Talk:Exponentiation/Archive 1

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Archive 3

What is arithmetic?

The present Arithmetic page claims exponentiation as an arithmetic operation. I may be getting it wrong, but I class arithmetic as the art of manipulating numerals - the symbols - in manners isomorphic to the algebraic ops. eg/ie one does not multiply the platonic six hundred seventeen by the platonic nine hundred two, but writes down "617" and "902" and after a bunch of ciphering (the exact word) ends up with another symbol, "556534", representing the number which would come out of the field op on the two inputs. (I'm coming to the point.)

I was taught the four basics plus the extraction of roots (square and cube; I know how it extends to higher degrees but hope never to have to do it). But I've never encountered exponentiation in arithmetic (other than by repeated multiplication, obviously). Is there an actual symbol-manipulation procedure specialised for exponentiation? Please add it if there is one ('cause I'm dying to know what it is). Please change Arithmetic if there is not. 142.177.23.79 23:01, 14 May 2004 (UTC)

Arithmetic is generally understood by people who work with numbers (that I have met) as working with numbers, not numerals. Rules (algorithms) for manipulating symbols are means to help us obtain the correct numbers, but they are just techniques. Different techniques (such as, the use of decimal numerals vs. binary numerals) do not change the facts about the numbers. Thus, facts about numbers are often regarded as legitimate parts of arithmetic even when they do not have symbol-manipulation procedures associated with them. Number theory is often called "higher arithmetic" for this reason. I believe it is fair to say that elementary arithmetic, as in teaching young people how to add and multiply, is heavy on the symbol manipulation, but that is just to learn how to get answers and is not the only part of arithmetic. Zaslav 23:25, 3 February 2006 (UTC)

It is likely possible to manipulate exponents, however you would have to memorize the numbers up to 9^9, though you could disclude stuff like 9^1 if you know 1^9...This is, however, quite difficult.

No. a^b != b^a. Lanthanum-138 (talk) 07:37, 24 March 2011 (UTC)

Exponentiation or Involution??

Go to Talk:Super-exponentiation. It says there that exponentiation is properly called involution and that exponentiation is just an awkward term. Any votes to move this page?? 66.245.22.210 17:46, 3 Aug 2004 (UTC)

Our heroes at Encyclopedia Britannica use the balanced terms "involution" & "evolution" as the standard to refer to the 3rd binary operation and its inverse. ----OmegaMan

Every mathematician I know (and I am one and know many) calls this operation "exponentiation". Not one of them ever said "involution" in my hearing, or in writing that I read. Zaslav 23:13, 3 February 2006 (UTC)

"Extracting a root used to be called evolution." This is as uncommon as calling exponentiation for Involution. The article on Evolution does not mention that meaning of the word. I will delete the sentence. Bo Jacoby 10:57, 24 May 2006 (UTC)

I think this article needs a History section; obsolete terminology (with citation) might be appropriate there.--agr 11:35, 24 May 2006 (UTC)

(x+h)^3

Where can I find the page that tells how to cube two variables that are being added or subtracted? I can't find it anywhere... pie4all88 22:48, 26 Sep 2004 (UTC)

Try binomial theorem. Revolver 04:00, 27 Sep 2004 (UTC)

1

Exponentiation in abstract algebra

The stuff about power-associative magmas in the abstract algebra section of the article is wrong. Power associativity is not enough to get this sort of thing to work, as the following 3-element magma shows:

   | 1 a b
 --+-------
  1| 1 a b
  a| a a 1 
  b| b 1 b

The above magma is power-associative, it has an identity element, and every element has a unique inverse. Yet a²a^-1 is not equal to a.

I'm going to change the article to use groups instead. It's possible to do it in somewhat greater generality than that (e.g., Bol loops), but it's probably not worth it. --Zundark 19:58, 17 Dec 2004 (UTC)

An excellent idea. Generality can be excessive. Zaslav 23:15, 3 February 2006 (UTC)

Decimals

This article doesn't say anthing about powers with non-integer exponents (i.e. decimals, fractions, etc.) I don't know much about it, so could somebody put it up? →[evin290]

merging

It has been suggested that this article or section be merged with exponential function.

I think this is a bad idea. The (real of complex) exponential function is just one aspect of exponentiation in general, and the variety of topics in the article suggests this. Revolver 20:48, 14 August 2005 (UTC)

I am a high schooler, with not so good math skills. This article is very helpful for studying and basic learning about the math that I have trouble with. Making this merge with another would only make it more complictaed for me. Please keep them seperate so I can get the basics down. -Thanks Bigfoot

"powers of one"

This § does not in fact appear to be about powers of one. Can anybody propose a more sensible title for it? Doops | talk 23:19, 9 November 2005 (UTC)

Hi Doops. I wrote it and used the notation 1^x meaning e^2πix, because e^2πi = 1. However, this nice and useful notation was deleted by other wikipedians who considered it 'original research'. Only the title survived. Bo Jacoby 15:31, 11 November 2005 (UTC)

Fractional exponents

Hello Arnold. § Fractional exponents depend on § Powers of e. Please repair your reorganisation. Bo Jacoby 15:06, 1 February 2006 (UTC)

Better now? --agr 17:12, 1 February 2006 (UTC)

Thank you. Its better, but still I am not happy. Fractional , real, and complex exponents are more advanced than integer exponents and these §s stops the uninitiated reader from continuing reading. The logical sequence is: integer exponents, powers of e, complex exponents. Then fractional and real exponents are special cases of complex exponents. You might have a § on exponent one. Why not move the advanced §s below the heading 'advanced' ? Bo Jacoby 07:20, 2 February 2006 (UTC)

The short answer is that I am still working on it and I think we have a similar view point, particularly about not stopping the uninitiated reader from continuing reading. I do think the sequence integer exponents, fractional exponents, real, complex has merit. I believe nth-roots are easier to understand for non-mathematicians than e^x and we do have a separate article on the exponential function. --agr 14:28, 2 February 2006 (UTC)

The n'th root is multivalued, and that cannot be helped. The contradiction of terms, multivalued function, paralyses the uninitiated reader who is sensible to contradictions. The interest in powers with fractional exponents faded when it became clear that they cannot be used for solving polynomial equations after all. It was the wrong path to walk. The reader must unlearn fractional exponents on positive reals and learn about finding complex roots in polynomials. Do you intend not to talk about e^x at all? If yes, do that. If no, utilize it to the limit. I strongly believe in the path: xⁿ --> e^x --> e^z . Bo Jacoby 16:32, 2 February 2006 (UTC)

Both approaches have merit. The advantage of the fractional exponent approach is that it does not require limits or calculus. In my experience, many people find it easier to grasp. I've added text to the section on "Defining exponentiation" referring readers to the exponential function article for an alternate explanation and there is a discussion of exp in the section on real exponents.--agr 19:48, 2 February 2006 (UTC)

The power function f(x)=x=x¹=e^{ln x} is not multivalued, even if ln x is multivalued. Bo Jacoby 16:45, 2 February 2006 (UTC)

That is a special case. --agr 19:48, 2 February 2006 (UTC)

Surely, any integer is a special case of a complex number, but when there are special cases where your conclusion is wrong, then your argument is wrong too, and that is confusing for your reader. It is not true that the fractional exponent approach does not require limits, because you cannot even compute the square root of two without taking limits. The fractional exponent approach requires the reader to solve a polynomial equation, which is advanced stuff compared to the original level of this article. It is true, however, that the fractional exponent approach does not require calculus, but the definition of e^x given in the § powers of e does not require calculus either. Your § on fractional exponents rely on the concept of n^th root, which has not yet been defined. Your example, 5^1.732, requires the reader to first solve the equation x¹⁰⁰⁰=5 and then compute x¹⁷³². Some people might not find it easy to grasp. But in practice you use the exponential function, which you do not define although it is easier to define than the fractional exponent. Your information that e is transcendental is confusing and useless, because a reader of the elementary article exponentiation cannot be expected to know the more advanced theory of transcendental numbers. So the fractional exponent approach is no good in practice, you admit, and it is no good in theory either, because when restricted to positive real radicands it is insufficient, and when extended to complex radicands it is multivalued. The picture from logarithm belong there and not here because this article is about exponentiation. Articles on related subjects should be referred to for more advanced reading but not for foundation. This article should be for beginners. Bo Jacoby 10:10, 3 February 2006 (UTC)

Children typically learn about square roots and solving polynomials in elementary school. They learn about limits in calculus, traditionally in freshman calculus, though often in AP high school classes. It's true that to rigorously define the real numbers, one needs limits, but such a rigorous exposition is usually not undertaken until calculus or a real analysis course. This is a pedagogical question and the fractional exponent approach is widely used, has been for generations, and deserves to be in Wikipedia. It was the original basis of this article. I have tried to make it more approachable for beginners. Going from no knowledge to complex exponentiation in a single reading is a bit much to expect, however. So things necessarily get more technical as on goes on. I added more about using e^x at your suggestion. Mentioning that e is transcendental is a judgement call. It is linked however. As for the graph, it shows the exponential function, not the log. --agr 11:48, 3 February 2006 (UTC)

This article on exponentiation need not depend on knowledge on square roots. To understand exponentiation with integer exponents you only need to know about multiplication and division. To understand e^x you also need to know a little about limits, but not much. (As ((n+1)/n)ⁿ is almost constant for big values of n, just take some big value as an informal definition of e). On this very modest prerequisite the theory of exponentiation with complex exponents is erected. Things do not necessarily get more technical and should not get technical just because they got technical in school. Going from no knowledge to complex exponentiation in a single reading should of course be done if possible, and it is possible, and it was done. The fractional exponent approach has been used for generations, true, but that is no excuse to torture the generations to come. You tried to make it more approachable for beginners, but you failed completely. Your text is a disaster, illogical and incomprehensive. I didn't suggest that you add more about using e^x. I repeat: Do you intend not to talk about e^x at all? If yes, do that. If no, utilize it to the limit. You cannot avoid e^x, and it is sufficient for defining exponentiation. If one of two methods is sufficient, then the other method is unnecessary. Radicals and square roots are unnecessary concepts in an article on exponentiation, irrespective of what you learned in school. Get rid of it. Bo Jacoby 14:29, 3 February 2006 (UTC)

The fact that the fractional exponent route is widely taught is a strong reason to include it here, if for no other reason than to support students who are learning it. If you'd like to add an alternate explanation based on e^x, feel free. --agr 18:40, 3 February 2006 (UTC)

It seems dogmatic to say that there can only be one approach to explaining a subject. If you want to be understood by people who don't already know what you're talking about, look for the most effective means of explanation. Sometimes that means giving more than one approach. Sometimes it means shifting the approach as the people learn more.

The approach to real number exponentiation via fractional exponents is a standard one, both pedagogically and in advanced theory. Combined with taking limits of exponents to get irrational exponents, it is often (I think usually) used as the definition of exponentiation in calculus classes. (I speak from much personal experience teaching those classes.) This does not require calculus. Some professors prefer to introduce exponentiation as the inverse of logarithms, which then would be introduced as the integral of 1/x using integral calculus. Another alternative in developing the mathematical theory is to use power series to define e^x; this works especially well for complex exponents, but it is more advanced mathematically and is not normally considered suitable for beginners.

Thus, there are several logically valid approaches; some are more elementary and some belong to higher mathematics. Some might be most appropriate for treatment in the article on the exponential function. The one via fractional exponents is certainly highly suitable to an article on exponentiation, but it is surely reasonable to mention (and possibly develop) other approaches. Zaslav 00:11, 4 February 2006 (UTC)

I agree that there are many ways. Now I made a restructuring of the article. I used one of the possible ways, but included the other computation of fractional exponents as a possibility. I don't use calculus or continuity, but one limit. I look forwards to your comments. Bo Jacoby 12:50, 4 February 2006 (UTC)

I made another pass to present both methods on an equal footing. As Zaslav points out the fractional exponent method is standard and must be here. I think Bo Jacoby's approach has merit and I tried to make it a bit clearer. I will add a little about a third method, defining ln first. --agr 21:33, 9 February 2006 (UTC)

In the present version the ln is used without being defined. Bo Jacoby 12:13, 22 February 2006 (UTC)

Thanks, I think I've fixed it. Did I get the place you were referring to?--agr 13:39, 22 February 2006 (UTC)

When you write any base is seems as if you mean any positive real base, although it is not obvious what you mean. That case is treated in a § on complex powers of positive real numbers. The problem of multivalued logarithm is not taken care of any more. You just refer to the inverse as if it was well defined. I think that an elementary article on exponentiation should not talk about logaritms at all, but that is too late now. I have a minimalistic taste, and you don't. It's OK. Bo Jacoby 14:09, 22 February 2006 (UTC)

It's a good point. Perhaps we should make it clear we are discussing real numbers only at first and then introduce complex numbers in the later sections. Also, I noticed a sticky point in your explanation of exponentiation. When you say "let n = mx", a reader might miss the point that m is not an integer in general. You then form (1 + 1/m)^m but it is non clear what the mth power means here. --agr 18:39, 22 February 2006 (UTC)

Yes, the definition is e^x = lim_n→oo(1+x/n)ⁿ where n is integer, so this point is not sticky at all. I'd like to emphasize at this stage that x might be a complex number or even a matrix, because neither of the two other methods for fractional exponents, (the one assuming knowledge of continuity of real functions, the other assuming knowledge of integration along the real axis), work for non-real exponents. When x is an integer it should be shown that the two definitions e^x = lim_n→oo(1+x/n)ⁿ and e^x = (lim_n→oo(1+1/n)ⁿ)^x coincide. This is easy using n = mx . When x is not an integer, the definition e^x = lim_n→oo(1+x/n)ⁿ is a generalization of the old definition of exponentiation with integer exponent. It cannot be proved, because it is a definition, and so the 'proof' is sticky. Note that the following real powers of one: 1^x = e^i·2π·x , are fundamental for the description of circular motion, and oscillation, and waves. Complex exponents are very useful and should not be considered advanced. Also I took care not to mention ln(a) but just: let b be any solution to the equation e^b=a . Bo Jacoby 13:34, 23 February 2006 (UTC)

I've tried to clarify the case where x is an integer. Saying n = mx is confusing, because x won't divide n in general. The point is mx is an integer if m is and one can take the limit on m. --agr 16:06, 27 February 2006 (UTC)

Well done. Bo Jacoby 16:50, 27 February 2006 (UTC)

The inverse

"The inverse of exponentiation is the logarithm" is not quite true. Exponentiation y^x defines two functions: the exponential function x→y^x and the power function y→y^x. The logarithm is inverse to the exponential function, but not to exponentiation. The radicals are inverse functions to the power functions. Bo Jacoby 14:05, 6 March 2006 (UTC)

Keep the Reader in mind

This is a cross post to user talk: Pol098#Exponentiation_and_Metric (Section Link) on the Intro change in this article, and the need for a better expanded intro:

You refer to the introductory tying the article on exponents to the metric system as pointless. I don't believe in reverts in general, so let me state my reasoning on hopes of achieving a conversion by thou heathen sinner <G>:

If you would be so kind as to consider that these articles are tied into others by links, you will quickly realize the numbers of articles dealing with measurements (which have a generally pragmatic utility to the lay reader) far outnumber the articles that are simply dry math. Since these articles tie into this topic by exactly that corespondence I addressed the tie in the intro as an appitizer of sorts for people linking in that manner. Burying the information way down in the body makes no sense... they are reading about measurements for their own purposes, and not interested in a sub-topic of math as a general topic... unless perhaps my little sentence wets their interest thus making your article experience much more traffic. Wouldn't that be a good outcome?

Now I cannot say that the quick and dirty change I posted was worded perfectly, I was nested deep within six to seven related edits at the time, and there are others that will always fiddle with sentence structure... but I do object to arbitrary removal of material that is certainly not off point or topic. You extended that so as to misconstrue it to multiples of other integers, so why not just reword it to qualify it better to the set of 1 X 10^x form.

I think you should revert and revise if you like to incorporate the sentence under the principle of the most utility to the most people. If we aren't striving for that, why are we bothering to donate our time when the media is so perfect for such a cross-link of knowledge.

Moreover, WP:MOS wants introductions to articles to recap a sense of the article as a whole. This phrase did that, though obviously, the whole article needs such a recap so the whole is more reader friendly. That is a expansion that is worthy of your time, not chopping down the seedling of knowledge I planted. It is afterall for the benefit of the housewife, child, or businessman that we write, not solely for the someones with a technical background like an engineer such as myself, or whatever field gainfully employs you. If you are writing for a technical audience alone, I submit you need a professional journal, not this venue. FrankB 17:53, 31 March 2006 (UTC)

These things we who write in technical topics need to keep in mind. Best wishes all FrankB 19:10, 31 March 2006 (UTC)

I added a short paragraph on the importance of exponentiation to the intro. --agr 22:37, 31 March 2006 (UTC)

any number to the power 0 is 1

The article states that "any number to the power 0 is 1". Should we write something about 0^0 which isn't generally defined (but sometimes, I've been told, defined as 0 or 1)? – Foolip 14:27, 27 April 2006 (UTC)

I think not. There is a fine discussion on the subject in Empty Product. Bo Jacoby 12:46, 28 April 2006 (UTC)

"Real powers of unity"

I have removed this section and related discussion, apparently inserted by User:Bo Jacoby, as a totally misleading invented notation.

This was already discussed ad nauseam in Talk:Root of unity, where Bo tried and failed to get the article to include his own invented notation.

The problem is this: the expression $1^{x}$ by convention always denotes the principal value, i.e. it always equals 1 in standard notation.

Moreover, the section explaining that multi-valuedness arises for complex exponents of complex numbers is also misleading, for two reasons. First, the same ambiguity arises in general for any non-integer exponent...it is false to imply it only happens for complex exponents or powers of unity when it wasn't mentioned earlier. Second, the ambiguity is resolved by the standard convention for the principal value.

I've changed the article to mention principal values at the beginning of the section on real exponents, giving the well-known example of square roots, and linked to the appropriate articles. I changed the section on complex exponents to explain that their multivaluedness is not really any different than that for real exponents, and gave the classic i^i example. I removed the sectioon on powers of unity entirely, as this is misleading for the reason noted above and is no different in any case than powers of any other real number. The Root of unity article is already linked.

—Steven G. Johnson 17:08, 17 May 2006 (UTC)

The power e^x is never considered multivalued, even if x is noninteger, so one symbol, e, is actually treated differently from the other numbers. The primitive square root of one is −1, while the principal square root of one is +1. Steven insists that the symbol 1^1/2 'by convention' must mean the principal and not the primitive square root, even if the principal exponential function 1^x=1 is utterly unimportant while the primitive exponential function 1^x = (e^2π·i·)^x = e^2π·i·x = cos(2π·x)+i·sin(2π·x) is fundamental for describing circular motion, harmonic oscillation, and fourier transform, We badly need a shorthand notation for e^2π·i·x , and 1^x is readily at hand, having no other sensible interpretation, and not being polluted with the confusing and irrelevant ingredients: e, i, and π. Steven prevents us from this simplification of the formulas, and the readers suffer. I regret the backwards steps that Steven makes, but I cannot help it. Bo Jacoby 13:37, 19 May 2006 (UTC)

If you think the standard notations of mathematics are broken, you are free to write you own textbooks, articles, etcetera trying to convince people that $1^{1/2}={\sqrt {1}}=-1$ is a better convention. But for the umpteenth time, Wikipedia is not the place for it. Please stop trying to sneak your personal notations back in, after the consensus (not just me, see Talk:Root of unity) has resoundingly rejected it as inappropriate and in violation of policy. —Steven G. Johnson 23:54, 19 May 2006 (UTC)

And, by the way, $x^{y}$ is never considered multivalued, either, in standard notation, especially for positive real x and real y...it always denotes the principal value unless otherwise noted. There is nothing special about e in this sense. —Steven G. Johnson 23:57, 19 May 2006 (UTC)

Steven, when you promote the principal value, you should emphasize that the fundamental rule (a^b)^c=a^bc becomes invalid. For example: 1=1^x=(e^2πi)^x is now not equal to e^2πix for noninteger x. Also note that the word complex means nonreal or real, while you use it meaning nonreal, when you claim that "it is false to imply it only happens for complex exponents or powers of unity". Actually it was correct what I wrote. Your crusade is degrading Wikipedia. Bo Jacoby 14:06, 22 May 2006 (UTC)

If you don't like standard notations, you are free to convince the rest of the mathematical world. This is not about me, and please stop making it about me...as you saw on Talk:Root of unity, I am not alone—your promotion of nonstandard personal notations is against Wikipedia policy. When people write $x^{y}$ without qualification, they mean the principal value in standard notation; this is not my choice. If you want to try to change Wikipedia policy on this, feel free to post a proposal on Wikipedia:Village pump (policy) that Wikipedia should adopt the nonstandard notation $1^{1/2}=-1$.

Note that the problem you point out arises for any choice, not just the standard branch cut. For example, with your nonstandard choice $1^{1/2}=-1\neq (e^{4\pi i})^{1/2}=e^{2\pi i}=1$. But I'm not going to argue whether your notation is good or bad—that's irrelevant. The only point here is that it is nonstandard.

On a separate note, it was misleading to put the multi-valued discussion only in the section on complex exponents, but not in the section on purely real exponents. Of course the complex case includes the real case; that is not the issue. I didn't say it was false per se, only that its implications were false: the multi-valued discussion should come as soon as the issue arises.

—Steven G. Johnson 16:38, 22 May 2006 (UTC)

You wrote the inequality sign on the wrong place. You meant to say $1^{1/2}=-1=(e^{4\pi i})^{1/2}\neq e^{2\pi i}=1$. The rule is saved only by letting the exponential be multivalued. $\ 1^{1/2}=(e^{2\pi in})^{1/2}=e^{\pi in}=(e^{\pi i})^{n}=(-1)^{n}=\{+1,-1\}$. The multivalued interpretation is quite common. It is neither invented by me, nor "totally misleading" as you claim. When you say: "When people write $x^{y}$ without qualification, they mean the principal value" you need documentation. Your argument doesn't get any stronger by being repeated. When you claim that the principal value is 'standard', to which standardization document are you referring ? And what about writing $x^{y}$ with some qualification ? When talking about roots of unity, you are not restricting yourself to the principal value, which is 1. How come you are not attacking there? Do you consider a primitive n'th root of unity ? Do you consider the formula ${\sqrt[{n}]{a}}=a^{1/n}$ to be true ? Are you allowed to substitute a = 1 ? Bo Jacoby 08:15, 23 May 2006 (UTC)

You use the fact that i=e^iπ/2, but you have deleted the explanation, so you made the article completely incomprehensible to the uninitiated reader for whom it was intended. (What is the word for doing that? I think the word is 'vandalism'). Whether or not you use the notation 1^x, the article must explain the fractional powers of unity and define π before it is used. Bo Jacoby 11:08, 23 May 2006 (UTC)

This is what happens when you force other editors to repair the damage after you fill a section with inappropriate material. You're right, it's more work than I first thought. I had to basically rewrite the whole section. Thanks. —Steven G. Johnson 18:00, 23 May 2006 (UTC)

Steven. In Eulers formula, e^ix=cos(x)+i sin(x), the left hand side is already defined as lim(1+ix/n)ⁿ, (where the limit is for large values of n). That has been explained to the reader of the article. However you use Euler's formula the other way round, now supposing the reader to know trigonometry and not exponentials. If you want to define trigonometry, then use cos(x)=(e^ix+e^−ix)/2 and sin(x)=(e^ix−e^−ix)/(2i). Why not start reading the article in the version it had before you began vandalizing it, then as step 2: understand the flow of logic, which by now you obviously do not understand at all, then, as step 3, do some serious thinking. Then, as step 4, write your suggested improvement on this discussion page. Then, as step 5, read the comments of your fellow wikipedians. That approach will do you honor and no shame. Bo Jacoby 08:43, 24 May 2006 (UTC)

Bo, you're being absurd: this article is not going to develop all of mathematics starting from the basic axioms of arithmetic. A reasonable encyclopedia article has to summarize, and rely on other articles for more details. It is unreasonable to talk about complex exponentiation and not to assume some understanding of trigonometry (or to rely on other articles to explain it). If the reader does not know what cosine and sine are, defining them from complex exponentials, while perhaps logically pleasing to you, is not going to help them. For most readers, trigonometry will be a more basic concept than complex exponentials, and it makes sense to use the former to describe the latter. —Steven G. Johnson 16:27, 24 May 2006 (UTC)

Summarizing does not mean neglecting logic. The intelligent reader deserves better than that. Use my comments as an opportunity to improve on your writing. When you consider yourself perfect then you prevent yourself from improving the article. I do not suggest that sin and cos should be defined nor used in this article, because I completely agree that "this article is not going to develop all of mathematics".

It is perfectly reasonable to explain exponentiation without referring to trigonometry. If z is a nonzero complex number, and n is a positive integer, then by now we know that 0⁰=z⁰=1 ; 0ⁿ=0 ; 0⁻ⁿ remains undefined ; zⁿ=z·zⁿ⁻¹ ; z⁻ⁿ=1/zⁿ ; e=lim(1+1/n)ⁿ and e^z=lim(1+z/n)ⁿ. Everything is well-defined and single-valued, and the rules e^z+w=e^ze^w and z^n+m=zⁿz^m and (zⁿ)^m=z^nm apply. You may continue like this:

Bo, I know that it's perfectly possible to define $e^{z}$ for general z in that way, and the article already does this. While pleasing from a minimalist viewpoint, however, this definition is not especially useful in telling someone how to perform complex-number exponentiation. A reader familiar with trigonometry will be able to immediately use a definition for complex exponentials based on sine and cosine. A reader unfamiliar with sine and cosine is not going to be helped at all by the limit-based definition, and will probably be totally confused. (How is the reader to prove that $e^{2\pi i}=1$ from that definition, for example? Yes, it is possible, but very few readers will be capable of the proof. Any reader who is capable of the proof does not need the Wikipedia article.) Such a reader is well-advised to learn basic concepts like trigonometry, then proceeding to Euler's identity, before taking on general complex exponentiation. This is how the subject is invariably taught, for good reason. (I hardly claim that the presentation style is original to me.)

As for "neglecting logic", there's nothing illogical about relying on trigonometry here, since trigonometry can be perfectly well defined without requiring complex exponentials, and is so defined elsewhere on Wikipedia. I think you're confusing minimalism with logic.

I have, however, added a short section noting that the trigonometric functions can be defined in terms of complex exponentials (via the limit definition), rather than vice versa. Does that satisfy you?

—Steven G. Johnson 18:12, 26 May 2006 (UTC)

The challenge is to generalize to a^z where a is a (real or nonreal) complex number. The answer is a^z= e^xz where e^x=a. However the equation e^x=a has many solutions. So a^z is multivalued. There exists a positive real number, π, such that 2π·i·n (where n assumes all the integer values, and i²=−1) are all the solutions to the equation e^x=1. If x is one of the solutions to e^x=a, then all the values of a^z are e^{(x+2π·i·n)z}

It is costumary to select one of these values as the principal value, but the principal values do not satisfy the rule (a^b)^c=a^bc. For example: 1=1^1/2=(e^2πi)^1/2 is not equal to e^πi=−1.

Bo Jacoby 14:26, 26 May 2006 (UTC)

As I've replied before, any choice of a specific value for $x^{y}$ leads to $x^{bc}\neq (x^{b})^{c}$ violations for non-integer exponents. In fact, this happens for purely real exponentiation as well: $(-1)^{2\cdot {\frac {1}{2}}}\neq ([-1]^{2})^{1/2}$. The rest of the mathematical world has learned to live with this. In practice, I think you're overstating the problem; one does not typically convert back and forth between polar and rectangular notations in the middle of a sequence of exponentiations, or take the square root of the square without knowing that this results in the absolute value. Also, anyone who includes explicit $e^{2\pi i}$ terms is deviating from the principal value, and knows it. —Steven G. Johnson 18:12, 26 May 2006 (UTC)

Thanks for asking my opinion. The goal is not to satisfy me, but to make a useful article. The reader must have some previous knowledge to understand any article. The more knowledge required by the reader, the fewer readers understand the article, and the less useful is the article. Perhaps you and I have different readers in mind, you're thinking of "The rest of the mathematical world" and I'm thinking of an intelligent young student? That is why I have this minimalistic point of view regarding assumptions on the part of the reader.

The algebraic definitions in the Trigonometric functions article depend on power series, which this article does not (and should not) rely on. That is why I would like this exponentiation article not to depend on trigonometry. My reader does not understand power series, and so talking about power series does not help him understanding exponentiation.

Who is talking about power series here? An "intelligent young student" is most likely to have learned of sine and cosine via the classic geometric definitions, as I mentioned, which is how they are primarily defined in the WP article. Giving the limit definition of e^x to the same student will leave him/her helpless, because proving anything from that definition is hopelessly difficult from an elementary perspective. —Steven G. Johnson 03:23, 29 May 2006 (UTC)

The sudden introduction of geometry into a purely algebraic context is confusing to the beginner. It is a sad fact of history that the use of complex numbers came later than trigonometry, and so many books still teach trigonometry using only geometry and real numbers. I disagree that "proving anything from the limit definition is hopelessly difficult from an elementary perspective". The algebraic derivation of the differentiation formula de^x = e^xdx is straightforward: de^x = d lim(1+x/n)ⁿ = lim d(1+x/n)ⁿ = lim n(1+x/n)^n-1d(1+x/n) = lim n(1+x/n)^n-1dx/n = lim (1+x/n)^n-1dx = lim (1+x/n)ⁿ(1+x/n)⁻¹dx = (lim (1+x/n)ⁿ)(lim(1+x/n)⁻¹)dx = (e^x)(1)dx = e^xdx. This leads to Taylor's power series for e^x, and from that the trigonometric power series are derived. Note that the WP article on Trigonometric function does not actually derive the power series from the geometrical definition. It is not that elementary. Bo Jacoby 09:11, 29 May 2006 (UTC)

Understanding the formula a^bc=(a^b)^c requires the multivalued interpretation of the exponentiation: 1^4(1/4) = 1 while (1⁴)^1/4 = 1^1/4 = {+1,+i,-1,-i}.

I'm not sure what your point is. The article states that non-integer exponents are multivalued, and gives examples, and also says that there is a conventional choice of the principal value which is the standard meaning of any "x^y" expression unless otherwise noted. All of this is true. It doesn't mention the identity you keep harping on except in the context of integer exponents, the only case where it is always valid. I have no objection to adding a brief discussion of the ramifications of non-integer exponents on this identity somewhere, but this is perfectly possible to do while sticking to standard notation and terminology. —Steven G. Johnson 03:23, 29 May 2006 (UTC)

You are right. My point was that the multivaluedness of noninteger powers y^x should not be swept under the carpet by introducing a conventional principal value as a solution, which it is not. Note the conventional exception y=e: y^x=e^x is singlevalued by convention even if x is noninteger. It seems in the article as if you consider e^2πixn multivalued? It is not. You write "the number of possible values" meaning the number of different integer powers of e^2πix. The very important special case e^2πix still deserves a treat of its own, even if you don't accept the concept of a primitive real power of unity. Bo Jacoby 09:11, 29 May 2006 (UTC)

On 0⁰ you added the comment:

(This depends on context, however, and in some contexts 0⁰ is considered indeterminate.)

This should be omitted, as it increases confusion and decreases clarity. The link to empty product explains:

A consistent point of view incorporating all of these aspects is to accept that 0⁰ = 1 in all situations, but the function h(x,y) := x^y is not continuous.

Also the remark that e can be also defined in other ways, goes without saying. Some readers gets confused by the remark, having plenty to do understanding a single definition. Some readers don't get confused, but nobody gets happier. Bo Jacoby 22:33, 28 May 2006 (UTC)

I disagree. First, readers are well-advised to be cautious about 0^0, since it often arises from limits of x^y expressions and similar; if a reader simply memorizes the rule that 0^0 = 1 as an absolute, they can easily be lead astray. Second, in my teaching experience there is no harm in mentioning to students that a subject has a depth that is not plumbed at the moment, but to give a hint of where to go for more information; on the contrary, this tends to excite the curiosity. Moreover, this is well in keeping with hyperlinked nature of Wikipedia, whose spirit is to give plenty hints of ways for the reader to branch off into other directions. —Steven G. Johnson 03:23, 29 May 2006 (UTC)

Yes, the link to empty product is necessary, but it is also sufficient. How are anybody lead astray by 0⁰=1 except if believing that x^y is continuous? Bo Jacoby 09:11, 29 May 2006 (UTC)

Basic question

Does anyone know how you could $x^{y}=z$ when you know both x and z? Example: $5^{y}=390625$ y can only be 8, but is there any way to figure that out without simply guessing? It may also be near impossible to guess for solving things such as $15.3^{y}=401.3$ or something like that.

You can actually find a use for this in probability, if you were finding the number of occurances necessary to create a certain odd e.g., if the probability of rolling a one on a die = ${\frac {1}{6}}$, the probabilty of it not happening = $1-{\frac {1}{6}}$, or ${\frac {5}{6}}$, so the number of times you must roll a die to make the odds of rolling a one in a round of rolls $\approx {\frac {1}{4}}=x$ when $\left({\frac {5}{6}}\right)^{x}={\frac {1}{4}}$

...but how do you find x?

That's what the log button on your calculator is for: if $x^{y}=z$, then y = log(z)/log(x). Try it.--agr 21:02, 21 May 2006 (UTC)

Well yeah, I know that... let me make it clearer (I'm bad at stating questions) what does "log" exactly do? Heh... I annoyed my teachers the same way asking how trig. ratios were found.

How about this: $x^{x}=4812$(or some other number)

Solve the equation x log(x)=log(4812) by some root-finding algorithm. For example by substituting x:=log(4812)/log(x). Starting with x=5, ending with the solution x=5.1644 . Bo Jacoby 14:24, 22 May 2006 (UTC)

Re "what does "log" exactly do?": do us a favor and take a look at the logarithm article and let us know what isn't clear. (Asking those questions is good, by the way.)--agr 19:57, 23 May 2006 (UTC)

$x^{x}=4812$ can be solved by the Lambert's W function.

Redirected from Prisoner Of War (POW)!

I think its a dumb redirect for my entry of POW (prisoner of war) to be redirected here.

Simplified intro

I agree that the intro needed to be simplified, but I think it has gone a bit too far by saying "exponentiation is repeated multiplication". The reader shouldn't have to wade through 3-4 screens full of math to learn that exponents do not have to be integers. --agr 13:23, 19 June 2006 (UTC)

The beginner should have a chance to read about positive integer exponents in peace. The advanced reader should eventually read the Taylor formula in exponential form, e^D=F, where the exponent is the differentiation operator and the result is the displacement operator. The reader can click inside the table of contents to skip the elementary screens. But of course it can be improved. Be bold and do it. Bo Jacoby 10:35, 20 June 2006 (UTC)

"Fractional exponent" section: nth root of a?

In the "Fractional exponent" section of the article, the following...

For a given exponent, the inverse of exponentiation is extracting a root.

If $\ y$ is a positive real number, and n is a positive integer, then the positive real solution to the equation

$\ x^{n}=y$

is called the n^th root of $\ a$

$x=y^{\frac {1}{n}}$

For example: 8^1/3 = 2.

This may sound ignorant, but...there's no "a" in either of those equations. Does it mean to say "the nth root of x"? --zenohockey 20:47, 20 June 2006 (UTC)

Thanks. I'll correct it immediately. It means to say the n'th root of y. Bo Jacoby 07:35, 21 June 2006 (UTC)

Non-integer exponent

If the expression for $10^{3}$=10 x 10 x 10, then what is the expression for $10^{3.2}$, or how do we calculate it without using the $x^{y}$ button on a calculator?

As a consequence, do non-integer exponents also deserve a slightly more detailed sub-section?

Answer to Lars-Erik: Using the general formula e^x=lim_n→∞(1+x/n)ⁿ, first find b as the solution to the equation e^b = 10 , then compute 10^3.2 = (e^b)^3.2 = e^3.2b . There are faster methods, but this one shows the principle. Bo Jacoby 10:09, 20 July 2006 (UTC)

Exponents redirect

Why does the Exponents page redirect to some band? I think most people searching for exponents will be expecting math. If no one objects, I'm going to fix it. Alex Dodge 10:25, 3 September 2006 (UTC)

Perhaps it should redirect to List of exponential topics and maybe mention the band there.--agr 11:21, 3 September 2006 (UTC)

That sounds good, and I'm doing it now. (Sorry about the delay. I just moved into college.) However, wouldn't a disambiguation page be more standard? I've never witnessed this "List of X Topics" construct before. Alex Dodge 18:23, 18 September 2006 (UTC)

That's a horrible idea. Make a disambiguation page at Exponents instead of a redirect. --Raijinili 07:39, 23 September 2006 (UTC)

Yes. I agree. And, as such, I have made a simple disambiguation page. Does this look acceptable to everybody? Alex Dodge 21:24, 23 September 2006 (UTC)

when the power is a vector

in e^X, when X is a vector {x1, x2, ...}, then e^X is a vector (e^x1,e^x2,...} Am I right? I think that we had better list some formula for this such as: e^X*e^XT=e^X*XT? ... Jackzhp 20:09, 3 September 2006 (UTC)

Dimension and exponentiation

There is a discussion at the ref desk about whether raising to a different power expresses a different dimension. If you want to contribute, be quick, because these discussions die out in a few days. DirkvdM 08:59, 4 September 2006 (UTC)

Terminology

I would like some clarification on terminology, especially the word "power".
In the expression:
$p=b^{x}$;
is p the "power", or is x? Some dictionaries define the "power" as the exponent, others as the result of multiplying a number by itself a specified number of times. The latter conforms to a common usage, e.g., if we list the "powers of two", the answer would be 2, 4, 8, 16, ...rather than 1, 2, 3, 4... So I would prefer to call p the power. But if x is the "power", what is the term for p?

In any case, a complete list of terms would be helpful. Given the expressions:
$p=b^{x}$; $b={\sqrt[{x}]{p}}$; $x=log_{b}{p}$ ;
is the following correct?
b is the BASE
x is the EXPONENT
p is the POWER [(x^th) power of base b ]
b is the ROOT [(x^th) root of p]
x is the LOGARITHM of p to base b

Are there any other words used to describe the elements of this equation?

Drj1943 02:06, 26 November 2006 (UTC) revised for clarification Drj1943 01:46, 27 November 2006 (UTC)

An invalid proof

I was recently presented with this interesting 'proof':

$e^{ix}=e^{\frac {i2\pi x}{2\pi }}=(e^{i2\pi })^{\frac {x}{2\pi }}=1^{\frac {x}{2\pi }}=1$

The poser of this problem told me the invalid part was that $e^{ab}=(e^{a})^{b}$ only holds for real exponents, but this is contradicted by this article. Can anyone find the flaw and explain it to me? I think it might be a useful addition to this article and to Invalid proof... plus it'll stop the stupid thing bugging me! -- Perey 19:05, 11 December 2006 (UTC)

$1^{\frac {x}{2\pi }}$ is multivalued. One of the values are 1 and another of the values are $e^{ix}$. Bo Jacoby 23:31, 11 December 2006 (UTC)

Thank you. I can't see anywhere where this information is present on the current page. The long-since deleted section Powers of one kind of hinted at it in a roundabout way, where it said that for the xth powers of one, $e^{i2\pi x}$ (after stating $e^{i2\pi }=1$), if x is integer then the result is 1, if x is rational then the result is a root of unity, and if x is real then the result is a 'direction' (meaning according to prior definition that it's a complex number on the unit circle). While this does suggest special cases for irrational exponents of one, it doesn't say that they're multivalued. Should it have? I really think a 'Powers of one' section should be reintroduced—without the confusing start from $e^{i2\pi }=1$. -- Perey 06:17, 12 December 2006 (UTC)

The subsection on Multivalued power says: "If e^b = a, then e^(b+2πi·n)x are the values of a^x ". As e⁰ = 1, then e^{(0+2πi·n)x/2π} are the values of 1^x/2π. For n=0 you get 1. For n=1 you get e^ix. I am pleased that you recall and miss the section of powers of one. Alas the notation 1^x was very strongly opposed by other WP editors. The formula $e^{i2\pi }=1$ should not confuse you. It merely says that the circumference of the unit circle is 2π. Bo Jacoby 08:44, 12 December 2006 (UTC)

Oh, yes, of course. Somehow I missed that whole section, seeing only the 'Multivalued logarithm' section—so I guess I inferred that logarithms could be multivalued ($log(e^{x})=x+2\pi n$), but not the reverse. And it's not so much that $e^{i2\pi }=1$ is confusing; but then, in my studies Euler's formula is bread and butter—to others it might be quite understandably confusing. But even to me, it seems to obscure the point that this is unity we're talking about. I would tentatively suggest stating the cases to begin with in terms of $1^{x}$ (the cases being integer x, rational x, real x, and—heaven help us—complex x). Then recall to the reader that $e^{i2\pi }=1$, and explore these results further using powers of e.--Perey 16:07, 12 December 2006 (UTC)

I agree that the expression e^2·π·i·x is spooky, involving transcendental numbers e and π. It could simply be called 1^x with a caveat that it is the primitive value rather than the principal value. But I am overwhelmingly opposed. (See Talk:Function_(mathematics)#warning_to_editors). Bo Jacoby 23:03, 12 December 2006 (UTC)

Powers of e

The actual page on exponentiation claims:

A non-zero integer power of e is

$e^{x}=\left(\lim _{m\rightarrow \pm \infty }\left(1+{\frac {1}{m}}\right)^{m}\right)^{x}=\lim _{m\rightarrow \pm \infty }\left(\left(1+{\frac {1}{m}}\right)^{m}\right)^{x}=\lim _{m\rightarrow \pm \infty }\left(1+{\frac {1}{m}}\right)^{mx}=\lim _{mx\rightarrow \pm \infty }\left(1+{\frac {x}{mx}}\right)^{mx}=\lim _{n\rightarrow \pm \infty }\left(1+{\frac {x}{n}}\right)^{n}$ .

The right hand side generalizes the meaning of e^x so that x does not have to be a non-zero integer but can be zero, a fraction, a real number, a complex number, or a square matrix.

Would someone care to explain this particular step: $\lim _{m\rightarrow \pm \infty }\left(1+{\frac {1}{m}}\right)^{mx}=\lim _{mx\rightarrow \pm \infty }\left(1+{\frac {x}{mx}}\right)^{mx}$ which confuses me profoundly?

I especially can't see how this motivates using $e^{x}=\lim _{n\rightarrow \pm \infty }\left(1+{\frac {x}{n}}\right)^{n}$ even for for x=0 as $mx\rightarrow \pm \infty$ hardly will hold in that case. -- Qha 00:44, 12 December 2006 (UTC)

Thanks for asking. The fraction 1/m is multiplied in numerator and denominator by x, which is not zero. When m goes towards plus or minus infinity, the so does mx. Then mx is renamed to n. Bo Jacoby 08:15, 12 December 2006 (UTC). PS. The formula also holds for x=0 because e⁰=1. Bo Jacoby 23:11, 12 December 2006 (UTC)

0^0

Trovatore, your own reference [1] concludes: "Consensus has recently been built around setting the value of 0^0 = 1". So you cannot use it for arguing otherwise. Bo Jacoby 23:52, 15 December 2006 (UTC)

Actually that was Carl's reference, not mine. I don't think he was using it for precisely that point. --Trovatore 00:02, 16 December 2006 (UTC)

Sorry for doing you injustice, sir. If you insist that "0^0 undefined" be included, you must provide modern references saying so. Bo Jacoby 00:08, 16 December 2006 (UTC) Don't fight an edit war. You claim that "others consider it undefined" but both references say that 0^0=1. Please understand that you must provide proof of your claim. Who are the authors that consider 0^0 undefined? Bo Jacoby 00:16, 16 December 2006 (UTC)

Trovatore, who are the authors that consider 0^0 undefined? Bo Jacoby 00:29, 16 December 2006 (UTC)

See the comment at Talk:Empty product signed "VectorPosse 01:08, 16 December 2006 (UTC)". Two contemporary calculus books are cited there - Stewart and Larson, Hostetler, Edwards - that state 0^0 is an indeterminate form. Treating 0^0 as undefined is completely standard in complex analysis, as well, because of the problems with a singularity of the exponential at the origin. Please don't revert well-sourced additions to the article merely because they disagree with your interpretation of the facts. The point of the Drexel math reference is the quote "Other than the times when we want it to be indeterminate, 0^0 = 1 seems to be the most useful choice for 0^0 ." This underscores the point that there are many situations in which 0^0 is best left undefined. CMummert 02:05, 16 December 2006 (UTC)

By the way, there seem to be two or three different places that 0^0 is discussed in this article. We should add a section on 0^0, probably the one that is mistakenly at empty product right now, instead of duplicating the same material several times. CMummert 02:08, 16 December 2006 (UTC)

When do we want to be indeterminate? What is the point of writing something meaning nothing? What are the "situations in which 0^0 is best left undefined"? Yes, there is a place for integer exponents and another for complex exponents. The reader should consider integer exponents in peace before being bothered by advanced stuff. See Talk:Empty product. Bo Jacoby 09:11, 16 December 2006 (UTC)

Here is an interesting quote that will be useful in rewriting the part on 0^0.

"It is not surprising that many students suspect the indeterminate form 0⁰ to be equal to 1, believing that the elementary rules of algebra will apply. The example $x^{\alpha /\log x}$ immediately dispels this myth." L. J. Paige, A note on indeterminate forms, American Mathematical Monthly v. 61. n. 3 (March 1954), p. 189-190.

CMummert 15:04, 16 December 2006 (UTC)

Sorry to be dense, but what is the significance of $x^{\alpha /\log x}$ here? --EdC 01:12, 17 December 2006 (UTC)

The indeterminate form for $\lim _{x\to 0}x^{\alpha /\log x}$ is $0^{0}$. If we naively replace $0^{0}$ with 1 then $\log \lim _{x\to 0}x^{\alpha /\log x}=\log 1=0$ but this second limit simplifies to $\alpha$ using continuity and logarithm rules. So if $0^{0}=1$ in this situation then every number $\alpha$ equals 0. CMummert 01:29, 17 December 2006 (UTC)

Oh, right. Of course, the better solution is to note that x^y is discontinuous at (0, 0) and so $\lim _{x\to 0}x^{\alpha /\log x}={\left(\lim _{x\to 0}x\right)}^{\lim _{x\to 0}{\left(\alpha /\log x\right)}}$ does not hold. And given that students will have to learn about continuity eventually, anyway... --EdC 02:47, 17 December 2006 (UTC)

By the way, there's a passage in the "Ask Dr. Math" ref that no one seems to have commented on:

As a rule of thumb, one can say that 0^0 = 1 , but 0.0^(0.0) is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0) ; but Kahan has argued that 0.0^(0.0) should be 1, because if f(x), g(x) --> 0 as x approaches some limit, and f(x) and g(x) are analytic functions, then f(x)^g(x) --> 1 .

This could actually be a reference for the "0.0^0.0 is undefined" formulation, depending on where it comes from, which is unclear. It's indented the same as the Knuth passage, making it seem a quote, but I don't know who's being quoted. Possibly it's Alex Lopez-Ortiz, the maintainer of the sci.math FAQ. Kahan is probably William Kahan. --Trovatore 20:09, 16 December 2006 (UTC)

---

The article now says this:

“ There are two differing conventions about whether the value of 0⁰ should be defined to be 1 or left as an indeterminate form. ”

I'm really not comfortable with this phrasing. That this is an indeterminate form when construed in the way that is necessary in analyis is a demonstrable fact, not a convention. I think I could argue for the same conclusion in the contexts where it makes sense to consider it an empty product, but not in a brief comment like this. I'm going to think about ways of rephrasing this. Michael Hardy 02:30, 17 December 2006 (UTC)

Please feel free to improve the phrasing, but please try to maintain neutral POV. CMummert 02:49, 17 December 2006 (UTC)

---

Here is the requested comment on the "Ask Dr. Math" passage above. The premise: "when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0)" , does not imply the conclusion that "0.0^(0.0) is undefined". That (lim x^y) is undefined implies only that no definition, or lack of definition, of (lim x)^{lim y} can make x^y a continuous function. So the logical implication of the passage is not valid, and it doesn't really matter that the premise itself is not quite valid either: When approaching from any nonvertical direction we have y=ax, and lim x^y =lim x^ax = 1. Only when approaching from a vertical direction do we have lim x^y = lim 0^y = 0. In order to obtain a limit k≠0 and k≠1 the non-analytical curve, y = log k/log x, having vertical tangent at (0,0) must eventually be followed. WP deserves better logic than this. Bo Jacoby 14:31, 17 December 2006 (UTC).

Summary 0^0

The three points of view regarding the values of 0⁰ and 0^0.0 seems to be these.

Position 1: 0⁰=1 and 0^0.0=1 (Knuth, Euler, Laplace, Kahan)

Position 2: 0⁰=1 and 0^0.0 is undefined (Trovatore)

Position 3: 0⁰ and 0^0.0 are both undefined (Cauchy, many textbooks)

As x^y is discontinuous for (x,y)=(0,0), the (lack of) limit give no definition of 0^0.0.

The subsection Exponentiation#Powers_of_zero says on 0⁰: "If the exponent is zero, some authors consider that 0⁰=1, whereas others consider it undefined or indeterminate, as discussed below".

The article says on 0^0.0: "The zeroth power of zero is usually left undefined in complex analysis; this is discussed below". This seems to be position 3.

The subsection Exponentiation#Zero_to_the_zero_power contains many arguments for position 1, one, perhaps, for position 2: ("In discrete mathematics, the convention is often adopted that 0⁰ = 1. In continuous mathematics such as calculus and complex analysis, the indeterminate form 0⁰ is often left undefined"), and none for position 3.

We don't know any authors besides Trovatore in favour of position 2, and we don't know any authors since Cauchy in favour of position 3. (The words "some", "others", "usually" and "often" do not qualify as references).

Bo Jacoby 11:16, 18 December 2006 (UTC).

I have reverted several of your changes, but left the ones that improved the article. Please find consensus here before adding unqualified claims that 0^0 = 1. It is inappropriate for this article to make a choice between the two different positions, since both are common. The present state of the article describes both conventions in a relatively neutral manner.

If you would like, I will make a list of textbooks where 0^0 is left undefined; VectorPosse listed two (Stewart and another) earlier. I will also look at a complex analysis book to verify that 0^0 is undefined in it, and give you a reference. Your distinction between "authors" and "textbooks" is moot; textbooks are written by authors. CMummert 12:53, 18 December 2006 (UTC)

Surely we need far better references that "some authors" and the like. The author of a textbook is not necessarily a reseach mathematician. He takes most of the material for his textbook from other textbooks or from articles. There is nothing wrong in that. A textbook which does not define 0⁰ does not necessarily argue that 0⁰ is not or should not be defined, but only indicates that no definition is important in the context of the present textbook. We need the argumentation af the authors of the textbooks. So far we have found absolutely no argumentation in favour of leaving 0⁰ undefined, only that "it is written in the scriptures" - a religious style of argumentation. You should not call the claim 0⁰ = 1 unqualified. I repeat from Concrete Mathematics: "Some textbooks leave the quantity 0⁰ undefined, because the functions x⁰ and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x⁰ = 1, for all x, if the binomial theorem is to be valid when x=0, y=0, and/or x=−y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant". This means that the claim 0⁰ = 1 is qualified, while the claim "0⁰ is undefined" is unqualified. If we want to qualify that claim we need a mathematician saying the opposite, that "0⁰ = 1 is a mistake". How could he possibly continue? "0⁰ must be left undefined because the functions x⁰ and 0^x have different limiting values when x decreases to 0" ?. That is simply not a valid reason. Leaving 0⁰ undefined does not help at all. I do look forward to see some mathematical argumentation rather that religious argumentation in this matter. Bo Jacoby 14:21, 18 December 2006 (UTC).

This is not an appropriate forum for mathematical argumentation. WP keeps a neutral point of view, and does not favor one author over another when each has the support of a large group of mathematicians. You are trying to discuss what 0^0 should be defined as; I have no desire to discuss this and WP is not the correct place to do so. These articles should discuss how 0^0 is handled in practice. Knuth does make strong statements; perhaps you should ask why he feels them to be necessary, since he doesn't make similar statements about why 2+2 = 4. CMummert 14:32, 18 December 2006 (UTC)

What support are you talking about? I am requesting references, so far in vain. "a large group" is not a proper reference. These WP-articles were supposed to make sense to young people, but we are doing a bad job. Why does Knuth feel strong statements necessary? Obviously because "Some textbooks leave the quantity 0⁰ undefined". Why doesn't he make similar statements about why 2+2 = 4 ? Obviously because no serious textbook leaves 2+2 undefined. Why do you feel strongly about undefining what has been successfully defined? Bo Jacoby 14:48, 18 December 2006 (UTC).

I have added two references to books where I looked up the convention that 0^0 is undefined. These are just two books that happened to be on my shelf; I am sure that many more can be found. WP doesn't define or undefine things, it attempts to describe practice as it stands. CMummert 14:56, 18 December 2006 (UTC)

Quote: "The convention that 0⁰ is 1 is not necessary here, because the series can be rewritten so that the first term is explicitly 1 rather than $0^{0}/0!$". Of course no evaluation of any expression is ever necessary if you just replace the expression with the proper value. Bo Jacoby 15:02, 18 December 2006 (UTC).

No, WP doesn't define or undefine things, but obsolete ideas are in articles on history, not on mathematics. I look forward to seeing your references, though. Please quote the argumentation from the books. Bo Jacoby 15:06, 18 December 2006 (UTC).

logarithmic branch point

The quote: "The function z^z, viewed as a function of a complex number variable z and defined as e^a ln z, has a logarithmic branch point at z = 0" is neither correct nor an argument for leaving 0^0 undefined. Please improve. Bo Jacoby 00:04, 19 December 2006 (UTC).

That error was introducted (inadvertenly) by M Hardy when cleaning up the math notation [2] . Thanks for pointing it out; I fixed it. CMummert 00:44, 19 December 2006 (UTC)

Thanks. The logarithm log(z) is discontinuous for z=0 as lim log(z)=∞ while z^z is continuous for z=0 as lim z^z=1, because lim z·log(z)=0. So z=0 is not a logarithmic branch point of z^z. And still it is not an argument for leaving 0⁰ undefined. Bo Jacoby 08:07, 19 December 2006 (UTC).

quote in the subsection or in the reference section

Trovatore: "While edits made in collaborative spirit involve considerably more time and thought than reflexive reverts, they are far more likely to ensure both mutually satisfactory and more objective articles." Wikipedia: edit war. Bo Jacoby 00:14, 19 December 2006 (UTC).

The sentence after that is: "In the case of less experienced contributors, who have unknowingly made poor edits, reversion by two or more people often demonstrates that such reversions are probably not fundamentalistic or in bad faith, but instead closer to an objective consensus." This is more applicable to you than your quote applying to Trovatore. --C S (Talk) 00:33, 19 December 2006 (UTC)

The edit in question is Trovatore's insisting in moving the important Knuth quote from the subsection (where in my opinion it belongs) to the reference subsection (which in my opinion is for references, not for quotes), without discussing the matter. I am not a less experienced contributor. Bo Jacoby 08:07, 19 December 2006 (UTC).

discrete and continuous

The subsection is close to position 2, except that one distinguishes between discrete and continuous math rather than between integers and reals. In consequence of that peculiar point of view one should state 0⁰=1 in the subsection on discrete mathematics, and allow 0⁰ to be left undefined in the subsection of continuous mathematics. Please comment on that. Bo Jacoby 08:07, 19 December 2006 (UTC).

convention or definition

Hi CMummert. Please comment on your edits and reverts to Exponentiation#Zero_to_the_zero_power. The mathematical term for assigning meaning is: "definition", not "convention". The reference to "empty product" need not be repeated. The statement "Other power series identities are similar in this respect" is sloppy: either we tell the story or we don't. The message "Knuth in particular has used this to justify putting 0⁰=1" is contained in the reference. I appreciate the new references to programming languages. Bo Jacoby 20:33, 19 December 2006 (UTC).

The term convention is well established in mathematics and this is exactly the sort of situation that it applies to. I put the name Knuth back into the article because I like to have authors names near references to their works; I don't care if it is removed. 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 $\sum _{n=0}^{\infty }x^{n}/n!$. I like repeating the link to empty product, because that bullet is specifically related to empty products.

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)

Thanks. The WP article on convention does not seem to explain your use of the word. I feel that the word 'convention' is implicating that this is not a matter of mathematical necessity but that anyone may pick the choice of his liking. Perhaps this is where we disagree. Note that $1+\sum _{n=1}^{\infty }x^{n}/n!=\sum _{n=0}^{\infty }x^{n}/n!$ only if $\ 1=x^{0}/0!$ . If we don't adopt the 'convention', then the left hand side is well defined for all values of x but the right hand side is undefined for x=0. If we do adopt the 'convention' then the equality holds everywhere. Psychologically I understand the opposition against the 'convention'. The different limits of x⁰ and 0^x seems to pinpoint the troublespot: 0⁰. So stay out of trouble by avoiding 0⁰. But the discontinuity remains whether or not 0⁰ is defined. Bo Jacoby 21:39, 19 December 2006 (UTC).

The closest entry on the convention disambiguation page is:

Convention (norm), a set of agreed, stipulated or generally accepted social norms, norms, standards or criteria

This is a quite accurate description of the definition 0^0 = 1. It is, indeed, not a matter of necessity to define 0^0, only a convenience to make it simpler to state certain theorems. CMummert 21:50, 19 December 2006 (UTC)

Yes, convention implies that "some alternative convention could be equally good but we accept this one for social convenience". That is not the case here. Therefore the word convention is misleading. It is far more than "a convenience to make it simpler to state certain theorems". It relieves computer programs and hand computations from special cases and provides a huge simplification. If the computer system had 0⁰ undefined every programmer would have to declare a subroutine power(x,y) by "if x=0 and y=0 then 1 else x^y" and use this subroutine rather than the standard routine x^y. No alternative definition does the job. There is no freedom of choice here. It is really not a convention. In Exponentiation#Powers_of_e we have a similar example: A formula is proved true for all integers, and the right hand side is defined for all complex numbers. Then this formula becomes the unique definition of the left hand side for complex numbers. There is no freedom of choice, and we would not call it a convention. The discussion on 0⁰ is even simpler, because 0⁰ is defined for integer 0, and the discussion is about whether this definition should also apply to real and complex 0. No actual generalization is called for. Bo Jacoby 22:22, 19 December 2006 (UTC).

I don't think it's mere convention. I was surprised when I learned, not long ago, that some respectable mathematicians do think it's mere convention. But maybe this isn't the best place to argue the point. Michael Hardy 02:30, 20 December 2006 (UTC)

textbooks for one, textbooks for all

0⁰=1 is supported by Euler, Laplace, Libri, Möbius and Knuth, while Cauchy leaves it undefined. These are mathematicians, not textbooks. If the authors of the textbooks following Cauchy shall count as authors we must quote their argumentation. Bo Jacoby 22:37, 19 December 2006 (UTC). PS. See this. Remember to comment on this.

Bo Jacoby, you don't have consensus

Do not make changes without consensus. We all tire of this. VectorPosse 08:47, 20 December 2006 (

There is consensus that in discrete mathematics 0⁰=1. Even Trovatore wrote: "If n is a nonnegative integer, there is no problem. As I said, I find 0.0⁰=1.0 to be completely convincing and unproblematic". (Talk:Empty_product#Remarks_on_.22binomial_theorem.22_argument). The controversy is for continuous mathematics. My change was for integer exponents. See also Exponentiation#Exponentiation_in_abstract_algebra. I requested your comment here and again here and you didn't respond. Bo Jacoby 09:44, 20 December 2006 (UTC).

To Bo Jacoby: Look, I'm trying really hard not to resort to knee-jerk reverts. As CMummert correctly pointed out earlier, many changes you make do improve the articles. But then you do some weird stuff too. You last edit summary was, "The examples from 'polar form' are redistributed". It appears that the whole section on polar forms was just deleted. So I don't know what you mean by "redistributed". I think that section was crap, so I don't mind that it was deleted, but at least state what it is you are doing. And better yet, why not improve that section instead of just deleting it? There was useful information there, even if it wasn't presented very well. Correct me if I'm wrong, or let me know to where you intend to "redistribute". VectorPosse 09:22, 20 December 2006 (UTC)

The examples from the polar form subsection are now found in Exponentiation#Multivalued_power and Exponentiation#Singlevalued_power. The rest belong in complex number but not in exponentiation and that's why I don't improve it. Bo Jacoby 09:44, 20 December 2006 (UTC).

Well, a few of them were relocated to the sections you mentioned. I still think your edit summary was a bit disingenuous. Nevertheless, I agree that the content could go elsewhere. VectorPosse 09:52, 20 December 2006 (UTC)

Discuss first and revert later. See Exponentiation#Primitive_and_principal_roots_and_powers_of_unity for the concept that confused you. Bo Jacoby 09:48, 20 December 2006 (UTC).

Don't pretend like your opinions on 1^x have not been discussed ad nauseum before. It didn't confuse me. It was "confusing" terminology and notation. And I didn't revert; I removed one sentence that violated the parallel structure of the two sections in question and said nothing about single-valuedness. VectorPosse 09:54, 20 December 2006 (UTC)

No I don't pretend. Here I stated standard interpretations of 1^x : the complete set of values (e^2πixn), a primitive value (e^2πix), and the principal value (1). My controversial observation, that the primitive value is the useful one, was not promoted here. Bo Jacoby 10:05, 20 December 2006 (UTC).

branch cut

"The principal value has the advantage of being singlevalued, but the price to be paid is that it ceases to be continuous". CMummert continues: "A branch cut for the logarithm must be defined in order to make it an analytic function". No, that is not correct. A branch cut was needed in the first place in order to define the principal value, but the branch cut does not make the function neither analytical nor continuous on points on the branch cut. When walking around the singularity, you experience either entering another branch (multivalued) or a discontinuity (singlevalued). The better solution is to consider the function value to be a multiset. Then after walking around the singularity the value returns to the same multiset as before, even if each point in the multiset has moved continuously into another point in the multiset. Example. The square root of x for x=1 is the multiset {+1,−1}. After moving x once around the unit circle from 1 via i, −1, −i and back to 1, the square root moves half a turn and ends in {−1,+1}, which is the same multiset, even if it is not represented by the same ordered pair of numbers. Please improve. Bo Jacoby 15:04, 20 December 2006 (UTC).

is often left undefined.

CMummert prefers "is often left undefined" rather than "some textbooks". The word "often" is unprecise and subjective. So far we have two textbooks in the reference list, and no quotations from them to support your claim. You too should keep NPOV. Bo Jacoby 15:13, 20 December 2006 (UTC).

If you want, the article could say "some discrete math textbooks define 0^0 to be 1" and "some continuous math textbooks leave 0^0 undefined". It isn't neutral to add the "some textbooks" to one side with no caveat at all added ot the other side. In texts that leave 0^0 undefined, there usually will be no comment on it - they just leave it undefined.

Here are two quotes, however, since you have asked for them several times.

"Although a^0 = 1 for any nonzero constant a, the form 0^0 is indeterminate—the limit is not necessarily 1." Edwards and Penny (sourced in article) p. 467. Emphasis is from the original.

"It is not surprising that many students suspect the indeterminate form 0^0 to be equal to 1, believing that the elementary rules of algebra will apply. The example $x\alpha /\log x$ immediately dispels this myth." L. J. Paige, A note on indeterminate forms, American Mathematical Monthly v. 61. n. 3 (March 1954), p. 189-190.

CMummert 17:30, 20 December 2006 (UTC)

About the latter quote – what does that have to do with the value of 0⁰ (not the indeterminate form)? –EdC 18:19, 24 December 2006 (UTC)

Indeed, the former quote appears to be about indeterminate forms as well. Do you have any quotes that actually support your position? –EdC 18:20, 24 December 2006 (UTC)

I think this distinction between "form" and "value" is an after-the-fact reinterpretation. The traditional understanding of an indeterminate form is that it does not have a univocal value. It is not surprising that the authors did not bother to say "it's an indeterminate form and its value is also undefined"; that's part of what "indeterminate form" was understood to mean, before some writers decided that it was useful to assign it a value, but nevertheless keep the "indeterminate form" terminology around. --Trovatore 19:47, 24 December 2006 (UTC)

Well, unfortunately I don't have access to any of these old textbooks so I don't have any way to see what the authors actually meant; the impression I get, though, is that it's the limit, not the value, which is undefined. –EdC 22:15, 24 December 2006 (UTC)

Certainly, if you want to interpret those quotes as saying that the fact that 0⁰ is an indeterminate form means that it doesn't have a defined value, you're going to have to show (i.e. provide a quote) that those authors thought of "form" and "value" as the same thing. –EdC 22:19, 24 December 2006 (UTC)

Wrong. You have the burden of proof here. The natural reading is that it's undefined. --Trovatore 02:23, 25 December 2006 (UTC)

The WP article on indeterminate form says that "an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions", which does not imply that the indeterminate form has no value. And further: "0⁰ is less indeterminate than the other indeterminate forms, and this is one reason why 0⁰ is usually not left undefined (but instead defined to be 1)". Bo Jacoby 07:49, 25 December 2006 (UTC).

The WP article is revisionist. The two sources Carl quotes above, especially the Paige reference, are clearly using "indeterminate" in a sense that is distinctive from the notion of having a well-defined value. It appears that some contemporary authors wish to assign a value to 0⁰ and nevertheless preserve the terminology of "indeterminate form" for exposition of the associated methodology (L'Hospital etc). In effect it makes "indeterminate form" a synonym of "point of discontinuity" (including the ones that involve ∞ -- there you just have to use the extended reals, or perhaps the real projective line, as appropriate). But this is not what has been traditionally meant by the term. --Trovatore 08:02, 25 December 2006 (UTC)

"clearly" – that's not how it appears to me. Where Paige refers to the "elementary rules of algebra", I read that as his saying that 0^0 is 1, but that the elementary rule that the limit of an expression is determined by the limits of the subexpressions does not hold. Otherwise, how do you explain the reference to the "elementary rules of algebra"?

And no, an indeterminate form isn't quite the same as a point of discontinuity. The defining feature of an indeterminate form is that limits of approaches to that point may take any of a range of values ((-∞, ∞) for 0/0; [0, π/2] for atan 0⁰). The similarity is to an essential singularity in complex analysis. –EdC 10:39, 25 December 2006 (UTC)

Resp. to Trovatore's comment. I have never seen in print the claim that 0^0 is an indeterminate form but that 0^0 as a number is equal to 1. As far as I can tell, this idea that it could be indeterminate but still defined was suggested by EdC in this edit [3]. My understanding of the word "form" is that it is used instead of the word "number" in situations where a numeric expression does not have a well defined value. Answers.com shows that I am supported by the McGraw-Hill Dictionary of Scientific and Technical Terms [4]. CMummert 22:41, 28 December 2006 (UTC)

I was following the Wikipedia article on indeterminate forms, which (along with most other sources I am aware of) makes the fine point that an indeterminate form is a type of limit (actually, a well-formed but content-free formula naming and describing a family of limits), not a numeric expression. Of course, most indeterminate forms would not have a well-defined value if evaluated as numeric expressions, which is presumably why McGraw-Hill erases the distinction (i.e. to save space in a dead-tree work) but to treat an indeterminate form as a numeric expression is to make a category mistake.

For a source that is able to treat the concept in sufficient depth, I suggest MathWorld:

…A mathematical expression can also be said to be indeterminate if it is not definitively or precisely determined. Certain forms of limits are said to be indeterminate when merely knowing the limiting behavior of individual parts of the expression is not sufficient to actually determine the overall limit.…There are seven indeterminate forms involving 0, 1, and ∞:…"Indeterminate." From MathWorld--A Wolfram Web Resource.

As I read the MathWorld quote, "indeterminate form" means "indeterminate form of a limit". McGraw-Hill's definition clearly conflicts with this; I believe it is an oversimplification. A dictionary is not a suitable source in an argument over terminology. –EdC 01:32, 29 December 2006 (UTC)

Indeed, McGraw-Hill is hopelessly wrong; $\lim _{x\to 0}{\frac {\sin x\sin 1/x}{x}}$ is an indeterminate form (0/0) that does not have a limiting value. –EdC 01:38, 29 December 2006 (UTC)

(unindenting) The article on indeterminate forms is quite bad, but it doesn't matter here because one WP article can't be used as an authoritiative source for another WP article. I will add it to my list of articles that need to be improved.

My understanding of the term "indeterminate form" doesn't come from the McGraw Hill dictionary; I merely pointed out that it is a published source showing that my understanding of the term is not unique. Trovatore says above that his understanding is similar to mine: I would define an indeterminate form as an algebraic expression that does not represent a definite number, although similar algebraic expressions do represent numbers. I am sorry that you feel it is wrong, but it is at least one reliable source. Do you have any source that claims that 0^0 is an indeterminate form but yet 0^0 is defined to equal 1?

It is true that 0^0 is usually left undefined when it is viewed as an indeterminate form (of a limit, if you wish), and defined to equal 1 when it is viewed as an empty product. But the mere two symbols 0⁰ are just two symbols - you can't tell which of the two they might refer to, except by context. That is why 0^0 is not universally held to be equal to 1. CMummert 02:02, 29 December 2006 (UTC)

Actually, I find the Wikipedia article on indeterminate forms to be quite good. I guess I'll have to watch it to make sure you don't introduce any glaring errors.

As for McGraw-Hill: sure; but that you advanced the McGraw-Hill definition without spotting immediately that it is wrong indicates that your understanding of the term cannot be the same as that presented in the McGraw-Hill definition. Oh, and it may be published, but it is certainly not a reliable source; for one, it is a dictionary (not even an encyclopedia), and for another, it is wrong.

"I would define an indeterminate form as an algebraic expression that does not represent a definite number" – also wrong, or at the very least incomplete; 1/0 is an algebraic expression that does not represent a definite number, but is not an indeterminate form.

"I am sorry that you feel [McGraw-Hill] is wrong" – I don't feel that it is wrong; I proved above that it is wrong. Or is there something wrong with my counterexample? This is mathematics, not exegesis.

"Do you have any source that claims that 0^0 is an indeterminate form but yet 0^0 is defined to equal 1?" – I don't have any new sources to add as of now. Every source I have come across that defines an indeterminate form as anything other than a type of limit has proved to be a misunderstanding or an oversimplification.

"the mere two symbols 0⁰ are just two symbols - you can't tell which of the two they might refer to, except by context" – OK, you're getting at something there. Absent any other indication, the default content of the expression "0⁰" is the arithmetic expression "0⁰", which has the value 1. However, in analysis, arriving at the expression "0⁰" in an answer or intermediate result often indicates that at a previous stage one made the mistake of taking limits of subexpressions. –EdC 15:34, 29 December 2006 (UTC)

I left a comment at Talk:Indeterminate form which you may want to comment on, if you are interested in that article. CMummert 16:49, 29 December 2006 (UTC)

Thanks. –EdC 17:27, 29 December 2006 (UTC)

pros and cons

Hi CMummert. Thanks a lot for the quotes.

When we, due to this controversy, cannot provide a clear article on exponentiation, we should at least provide clarity on the logic of both points of view, and elaborate on the consequences.

I see a danger in not defining, because the formulas must then be supplemented by an exception for zero. Alone in the short article on power series there are 14 formulas to correct, and we must expand the article with about the same number of lines, decreasing the clarity and quality of the article. It is hard work. You do it - not me. There must be an easier way. Why not add a note saying: "in this article 0⁰ means 1". That must be done in some hundred mathematical articles in wikipedia. It could be centralized, though, saying in one article: "Note that in wikipedia 0⁰ means 1". Which article? Exponentiation of course. Then we could add a note to the wikipedia articles where this is not the case, that "in this article 0⁰ is undefined". That is not many articles because undefined expressions are not used. It doesn't make sense to say something that means nothing.

But you are the one to see a danger of defining. Tell me. What can possibly go wrong by defining what is otherwise undefined? There is no contradiction, as if we were aiming at defining 1/0. The definition 0⁰=1 gives an expressive power which is otherwise lost. If you at least admit that 0⁰=1 in discrete mathematics, then please stop supporting VectorPosse. It will save you years of tedious editing work.

Bo Jacoby 19:18, 20 December 2006 (UTC).

I'll take a shot at this. First, Bo, I want to say I appreciate your toning down of your earlier style. It's much more pleasant now to work with you.

Let me preface this by pointing out that these are non-editorial discussions we're having now; strictly speaking, they're off-topic. Even if we were to come to a consensus about what the definitions ought to be, it would not justify reporting that consensus in the article.

So, what's the possible downside of giving 0.0^0.0 a definition? Well, would you agree that there would be a downside to defining 0/0 to be 1? Or defining it to be 17, or George W. Bush? I hope you would agree with that; otherwise there's not much more to talk about.

What's the downside? The definition 0/0=1 would make some formulas true over a wider range of variables, after all. But it's unmotivated. I think that's the real "downside". And 0.0^0.0=1 is similarly unmotivated. --Trovatore 20:10, 20 December 2006 (UTC)

Thanks for the nice words. I am not aware of any change in style. Of course I don't intend to offend anybody and I apologize if I did. I agree that dividing by zero is a disaster. The expression x=a/b means exactly the same thing as the equation a=bx. If a=0 and b=0, then the equation is true for all values of x. If a≠0 and b=0, then the equation is false for all values of x. In neither case is the equation suitable as a definition. But that x⁰=1, even for x=0, is the de facto standard in power series and in many more places. The motivation is explained: empty product, empty function, even continuity is valid when the exponents are integers only. Now 0=0.0 is also de facto standard: nobody cares whether 0 is integer or real. But while undefined 0⁰ is a disaster, an undefined 0^0.0 would pass virtually unnoticed, because the function 0^x is quite unimportant and almost never used, while x⁰ is used all the time. Nevertheless, no harm is done by setting 0^0.0 = 0⁰ . It is the right thing to do. You set 2+2=2+2.0=2.0+2.0 without hesitation.

An editor of a WP mathematics article should be a mathematician. It is neither sufficient nor necessary to copy a textbook from the shelf. The information should be discussed and criticized from a mathematical point of view and accepted or rejected based on that discussion. Perhaps this is where CMummert disagrees. Trovatore's original objection against defining 0^0.0 was not the textbook from the shelf, but the lack of interpretation of the expression 0^0.0 . That is a philosophical argument, not a mathematical argument. The same kind of argument has been made against x⁴, because space has only 3 dimensions and so x⁴ has no interpretation; and against 1/2, because you cannot count to 1/2, and against negative numbers, and against complex numbers. It is hard for me to explain why, but the argument is completely invalid mathematically. Mathematics does not depend on interpretation, but on logical consistency. This is why I disagree with Trovatore. I am completely convinced that the undefining of 0^0.0 and 0⁰ is a bad mistake. It is OK to leave an expression undefined as long as there is no clue to a definition, but when a constructive definition has emerged, then there is no point in going backwards to the expression being undefined. Who are we to prevent people from the benefit of a good definition? We are free not to use it ourselves if we don't want to.

Bo Jacoby 23:22, 20 December 2006 (UTC).

OK, here we see the style issue cropping up again. I am a mathematician, and Carl is, and VectorPosse is, and we haven't yet seen your publication list. Don't bother telling me that that's not relevant, as you are the one bringing it up, and making bold pronouncements about what is and isn't "valid mathematically".

Anyone can use any definition he finds useful, of course, provided he makes it clear to anyone he wants to communicate it to. What's not clear is that your preferred definition is in fact a "good definition" by default, in the continuous context. Yes, philosophical points need to be considered here; there's nothing strange about that. There's not even a clean boundary between philosophy and mathematics, just a difference in emphasis. (There's also no clean boundary between mathematics and natural science, or philosophy and natural science, but that's a discussion for another day.) --Trovatore 00:28, 21 December 2006 (UTC)

"The information should be discussed and criticized from a mathematical point of view and accepted or rejected based on that discussion. Perhaps this is where CMummert disagrees." As far as I can tell, this is where nearly everyone here disagrees with you. We don't idealize the presentation of math here. We present what has been established and what has consensus. Nobody cares if you disagree with virtually every calculus textbook out there. Nobody cares if virtually every calculus textbook out there is "wrong". It's irrelevant. We understand your argument; you don't need to keep repeating it as if it's just a matter of time before we "get it". It's irrelevant. Your ideal vision of mathematics is not what dictates the content of this page. I have said it before and I'll say it again. I will not be sucked into your vortex of "argumentation" and I refuse to continue to address points which have been discussed plenty. I will give you (and the community) the courtesy of justifying changes I make in a brief entry here in the talk page. You are welcome to reply. You are welcome to find consensus and revert my change. But no more argument. VectorPosse 00:36, 21 December 2006 (UTC)

CMummert wrote "Let me preface this by pointing out that these are non-editorial discussions we're having now; strictly speaking, they're off-topic". To that I answered that "The information should be discussed and criticized from a mathematical point of view and accepted or rejected based on that discussion". I am not implying that we are not all mathematicians, but that our mathematical discussion is important in our roles as WP editors and not off-topic. I am sorry if that was misunderstood. The authors of polynomial and power series and binomial theorem and many others assume the definition 0⁰=1. Where do CMummert, Trovatore and VectorPosse expect our readers to find this definition if not in exponentiation ? Bo Jacoby 12:27, 21 December 2006 (UTC).

It is easy to put a footnote or parenthetical remark into an article describing the conventions it uses. Or the article could leave the convention to context, and we can add a description of the convention only when someone asks. If there is no link to exponentiation in some article, there is no reason for someone reading that article to read exponentiation first to find out what conventions WP uses; how would they know to do so?

Your argument that this article must unequivocally "define" 0^0 is weak for a second reason, as well. The present exponentiation article does say that the identity 0^0 = 1 is used in writing power series, the binomial theorem, etc. So if a reader wondered about 0^0 when reading about these subjects, and came here through divine insight, they would find out about the standard convention. Overall, I don't believe it is a serious issue. CMummert 13:19, 21 December 2006 (UTC)

The situations where you must take 0⁰ = 1 vastly outnumber the cases where you mustn't. In fact, I can't think of a single Wikipedia article except this one where a non-unit limit of x^y is encountered. I'd rather put a footnote on each of those instances (if they exist) than on all power series and references to the binomial theorem.

This article should state that 0⁰ = 1 but with specific caveats. Writing that we can define 0⁰ = 1 in the context of discrete mathematics but not in continuous mathematics is too vague, because many readers probably come here after encountering power series in calculus. Fredrik Johansson 14:26, 21 December 2006 (UTC)

How can two things be equal, but with caveats? If they are equal, they are equal, and if they are not they are not. I think that the "caveat" you mean is that 0^0 is used to represent 1 in some contexts but not others. This is exactly what the current article says. CMummert 13:01, 22 December 2006 (UTC)

We have seen no examples of people using 0⁰ meaning anything but 1. The authors leaving it undefined don't use it, of course. Bo Jacoby 12:45, 24 December 2006 (UTC).

Not all power series have a nonzero constant term.

Not all power series have a nonzero constant term. Not log(1-x)=x+x²/2+x³/3+x⁴/4+... I have fixed it. Bo Jacoby 15:17, 20 December 2006 (UTC).

VectorPosse lacks support

There is no consensus, not even support, for VectorPosses point of view that 0⁰ is undefined in discrete mathematics. Bo Jacoby 15:38, 20 December 2006 (UTC).

True, but it doesn't matter. I don't believe there is a consensus in the literature that it is defined. We are not empowered to create our own consensus here. We are not slavish stenographers; certainly we're informed in our choices by our own experience and expertise, but we can't make up a consensus that doesn't exist. --Trovatore 20:31, 20 December 2006 (UTC)

I have already explained why I made that change. I will say no more on the matter. If and when you get consensus, Bo Jacoby, you are welcome to change it back. VectorPosse 21:20, 20 December 2006 (UTC)

Original article, computer languages

The original discussion of 0⁰ in the "empty product" article [5], written back in 2003 mainly by Michael Hardy and Toby Bartels, was perfect. I want it back!

A particularly silly claim of the present article is that we should define 0⁰ = 1 because that's what programming languages do. Programming languages are based on math, not the other way around.

Besides, the list is not comprehensive. Mathematica, for example, takes 0⁰ to be an indeterminate form. Curiously, Mathematica simplifies to a⁰ to 1 even if you don't put any restraints on a, but does not attempt to simplify 0^a. It presumably takes 0⁰ to be indeterminate for the purpose of symbolic limit evaluations.

The conclusion in the 2003 edition of the "empty product" article got it right:

A consistent point of view incorporating all of these aspects is to accept that 0⁰ = 1 in all situations, but the function h(x,y) := x^y is not continuous.

Then 0⁰ is still an indeterminate form, because we do not know the value of the limit of f(x)^g(x) (in the example above), but that is a statement about limits, not about the value of 0⁰, which is still 1.

(More nuanced approaches are possible, but this view is simple and will always work.)

I read the "empty product" article a few years ago when I first needed to know the deal with 0⁰, and found it extremely helpful. The article we have here is not helpful the way it is presently written. It seems that certain editors (no names) are more interested in coming up with "more nuanced approaches" for their own pleasure than to actually serve the readers. Fredrik Johansson 13:56, 21 December 2006 (UTC)

The "consistent point of view" mentioned at the end of that article is indeed consistent, but I don't think it is common in the real world - in the situations in which 0^0 is an indeterminate form, it is typically not considered to have the value 1. Another way of writing the final paragraph of the previous version would have been:

A consistent point of view incorporating all of these aspects is to accept that 0^0 is not actually equal to any number, but when 0^0 appears in a combinatorial identity or power series it may be replaced by 1 for the purpose of calculation. More nuanced approaches are possible, but this view is simple and will always work.

There are lots of consistent views. But I think your suggestion has some merit; we can add a summary paragraph to the end of the section that gives these simple interpretations. CMummert 14:22, 21 December 2006 (UTC)

An expression that "may be replaced by 1 for the purpose of calculation" has the value 1. A consistent no-nonsense point of view is that

• lim_x,y→0+ x^y is an indeterminate form having no value,
• lim_x→0 x⁰ = 1,
• lim_y→0+ 0^y = 0,
• 0⁰ = 1.

Bo Jacoby 21:01, 21 December 2006 (UTC).

Yes, that viewpoint is consistent. It would be equally consistent if you replaced the last bullet by 0^0 = 5. The viewpoint that 0^0 has no true meaning at all but is a convenient notation for 1 in some settings is also consistent and no-nonsense. The point is that there is not a unique consistent viewpoint here. CMummert 13:06, 22 December 2006 (UTC)

The viewpoint 0⁰ = 5 is consistent but nonsense: When restricting the exponent to integer values, the definition 0⁰ = 1 is in harmony with interpretation as well as with continuity: x⁰ = 1 for x≠0, and this function is continuous for x=0. When the variables are generalized to nonnegative reals, continuity does not survive. When generalized to complex numbers, singlevaluedness does not survive. This is not a reason why the definition of xⁿ should not survive. There is no improvement in removing a definition, unless you want to use the now undefined expression for something else, which you might do if the expression is not already used. But you don't want to redefine 0⁰. You want to undefine it. And for no reason but that there are some books on your shelf. Look at Derivative#Rules_for_finding_the_derivative.

• Power rule: If $f(x)=x^{r}$, for some real number r;

$f'(x)=rx^{r-1}\,$.

That is continuous mathematics, well within the realm where you insist that 0⁰ be undefined. Now insert r=1 and get the result that the derivative of x is the discontinuous function x⁰, which equals 1 everywhere except for x=0. That is an incorrect result. The 'consistent' CMummert-definition 0⁰ = 5 gives another incorrect result. The definition 0⁰ = 1 gives the correct result. You are doing everyone a disservice by you crusade against the definition 0⁰ = 1. Please do some serious thinking before you answer by knee-jerk reaction. You claim to be a mathematician. Show me. Bo Jacoby 14:52, 22 December 2006 (UTC).

Here is a parallel example to illustrate the mistake in your argument. Assuming that x^0 equals 1 everywhere, as you would like to do, means that the derivative of f(x) = x^0 should to be 0/x according to the "rule" you stated above. Now f'(0) = 0, because f is constant; thus your argument would also say that we should define 0/0 to be 0 so that the derivative rule is correct. This is clearly bogus reasoning, and your reasoning that 0^0 must be 1 so that the derivative of g(x) = x^1 is correct is similarly flawed. In any case, the derivative of x^1 being x^0 is just an example of an empty product, which is already covered by the article. CMummert 22:53, 28 December 2006 (UTC)

The power rule doesn't hold for r=0; implicit in any proof of the power rule is the assumption that r≠0. –EdC 01:56, 29 December 2006 (UTC)

I know. I was pointing out the error in the previous post, where it was claimed that the power rule holds for all values of x and all values of r. The power rule is only valid when both sides are defined, which (to me) excludes the cases 0/0 and 0^0. We could define 0/0 = 0 and 0^0 = 1, and remove this restriction. But this is not usually done, at least not in the calculus books I have seen. CMummert 02:05, 29 December 2006 (UTC)

Actually, yeah, let me correct the above. The power rule does hold for r=0, but only for x≠0; both sides need to be defined. However, the power rule for r=1 is (must) be proved elementarily, without the exponential function, and that proof is considerably simpler if one accepts 0⁰=1 (not that that will make any difference to you). By the way, the question isn't about defining 0⁰=1, it's about retaining the definition from discrete mathematics. –EdC 14:46, 29 December 2006 (UTC)

Retaining the definition 0⁰ = 1 is not a mathematical necessity, only a mathematical convenience, as it avoids complications. Undefining it leaves a lot of formulas - in discrete as well as continuous mathematics - to be rewritten, showing that the authors of these formulas did assume the definition. While x=0/0 says nothing about x, (because 0x=0 for all values of x, so x might even be equal to 5), the statement x=0⁰ says that x is an empty product. So there is a profound difference between 0/0 and 0⁰. Your 'counterexample' is nevertheless to the point: One doesn't prove that 0⁰ = 1. It's a definition. Bo Jacoby 17:10, 29 December 2006 (UTC).

confusion links in zero to the zero power

The links to discrete mathematics and to continuous mathematics are incorrect. We are talking about integer exponents rather than about finite mathematics, and about non-integer exponents rather than about numerical analysis. Somebody please correct it. Bo Jacoby 09:34, 22 December 2006 (UTC).

I think the dichotomy is essentially correct, although I would say "discrete mathematics and power series" versus "continuous mathematics". The article used to say this. CMummert 12:57, 22 December 2006 (UTC)

The dichotomy is correct, but the links were misleading. Did you ever click on those links? Now I removed them from the article. Bo Jacoby 14:06, 22 December 2006 (UTC).

I did click them. There is an article entitled discrete mathematics. The fact that continuous mathematics redirects to numerical analysis doesn't mean we shouldn't link to it; it means that the redirect will someday need to be fixed. In the meantime, there is no harm in linking to the redirect from this article, because the redirect is supposed to point to the best article currently available on continuous mathematics. CMummert 22:23, 28 December 2006 (UTC)

Fractional exponent

Isn't it true that fractional exponents $a^{\frac {m}{n}}$ are only defined for $a>=0$?

That is quite at tricky question, actually. The answer depend on the book you are reading, because different books have different definitions. It is not entirely objective.

That x=a^m/n means that x is a solution to the equation xⁿ=a^m. (Assume that m/n cannot be reduced).

If a is a positive real number, then some positive real number solves the equation. So the fractional power is defined. However, if n is even, then there is also a negative solution. So the fractional power is not uniquely defined. People don't like that. So they arbitrarily discard the negative solution, and then the fractional power is defined for nonnegative reals.

If a is a negative real number and m and n are odd numbers, then some negative real number x solves the equation. In that case the fractional power is defined for a<0. This is unimportant.

Using complex numbers, the equation has n solutions. So the fractional power is a set consisting of n complex numbers. From this point of view the answer to your question is 'no'. Bo Jacoby 20:34, 22 December 2006 (UTC).

I appreciate your answer. Don't you think there should be some reference to these remarks in the article itself? —The preceding unsigned comment was added by 83.130.76.97 (talk) 11:29, 23 December 2006 (UTC).

Yes. Bo Jacoby 12:23, 24 December 2006 (UTC).

More unsupported Bo Jacoby changes

There is plenty of discussion in the section "Zero to the zero power" to support both points of view. Bo Jacoby needs to stop making changes to this section and others without consensus. People have worked extra hard to make sure the wording is neutral. VectorPosse 20:16, 28 December 2006 (UTC)

There was support from several editors, and only one of the quotes supposed to argue for 'undefined' actually did so. The article should be readable for beginners, and that is not the case if it is not clear. Both points of view are clearly stated and argued. The discussion, which VectorPosse did not take part in, prepared for the additions: the derivation formula, the polynomial rather than the power series. If you want to take part in the developement, then make forward steps rather than backwards steps. Bo Jacoby 21:12, 28 December 2006 (UTC).

There is certainly not consensus here to revert all the places where 0^0 is discussed to their previous claims that 0^0 = 1. I disagree with the edit comment that claims VectorPosse's reversion is vandalism. I reverted it again.

Here are some specific problems the reverted version had:

1. You added the sentence "The definition 0⁰ = 1 is usually assumed in mathematics." I doubt that this sentence is true; most mathematics has no relationship to the value of 0^0 and few books will even bother to mention it.
2. Whether certain polynomial expressions or power series identities require 0^0 to be 1 in order to be correct is not relevant to whether 0^0 must be defined, because these identities can always be rewritten so that the summation starts at 1 and the 0 term is explicitly stated. For example, replace $\sum _{n=0}^{k}x^{n}$ with $1+\sum _{n=1}^{k}x^{n}$. It is not necessary to write these identities in the shorter form, and thus not necessary to define 0^0 = 1 just so that the shorter form has a well-determined meaning. Similarly, the case of the binomial theorem when n = 0 can be stated as a special case; it isn't necessary to use the same equation for this case as for the cases when n is positive.

CMummert 22:21, 28 December 2006 (UTC)

x^0=1 is assumed in calculus too

The very replacement of $\sum _{n=0}^{k}x^{n}$ with $1+\sum _{n=1}^{k}x^{n}$ actually assumes the definition x⁰ = 1. Whether necessary or not, the culture of mathematics is that $1+\sum _{n=1}^{k}x^{n}=\sum _{n=0}^{k}x^{n}$ for all x, and so that x⁰ = 1 for all x, even for x=0. The replacements are not actually made, not in wikipedia, not anywhere. The use of x⁰ = 1 is not limited to discrete mathematics. In calculus the formula ${\frac {dx}{dx}}=1$ is valid, and x=x¹ is valid too. Now the general formula ${\frac {dx^{r}}{dx}}=rx^{r-1}$ leads to ${\frac {dx^{1}}{dx}}=1x^{0}=x^{0}$ . For consistency, calculus has to accept the definition x⁰ = 1 for all values of x. Even if "few books will even bother to mention it", it is assumed by all the mathematicians. So the claim of the article that "In continuous mathematics such as calculus and complex analysis, the indeterminate form 0⁰ is often left undefined" is simply not correct. Even in calculus the definition 0⁰ = 1 is assumed. Bo Jacoby 07:46, 29 December 2006 (UTC).

You have already presented this argument above, and I have already pointed out the flaws in it. Here they are again.

1. The expression $\sum _{n=0}^{k}x_{n}$ is simply shorthand for $1+\sum _{n=1}^{k}x_{n}$. This is not because 0^0 is "defined" to be 1; it is because the expression on the left is defined to be shorthand for the expression on the right. One way to make the expressions equal would be to unilaterally decalre that 0^0 = 1, but this is not a universal definition that is universally made. And this expression's value at 0 is just another example of an empty product, which the article already covers.
2. The symbolic forms of the power rule are only valid when both sides are defined. It would be possible to declare 0^0 = 1 and 0/0 = 0 to extend the power rule to include the two cases where it is undefined, but I have never seen this done in practice. I explained above how the same argument you are presenting says we must define 0/0 = 0.
3. Several mathematicians here have pointed out that they, personally, find it possible to do research in mathematics without assuming 0^0 =1. It is not part of the "culture of mathematics", is not "assumed by all mathematicians" and is not "necessary for consistency". It is just a shorthand that some authors use.

I have no desire to debate the issue with you, so I will not respond again to these issues. This lack of responses does not mean that you have found consensus to edit the article to remove the neutral point of view about 0^0. CMummert 13:17, 29 December 2006 (UTC)

There is no difference between saying that 0^0 is a shorthand for 1 and saying that 0^0 equals 1. The empty product applies to integer exponents, and you say you agree that an empty product is one. Why not accept it at least in the subsection on integer exponents? I don't need consensus, nor do you, but we both need support. I count Fredrik Johansson and Hardy and EdC amoung my supporters, and you count VectorPosse and Trovatore amongst yours. You have lots of editing to do, in this article and in others, to prevail in your pointless crusade against a commonly used definition. Bo Jacoby 17:40, 29 December 2006 (UTC).

involution again

The sentence, (Exponentiation used to be called "involution".), has popped up once more. See above for the earlier discussion on the subject. The sentence adds more to confusion than to clarification, and the article on involution says something completely different. Bo Jacoby 13:08, 29 December 2006 (UTC).

Both m-w.com and oed.com say that one meaning of the word "involution" is exponentiation. The article on involution is describing a different concept with the same name. Ther eis no reason this article shouldn't include a sentence pointing out former terminology. CMummert 13:24, 29 December 2006 (UTC)

Nor is there any reason why an obsolete word shall be the first one to meet the uninitiated reader. I'll move the sentence down to 'advanced topics'. Bo Jacoby 13:21, 31 December 2006 (UTC).

censorship

CMummert, please note the following surviving piece of heresy from Exponentiation#Exponentiation_in_abstract_algebra:

suppose that the operation has an identity element 1. Then we can define x⁰ to be equal to 1 for any x. Now xⁿ is defined for any natural number n, including 0.

Why didn't you purge it and placed it on your index librorum prohibitorum? My friend, you have plenty of work to do. Happy new year! Bo Jacoby 18:34, 31 December 2006 (UTC).