Talk:Exponentiation/Archive 3

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This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Contents 1 Non-integer exponent 2 Exponents redirect 3 when the power is a vector 4 Dimension and exponentiation 5 Terminology 6 An invalid proof 7 Powers of e 8 0^0 9 Summary 0^0 9.1 logarithmic branch point 9.2 quote in the subsection or in the reference section 9.3 discrete and continuous 10 convention or definition 11 textbooks for one, textbooks for all 12 Bo Jacoby, you don't have consensus 13 branch cut 14 is often left undefined. 15 pros and cons 16 Not all power series have a nonzero constant term. 17 VectorPosse lacks support 18 Original article, computer languages 19 confusion links in zero to the zero power 20 Fractional exponent 21 More unsupported Bo Jacoby changes 22 x^0=1 is assumed in calculus too 23 involution again 24 censorship

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)

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; ; x = log_bp ;
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':

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)

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^i2πx (after stating e^i2π = 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^i2π = 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^i2π = 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 + 2πn), but not the reverse. And it's not so much that e^i2π = 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^i2π = 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

.

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: which confuses me profoundly?

I especially can't see how this motivates using even for for x=0 as 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^{α / logx} 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^{α / logx} here? --EdC 01:12, 17 December 2006 (UTC)

The indeterminate form for is 0⁰. If we naively replace 0⁰ with 1 then but this second limit simplifies to α using continuity and logarithm rules. So if 0⁰ = 1 in this situation then every number α 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 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 0⁰ / 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. --Chan-Ho (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 or . 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 only if . 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 xα / logx 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; 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;

.

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 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 with . 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 with actually assumes the definition x⁰ = 1. Whether necessary or not, the culture of mathematics is that 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 is valid, and x=x¹ is valid too. Now the general formula leads to . 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 is simply shorthand for . 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).