Talk:Exponentiation

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A vital article.

[edit] 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)

[edit] 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)

[edit] (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)

[edit] 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)

[edit] 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]

[edit] 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

[edit] "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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] "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. $\ 1^{1/2} = (e^{2\pi i n})^{1/2} = e^{\pi i n} = (e^{\pi i})^n = (-1)^n = \{+1,-1\}$ . 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 2π 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 2π 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)

[edit] 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.

[edit] Redirected from Prisoner Of War (POW)!

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

[edit] 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)

[edit] "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)

[edit] Non-integer exponent

If the expression for $103$ =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)

[edit] 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)

[edit] 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)

[edit] 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)

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