, that section will clarify the issue. I will not continue to discuss the issue on this talk page. CMummert · talk 22:03, 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).

Are you talking about the statement in the introduction? If so, then, no, there is no need to be super-technical or precise. Typically, the intro provides an informal definition. The technical case of 0^0 is covered in the body of the article, as is appropriate. No need to be obtuse just to make your point. VectorPosse 01:10, 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).

CMummert just addressed this in the paragraph right above your complaint. (In fact, many of us have spoken on the matter over and over again.) Anyway, you can understand why I thought you were referring to the intro given that you used the letters "a" and "n" just like in the intro, and not the subsection "Exponents one and zero". Why be intentionally misleading?
The exposition is correct in the subsection since it resticts to nonzero numbers. And I don't trust your judgments about any supposed confusion on the part of the reader. My argument that my calculus students would be confused if the whole article were written your way is just as valid (and just as irrelevant in the end). VectorPosse 09:11, 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).

If they are confused, there is a link right there to the section on 0^0 that will clarify the issue. It's not a problem. CMummert · talk 14:19, 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).

The problem is that the various conventions and definitions are inconsistent in the real world. Nothing this article says will change that. The point of WP articles is try to describe the way that things actually are, not to recreate them to be more consistent. Please stop trying to microanalyze the definitions and their consequences; you will run into contradictions involving 0^0 because these appear in the real world. None of the other editors watching this page feel that there is a problem; I will not continue discussing this issue. CMummert · talk 14:47, 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).


Trovatore's revert

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's revert

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).

Quote from User_talk:DavidCBryant: I read both versions of the article before I reverted your edit of exponentiation on 2 Feb, 2007. Before you chopped it all up, the article read fairly smoothly. After you introduced a lot of new headers, and switched the order of presentation all around, the article was jumbled and confused. I put it back in order, thereby improving it. DavidCBryant 22:23, 6 February 2007 (UTC)

I tried to solve the following problems in the subsection on complex powers of complex numbers.

  1. The number pi was not explained.
  2. Euler's formula is unknown to the elementary reader.
  3. The complex logarithm function was not explained.
  4. The use of the exponential function of imaginary numbers was not explained.
  5. The use of the exponential function of real numbers was not explained.

Bo Jacoby 13:28, 7 February 2007 (UTC).

These issues have already been discussed in the past month. CMummert · talk 13:33, 7 February 2007 (UTC)

Certainly, but the problems remain unsolved. Bo Jacoby 23:48, 7 February 2007 (UTC).

EdC's edit, Zero_to_the_zero_power

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).

I don't think the exact wording currently in the article is optimal. The idea is correct, though - the reason that 0^0 is left undefined in calculus settings is that with this convention exponentiation does commute with limits whenever it is defined. This makes it very similar to the operation x \mapsto 1/x, which also commutes with limits whenever it is defined. CMummert · talk 14:37, 8 February 2007 (UTC)
For purposes of keeping the article elementary, I would be inclined to remove this part, although I understand what CMummert is saying. I'm not sure that commutativity of operations is easy enough a concept to be casually introduced here without any further explanation. Even to make sense out of the idea requires certain assumptions on the form of the exponential. For example, if we are thinking of calculus, then most 0^0 indeterminate forms involve an expression like f(x)g(x). Unless f(x) does not depend on x (i.e., it is constant), then it doesn't even make sense to commute a limit involving x past it. VectorPosse 17:14, 8 February 2007 (UTC)
The property that holds of exponentiation, and that I understand the major justification for holding 00 undefined to be, is that if f(x) \to a and g(x) \to b nonnegative constants not both 0, then f(x)^{g(x)} \to a^b. Without this it's not obvious why the fact that xy is discontinuous at (0, 0) but continuous in the remainder of the positive quadrant has any bearing on whether 00 is defined. –EdC 18:20, 8 February 2007 (UTC)
The problem I had with the passage is that in
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.
the second sentence is a non sequitur. –EdC 18:34, 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:

  1. A real function f, defined for x>0 and y>0 by f(x,y)=xy , is discontinuous for x=y=0, no matter whether f(0,0) is being defined or not.
  2. A real function f, defined for y≠0 by f(x,y)=x/y , is discontinuous for y=0, no matter whether f(0,0) is being defined or not.
  3. A real function f, defined for x≠0 by f(x)=x/x , is continuous for x=0, no matter whether f(0) is being defined =1 or left undefined.

So EdC is right: non sequitur. Bo Jacoby 22:32, 8 February 2007 (UTC).

Actually your case (3) is wrong. A function is neither continuous nor discontinuous at a point not in its domain. You might as well ask whether it's continuous at green. --Trovatore 00:06, 9 February 2007 (UTC)
And by the same token, your points (1) and (2) are also wrong. --Trovatore 00:13, 9 February 2007 (UTC)
Now I'm confused as to what this is about. If EdC objects to the "non sequitur" caused by using the phrase "For this reason" then we can try to re-word that. (Also, it's not really a non sequitur.) But the stuff about commutativity doesn't fix that; it just makes the sentence unnecessarily complicated. VectorPosse 02:03, 9 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 Trovatore. A function may be continuous or discontinuous at a limit point of the domain. See closure (topology). The number zero is a limit point, but the color green is not.
To VectorPosse. The concepts of a ' natural domain' or ' natural way to define' are non-mathematical. Your position is contradictory: First you state: 'being continuous at a point and being defined at a point are not related'. Secondly you state that 'it is not a non sequitur to connect the ideas of continuity of a function to the definition'. So you have made clear that you have got no clear position. Your ad hominem comments are out of order according to Wikipedia:Resolving_disputes. Disagreement is no excuse for being disrespectful or for not assuming good faith. 'The benefit of the doubt' is about the relationship between a judge and an accused person, not about the relationship between wikipedia editors. If you regret what you have written, the solution is to delete it, rather that to kick yourself. Bo Jacoby 08:57, 9 February 2007 (UTC).
I hold my tongue a lot here to try to keep this civil, but I must register my objection. Bo, your edits are far more disrespectful to the editors here than my alleged ad hominem attacks. How else can one explain this:[1]? After weeks and weeks, to make such an edit in blatant disregard for everything you've been told here--it's shameful. I am tired of it. I am taking this page off my watchlist. In the grand scheme of things, 0^0 is not worth my time. I shouldn't let this frustrate me as much as it does. But alas, it does, and I have better things to do with my time. My apologies to Trovatore and CMummert and a number of others who are trying to watch this page and keep it NPOV. Bo, you lost your right some time ago to have anyone "assume good faith" in your edits. EdC, I hope I haven't judged you harshly, as I truly do assume good faith on your part. The "benefit of the doubt" to which I referred was simply meant to suggest that you are genuine in your efforts to clarify your point, and you don't edit recklessly, even if I don't agree with you. My sincerest apologies if it came across otherwise. VectorPosse 09:40, 9 February 2007 (UTC)
The explanation is that the definition an=1·a····a (n multiplication by a), leads to a0=1 for all a without exception. I did not suppress the fact that some authors do not want to define 00. It is NPOV and respectful. Bo Jacoby 10:35, 9 February 2007 (UTC).
In the immediate context, yes; in the wider context that edit was misleading and inflammatory, and you knew that. –EdC 14:33, 9 February 2007 (UTC)

To VectorPosse, above:

...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.

I agree with all of the above (apart from the implication that (0, 0) is not in the natural domain of xy). 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 xy. 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 2^{\aleph_0}, 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)


EdC: The notion of continuity is not defined at a point not in the domain, period. What you can ask is whether the function has a continuous extension to the larger domain, and, if so, whether that extension is unique. But there is no notion, in standard mathematical usage, of the function itself being continuous at a point not in the domain. --Trovatore 17:44, 9 February 2007 (UTC)
Hi EdC--I agree with you with respect to 2^{\aleph_0}. Calculus does not demand anything in this case because it is rarely (if ever) tied to a calculation in calculus. On the other hand, 0^0 comes up all the time. So it is more justified to make a cboice about its definition or non-definition. VectorPosse 18:48, 9 February 2007 (UTC)
So this is an interesting contraposition, but maybe not for quite the reasons thus far brought up.
There is no mathematical issue yet to be decided that would tell us whether or not we should allow the point (0,0) into the domain of the exponential function. It is a matter of convention. There are reasons for conventions, and we can discuss them as much as we like, but it doesn't affect the article–our role here is simply to report the conventions in use, not to prescribe.
The case of 2^{\aleph_0} is quite opposite. There is no conventional issue to be decided; the setting is the von Neumann universe, and if you aren't using that setting, then you're just not talking about the same problem. The question, for mathematical realists anyway, is one of fact: Does there in fact exist a wellorder of the reals whose every proper initial segment is countable? There either is or there isn't, ZFC's failure to decide the question notwithstanding. --Trovatore 19:00, 9 February 2007 (UTC)
Realist? Isn't that a euphemism for Platonist? Regardless, what I'm trying to establish is why calculus is required to take a position on the definedness of 0^0. –EdC 00:02, 10 February 2007 (UTC)
I don't know that the word "Platonist" needs euphemisms. "Realist" is just more accurate; mathematical realism doesn't have that much to do with the philosophy of Plato, per se.
I don't think anyone has claimed that calculus requires us not to define a value for 00. In my view there is no natural value for 0.00.0, in the real-continuous (or even real C∞) category, but we don't have to say that. We just report the conventions. That's all we have to do. Let's avoid straining to provide explanations that go beyond the sources. --Trovatore 03:19, 10 February 2007 (UTC)
We're not just reporting convention, though; we're reporting justification for those conventions. Currently the article says that xy is continuous in the upper right quadrant less (0, 0); it doesn't say why this would be considered a desirable property. –EdC 12:36, 10 February 2007 (UTC)
So then remove all reported justifications, unless they come from a reference. --Trovatore 18:56, 10 February 2007 (UTC)
I don't see any need to remove the things that are currently in the article. The part about commuting limits could stand to be rewritten, but I don't think it is compeltely out of place. It is easy enough to say that all the other arithmetical operations are continuous whenever they are defined, but x^y has no continuous extension to (0,0). CMummert · talk 22:29, 10 February 2007 (UTC)
Thanks. Yes, removing passages from that section would be horribly disruptive. Also, I don't think the Paige quote could stay under those conditions (don't get me wrong, I think it should stay); without a reason to want x^y to be continuous, the fact that x^y has no continuous extension to 0^0 has no bearing on whether 0^0 should be taken to be defined. –EdC 22:48, 10 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).

Myrizio's addition

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 \varphi_x : \mathbb{R}^+ \rightarrow \mathbb{R}^\times such that φx(1) = x, this condition defines directly the values of φx(y) for y \in \mathbb{Q}, and the function is extended to \mathbb{R} by continuity. But of course definition of 00 would break homomorphism axioms. Myrizio 01:42, 19 May 2007 (UTC)

You wrote:
  • Defining 00 = 1 breaks algebraic properties of exponentiation, because 00 = 01-1 = 010-1 = 0/0.
At face value, that statement is evidently false; the same statement applies to all powers of 0. I don't understand how the above is related; would you care to elucidate? –EdC 15:40, 19 May 2007 (UTC)
Mh, ok. You're quite right too, because if one pretends that x-y = 1/xy then it is impossible to define 0y for any negative y too. The big problem is that \mathbb{R}^\times is not a group (\mathbb{R}^*, without 0, is), so the definition as continuous homomorphism do not actually make a lot of sense. Myrizio 02:13, 21 May 2007 (UTC)

CMummert's edit

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)

It is true that these are not explained in great depth, but that was somewhat intentional from the beginning. Paige (the reference) does not go into great depth about why 0^0 is undefined, but he calls it a "myth" that it is defined. The justification from complex analysis probably requires the reader to understand branch points, but given that much background it is pretty clear that in the context of complex analysis there is no reason to define a function to have a particular value at a branch point.
An argument in favor of keeping the justifications short is that this section is not intended to convince any that 0^0 should be defined, or that it should not be defined. The point of the section overall is to report the fact that in some contexts it is defined and in some contexts it is not. The explanation, such as it is, is meant to expand on that basic fact a little without making the section into a soapbox. CMummert · talk 02:18, 11 February 2007 (UTC)
If it's not intended to convince, or at least explain why a belief is held, then it's not a justification; it's an assertion. In that case, the section title is misleading.
Also, I know what a branch point is, and nothing in my understanding of complex analysis precludes a function with a branch point having a value at that point.
Finally, the references at present don't support an assertion that "in some contexts it is defined and in some contexts it is not"; they indicate that some mathematicians have one opinion and other mathematicians have another. That's not the same as demonstrating a context-dependent difference. –EdC 23:14, 11 February 2007 (UTC)
Yes, nothing precludes a function with a logarithmic branch point from being arbitrarily assigned a value at that point, but such an assignment is not in any way normal in complex analysis. Similarly, nothing precludes an "indeterminate form" from being defined to have a particular value, but such an assignment is never to my knowledge made in the calculus texts that discuss indeterminate forms.
The bullet points do (are intended to) explain why 0^0 is usually taken as defined in particular contexts and undefined in others, but in a brief manner. I just don't want to see the 0^0 section blow up to twice its size with long explanations of each bullet point. I added a sentence to each of the last two bullet points in an attempt to compromise. What I mean by calling something "uncommon" is that nobody here has presented even a single reference that discusses it. CMummert · talk 02:29, 12 February 2007 (UTC)
Yes, definitely. I still think that something more practically oriented could help, akin to the examples above in discrete mathematics. Something like:
  • 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 f(x) \to a and g(x) \to b, f(x)^{g(x)} \to a^b, and so would be avoided.

Invented justifications

  • 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. It is uncommon in the context of complex analysis to define a function at a logarithmic branch point.Many results in complex analysis assume that functions are undefined at logarithmic branch points.
Does that look OK? –EdC 14:38, 12 February 2007 (UTC)
Done.–EdC 03:14, 21 February 2007 (UTC)

'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).

It is not correct to claim that 'Many results in complex analysis assume that functions are undefined at logarithmic branch points', because no single result of this kind is quoted. Nor is it correct that 'A discontinuous extension would cause the function to lose a number of desirable properties, for example that when f(x) \to a and g(x) \to b, f(x)^{g(x)} \to a^b, and so would be avoided', because limx,y→0 xy is undefined no matter whether 00 is defined or not, so the medicine does not cure the disease. The supporters of leaving 00 undefined are evidently unable to justify their point of view. Bo Jacoby 10:16, 21 February 2007 (UTC).
As several people have said before, there are already plenty of references showing that the view that 0^0 is undefined exists in contemporary mathematics. There is no need for the article to "prove" that this view is reasonable beyond providing these sources. I think that EdC said he was concerned beause there were justifications listed for many of the 0^0 =1 bullets, and so he wanted to add the informal justifications for leaving it undefined to make that section more parallel. Still, I'll reword them a little. CMummert · talk 14:11, 21 February 2007 (UTC)
It is not disputed that some references do support the point of view that 00 is, or should be, undefined, but it is remarkable that none of these references are able to justify their point of view. WP editors should report this fact, rather than invent insufficient justification, just for the sake of symmetry. NPOV does not mean that unjustified points of view must have the same weight as justified points of view. Bo Jacoby 15:03, 22 February 2007 (UTC).
Well, a reference can't "support" a view if it's unjustified; it can only "assert" it. I don't think it's really a question of justification, though, rather of filling in the gaps in the argument that would have been obvious to the intended audience. I'm sure that to a mathematician working in analysis it goes without saying that x^y needs to be continuous; the reason the full justification is needed is that the same is not at all obvious to a discrete mathematician or computer scientist. –EdC 16:40, 22 February 2007 (UTC)
(responding to Bo Jacoby above) "...because limx,y→0 xy is undefined no matter whether 00 is defined or not..." There's no such thing as limx,y→0 xy; what did you mean here? –EdC 16:43, 22 February 2007 (UTC)
I ment to write lim(x,y)→(0,0) xy. Sorry for being unprecise. lim(x,y)→(1,0) xy=10 and lim(x,y)→(0,1) xy=01 because of continuity, but lim(x,y)→(0,0) xy is undefined because of discontinuity. EdC is probably right that 'it goes without saying that xy needs to be continuous', but the hard fact is that xy is not continuous for (x,y)→(0,0), no matter whether xy is defined for (x,y)=(0,0) or not. Bo Jacoby 22:14, 22 February 2007 (UTC).
And that was me being imprecise; when I said "x^y needs to be continuous" I meant "x^y needs to be continuous everywhere it is defined". The thing is, removing (0,0) from the domain of x^y might look like a nasty hack but it does work; if (0,0) is removed from the domain then topologically (x,y)→(0,0) x^y has no more significance than (x,y)→(∞,∞) x^y. –EdC 05:03, 23 February 2007 (UTC)
- which also has nothing to do with whether ∞∞ should be defined or not. There is no topological reason for defining 00, but there are algebraical and combinatorial reasons for defining 00. There is no topological reason for leaving 00 undefined either. In short: topology has nothing to do with the case. Bo Jacoby 12:33, 23 February 2007 (UTC).
OK, maybe I didn't express myself all that adequately. I argue that f(x, y): \mathbb{R}_{\ge 0}\times\mathbb{R}_{\ge 0} \setminus (0, 0) \to \mathbb{R}: (x, y) \mapsto x^y is a continuous function. The fact that it has no continuous extension to \mathbb{R}_{\ge 0}\times\mathbb{R}_{\ge 0}, nor that its natural extension to \mathbb{R}_{\ge 0}\times\mathbb{R}_{\ge 0} is discontinuous, is not relevant to its continuity. Consider: \_\times\_ has no continuous extension to the one-point compactification of \mathbb{R}^2, but that doesn't change the fact that it is continuous on \mathbb{R}^2.
Thus, there is a topological reason to remove (0, 0) from the domain of x^y: it leaves the remnant function continuous. –EdC 17:52, 23 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).

That sentence is, as you say, inconsistent; indeed, it's utterly incorrect. Removed. As you say, x/x is continuous; actually, sign(x) is properly discontinuous if sign(0) = 0.
And - right. If continuity is all-important, then it is better that a function be continuous everywhere (in its domain) than that it have maximal domain. –EdC 18:23, 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).

Yes... but talking about 'removing' (0,0) from the domain of x^y might not be seen as entirely NPOV. As for 1/x, going by definitions it is continuous; of course, it has an non-removable singularity at x=0, but that's not the same as a discontinuity. –EdC 21:53, 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).

To be honest, I think that's what we have already:
A discontinuous extension would cause the function to lose a number of desirable properties. For example, it is ordinarily taken as a rule in calculus that [some limit statement holds] whenever both sides of the equation are defined; this rule would fail if 0^0 were defined.
If you reverse the POV in that then you have essentially the same as your interpretation a few paragraphs above. The challenge is to present that statement in a way that doesn't imply "defining" or "removing" 0^0 from a "natural" exponential function. Perhaps comparing a "reduced-domain" and "extended-domain" exponential function would work? –EdC 23:00, 27 February 2007 (UTC)
As an aside, I can't agree that 00=1 is more elementary than topological discontinuity; it is possible to construct topology without number theory, and in fact taking the cardinality of 0^0 requires higher-order equipment (quantifying over functions) than testing continuity (testing preimages of members of the topology). –EdC 23:00, 27 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).

The word-alphabet model isn't elementary, though; the naïve motivation for exponentiation is repeated multiplication, and it's not immediately apparent that that coincides with the number of words. Topology employs concepts that can be expressed intuitively if non-rigorously using language such as "smooth", "connected", "limit" etc.
As for the structure of the article - we shouldn't make too much of that. If some mathematicians don't see leaving 0^0 undefined as removing it from the domain, we shouldn't present that as fact.
The thing with x^y is that it is an elementary function; it is composed with other functions to produce functions used almost everywhere in analysis, science and engineering. The sawtooth function is motivated by having discontinuity, whereas for exponentiation its singularity is "surprising". EdC 12:50, 3 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

C5,2 = | { {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5} } | = 10.

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

32 = | { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } | = 9.

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).

Elementarity, conventions, limit statements

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)

Children count before they multiply. Therefore counting is more elementary that multiplication. The combinatorial explanation is often omitted when explaining exponentiation, but not when explaining binomial coefficients. I see no good reason for this difference. The two concepts are otherwise very similar.
The limit value of the fourier series of the sawtooth equals the function value at all points of continuity, and the limit value is also defined at the singular points, where all the terms of the series equal zero. It is illustrated by the nice animation: sawtooth wave.
Alas the exponential does not 'preserve limits', because
limy→0+ limx→0+ xy ≠ limx→0+ limy→0+ xy.
These two limits are both defined, even if 00 is not defined, but they have different values. The left hand side is
limy→0+ limx→0+ xy = limy→0+ 0y = limy→0+ 0 = 0,
while the right hand side is
limx→0+ limy→0+ xy = limx→0+ x0 = limx→0+ 1 = 1.
Bo Jacoby 00:47, 6 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; \lim_{x\to 0+}{x^y} \neq 0^y since the latter is undefined at y = 0; \lim_{y\to 0+}{x^y} \neq x^0 since the latter is undefined at x = 0. 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)

  1. Certainly there are other motivations for exponentiation than exponential sets, but no explanation of exponentiation is more elementary than counting, and counting leads to 0n=0 for n>0, and n0=1 for n≥0.
  2. The fourier series of the sawtooth wave shows that a discountinuous extension is not unseen in calculus, such as the article says. A discountinuous extension does not 'cause the function to lose a number of desirable properties'. The point is that the 'Justifications for leaving 00 undefined' are incorrect.
  3. The limits do not depend on the values on the axes. For y>0, \lim_{x\to 0+}{x^y} = 0 , and for x>0, \lim_{y\to 0+}{x^y} = 1 . But (except for 00) the axes values x0 and 0y are determined by continuity; for x>0, x^0=\lim_{y\to 0+}{x^y} , and for y>0, 0^y=\lim_{x\to 0+}{x^y}. The value of 00 is not determined by continuity, but by other means. Bo Jacoby 14:51, 7 March 2007 (UTC).
  1. Right. You try to explain to an 8-year-old (or a 28-year-old) about counting exponential sets. I guarantee the results will be as to something out of the worst nightmares of the New Math. Repeated operations, on the other hand, are something humans are equipped to grasp.
  2. Reduced-domain exponentiation has the property that \lim_{x\to a}{{f(x)}^{g(x)}} = {\lim_{x\to a}{f(x)}}^{\lim_{x\to a}{g(x)}}. That is desirable, in particular because elementary functions (used heavily in applied mathematics) often feature composition through exponentiation.
  3. Well, yes. 100% correct. I don't see what that has to do with the above, though. –EdC 22:38, 7 March 2007 (UTC)
  1. Binomial coefficients are often explained using combinatorics without anybody complaining about New Math, and exponentiation is even simpler.
  2. Even reduced-domain exponentiation does not have the desirable property that \lim_{x\to 0+}\lim_{y\to 0+}x^y = \lim_{y\to 0+}\lim_{x\to 0+}x^y . (I see no point in complicating matters by introducing f and g).
  3. The relevance is to clarify that both the above double limits are defined for reduced-domain exponentiation. Bo Jacoby 23:11, 7 March 2007 (UTC).
  1. The primary motivation for binomials is in combinatorics. The same is not the case for exponentiation.
  2. The two are entirely distinct. With f and g both parametrised by x, we have a simple limit statement. Your double limit is a far more complex construct, and does not even exist.
  3. It does not accomplish that purpose, in that case. It is not legitimate to substitute functions inside a limit expression if those functions have different domains. It might be possible to repair the limit expression in other ways, but your statement that "the limits do not depend on the values on the axes" is mere handwaving. –EdC 22:49, 8 March 2007 (UTC)
  1. True, but irrelevant. Your argument was that a combinatorial explanation leads to a New Math disaster. My answer is that it did not in the case of binomial coefficients. This does not depend on primary motivations. Still the combinatorial interpretation of exponentiation is very important.
  2. Assuming that the nonnegative functions f and g are continuous at a, the equation \lim_{x\to a}f(x)^{g(x)} = f(a)^{g(a)} holds for \left(f(a),g(a)\right)\ne(0,0). When (f(a),g(a))=(0,0) \ , the left hand side is defined, but in the restricted domain where 00 is undefined, the right hand side is undefined, and so the equation is not true. In the unrestricted domain where 00=1, the right hand side is defined but may differ from the left hand side, and so the equation is not necessarily true in this case either. So, restricting the domain solves no problem.
  3. The double limit equation \lim_{x\to a}\lim_{y\to b}x^y = \lim_{y\to b}\lim_{x\to a}x^y is true at all points of continuity, but for (a,b)=(0,0) the equation is not true, even if each of the double limits exists. \lim_{x\to 0+}\lim_{y\to 0+} x^y = 1 and \lim_{y\to 0+}\lim_{x\to 0+}x^y =0 . Why do you think that they do not exist? Even when the domain is further restricted by removing the entire axes from the domain, these limits still exist. Bo Jacoby 00:02, 9 March 2007 (UTC).
  1. It's relevant to your statement above that repeated multiplication is "more advanced" than the combinatorial interpretation. By general pedagogical experience (at least, outside the now discredited New Math tradition), since the motivation in repeated multiplication comes before any experience of combinatorics, it is difficult to see any metric by which repeated multiplication is more advanced than taking cardinalities of exponential sets.
  2. Two sides having different values is "worse" than one side being defined and the other not.
  3. You used an illegitimate operation in your proof; two functions must be equal and have the same domain for substitution to be valid. –EdC 19:09, 11 March 2007 (UTC)
  1. Counting the number of two-letter words from a three-letter alphabet can be done in preschool while multiplication may be learned about three years later. But it is not crucial which one is more elementary. None of them are too advanced.
  2. Why is that? None of the two cases provide a true equation. In both cases one has to be alert that a dangerous singularity invalidates the usual continuity.
  3. Which operation is illegitimate? And here we have a false equation where both sides are defined, even in the restricted domain situation. Bo Jacoby 19:32, 11 March 2007 (UTC).
  1. Yes - although I wonder quite what the preschooler will gain from the exercise, and would point out that multiplication through Cartesian products is of the same difficulty.
  2. I suppose the answer is that "undefined" popping out serves as a clear warning that an illegitimate assumption of continuity (or continuous extensibility) has been made.
  3. Substituting nonidentical functions. It can be rescued quite trivially, I think, but I will not discuss the issue until you provide a rigorous proof. –EdC 22:26, 13 March 2007 (UTC)
  1. ok
  2. For the function f, defined for nonzero x by f(x)=0/x, the answer "undefined" for f(0) has the quite opposite effect than that of a warning, namely of triggering the obvious extension f(0)=0. So "undefined" does not generally serve as a warning, but rather as a challenge. The warning "singularity!" must be issued otherwise, and leaving 00 undefined serves no purpose.
  3. The function f, defined for positive x and y by f(x,y)=xy, satisfy the equation \lim_{x\to a}\lim_{y\to b}f(x,y) = \lim_{y\to b}\lim_{x\to a}f(x,y) for nonnegative a and b satisfying (a,b) ≠ (0,0), but not for (a,b) = (0,0). Rigorous proof: For a>0, b>0, \lim_{x\to a}\lim_{y\to b}f(x,y) = f(a,b) and \lim_{y\to b}\lim_{x\to a}f(x,y)=f(a,b) . For a>0, b=0, \lim_{x\to a}\lim_{y\to b}f(x,y) = \lim_{x\to a}1 =1 and \lim_{y\to b}\lim_{x\to a}f(x,y)= \lim_{y\to b}f(a,y)=1 . For a=0, b>0, \lim_{x\to a}\lim_{y\to b}f(x,y) = \lim_{x\to a}f(x,b) = 0 and \lim_{y\to b}\lim_{x\to a}f(x,y)= \lim_{y\to b}0=0 . For a=0, b=0, \lim_{x\to a}\lim_{y\to b}f(x,y) = \lim_{x\to a}1 = 1 and \lim_{y\to b}\lim_{x\to a}f(x,y)= \lim_{y\to b}0=0 .

Q.E.D. Bo Jacoby 09:00, 16 March 2007 (UTC).

  1. Yes, to a mathematician. However, physicists and others who simply apply the mathematical tools they are taught are less likely to view undefinedness as a challenge.
  2. OK, yeah; once you restrict x^y to \mathbb{R}_{>0}^2 the double limits exist. Good. Right; now the issue is whether this matters. Note that the initial limit is a pointwise limit, not uniform; the resulting intermediate limit cannot be expected to behave. –EdC 20:32, 18 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).

I added two more references today. Now both of the justifications are backed up by sources; they do not need to be direct quotes. Also, I found a recent book that directly states that the definition of 0^0 is a convenience rather than a necessity. CMummert · talk 19:01, 22 March 2007 (UTC)

2007-3-28

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)

The recapitulation from 22 March 2007 was not argued. The definition 35 = 1·3·3·3·3·3, (5 multiplications by 3), has not been argued either. It leads unavoidably to xn = 1·x···x, (n multiplications by x). A special case of this is 00 = 1. The absolute claim that 00 = 1 is a logical consequence of the definition that xn = 1·x···x, (n multiplications by x), which has not been argued, although I have encouraged CMummert to change the definition to reflect his point of view. Within each tiny subsection we must stick to logic. Bo Jacoby 13:07, 28 March 2007 (UTC).
All of these concerns have already been addressed, as you must be aware. There is no requirement that others must continue to respond to your comments when you raise no new issues. CMummert · talk 13:14, 28 March 2007 (UTC)
You must be consistent. The definition implies the conclusion 00 = 1. You have not modified the definition, although you have had plenty of time to do so. Bo Jacoby 15:00, 28 March 2007 (UTC).

Reason for linking to C Rationale?

Many languages, including Java, Python, Ruby, Haskell, ML, Scheme, MATLAB, Microsoft Windows' Calculator, and others (especially when using IEEE floating-point arithmetic, but also for integer arithmetic), evaluate 00 to be 1. [7]
7. For example, see John Benito (April 2003). "Rationale for International Standard — Programming Languages — C". Revision 5.10.

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)

In that case, we should have "By contrast, C allows implementations to choose whether 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)

Location of Powers of i and e

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)

We are talking about integer powers of i and e. So they are correctly placed where they are under exponentials with integer exponents. Bo Jacoby 01:03, 18 March 2007 (UTC).

Image of x^y

x^y for positive x
x^y for positive x

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)

I've now replace Sam's image and added some more curves to it. What you find is that its only curves which are tangent to the y-axis which give a limit different to 1. The image of any curve in the x-y plane with a different tangent will have a limit of 1. --Salix alba (talk) 11:48, 3 April 2007 (UTC)
Yes, this is currently footnote 7. CMummert · talk 13:25, 3 April 2007 (UTC)
Thinking about this in singularity theory terms is amusing. Consider the space of all (smooth/analytic) functions f(t) = (fx(t),fy(t)) from R to R^2 and compose it with the exponation function. We can consider the 1-jet of these fuctions, which give a two dimensional set of functions f(t)=(a t,b t). Those funtion for which the limit is not 1 form a sub-space of dimension 1, that is co-dimension 1. We can then expand this to consider the full space of germs and again find that its a co-dimension one set of functions with a non-unitray limit. --Salix alba (talk) 17:02, 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).

Exponential map in differential geometry?

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)

See the article Exponential map. Bo Jacoby 20:57, 5 April 2007 (UTC).

GA-status granted

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)

Definition for negative integers

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,

a^{-n} = \frac{1}{a^n} = \frac{1}{\underbrace{a \times \cdots \times a}_n},

but leaving the sentence on its own can create some confusion for those who don't know basic index laws. Thanks GizzaChat © 12:00, 7 April 2007 (UTC)

The sentence in question covers more than negative integers. That definition is given later in the article. Indeed most of the article is dvoted to explaining the thired sentence. It could be worded more clearly, however.--agr 00:02, 8 April 2007 (UTC)
I expanded it this morning (before the comment signed agr was left here). If it is still too vague, be bold and rewrite it. Please keep in mind that the lead is supposed to summarize but not duplicate what is located in the rest of the article. CMummert · talk 00:34, 8 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)

Order of Operations

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?

In traditional math typesetting, 234 is 2^81, and the 4 would be in a smaller font than the 3, which would again be smaller than the 2. CMummert · talk 11:13, 9 April 2007 (UTC)
2^3^4 = 2^(3^4) ≠ (2^3)^4 = 2^(3·4). Yes, the article could be explicite about it. Bo Jacoby 19:34, 9 April 2007 (UTC).

1^infty

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)


The above discussion is preserved as an archive. Please do not modify it. Subsequent comments should be made on the appropriate discussion page, such as the current discussion page. No further edits should be made to this page.