Talk:Exponentiation/Archive 2014

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Knuth's paper

I see that Knuth's paper (Donald E. Knuth, Two notes on notation, currently reference 15) is available here on arXiv.com. I presume that this is not a copyright violation, so should we not link to it? — Quondum 16:26, 27 July 2013 (UTC)

Can't see why not. I think Knuth has caused a lot of trouble with his 'But no, no, ten thousand times no!' and 'it has got to be 1' business about 0⁰ rather than being very clear about when a rule like that is okay to use. He could have done the whole business much better with his knowledge of computing. But yes that seems a good link to me. Dmcq (talk) 17:09, 28 July 2013 (UTC)

I've added the link, and softened the interpretation in the text and associated footnote. I personally find it easy enough to interpret him as merely objecting (albeit loquaciously) to the view that that 0⁰ must never be defined. Yes, in retrospect it would have helped if he'd been clearer on this, but perhaps we should not persecute him for not crafting his paper to that degree. — Quondum 01:15, 29 July 2013 (UTC)

So that we aren’t taking quotes out of context, I’m going to go ahead and paste the relevant few paragraphs from Knuth’s paper here (emphasis original):

Some of Libri’s papers are still well remembered, but [32] and [33] are not. I found no mention of them in Science Citation Index, after searching through all years of that index available in our library (1955 to date). However, the paper [33] did produce several ripples in mathematical waters when it originally appeared, because it stirred up a controversy about whether $0^{0}$ is defined. Most mathematicians agreed that $0^{0}=1$, but Cauchy [5, page 70] had listed $0^{0}$ together with other expressions like $0/0$ and $\infty -\infty$ in a table of undefined forms. Libri’s justification for the equation $0^{0}=1$ was far from convincing, and a commentator who signed his name simply “S” rose to the attack [45]. August Möbius [36] defended Libri, by presenting his former professor’s reason for believing that $0^{0}=1$ (basically a proof that $\textstyle \lim _{x\to 0^{+}}x^{x}=1$). Möbius also went further and presented a supposed proof that $\textstyle \lim _{x\to 0^{+}}f(x)^{g(x)}=1$ whenever $\textstyle \lim _{x\to 0^{+}}f(x)=\lim _{x\to 0^{+}}g(x)=0$. Of course “S” then asked [3] whether Möbius knew about functions such as $f(x)=e^{-1/x}$ and $g(x)=x$. (And paper [36] was quietly omitted from the historical record when the collected works of Möbius were ultimately published.) The debate stopped there, apparently with the conclusion that $0^{0}$ should be undefined.

But no, no, ten thousand times no! Anybody who wants the binomial theorem

$(x+y)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{k}y^{n-k}$

to hold for at least one nonnegative integer $n$ must believe that $0^{0}=1$, for we can plug in $x=0$ and $y=1$ to get $1$ on the left and $0^{0}$ on the right.

The number of mappings from the empty set to the empty set is $0^{0}$. It has to be $1$.

On the other hand, Cauchy had good reason to consider $0^{0}$ as an undefined limiting form, in the sense that the limiting value of $f(x)^{g(x)}$ is not known a priori when $f(x)$ and $g(x)$ approach $0$ independently. In this much stronger sense, the value of $0^{0}$ is less defined than, say, the value of $0+0$. Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side.

— Donald Knuth, Two Notes on Notation

As a mathematician, it is obvious to me that Knuth was drawing a clear distinction between the expression $0^{0}$, which has defined value $1$, and the limiting form $0^{0}$, which is indeterminate. The corresponding equations are $0^{0}=1$ and $\textstyle \lim _{a,b\to 0}a^{b}$ DNE. These statements are not in contradiction; they are talking about different mathematical objects. That is Knuth’s point. We don’t need to live in a confusing world where the expression $0^{0}$ is undefined or “partially defined” or “defined in some contexts” just because the corresponding limiting form is indeterminate. The Wikipedia article in its current state, which cherry-picks just the bits of historical exposition that Knuth is arguing against out of context to support an undefined $0^{0}$, is highly misleading and not representative of the source nor of the modern mathematical convention. —Anders Kaseorg (talk) 02:11, 27 December 2013 (UTC)

Other mathematicians disagree with you and think Knuth was wrong to put that in a place undergraduates would read and be swayed by his reputation into this messianic sort following. It is like the silly business where they tried teaching children calculus using non-standard arithmetic. Knuth is just another person like any other and his maths is not gospel. Dmcq (talk) 08:47, 27 December 2013 (UTC)

In fact if you like the computer side of things why not try and actually write something that does differentiation and uses Knuth's idea as you see it. As far as I could make out his idea is not easily usable, in practice, it is a handwaving notion for simple cases not something that mechanizes anyway easily - it makes various other things rather difficult. Dmcq (talk) 09:10, 27 December 2013 (UTC)

Contrariwise, I think Anders has described it perfectly, and Knuth described it very well. I think undergraduates who are unable to understand such clear language should not be the basis of such a criticism. And I fail to see what programming has to do with it. —Quondum 04:40, 29 December 2013 (UTC)

Being able to program some reasoning is proof that it works in a reasonable way. If you want to try and approach Knuth's idea and support differentiation you have got to treat powers as proper functions and not do many of the simplifications one normally applies without thinking. At least that's how it appeared to me when I tried it. Distinguishing between integer and real powers works out as pretty straightforward and if you notice Knuth talked about 'to hold for at least one nonnegative integer'. Dmcq (talk) 11:05, 29 December 2013 (UTC)

Well actually there was one little problem with distinguishing integer and real and saying x^0 is 1 but x^0.0 is only defined for x > 0. Is x^(0×0.0) = 1 or undefined at zero? Treating the product as real which is the obvious widening makes the result undefined even though there's a reasonable argument for it to come out as 1. Personally I'm happy treating it as undefined though. What one has to add to keep things consistent is undefined^0 = undefined. Dmcq (talk) 12:10, 29 December 2013 (UTC)

Agreeing with Anders and Quondum, I fail to see the purpose of undefining 0⁰. Which 'various other things' are made difficult by Knuth's idea? Bo Jacoby (talk) 07:49, 29 December 2013 (UTC).

@Dmcq: You seem to be implicitly conceding through your treatment the conclusion that the functions are in fact distinct (overloads, in computerese) despite similarity of notation, which I've already stated removes the issue. But this involves recognizing that the notation is ambiguously used for distinct functions, which seems to be something many are shying away from. —Quondum 17:14, 29 December 2013 (UTC)

Yes I agree with that. I think Knuth is right that we need to cater for 0^0 in some cases especially series, but I think the best way is to think of exponentiation as being distinct depending on the types of the base and exponent rather than trying to distinguish between it and limiting cases. I see no need to assign a value to something in the, well tautological, case of when there is no need to, and especially when it is liable to cause people to come to the wrong conclusion when actually evaluating limits. Having the result 1 when dealing with an exponent which is a natural number seems to work okay but if I'm being careful I still check it is giving the right result rather than depending on it. Just because the case of 1 is okay and a recursion rule gives all the following numbers doesn't mean that a formula will always work for zero, that way lies magical thinking. Dmcq (talk) 21:31, 29 December 2013 (UTC)

Even though I do not agree that a function must be continuous at every point of its domain, the problem does not arise, since for that case the function is not defined at the point (0,0) in its domain. And however you use context to disambiguate which function is being used, it would make far more sense to clarify distinctions between the functions and domains of each function than to try to idiot-proof a one-size-fits-all function. From which springs an idea for the article: should we not be emphasizing that despite their similarity and domains, the best way to treat the topic is as a group of distinct functions? Surely this idea must have notability? —Quondum 01:01, 30 December 2013 (UTC)

I consider myself very good at both mathematics and programming (I didn’t come here to brag, but if the context helps, I graduated from MIT in course 18C, and you’re welcome to look up my other achievements on Google). Can you explain exactly what the problem is that you think $0^{0}=1$ makes more difficult? If symbolic differentiation is what you have in mind, $0^{0}=1$ makes that much easier. For example, given $0^{0}=1$, you can compute ${\tfrac {d}{dx}}x^{y}=yx^{y-1}$, which holds for all $x,y$; but if $0^{0}$ were undefined, you would have to output some kind of conditional expression for the special case $(x,y)=(0,1)$. My main concern, in any case, is to clarify what Knuth’s paper really advocates—the current article is not at all an accurate reflection of its sources, which is the important thing as far as Wikipedia is concerned. —Anders Kaseorg (talk) 04:14, 30 December 2013 (UTC)

As it says at indeterminate form 'if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form'. If one defines 0^0 as 1 then you do have enough information - it is 1 and is not indeterminate. How does one know the form 0^0 has arisen because of the limit? It may have been in the original expression and so should be 1 rather than the limit value. Therefore one needs to distinguish between powers that are there because of the original expression and those that arise in the course of the evaluation and are a limit. And one needs to propagate the differences through simplifications. One can't for instance replace x^y by y*log x because that is undefined at 0,0. However by just assuming 0^0 is undefined none of these problems arise and that's what has been done for quite a while now and it works fine. Dmcq (talk) 14:58, 30 December 2013 (UTC)

I looked at the paragraph in the article here that references Knuth, and I don't see anything inaccurate. Could you say how to improve it? As to "exactly what the problem is", the problem is sometimes overstated. When $0^{0}$ is an empty product, there are many reasons to define it to be 1, and basically everyone does. But when $0^{0}$ is an abbreviation for $\operatorname {exp} (0\log 0)$, it is undefined, because $\log 0$ is undefined. So one has to pay attention to exactly which definition of exponentiation is being used; that (the definition) is what varies by context. — Carl (CBM · talk) 13:30, 30 December 2013 (UTC)

I fully agree. It would be nice to distinguish the cases even better in the article but the sources do call them both exponentiation and treat them as instances of the same thing. And treating 0^0 as 1 works mostly even for the reals, so as far as an engineer is concerned it might as well be 1 as save them some bother. You can see that with William Kahan's definition of pow in computing libraries where he has it as 1, and I really can't accuse him of not being careful. I think we're stuck with this business of people trying to force the article to say it is 1 and quoting Knuth even though he was nowhere so dogmatic. And I blame his 'a thousand time no' for much of the mess now. Dmcq (talk) 14:58, 30 December 2013 (UTC)

One cannot replace $x^{y}$ by $e^{y\log x}$ either way because the latter is undefined at (0, positive)! Undefining $0^{0}$ does not help at all, unless you also want to undefine $0^{1}$, $0^{2}$, $0^{\pi }$, etc. You have not gained anything from the extra complexity of dealing with two different definitions of exponentiation. Also, symbolic differentiation is not done by explicit manipulation of limits, so your supposed problem does not arise with symbolic differentiation in the first place. —Anders Kaseorg (talk) 21:37, 30 December 2013 (UTC)

The function is continuous at those points so there is no problem with taking a limit from positive x, you get the same value. Dmcq (talk) 22:05, 30 December 2013 (UTC)

In fact I don't regard the values with x=0 as part of the basic function but as part of the usual business of extending a function to any points on its closure where the limit is defined. I've been trying to think of the right words, is there a good word for that for real functions as opposed to complex ones? 23:16, 30 December 2013 (UTC)

I’m glad you don’t want to undefine $0^{x}$ for positive $x$, but the point stands that if you don’t, then undefining just $0^{0}$ doesn’t gain you the ability to replace $x^{y}$ by $e^{y\log x}$—so you’ve still shown no evidence of a “mess” for which to blame $0^{0}=1$. —Anders Kaseorg (talk) 10:33, 31 December 2013 (UTC)

I explained why above in the bit I wrote starting 'Being able to program some reasoning is proof that it works in a reasonable way' and again after 'As it says at indeterminate form'. Indeterminate forms are not just a form, they are part of a method that depends on the value not being defined. You have to drop that method and fall back to being careful between distinguishing between a limit and the value at a point and which instances of a power falls in one category and which in the other. As to 0^x it doesn't matter much one way or the other for finite x greater than 0 whether it is defined or not. Why are you so keen on this business about 0^0 for real powers? Knuth is quoted for what he said but what weight should be given to him? He is good at maths but certainly not an analyst never mind another Cauchy. And I repeat, if you check what Knuth said, and I underline the relevant part, it was 'to hold for at least one nonnegative integer'. Why do you feel the need to go beyond that? Dmcq (talk) 12:25, 31 December 2013 (UTC)

The problem with the article is that, although it presents Knuth’s quote “it has to be 1”, it immediately backs away from that statement and concludes the section with out-of-context quotations: “Cauchy had good reason to consider $0^{0}$ as an undefined limiting form, … In this much stronger sense, the value of $0^{0}$ is less defined than, say, the value of $0+0$. [omitted: Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side.]”, and “This quieted the debate for some time” paraphrasing “The debate stopped there, apparently with the conclusion that $0^{0}$ should be undefined. [omitted: But no, no, ten thousand times no!]” The article’s interpretation makes it sound like Knuth didn’t really mean what he actually said. Nonsense. He really did mean it, and he was right. The article should at least acknowledge that Knuth is a strong supporter of $0^{0}=1$, and if the article doesn’t itself conclude that $0^{0}=1$, it should cite other sources for the supposed differing views. —Anders Kaseorg (talk) 21:37, 30 December 2013 (UTC)

The article does cite other sources; I count at least 5. Have you tried finding any complex analysis textbook that defines 0^0 to be 1? I have looked at quite a few for this purpose and I have yet to find one. They all have the same definition: x^y is exp(y log x). As you suggested above, many complex analysis texts do not define x^y for x=0 at all, regardless of the value of y. That is not to say that the "undefine it"; they simply never define it. — Carl (CBM · talk) 22:17, 30 December 2013 (UTC)

Is 0^2 also undefined?

Seriously Carl? Are you living in a world where 0² is never defined? Bo Jacoby (talk) 11:07, 31 December 2013 (UTC).

Many is not the same as either never or not never. As he says many do not define it. The section Exponentiation#Limits of powers gives other limit values like that. Dmcq (talk) 12:48, 31 December 2013 (UTC)

Carl wrote: " they simply never define it". If you want to undefine every definition of x^y except exp(y⋅log(x)), then you must also undefine 0². Is that what you want? If not, then why insisting in undefining 0⁰? It simply doesn't make sense to me. Bo Jacoby (talk) 18:17, 31 December 2013 (UTC).

How about finding a source that defines 0² for reals? That could form a basis for discussion. Dmcq (talk) 19:13, 31 December 2013 (UTC)

Is 0² undefined for reals? Shall our article also have a section about some authors not defining 0² ? Bo Jacoby (talk) 19:18, 31 December 2013 (UTC).

break

I’ve gone ahead and made some changes to the presentation of Knuth’s view, without weakening the presentation of counterarguments from other sources. I intend for these changes to make sense even if you don’t accept $0^{0}=1$ yourself. You are free to add additional arguments from other sources, but I hope we can at least agree on what Knuth actually said. —Anders Kaseorg (talk) 11:15, 31 December 2013 (UTC)

Looks good to me. — Carl (CBM · talk) 11:53, 31 December 2013 (UTC)

I think it emphasises Knuth's views too much and it leaves out an important part of the original, that it was $0^{n}$ when the nonnegative integer $n$ was 0 that he wanted to be 1. Dmcq (talk) 12:35, 31 December 2013 (UTC)

You have presented no sources to support your apparently original idea that $0^{0}$ has a different definition depending on whether the exponent has “integer context” or “real/complex context”. If you find such a source, you are welcome to add a third viewpoint, but this is not a distinction made by Knuth, even though he happens to use integers in his argument. (They are not fundamental to his argument. The generalized binomial theorem makes just as convincing a case for $0^{0}=1$ for an exponent in “real/complex context” as the standard binomial theorem does for “integer context”; so does the generalized power rule.) —Anders Kaseorg (talk) 13:43, 31 December 2013 (UTC)

Do you dispute that many books define the real form of exponentiation $x^{y}$ by $e^{y\log x}$ as described just above here? Are you saying 0^0 must be 1 for complex exponentiation? if people fill the hole in that it is normally with zero. Having $0^{0}$ be 1 is a convenience in the Binomial series, the series doesn't even have to be written in the form. If you have a look at the section generalised binomial theorem that you pointed at it doesn't even require it when the series is in the expanded form.

I do dispute that, because there seems to be no trouble understanding that $0^{x}=0$ for positive $x$ carries over just fine from the integers. If these books do not define $0^{0}$, then they are perfectly well categorized by the first bullet point in the article section. I’ve never seen a serious proposal for $0^{0}=0$ in any context; please provide sources. Are you suggesting mathematicians should abandon summation notation just to avoid stepping on $0^{0}$? And to claim that $0^{0}=1$ is merely “a convenience in the binomial series” is to ignore all the other arguments that have been presented in the article and on this talk page, including even a few that are specific to the continuous exponent case; see the result that $\textstyle \lim _{t\to 0^{+}}f(t)=\lim _{t\to 0^{+}}g(t)=0$ implies $\textstyle \lim _{t\to 0^{+}}f(t)^{g(t)}=1$ for analytic $f,g$ with $f$ nonzero. —Anders Kaseorg (talk) 23:02, 31 December 2013 (UTC)

Knuth used integers in his argument. You have expanded on what he said. Practically none of his work has been outside of the discrete cases, why do you take him as a guide about reals and complex numbers? He is not an analyst, he is a mathematician who went into computer studies and not numerical analysis even. What we actually know from what he wrote was that he wanted $0^{n}$ to be 1 when $n$ was the nonnegative integer 0. Dmcq (talk) 17:22, 31 December 2013 (UTC)

To claim that I have expanded on what Knuth said is because he used integers in his argument is like claiming that Newton's law of universal gravitation expands on what Newton said because he used apples in his argument. It’s not impossible to imagine a world in which apples fall according to different laws than peaches, but such a world is unnecessarily more complicated, and someone endorsing such a view on Wikipedia would need to have sources backing them up; it would be ridiculous to demand specific sources from the proponents of universal gravitation for all the different types of fruit that one might like to use as a projectile. In all standard presentations of mathematics, the complex numbers are an extension of the integers (if you’d like to imagine that they are distinguished by some type system, then the integers should still be substitutable for complex numbers). Any suggestion otherwise is nonstandard and needs sources. —Anders Kaseorg (talk) 23:02, 31 December 2013 (UTC)

Nobody is suggesting that the complex numbers are not a field extension of the reals. But there is no reason to think this means that the complex exponentiation function is an extension of the real exponentiation function; indeed they already differ about $-1^{1/3}$. Can you point to any complex analysis book that defines $0^{0}$ in the context of complex exponentiation? — Carl (CBM · talk) 23:53, 1 January 2014 (UTC)

Newton did not mention apples in the Principia. He talked about a projectile on the earth and about the moon and did some very careful calculations. And gravitation does not work the same as electrostatic attraction or magnetism even though they are inverse square to a first approximation. As to substitutable an integer is not a type of real, I don't know why you are pointing to some computer science thing as being something mathematicians have agreed to. Dmcq (talk) 23:59, 31 December 2013 (UTC)

The very concept of type is computer technology. In FORTRAN a numeric variable is declared as INTEGER or REAL or COMPLEX for efficiency reasons. Declaring all numbers COMPLEX makes the program bigger and slower, but still correct. But Dmcq (& al) thinks that the integer 2 and the real number 2 and the complex number 2 are different numbers. That is an error. They are merely different representations inside the computer. Just like the number 1+1 is equal to the number 2 even if the representation "1+1" is different from the representation "2". Dmcq's point of view is nonstandard and inconsistent, and it has corrupted our article for years. Bo Jacoby (talk) 02:06, 1 January 2014 (UTC).

More confusion on the domain(s)

(−2)^1/3 does have different defined values depending on the domains. If you let r denote the number 2^1/3, then

$(-2)^{1/3}=-r$ in R × Q_odd (the set of rational numbers with odd denominator).

$(-2)^{1/3}=re^{i\pi /3}$ in C × R.

As we have noted above,

$0^{0}=1$ in (R or C) × (Z or N, or even Q)

$0^{0}=\mathrm {undefined}$ in R × R.

— Arthur Rubin (talk) 18:06, 30 December 2013 (UTC)

(−2)^1/3 has nothing to do with 0⁰. Bo Jacoby (talk) 23:52, 30 December 2013 (UTC).

It does point out that complex exponentiation is not the same function as real exponentiation. even when the inputs are real numbers. — Carl (CBM · talk) 23:58, 30 December 2013 (UTC)

It points out that a non-integer power of a negative number should not be defined as a real number, because it is better defined as a non-real number. (−1)^x=e^iπx. Bo Jacoby (talk) 00:21, 31 December 2013 (UTC).

But in elementary algebra, most people learn that ${\sqrt[{3}]{-1}}=(-1)^{1/3}=-1$. Only in the context of complex analysis do they use a different definition of the exponential function in which the value is not -1. Similarly, one might learn in elementary combinatorics that 0⁰=1, but there is no reason to suspect this will hold with other definitions of the exponential function. — Carl (CBM · talk) 01:35, 31 December 2013 (UTC)

Carl undid my edit. Carl apparently does not accept that powers with nonnegative integer exponent can be defined by xⁿ=1 when n=0 and xⁿ=x⋅x^n-1 when n>0. Even hardcore undefiners like Trovatore agree that 0⁰=1 when the exponent is integer. But the undefiners are now a minority in this discussion.

There is no point in undefining (−1)² or 0⁰ even if these expressions are not evaluated by the formula x^y=exp(y⋅log(x)). They are calculated by other means and they are very widely used. But stop teaching innoscent children that (−1)^1/3=−1, because (−1)^1/3=(1+i3^1/2)/2 is more useful and more widely used. In J it reads

  _1^2
1
  0^0
1
  _1^%3
0.5j0.866025

Bo Jacoby (talk) 07:54, 31 December 2013 (UTC).

The Undefiners

@Bo – You should add a third category of opinions to your "definers" and "undefiners": those who consider exponentiation to be several distinct two-argument functions, some of which are defined on (0,0), and some of which are not. Count me in this third group. —Quondum 21:55, 31 December 2013 (UTC)

WP:OR. Write it up and have a secondary source review it and perhaps it will then be put in the article along with what the other sources say. Dmcq (talk) 22:21, 31 December 2013 (UTC)

@Quondum. Exactly how many distinct two-argument functions should in your opinion be considered in our WP-article on exponentiation? Bo Jacoby (talk) 02:40, 1 January 2014 (UTC).

I know that this is OR, but it seems to me that we have two distinct styles of function x^y:

• X × Z → X with X any power-associative magma and in which it makes sense to define x⁰ = 1 for all x ∈ X, including zero, if 1 ∈ X. Negative exponents must be excluded when x has no inverse.
• R₊ × Y → Y with Y any R-algebra (essentially the exp function with a logarithmically scaled x-axis).

From this it should be clear that there is little connection between the algebras of the two inputs. There is the extension to exponentiation of cardinal numbers. Almost all further generalizations (in the sense of expanding the domains) of each style appear to unavoidably get into the multi-value problem and hence Clausen's paradox, and should be presented separately. —Quondum 07:00, 2 January 2014 (UTC)

It seems as you agree with my above suggestion : Talk:Exponentiation#merging_three_definitions. Right? Bo Jacoby (talk) 10:51, 2 January 2014 (UTC).

There's four problems I see there. Firstly I don't know who 'you' is in that. Secondly some people don't consider 0^0 defined even for integers. Thirdly in your merging business you talk of different domains but we know you consider the integers to be reals so you're not really distinguishing in your terms. Fourthly Wikipedia is an encyclopaedia, this talk page is about improving the article, and we should follow what the main sources say, and not just some debatable interpretation of Knuth and the J computer language. Dmcq (talk) 12:56, 2 January 2014 (UTC)

Talking about J, do you support it giving 0 as the value of 0/0 instead of 1 like APL gave? Dmcq (talk) 16:16, 2 January 2014 (UTC)

Answering Dmcq's questions. Firstly, I was refering to Quondum who seemed to agree, but your opinion is welcome too. Secondly, people who don't consider 0⁰ defined are free never to write it, but people who do write xⁿ for x=n=0 refer to the value 1. Thirdly, even if I consider the integers to be reals too, the domains are well defined. Fourthly, a wikipedia article should make sense to the reader, and stuff like undefining 0⁰ doesn't make sense. Exponentiation with nonnegative integer exponent is elementary and must be explained without reference to advanced math. Talking about J, both x=1 and x=0 are solutions to 0x=0, and the discontinuity of x/y in the neighbourhood of x=y=0 cannot be helped by redefining or undefining 0/0, just as the discontinuity of x^y in the neighbourhood of x=y=0 cannot be helped by redefining or undefining 0⁰, (even if The Undefiners think so).

I too prefer integers to be treated as indistinguishable from a subset of the reals, but would like to see the distinct exponentiation functions being defined as distinct functions regardless of overlap in their domains; it is a pity that they use the same notation. This is in essence how it is used although it appears from this thread that few authors tackle this problem (very uncharacteristic for mathematicians). My approach succinctly covers a large range of domains for the single-valued instances, including matrices, Clifford algebras, finite fields, complex numbers and many more. The question is whether this approach is adequately documented (as opposed to merely used implicitly) in the literature, as we would want to be encyclopedic.

Bo's idea of defining a version on real numbers that is defined on the union of the domains has little appeal and does not generalize well, and I doubt whether it'd be notable. —Quondum 16:46, 2 January 2014 (UTC)

I agree fully with that. I have gone on about integers and reals being different, perhaps too much, but really it is just a way of distinguishing different functions and naming them different would be better. And as you say at the end of the day the sources dictate what we can put into the article despite what we might prefer. Dmcq (talk) 17:07, 2 January 2014 (UTC)

Naming the different exponentiation functions differently is OR. It is not what the mathematical community does. The functions are all called x^y. There is consensus in the literature that 2²=4, that 0²=0, that (−2)²=4, that 2⁻¹=1/2, that e^iπ=−1, and that 2^1/2=√2. The function is generalized to still bigger domains without renaming. Nor do we rename the plus-sign when addition is generalized from non-negative integers through integers, rationals, reals, complex numbers, vectors, matrices, et cetera. I wonder if The Undefiners, (Dmcq, Trovatore, Carl, Quondum), still think that 0≠0.0 and that 0≠0+i0 ? Bo Jacoby (talk) 09:09, 3 January 2014 (UTC).

Would you leave off the undefiners silliness please. I do not consider them the same things. I have explained my position quite clearly and agree with Quondum in his assessment. As you say we should follow the sources. That covers my position too. It is you who wants to stick things together that the sources don't. Many sources clearly say 0^0 is an indeterminate form and do not give it a determinate value. I have gone through before why saying indeterminate forms having actual values gives problems in some circumstances.

I'll go through why the thing you're pushing does not work well yet one more time using 0/0 for which you say above "Talking about J, both x=1 and x=0 are solutions to 0x=0, and the discontinuity of x/y in the neighbourhood of x=y=0 cannot be helped by redefining or undefining 0/0, just as the discontinuity of x^y in the neighbourhood of x=y=0 cannot be helped by redefining or undefining 0⁰, (even if The Undefiners think so)." The point is that not giving 0/0 a value does help. For instance consider ${\frac {\sin ^{2}x}{x}}$ Using the convention in J that 0/0 is 0 this has the value of 0 at 0 and has no discontinuity. When we differentiate it though we get ${\frac {2\sin x\cos x}{x}}-{\frac {\sin ^{2}x}{x^{2}}}$. Now using J's convention we do get a discontinuity at 0, the limit there is 1 but J says the value is 0.

Yes it is easy enough to spot the problem here, but you talk about "Fourthly, a wikipedia article should make sense to the reader, and stuff like undefining 0⁰ doesn't make sense. Exponentiation with nonnegative integer exponent is elementary and must be explained without reference to advanced math." The exact same argument should apply in spades to division. And in fact some early Indian mathematicians treated 0/0 as 1. But here we get a straightforward solution to differentiating a simple function with no observable discontinuities and in fact is quite smooth and our solution has a discontinuity with no telltale warning because of assigning a value to 0/0. What is the way out? To have two division operations one which gives a value and the other which doesn't? That's just mad for the case of division. For exponentiation though you want to do the equivalent, have one which might be a limit because a differentiation produced it and another which we say isn't because no differentiation was involved.

People agree that for exponentiation the equivalent of assigning a value to an indeterminate form without bothering to evaluate a limit is a worthwhile thing to do in some circumstances. Your idea of distinguishing limits and values gives the same problem as I show with the divide. We need two different functions if one is to be given a value. We don't have sources going into that and we have to follow the sources. Some sources say it should be 1. Other sources say it is an indeterminate form and give no value. We don't extrapolate and say indeterminate forms have values. That is OR and besides as I have shown it simply isn't a workable way of doing things. The article describes the different things that are in the sources so could you just stop going on about sticking in your interpretation of Knuth and that other mathematicians are wrong? If you want to change things write your own book and have it generally accepted by mathematicians and then it can be used as a source here. Dmcq (talk) 13:53, 3 January 2014 (UTC)

Answering Dmcq

The Undefiners prevent our article from improving. Dmcq claims that 0+i0≠0 which is OR. Trovatore claims that 0.0≠0, which is OR. Quondum claims that the different definitions of exponentiation should lead to different function names, which is OR. Carl reverts that xⁿ (for non-negative integer n) can be defined by x⁰ = 1 and xⁿ⁺¹ = x⋅xⁿ. Dmcq is correct that silliness is a better word for describing the positions of The Undefiners.

Many authors do not define 0⁰. That doesn't undefine 0⁰ once it is defined, and it is quite unimportant, and it is a nuisance.

Dmcq thinks that not giving 0/0 a value does help. It is not advisable to divide by zero because x/y is not continuous around x=y=0, no matter whether 0/0 is defined or not. Dcmq could write more carefully:

f(x)=sin²x / x for x≠0

f(x)=0 for x=0.

and differentiation gives

f'(x)=2 cos x sin x / x − (sin x / x)² for x≠0

f'(x)=1 for x=0.

"no telltale warning because of assigning a value to 0/0". Assigning a value to 0/0 does not remove the warning against dividing by zero. The discontinuity of x/y around x=y=0 is still there. Not giving 0/0 a value does not help.

Dmcq believes that lim_x→0 f(x) must be equal to f(0), such that the only way to avoid a discontinuity is to undefine f(0). This belief is a mistake. Undefining does not necessarity remove discontinuities. x^y is discontinuous around x=y=0, even if you undefine 0⁰.

We need two different functions if one is to be given a value. No, we don't. We just have to accept the discontinuity.

Some sources say it should be 1. Other sources say it is an indeterminate form and give no value. We don't extrapolate and say indeterminate forms have values. If a function f(x) is discontinuous for x=0, then the value f(0) is not defined by continuity as lim_x→0 f(x). Some sources call f(0) an indeterminate form when it is not defined as lim_x→0 f(x) . But f(0) may be defined by other means.

I want our article to be understandable. x⁰ is defined to be equal to 1 in the elementary section. Later it turns out that x^y is discontinuous around x=y=0. This is not helped by undefining 0⁰. Bo Jacoby (talk) 08:17, 4 January 2014 (UTC).

I have asked you to stop saying undefiners and using it like you do. You have not, instead you capitalised it. That is not WP:CIVIL. I will therefore just point you to WP:SYNTH, putting two things together to imply a result that is not supported by the sources. Write a book or paper and have it accepted outside Wikipedia first. Dmcq (talk) 12:39, 4 January 2014 (UTC)

If you are an undefiner no more, then I am not refering to you, and so you have no reason to be offended. If you are still an Undefiner, then take pride in being capitalized. You choose your words and I choose mine. But you are changing the subject. The question is: Do you by now understand that undefining serves no purpose and solves no problem? Bo Jacoby (talk) 17:58, 4 January 2014 (UTC).

I will answer in a week's time so this business of calling names costs you. Don't do it again in that time. Dmcq (talk) 19:58, 4 January 2014 (UTC)

Not I, but the readers of our article, suffer censorship from The Four Undefiners: Dmcq, Trovatore, Quondum, and Carl. This fine suggestion, put forward by Javalenok and supported by Mark van Hoeij, and 128.186.104.253, and myself, could not be implemented. Bo Jacoby (talk) 08:04, 5 January 2014 (UTC).

That some define 0⁰ as 1, some do not, and both "definitions" assist mathematics in different fields, would be fine. Going further than that requires evidence, which has not been forthcoming, even disregarding the personal attacks by the definitionoids. — Arthur Rubin (talk) 17:08, 5 January 2014 (UTC)

It is neither correct nor incorrect that 0 + i0 = 0. What is correct is that there are multiple functions on different domains called "exponential". Some may define 0^0, and others do not. Two of the different functions have different values for (-2)^(1/3). — Arthur Rubin (talk) 17:13, 5 January 2014 (UTC)

Answering Arthur Rubin

Stating that

It is neither correct nor incorrect that 0 + i0 = 0.

is original research, which is prohibited in wikipedia. (ps. It is also doublethink). The lead of our article on complex number says:

a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.

The complex number 0 + i0 has zero imaginary part and so it is a real number. As the real part is also zero makes is clear that 0 + i0 is the real number zero. So it is correct that 0 + i0 = 0. There is no doubt about it. Denying it leads to contradiction. How would you explain it to the uninitiated wikipedia reader? How would you rewrite complex number ?

That

both "definitions" assist mathematics in different fields

is not true. Undefining 0⁰ solves no problem whatsoever.

You write:

Going further than that requires evidence, which has not been forthcoming

From the fact "A", that some authors define 0⁰ = 1, and the fact "B", that other authors do not define 0⁰, The Undefiners conclude "therefore C", that 0⁰ is defined in some contexts and undefined in other contexts. Note the rule in Wikipedia:OR#Synthesis_of_published_material_that_advances_a_position:

Do not combine material from multiple sources to reach or imply a conclusion not explicitly stated by any of the sources. If one reliable source says A, and another reliable source says B, do not join A and B together to imply a conclusion C that is not mentioned by either of the sources. This would be a synthesis of published material to advance a new position, which is original research.[8] "A and B, therefore C" is acceptable only if a reliable source has published the same argument in relation to the topic of the article. If a single source says "A" in one context, and "B" in another, without connecting them, and does not provide an argument of "therefore C", then "therefore C" cannot be used in any article.

So The Undefiners' position is original research.

You write:

Two of the different functions have different values for (-2)^(1/3).

That is true and it is a nuisance, but it is a different problem from that of defining 0⁰. Right now I prefer to concentrate on: exponentiation with arbitrary base and nonnegative integer exponent, exponentiation with nonzero base and negative integer exponent, and exponentiation with positive base and arbitrary exponent. Bo Jacoby (talk) 19:18, 5 January 2014 (UTC).

Actually, you're the one engaging in original synthesis, by combining definitions from different places (say, the iterative definition of xⁿ, and the log-based definition of x^y, when the sources don't say that they're talking about the same function).

But Bo, all legalisms aside, this is the bottom line. Traditionally, through the 20th century, 0⁰ was generally considered undefined. Late in the 20th century, some workers, especially Knuth, made a counter-argument, one you happen to agree with. But you seem to want to present that argument as having closed the issue, as though it had achieved general acceptance among mathematicians, and in fact it has not. --Trovatore (talk) 22:03, 5 January 2014 (UTC)

Answering Trovatore

The notation $f(x)=\sum _{k=0}^{n}a_{k}x^{k}$ was used way before the 20^th century, and nobody doubts that f(0)=a₀, so it goes without saying that 0^k=0 for k>0 and 0^k=1 for k=0. Donald Knuth didn't invent 0⁰=1. In the 18^th century Leonhard Euler wrote e^x with the same exponential notation as x², so it is not original synthesis on my part. Bo Jacoby (talk) 23:03, 5 January 2014 (UTC).

No, in fact, it does not go without saying. It is quite common in mathematics to use the same notation for different things. The burden of proof is on you to show that the same exponential function was intended in both cases. --Trovatore (talk) 00:52, 6 January 2014 (UTC)

Do you really doubt that f(0) is ment to be equal to a₀ when f(x) is defined by ∑_k a_k x^k ?

It doesn't matter which exponential function was intended as long as the competing definitions produce the same result. 3² = 1⋅3⋅3 = 9 and 3² = exp(2⋅log(3)) = 9.

An analogous problem is well handled in Addition#Addition of natural and real numbers. Bo Jacoby (talk) 07:07, 6 January 2014 (UTC).

You're arguing the merits of how math should be. That's out of place. There is in fact no consensus among mathematicians that 0⁰=1, no matter how much you might want it to be. I've tried to explain why this is not as unreasonable as you think it is, which is an interesting discussion, but beside the point. --Trovatore (talk) 07:40, 6 January 2014 (UTC)

Thinking that 0⁰ is sometimes defined and sometimes undefined is not mathematics, it is doublethink. Bo Jacoby (talk) 08:33, 6 January 2014 (UTC).

Bo, that's complete nonsense, and we've tried to explain it to you but you won't listen. This isn't the right forum to discuss it anyway. --Trovatore (talk) 08:37, 6 January 2014 (UTC)

Show your good will by answering my question: Do you doubt that f(0) is ment to be equal to a₀ when f(x) is defined by ∑_k a_k x^k ? Bo Jacoby (talk) 17:55, 6 January 2014 (UTC).

On this page? No, I will not answer that question on this page, because no possible answer to it has any relevance to improving the article. You're arguing the merits. The merits are irrelevant. --Trovatore (talk) 08:03, 7 January 2014 (UTC)

Do The Undefiners doubt that e⁰ = 1 where e^x is defined by the power series $e^{x}=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}$ ? Bo Jacoby (talk) 08:02, 7 January 2014 (UTC).

Show your good will Undefiners? Get a grip on yourself. Dmcq (talk) 09:15, 7 January 2014 (UTC)

As The Undefiners are not willing or not able to explain their own position, I will give it a try. Everybody agree that $e^{0}=1$ and that $e^{0}=\sum _{k=0}^{\infty }{\frac {0^{k}}{k!}}={\frac {0^{0}}{0!}}$. So ${\frac {0^{0}}{0!}}=1$. Both the numerator and the denominator are empty products, and so 0⁰=0!=1. Recursive definitions are: 0!=1 and n!=n⋅(n−1)! for n=1,2,3,... , and x⁰=1 and xⁿ=xⁿ⁻¹⋅x for n=1,2,3,... . Carl is sensitive against 0⁰, but Trovatore and Dcqm agrees that 0⁰=1 for integer exponent 0. But, because some authors do not define 0⁰, it is important to The Undefiners to claim that 0⁰ is sometimes not defined. I don't follow this argument. Of course some authors do not define 0⁰. Most authors write about something else. Dmcq thinks that the discontinuity of x^y around x=y=0 is more spectacular when 0⁰ is undefined, than when 0⁰ is defined to be 1. Trovatore promotes the idea that 0⁰ is undefined when the exponent 0 is real, but defined when it is integer. This idea is given the formulation: 0.0≠0 , real zero is not the same thing as integer zero. I don't follow this argument. Trovatore and Carl thinks that the set of integers, ℤ, is disjoint to the set of reals, ℝ, while I consider ℤ to be a subset of ℝ, such that operations on ℝ generalize operations on ℤ. Trovatore has problems with 1+0.5=1.5, because 1 is an integer and 0.5 is a (non-integer) real. I don't have the same problem because integer 1 and real 1 to me is the same number: 1=1.0. I find support for this interpretation here and elsewhere. The Undefiners polluted our article only by undefining 0⁰ , but they should also undefine (−1)² which also cannot be defined by x^y = exp(y⋅log(x)). There is no real logarithm to −1. The fact that some authors do not define 0⁰ should not force our article to become unreadable. Bo Jacoby (talk) 22:36, 7 January 2014 (UTC).

And polluters too? I'll reset to delay a week from now before making any response. Dmcq (talk) 23:25, 7 January 2014 (UTC)

What would Dcmq want The Undefiners to be called? The Doublethinkers? Bo Jacoby (talk) 08:15, 8 January 2014 (UTC).

Bo, whether you follow the arguments or not, quite frankly, doesn't matter. In fact the merits of the argument don't matter either, not here. You are trying to push a view that, though it is certainly held by a number of workers and is correctly represented, is not by any means universal consensus and therefore cannot be represented as such. That's what it all comes down to. Therefore everything in your above long paragraph is completely beside the point. --Trovatore (talk) 07:04, 8 January 2014 (UTC)

Is it also besides the point that the article is incomprehensible? Exponentiation#Arbitrary_integer_exponents says: "The case of 0⁰ is controversial". In fact it is not at all controversial for integer exponents. Also the reader gets the warning: "Any nonzero number raised by the exponent 0 is 1". In fact x⁰ is always interpreted as 1. Nobody writes 0⁰ if it is supposed to be undefined. Why not omit this nonsense? Bo Jacoby (talk) 08:15, 8 January 2014 (UTC).

Again, you speak nonsense. There are multiple functions called "exponentiation". The ones from ℝ × ℤ → ℝ^* (the real projective line) or ℂ × ℤ → ℂ^* (complex projective plane) define $0^{0}=1$. The one from ℝ^≥0 × ℝ → ℝ^* usually does not define $0^{0}$. The one from ℂ × ℝ → ℂ^* almost always does not, and the one from ℂ\{0} × ℂ → ℂ always leaves all $0^{z}$ undefined. — Arthur Rubin (talk) 15:51, 8 January 2014 (UTC)

Also, a book that comes out and says directly how to the seeming issue with 0⁰ in power series: Complex Analysis for Mathematics and Engineering, John Mathews, Russell Howell, Jones & Bartlett, 2010, p. 151.

"Technically, this series [a general power series] is undefined if z=α and n=0 because 0⁰ is undefined. We get around this difficulty by stipulating that the series $\sum _{n=0}^{\infty }c_{n}(z-\alpha )^{n}$ is really compact notation for $c_{0}+\sum _{n=1}^{\infty }c_{n}(z-\alpha )^{n}$."

This could be useful for the article at some point. — Carl (CBM · talk) 20:13, 8 January 2014 (UTC)

Stipulating that $\sum _{n=0}^{\infty }c_{n}z^{n}$ = $c_{0}+\sum _{n=1}^{\infty }c_{n}z^{n}$ is actually defining that z⁰ =1. Bo Jacoby (talk) 07:00, 9 January 2014 (UTC).

Evidently the authors do not agree, given that they explicitly assert that 0⁰ is undefined. --Trovatore (talk) 07:22, 9 January 2014 (UTC)

These authors wrote that $\sum _{n=0}^{\infty }c_{n}(z-\alpha )^{n}$ is really compact notation for $c_{0}+\sum _{n=1}^{\infty }c_{n}(z-\alpha )^{n}$. Consider the case z=α, c₀=1, c_n=0 for n>0. What do you get? That 0⁰ is really compact notation for 1. Asserting that it is undefined is doublethink. They just defined it themselves. Bo Jacoby (talk) 07:55, 9 January 2014 (UTC).

Bo, you're welcome to hold that view. However, before it can go in the article as the consensus view of mathematicians, it needs to actually be the consensus view of mathematicians. All your arguments as to why it should be are irrelevant. --Trovatore (talk) 08:06, 9 January 2014 (UTC)

Oh, thank you for allowing me to hold my view. But how can you or any other mathematician think that the above identity does not imply that 0⁰ =1 ? Must wikipedia convey the impression that mathematicians are insane? Bo Jacoby (talk) 10:44, 9 January 2014 (UTC).

The first question is irrelevant and off-topic, and I take the second to be merely rhetorical and not in need of an answer. --Trovato re (talk) 18:46, 9 January 2014 (UTC)

I am not the only one to see that the stipulated identity defines 0⁰. The position of The Undefiners (including Mathews and Howell) is manifestly insane. So the obvious answer to my rhetorical question is that wikipedia should stop conveying the impression that mathematicians are insane. The Undefiners must leave in shame. Sensible suggestions on this talk page were turned down by The Undefiners, and my very innocent edit was undone by Carl. This must stop. Bo Jacoby (talk) 05:36, 10 January 2014 (UTC).

Answering Arthur Rubin

It is inconsistent that the article claims that 0⁰ is more controversial than, say, (−3)². Neither are defined by: x^y=exp(log(x)⋅y), and both are defined by the recursive definitions: x⁰ = 1, and xⁿ⁺¹ = xⁿ⋅x, and xⁿ⁻¹ = xⁿ/x for nonzero x. If only one definition apply, then that is the one to be used, such as 3^1/2=exp(log(3)/2). If both definitions apply, then the result is the same: 3² = 9 by both definitions. So yes, there are multiple functions called exponentiation, but that is not a problem, and that is not the point. Bo Jacoby (talk) 19:05, 8 January 2014 (UTC).

For (-1)^1/3, at least three definitions apply: one makes the value -1, one makes it a non-real complex number, and one makes it undefined. But I do not see any actual problem with the article as it is written; it keeps all the 0^0 stuff together in one place rather than spreading it out. The article seems to be quite direct about the actual status of 0^0, and not at all unreadable. I don't see an issue with it, apart from your (original research) idea that there is one "true" exponentiation function rather than a collection of several different ones. In general, unless some genuinely new idea comes up, I would recommend to everyone to stop responding to this discussion thread. — Carl (CBM · talk) 19:57, 8 January 2014 (UTC)

All the definitions define xⁿ (for n an integer) the same way except for x=0 and n≤0. If n<0, it's either ∞ or undefined. If n=0, it's either 1 or undefined, under the definitions I'm familiar with. Sometimes "undefined" is more appropriate. — Arthur Rubin (talk) 21:10, 8 January 2014 (UTC)

The two definitions for xⁿ are the analytic one: xⁿ = exp(log(x)⋅n), and the recursive one: xⁿ = 1 for n=0 and xⁿ = xⁿ⁻¹⋅x for n>0. These definitions produce identical results whereever they both apply. The analytic definition applies to positive x and general n. The recursive definition applies to general x and nonnegative integer n, and to nonzero x and negative integer n. "Undefined" is not appropriate when something is defined. Bo Jacoby (talk) 06:46, 9 January 2014 (UTC).

Answering Carl

Making something undefined is not a definition. I want to postpone the inconsistent definitions of (-1)^1/3 as the discussion is sufficiently complicated by now. The 0⁰ stuff is almost kept together in one place, except the warnings against 0⁰ while treating integer exponents. I did not say that there is one true exponential function. The exponential notation may refer to different functions, as explained above. You have not told why you disagree that "xⁿ can be defined by x⁰ = 1 and xⁿ = xⁿ⁻¹⋅x for positive integer n". You undid my contribution. Bo Jacoby (talk) 20:30, 8 January 2014 (UTC).

"Making something undefined is not a definition." is not always correct. — Arthur Rubin (talk) 21:13, 8 January 2014 (UTC)

Answering Arthur Rubin

You mean that it is sometimes correct and sometimes not correct? This is doublethink. Bo Jacoby (talk) 21:52, 8 January 2014 (UTC).

I should have said that making something undefined can be a definition. But, would you stop writing "Answering (editor)" for section headings. That makes it impossible to follow threads. — Arthur Rubin (talk) 01:54, 9 January 2014 (UTC)

I need managable sections. You are welcome to rename the headings for your convenience. But how can making something undefined be a definition? What do you mean? Bo Jacoby (talk) 06:20, 9 January 2014 (UTC).

Part of a definition of a function is the domain at values outside the domain, the function is undefined. Hence, part of the definition of the function includes reporting values at which the function is not defined. (Further down in the "domain" article, there is a note that domain of a partial function has two distinct meanings; I'm using the mathematical definition, rather than the category theoretical definition.) — Arthur Rubin (talk) 17:45, 9 January 2014 (UTC)

We must distinguish between function f and function value f(x). Specifying the domain is part of the definition of a function. Removing x from the domain of f makes f(x) undefined. This redefinition of f is not a definition of f(x). Making f(x) undefined can not be a definition of f(x). Bo Jacoby (talk) 05:07, 10 January 2014 (UTC).

Dmcq's edit

Dmcq changed "Non-negative" to "Positive" and "$b^{0}=1$" to "$b^{1}=b$". Any reader need to know why. Is it incorrect to say that: "Formally, powers with non-negative integer exponents may be defined by the initial condition $b^{0}=1$ and the recurrence relation $b^{n+1}=b^{n}\cdot b$"? Does the uninitiated reader benefit from making $b^{0}$ an exception from the definition? Bo Jacoby (talk) 18:25, 10 January 2014 (UTC).

Citation added for what's there. Dmcq (talk) 19:28, 10 January 2014 (UTC)

So you found a book that left 0⁰ undefined. The question still remains: Does the uninitiated reader benefit from making 0⁰ an exception from the definition? Bo Jacoby (talk) 21:46, 10 January 2014 (UTC).

See WP:NOTOPINION Dmcq (talk) 22:22, 10 January 2014 (UTC)

I know what wikipedia is. But now I also know why your source does not define b⁰. Your source is about general algebraic rings, not only rings with an identity. So the ring of even numbers, 2ℤ, is an example. In this ring b⁰ is not defined, because 1 is not an even number, 1∉2ℤ. You could and should have discovered this yourself. Our article is about the rings with identity, ℤ or ℝ or ℂ. One is a number. Now that this is clear, would you please either restore the definition b⁰=1, or carefully explain why not. Bo Jacoby (talk) 23:16, 10 January 2014 (UTC).

See the bottom of the page 'for each nonzero x..." Dmcq (talk)

if you want something simpler try [1] bottom of page 101. Perhaps I should add that aswell. Dmcq (talk) 23:50, 10 January 2014 (UTC)

Does your source tell why it leaves 0⁰ undefined? Bo Jacoby (talk) 06:40, 11 January 2014 (UTC).

Shouldn't the links to sources leaving 0⁰ undefined be collected here after the sentence "not all sources define 0⁰" ? Bo Jacoby (talk) 09:10, 11 January 2014 (UTC).

The first was to support the recursive definition of exponentiation starting from $b^{1}=b$. The second was to support that for all nonzero b that $b^{0}=1$. They do also mention about 0⁰ but that is mainly in a section you didn't edit. Dmcq (talk) 12:14, 11 January 2014 (UTC)

The section above #Sticking 0^0=1 into the article is the appropriate place to discuss the 0^0=1 or undefined business. Dmcq (talk) 15:55, 11 January 2014 (UTC)

Why restrict b⁰=1 to nonzero b at this stage? An empty product doesn't depend on the value of a factor that isn't there.

Shouldn't "0⁰=1 or undefined" be treated in the article rather than on the talk page? Bo Jacoby (talk) 16:45, 11 January 2014 (UTC).

It is treated in the article at Exponentiation#Zero_to_the_power_of_zero. You removed the reference to that from the section being discussed at #Sticking 0^0=1 into the article. Please go there to discuss what should be there if you disagree. Dmcq (talk) 17:03, 11 January 2014 (UTC)

OK, I went there. Why not wait until Exponentiation#Zero to the power of_zero with mentioning "nonzero b" ? Even Trovatore agrees that b⁰=1 is not restricted to nonzero b when the exponent is integer (which it is). Bo Jacoby (talk) 04:07, 12 January 2014 (UTC).

It would be better to put that at that other discussion. Dmcq (talk) 13:45, 12 January 2014 (UTC)

I am discussing Dmcq's edit right here. Please answer my questions. Bo Jacoby (talk) 14:16, 12 January 2014 (UTC).

Combinatorial interpretation

I moved that section "combinatorial interpretation" back to its original location. The article should certainly include that topic, but since this article is on an elementary subject, and that interpretation is slightly more advanced, it should not be placed so highly in the article. Compare WP:MTAA section 4.1. — Carl (CBM · talk) 13:11, 12 January 2014 (UTC)

I disagree. The combinatorial interpretation is counting. Counting is more elementary than computing. You must count before you can add or multiply. Bo Jacoby (talk) 14:14, 12 January 2014 (UTC).

It is counting, but it is counting arbitrary functions from one set to another. The idea of an arbitrary function comes much later in the curriculum than multiplication. Indeed most students learn the basic rules of exponentiation much earlier than they learn the idea of "an arbitrary function from a set of size a to a set of size b." — Carl (CBM · talk) 16:24, 12 January 2014 (UTC)

The 3-letter words from a 3-letter alphabet are: aaa aab aac aba abb abc aca acb acc baa bab bac bba bbb bbc bca bcb bcc caa cab cac cba cbb cbc cca ccb ccc. Count to 27. So 3³=27. This is elementary. There are no arbitrary function involved. That 3³=3⋅3⋅3 is a theorem rather than a definition. Bo Jacoby (talk) 17:00, 12 January 2014 (UTC).

Just produce a curriculum that talks about set theory and combinatorics before powers and you'll have a reasonable case. The arguments about why it is better earlier aren't really relevant. Dmcq (talk) 17:51, 12 January 2014 (UTC)

Cardinals of finite set is the elementary school explanation of natural numbers. I miss your explanation regarding arbitrary functions. Bo Jacoby (talk) 19:17, 12 January 2014 (UTC).

Sticking 0^0=1 into the article

We probably should be revising the integer section and saying a bit more about 0^0=1 there, but just sticking it at the start when that is not generally agreed is simply WP:OR. How about just trying to fix the article up rather than trying to push a point of view? Dmcq (talk) 10:58, 10 January 2014 (UTC)

My opinion is that it's better to keep all the 0^0 stuff together in one place. That makes it easier to present a clear and concise description of the situation, and it makes referencing easier. It makes it easier for a reader to compare the different points of view in the literature if those points of view are near each other. If the material is spread out, then it tends to end up being duplicated - new editors will only see part of the article, and duplicate the other part not realizing it is already present somewhere else. — Carl (CBM · talk) 12:33, 10 January 2014 (UTC)

By the way, the other reason I was concerned about the power series claim is that only some authors resolve the issue by defining 0^0=1. Others, like the ones I quoted above, resolve it by viewing the entire power series notation as an abbreviation which when unabbreviated does not contain 0^0 at all. — Carl (CBM · talk) 12:36, 10 January 2014 (UTC)

True, but the section about 0^0 doesn't mention that possibility. I think it really comes under them assuming x^0 means 1 in the context of a power series rather than actually defining 0^0. This is basically using the exponentiation notation where the start of the series is 1 and the next in series is got by multiplying by the last by x. Dmcq (talk) 13:33, 10 January 2014 (UTC)

Agreed. — Carl (CBM · talk) 13:36, 10 January 2014 (UTC)

I agree with Carl that it's better to keep all the 0^0 stuff together in one place. Assuming that x^0 means 1 in the context of a power series is actually the same thing as defining 0^0=1. (See #Answering_Trovatore). Bo Jacoby (talk) 03:59, 12 January 2014 (UTC).

Well it is the same as defining 0^0=1 in the context of a power series. Just because the one wall in a house is white doesn't mean all the walls in the house are white. Dmcq (talk) 13:48, 12 January 2014 (UTC)

How can the reader decide whether a problem is "in the context of a power series" or not? When and why and how does 0⁰ fade away from being 1 to being undefined? Bo Jacoby (talk) 14:24, 12 January 2014 (UTC).

Because an author says that's what it is supposed to mean is probably the easiest way. Dmcq (talk) 17:54, 12 January 2014 (UTC)

I don't understand what you mean. Bo Jacoby (talk) 21:18, 12 January 2014 (UTC).

I have a bit of a problem with the assumption that an author's intent is assumed without them being explicit. If an author says or implies that in the context, instances of 0⁰ are to be regarded as 1, this does not constitute defining the value of 0⁰ or making a caveat about the notation (e.g. special treatment in a power series); it merely clarifies that whatever interpretation is applied, the consequence in the instance is as implied. My own feeling is that in these contexts, the simpler hypothesis is that a version of the exponentiation function is used in which it 0⁰ = 1 is defined, as opposed to "notational shorthand". This does not imply that the same version of the exponentiation function is to be assumed uniformly elsewhere, even in the same work. All this mess and debate seems far too much cost compared to the alternative: acknowledging that in any (inherently single-valued) instance, we are dealing with one of two distinct functions and simply need to choose which, usually from the context. My gut feeling is that many authors are implicitly doing so, or at least thinking in ways consistent with this. I see very little thought being given in this debate to this interpretation, even though it seems to have been received with a degree of acceptance. This approach also becomes unavoidable and rather obvious when one considers the exponentiation function in more general rings, which is not uncommon, for example with matrix algebras and Clifford algebras. —Quondum 03:17, 13 January 2014 (UTC)

Stop The Undefiners of 0^0

Dmcq undid my edit. Carl's source actually did define 0^0 . See above. Bo Jacoby (talk) 11:02, 10 January 2014 (UTC).

You're going to get banned from this article the way you're going on. A much more nuanced way is needed than what you put in. Dmcq (talk) 11:20, 10 January 2014 (UTC)

I is a diplomatic challenge to negotiate with people who seriously deny that 0+i0=0. Step back and relaxe. You have still unanswered questions. Improvements are not made by your automatic undoing my contributions. I welcome your stated willingness to discuss. Bo Jacoby (talk) 11:36, 10 January 2014 (UTC).

It is not diplomatic to ignore a straightforward section just above and set up a title like this. You are being uncivil again. If you have something constructive to say then say it but this is not a forum for your histrionics. Dmcq (talk) 11:46, 10 January 2014 (UTC)

My edit was finished at 1047 o'clock and your straightforward section just above was made at 1058, so I didn't ignore it. I asked you what you guys want to be called instead of Undefiners, and you did not answer. I look forward to see your arguments on the subject matter, but not your comments on my histrionics. Your reversions are so quick that you haven't taken time to study my contribution in calm and detail. Bo Jacoby (talk) 12:09, 10 January 2014 (UTC).

I read your edit. I've read it again. I stand by the revert. You don't need to call people anything. You shouldn't be putting in section headers saying you're replying to particular editors. Address the issues not the editors. Dmcq (talk) 12:18, 10 January 2014 (UTC)

Well CBM chopped out the bit I stuck in the zero exponent section in aid of keeping all the zero stuff together. I could live with a little bit more there. Anyway if you'd like to propose a new contents for that section I'm sure people would be happy to review it. Dmcq (talk) 12:28, 10 January 2014 (UTC)

I found your section above on the talk page and responded there already. — Carl (CBM · talk) 12:37, 10 January 2014 (UTC)

Although there is not much progress in this discussion, I do see some light. Trovatore seems to have stopped denying that 0.0=0, and Dmcq seems to have stopped denying that 0+i0=0, and Carl seems to have stopped denying that ℕ⊂ℤ⊂ℝ⊂ℂ, and to realize that the equation $\sum _{n=0}^{\infty }c_{n}(z-\alpha )^{n}=c_{0}+\sum _{n=1}^{\infty }c_{n}(z-\alpha )^{n}$ is no different from defining z⁰=1. So some progress has been made, and The Undefiners no longer constitute quite the same mathematical madhouse as they used to. Dcmq does not like being called an undefiner, and I will stop calling Dcmq an undefiner as soon as Dmcq stops being an undefiner. But because some sources do not define 0⁰ it is important to Dcmq to claim that 0⁰ is not always defined. Dmcq did find a source which does not even define b⁰ for nonzero b. but for unknown reasons that did not urge Dmcq to claim that b⁰ is sometimes not defined for nonzero b. Trovatore knows that the definition b^y = exp(log(b)⋅y) does not cover the case b=0, but neither does it cover the case b=−1, which is no problem to Trovatore. The Undefiners' attitude of infallibility made collaboration difficult and unpleasant. I did my duty. I regret to the wikipedia readers that I was not successful in removing the nonsense from our article. The simplicity and clarity of Donald Knuth's approach should be used in the article, and the obscurity of undefining should be concentrated in the appropriate subsection. Bo Jacoby (talk) 15:17, 14 January 2014 (UTC).

Isheden's suggestion

I think the problem here is that, as noted above, exponentiation refers to several different functions depending on the permissible values of the base and exponent, respectively. Defining 0⁰ may or may not make sense, depending on which function is referred to:

$b^{n}=\underbrace {b\times \cdots \times b} _{n}$ (b positive, positive integer exponent n)

$b^{0}=1$ (exponent zero)

$b^{-n}={1}/{b^{n}}$ (negative integer exponent)

$b^{\frac {m}{n}}=\left(b^{m}\right)^{\frac {1}{n}}={\sqrt[{n}]{b^{m}}}$ (rational exponent m/n)

$b^{x}=(e^{\ln b})^{x}=e^{x\cdot \ln b}$ (real exponent x)

$b^{z}=e^{z\cdot \ln b}=e^{(x+iy)\cdot \ln b}$ (complex exponent z=x+iy)

$z^{n}=\underbrace {z\times \cdots \times z} _{n}$ (complex base z, positive integer exponent n)

$w^{z}=e^{z\log w}$ (complex base and exponent, ambiguous in the same sense that log w is)

$w^{z}=e^{z\log w}=e^{z(\log r+i\theta )}$ (with $w=re^{i\theta }$ in polar form)

In my view, the article should be structured so that it is always clear where a certain generalization is needed and what issues are related to using a certain generalization (e.g. that (−2)^1/3 may produce different results or that a certain limit may have different values depending on how you approach it). Isheden (talk) 10:39, 15 January 2014 (UTC)

Wow, how did we pass the seventh anniversary of this thread without celebrating? Congratulations to everyone (especially the inimitable Mr. Jacoby) for their stamina. — Steven G. Johnson (talk) 21:06, 15 January 2014 (UTC)

Thanks to Mr. Johnson for the congratulation! Isheden's suggestion needs proofreading:

$b^{n}=\underbrace {b\times \cdots \times b} _{n}$ (b is not necessarily positive, positive integer exponent n)

Defining 0⁰ = 1 makes sense. Undefining 0⁰ makes no sense.

Shorthand for this is formally

bⁿ = 1 for n = 0

bⁿ = bⁿ⁻¹⋅b for n > 0

and informally:

b⁰ = 1

b¹ = b

b² = b ⋅ b

b³ = b ⋅ b ⋅ b

etc

Be bold and make your edit. Undoing a contribution is not progress, and undefining a definition is not progress either. Bo Jacoby (talk) 22:01, 15 January 2014 (UTC).

Proofreading done. There is a general expression for a complex base z and positive integer exponent n and a simple expression with positive base b and positive integer exponent n that can be readily generalized to arbitrary integer, rational and real exponents, respectively. Allowing for other bases than positive reals in the first line more naturally leads to the other generalization with complex base and positive integer exponent. Isheden (talk) 22:18, 15 January 2014 (UTC)

To my understanding, 0⁰ may be regarded as

• $\lim _{b\rightarrow 0^{+}}b^{0}=1$, starting from $b^{0}=1$ for positive b.
• $\lim _{b\rightarrow 0^{+},x\rightarrow 0}e^{x\cdot \ln b}$, starting from $b^{x}=e^{x\cdot \ln b}$ and which doesn't exist. Isheden (talk) 22:42, 15 January 2014 (UTC)

The definition 0⁰ = 1 is always assumed in the usual expression for a polynomial or a series, where $f(0)=a_{0}$ when $f(x)=\sum _{k=0}^{\infty }a_{k}x^{k}$. The definition b⁰ = 1 need not be restricted to nonzero b (or to positive b). But do not assume that $\lim _{(b,x)\rightarrow (0^{+},0)}b^{x}=0^{0}$. The function b^x = e^log(b)⋅x is defined for b>0, but it is not continuous around b=x=0, so don't even try to define 0⁰ by continuity. Bo Jacoby (talk) 23:37, 15 January 2014 (UTC).

Arbitrary integer exponents

Hi, following part didn't seem right to me, but I hesitated to edit.

The following identity holds for arbitrary integers m and n, provided that m and n are both positive when b is zero:

• $b^{m+n}=b^{m}b^{n}.$

First, why write b like a variable if b is always zero. Why say arbitrary integers then say positive integers? What good is 0^(m+n), why not just say 0^m is zero when m>0? Why is there a dot after the equation? — Preceding unsigned comment added by 85.108.132.131 (talk) 06:16, 10 January 2014 (UTC)

It should have said nonzero b! I also simplified the wording as it is true for all exponents. The punctuation "," and "." is commonly used between expressions in maths but doesn't seem to be used elsewhere in the that section so I've left it out. Perhaps someone else will fix up the punctuation if it is standard in the the MOS (manual of style). Dmcq (talk) 07:26, 10 January 2014 (UTC)

The formula is true both for zero and nonzero b. This is just one more case of the superfluous restriction creating confusion. Bo Jacoby (talk) 08:22, 17 January 2014 (UTC).

The whole sentence was messed up talking about arbitrary integers and then about them being positive. It doesn't necessarily hold for b equal to zero for instance in $b^{2-1}$ as the right hand side would then have a $0^{-1}$. Dmcq (talk) 11:30, 17 January 2014 (UTC)

The identity also holds for b = 0 with positive exponents throughout. When manipulating expressions, it makes more sense to exclude specifically the combination b = 0 with negative (and potentially zero) integer exponents, but to retain the case of all positive exponents, since treating this as a special case is simply unnecessary. So replacing "provided that the base is non-zero" with "provided that the base is non-zero whenever the exponent is not positive" might be a useful improvement. Or a statement could be added, e.g. "When the base is zero, the identity still holds provided that all three exponents are positive". —Quondum 16:26, 17 January 2014 (UTC)

a formula that holds in one context doesn't have to hold in another

Bo, I have stopped talking about whether the 0 of the reals is identical to the 0 of the naturals, because it's off-topic here. At one time it was marginally on-topic, at least in a generous interpretation of "on-topic" in a relaxed environment, as a way to explain to you why a formula that holds in one context doesn't have to hold in another. It has become evident that using a relaxed interpretation of "on-topic" is not a productive course of action when you're around, so I had to drop that. --Trovatore (talk) 19:08, 14 January 2014 (UTC)

I'm amazed by your idea that a formula that holds in one context doesn't have to hold in another. 2+2=4 no matter whether the context is apples or oranges. That is the basis of mathematics. Bo Jacoby (talk) 04:32, 17 January 2014 (UTC).

You can be amazed, but it's off-topic on this page. --Trovatore (talk) 04:34, 17 January 2014 (UTC)

As Trovatore's idea leads to support to the aimlessly undefining editwarriers it is on-topic to ask if it is Trovatore's own brilliant original research, or if there is any support in the mathematical community that a formula that holds in one context doesn't have to hold in another. Bo Jacoby (talk) 15:04, 18 January 2014 (UTC).

You are misinterpreting Travatore. Presumably not deliberately, which suggests that you are implicitly assuming that a "formula" necessarily has a meaning that is independent of the context. I doubt that any experienced mathematician would deny that the choice of definitions in use depends on the context. —Quondum 17:53, 18 January 2014 (UTC)

No, Bo, that's off-topic too. That's part of the argument about whether 0⁰ should be defined, and that whole line of discussion is off-topic. --Trovatore (talk) 21:06, 18 January 2014 (UTC)

Trovatore, there is no question that 0⁰ is defined, the question is whether it should be undefined afterwards. Some have claimed that undefining has advantages, but without explaining what these advantages are. The discontinuity is not removed by undefining. Quondum, do you have sources or other examples that a formula that holds in one context doesn't have to hold in another?. Bo Jacoby (talk) 05:47, 19 January 2014 (UTC).

Bo, that's still arguing the merits, which is off-topic. The merits don't matter. What matters is, the mathematical community has not accepted your position. Some workers have, but not to the point that we can present it as consensus. Go convince the mathematical community, bring back proof, and then we have something to talk about. Until then, you don't have a leg to stand on. --Trovatore (talk) 07:28, 19 January 2014 (UTC)

From the fact that some authors do and others do not define 0⁰ you cannot conclude that 0⁰ is defined or not depending on context. There is not consensus that a formula that holds in one context doesn't have to hold in another. Go convince the mathematical community, bring back proof, and then we have something to talk about. Bo Jacoby (talk) 10:37, 19 January 2014 (UTC).

The first of the Peano axioms for arithmetic is that there is no natural number that has a successor equal to zero, i.e. $\forall x.\lnot (Sx=0)$. That does not hold for the integers. Agree with Trovatore all this is off topic though. Wikipedia needs evidence for inclusion from reliable sources, not lack of evidence so OR can be included. Dmcq (talk) 11:42, 19 January 2014 (UTC)

I'm pleased that Dmcq agrees that Peano's axioms are off topic. What is on topic is what is written in the article: "Any nonzero number raised by the exponent 0 is 1". The word "nonzero" is a warning. The Undefiners' concern is that $\lim _{(b,x)\rightarrow (0^{+},0)}b^{x}$ might mistakenly be identified with 0⁰. Well, they are not necessarily equal because the function b^x is not continuous around b=x=0. So fear not! There is no danger in the definition: "Any number raised by the exponent 0 is 1". There is no point in warning and confusing the readers. Bo Jacoby (talk) 17:34, 21 January 2014 (UTC).

FAQ entry on 0^0?

In many other Wikipedia articles Template:Round in circles and Template:FAQ are used to respond to long-running and often-repeated arguments that go nowhere.

The arguments over 0⁰ seem to fall into this category. Arguments have been going on for over seven years now, with no really new arguments being raised nor any substantially different sources being cited as far as I can tell. Wouldn't it be more productive to create a FAQ entry and simply refer to that when similar arguments are raised in the future?

— Steven G. Johnson (talk) 21:47, 15 January 2014 (UTC)

Go ahead and do it! Bo Jacoby (talk) 22:06, 15 January 2014 (UTC).

I think that it is possible to: (a) agree on what the page should say, while at the same time (b) disagreeing on which arguments carry the most weight. MvH Feb 7, 2014. —Preceding undated comment added 15:45, 7 February 2014 (UTC)

Powers via logarithms

What is the point of this section if not to justify the continuity of the exponential function through the continuity of the logarithm through its own definition? The fact is one textbook prefers to define the logarithm as a continuous function first, and later introduce the exponential function and the justification for the existence of real exponents on said definition:

$\ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,x>0;\ e^{x}=\ln ^{-1}x.$

Toolnut (talk) 07:25, 7 February 2014 (UTC)

I'm notr quite certain what you're getting at. The article is about exponentiation, not about the exponential function or the logarithm both of which have their own articles. The log is still the inverse of the exponential function however they are defined. The point of the section is to say powers can be defined using the exponential and log functions. The exponential function article has a section on different ways of defining he exponential function which points to a separate article on the subject Characterizations of the exponential function, perhaps that's what you were thing about? Dmcq (talk) 09:33, 7 February 2014 (UTC)

But the section containing this subsection is titled "Real exponents" and talks at length about the justification for the extension to real exponents from rational exponents. Of note, "Since any irrational number can be approximated by a rational number ...": if you use the logarithm definition, that kind of justification becomes unnecessary; therefore, it is a different approach well worth noting. So you appear to be saying the reader should look for the logarithm definition elsewhere and not become aware of this finer point here, where it has been brought up. The separate article on logarithms actually begins by defining a logarithm as the inverse of the exponential function, not independently of it; all the more reason to make this fine point here.Toolnut (talk) 10:32, 7 February 2014 (UTC)

There's two main ways exponentiation of a real is defined and this section is about the alternative to the definition in the previous section. As it says 'This can be used as an alternative definition of the real number power $b^{x}$ and agrees with the definition given above using rational exponents and continuity.' So it is explicitly noting it. If there is a problem with the article about logs then the article about the log is where it should be fixed. As far as I can see it does not define it as the inverse, it simply states that it is the inverse just like this article does. If you go down to the definition section it shows how it is defined. Dmcq (talk) 17:40, 7 February 2014 (UTC)

Frequently Made Arguments for 0^0=1

In response to the request for a FAQ. I started by making a summary of frequently made arguments for "1". The goal of this section is to save time (i.e. the goal is not to do the debate all over again!). Instead, the goal is to give a summary of arguments that have been made dozens of times, over hundreds of kilobytes, and compress it down to this one section.

Identities in basic arithmetic should not be undefined simply because they do not fit in a later formula such as exp(y log(x)). If we were to allow that, then we can undefine not only 0^0 and 0^2, but pretty much anything else as well.

0^0 = 1 is basic arithmetic, it follows from fundamental rules such as the empty product rule, and cardinal number arithmetic, both of which are far more fundamental than the exp(y log(x)) formula, or the technical issue regarding limits that is used to undefine 0^0.

The limit issue used to undefine 0^0 ignores the fact that the same argument also undefines f(0) for any function that is discontinuous at x=0, such as the floor function. It is not logically consistent to use this argument to say that 0^0 is undefined but floor(0) is not. Other common arguments for "undefined" are equally inconsistent (the most common one says that "0^x = 0 for all x", blissfully ignoring x=-1).

Efforts to portray 0^0 as undefined lead to bizarre wordings in the text, for example, right now we have a line "regarding b^0 as an empty product" which is a strange thing to say because b^0 is an empty product, just like an empty bag has 0 applies.

To justify this bizarre wording you need to hold a circular "definition" of 0; to believe that 0 is not a counting number (0 is the number of apples in an empty bag, or in more sophisticated words, it is the cardinality of the empty set) but instead, one somehow views 0 as a function that converges to 0, ignoring the circular nature of this view of numbers.

Efforts for "undefined" also leads to conclusions that we would normally not make. For example, if a book says f(x) = g(x) for all |x|<1, but does not say that f(0) = g(0), then in normal circumstances, the conclusion f(0) = g(0) would not be Original Research. Yet, if we apply this to 1/(1-x) = sum x^n then it suddenly becomes Original Research? To keep 0^0 undefined, one must ignore the fact that 0^0 = 1 is used throughout mathematics, and that even the calculus books that say "undefined" require 0^0 to be evaluated as 1 in their formulas, and that they give plenty of formulas that say 0^0=1 after a simple substitution. You have to pick one:

(a) simple substitutions are OR, (b) simple substitutions are OK, or (c) simple substitutions are OK but if I don't like the outcome, then they are OR.

Now you might say, if calculus textbooks (through the binomial theorem, or 1/(1-x), or power series notation in general) imply 0^0=1, they still do not mean to say 0^0=1 (the argument goes something like: the x^0 in power series is not unconditionally 1, it is just a nice convention that makes formulas shorter). The problem with that argument is that these books do not actually say that x^0=1 is a convention. In any decent math journal, if you imply a statement, it means that you claim it is true. There are two possibilities: either the calculus textbook authors do not follow the standards held by math journals, or, more likely, they do not know that their own writings imply 0^0=1. If our 0^0 section does not point out that many common formulas imply the value 1, then we allow textbook authors to remain ignorant about the implications of the formulas they write.

In the table of indeterminate forms, the "0/0" does not refer to an arithmetic operation. If it did, that would already be an error. Instead, it is an abbreviation of a lim f(x)/g(x) where f(x),g(x) both converge to 0 and l'Hopitals rule is needed to proceed. Likewise, "0^0" in the table of indeterminate forms does not refer to an arithmetic operation 0 to the power 0, instead, it is an abbreviation for a limit that should be evaluated with a transformation followed by l'Hopital.

The statement that "0^0" is an indeterminate form has no implications for the arithmetic operation 0^0. The indeterminate form "0^0" is an abbreviation of a statement about functions, while 0^0 = ... is statement about numbers. Confusing "abbreviation of limits of functions" with "an arithmetic operation" works in favor of the wrong conclusion, so it is best to clear up this confusion.

The sci.math FAQ says that 0^0 = 1 is now a consensus. Of course, you can find webpages that say something else. To determine consensus, status counts too (Benson vs. Knuth? Clearly these are not equals).

Note that the consensus can change, even in math: 1 used to be a prime number, but not anymore. Closely related is that the empty product rule is now standard in mathematics, but it was not always so. That's why old references should have little weight compared with modern conventions (if old texts counted equally, then "Pluto is a planet" would still be the consensus). Computer algebra systems have to deal with people from the 21'st century, that's one way to get a sampling of the current consensus.

Of course, computers should follow math rules, instead of the other way around. However, "following math rules" is only possible if the rules are consistent! The option "undefined" breaks many rules, and "1" does not. MvH —Preceding undated comment added 18:48, 9 February 2014 (UTC)

Benson's claim "the best value for 0^0 depends on context" is an urban myth. Yes, people do believe that there exist contexts where 0 is a better value than 1, but nobody has actually seen such a context.

Looking at limits of 0^x and x^0 it seemed so plausible to Benson that both 0 and 1 are useful as values of 0^0 in some contexts, that he wrote this without actually checking it. In doing so, he underestimated how consistent math is when you apply general principles consistently. Wikipedia repeats Benson's claim, again without checking it, presenting it as a valid argument instead of an urban myth.

Of all the arguments for "undefined", the only one with actual merit is an educational argument: If the calculator returns 0^0 = undefined instead of 1, this might help a calculus student to avoid a mistake on a test about limits. The value of "undefined" is that it can function as a warning flag when a student works on limits.

However, the educational argument for "1" is stronger than that for "undefined". After calculus, you are unlikely to work with functions of the form f(x)^g(x) because whenever the exponent varies continuously, you would be advised to rewrite the expression as exp(...) at which point "undefined" no longer serves a purpose. Moreover, students taught "undefined" still need to know that 0^0 is 1 in order to interpret numerous formulas. In addition, 0! and 0^0 give an excellent teachable moment. Students start out being very uncomfortable with empty sets/lists/products/sums (and the concept of zero in general). But by using these concepts you end up with shorter and cleaner theorems/proofs/algorithms due to having to consider fewer cases. Dijkstra argued this years ago, but I still see this on a regular basis when I give programming assignments to students; inevitably they start out writing unnecessarily long programs for simple tasks, almost all have to learn these things. MvH (talk) 23:23, 1 March 2014 (UTC)MvH

Please stop arguing the merits. There are lots of places to do that; this is not one. --Trovatore (talk) 23:34, 1 March 2014 (UTC)

Fairly obvious citations are what are required on Wikipedia. Long arguments by editors comes under WP:Original research. Dmcq (talk) 01:35, 2 March 2014 (UTC)

Trovatore is the guy who argued that 0.0≠0. Dmcq is the guy who argued that 0+i0≠0. Both points of view is OR, to put it mildly, or insane, to put it straight. We do have citations from books saying that 0⁰ is undefined, but we do not have citations claiming that 0⁰ is sometimes undefined and sometimes =1, such as WP says. The undefiners can not be taken seriously any longer. Bo Jacoby (talk) 20:34, 2 March 2014 (UTC).

Bo, you don't know what the heck you're talking about, and your third sentence is completely wrong. However it is not on-topic to explain why, here. I have fallen into that trap in the past. --Trovatore (talk) 20:42, 2 March 2014 (UTC)

You'll be glad to know I'm with Trovatore now and won't engage in anything you might consider as OR with you. I'll just look for improvements backed with citations. Dmcq (talk) 22:14, 2 March 2014 (UTC)

Bo, lets not write negatively about the people that disagree with us. Trovatore and Dmcq have read the wikipedia guidelines, it is reasonable to ask for citations instead of arguments. However, the guidelines don't really settle the matter. If we allow simple substitutions, e.g. plugging in x=0 in textbook formulas like sum(x^n), then we'll find 0^0=1 in almost any calculus book. Problem is that the guidelines won't explicitly tell us if a simple substitution is an acceptable way (i.e. standard practice and unoriginal) of reading the text, or, if it is original research. For us, a simple substitution is clearly acceptable standard practice (PS. the textbook proof of sum(x^n)=1/(1-x) assumes 0^0=1, if you drop that assumption, then we can only prove sum(x^n) = x^0/(1-x) for |x|<1. Textbooks implicitly say, and implicitly use, that x^0=1 for every x, including 0, and in math refereeing, if you imply something, then you've claimed it). Anyhow, I'm getting of course with that PS, what I'm trying to say is that unless the guidelines become so detailed that they settle this matter explicitly (substitution is OK or not OK), we can have good editors disagreeing with us about what the sources tell us (and, which sources carry more weight, etc.). I'll leave the issue now. MvH (talk) 14:58, 3 March 2014 (UTC)MvH

The problem is that "simple substitution" is a well defined concept only for single valued continuous functions. In all the other cases it may lead to errors that are difficult to detect for non-experts. For example ${\sqrt {x^{2}}}=|x|$ is a formula which is true when x is substituted by a real number and false when substituted by a complex number. A less trivial example occurs with the fundamental theorem of calculus: the fact that the integral on an interval of a continuous function is the difference of the values of an antiderivative at the endpoints of the interval supposes that the antiderivative is continuous. There are continuous rational functions such the antiderivative provided by a computer algebra system is not continuous (the arctan of a rational fraction is involved, which has a denominator with a zero inside the interval of integration), and thus the substitution of the values of the endpoints in the antiderivative gives a wrong result. Every "indeterminate form", such as 0⁰, may lead to such hidden errors. Therefore, trying to give a value to an indeterminate form, may be highly confusing for any non-expert. In other words, in an encyclopedia, 0⁰ must be defined as undefined. It may be said that in some contexts, some authors may attribute a value, but it must be clear that this is a marginal definition that must be used with great care. D.Lazard (talk) 16:32, 3 March 2014 (UTC)

Did not want to be pulled back in, but I do hope that you agree that when a textbook explicitly states: for all |x|<1, then you really are allowed to substitute x=0. Moreover, "0^0 may lead to a hidden error" is only true if there is an example. If there isn't one, then a textbook can't say that. MvH (talk) 14:54, 4 March 2014 (UTC)MvH

Trigonometric functions

The inclusion of trigonometric functions in this article seems to belong in Exponential function rather than here. I propose moving all mention of trigonometric functions there. —Quondum 03:07, 5 March 2014 (UTC)

Agree with that okay, I think the statemet of Euler's formula in the previous section should also be put on a line of its own rather than stuck inline. Dmcq (talk) 08:07, 5 March 2014 (UTC)

responses to the previous section

FAQ stands for Frequently Asked Questions. What you wrote has nothing to do with a FAQ, it is just your own thoughts on a topic that has bedeviled this article's talk pages. What a FAQ should be for something like this is

Q: But isn't 0^0 equal to 1 / undefined. It can't be both?

A: Wikipedia does not try and decide what is true, it tries to summarize what is said in reliable sources with due weight. Some sources say it is 1. Some say it is undefined. And others say it depends on the circumstances or don't give an opinion on the subject. The article summarizes the situation as it is. It would be against the Wikipedia policy on neutrality to try and decide between the alternatives when mathematicians are not decided on it and many think there is not really one true answer. Dmcq (talk) 21:20, 9 February 2014 (UTC)

Yes, nicely put. A FAQ must not become a venue to argue the merits of whether 0^0 should be defined as 1. There are lots of places to make that argument, but none on Wikipedia. --Trovatore (talk) 21:50, 9 February 2014 (UTC)

I agree with several of the points made. So I renamed the section: Frequently Made Arguments. It was written in response to a request from Steven G. Johnson, to make an overview of the arguments that are frequently made, in the hopes that this would reduce the "Round in circles" problem. The "Go ahead and do it!" line in the previous section gave me some hope that such a "FAQ"/"FMA" (one for "1" and one for "undefined") would save time in the long run so that people don't have to read hundreds of kilobytes to see the often-made arguments. I propose that arguments on one side go into one section, and arguments on the other side go in the other section.

The section should not simply be "my own thoughts", the goal is to make a summary of arguments that have been written here dozens of times, in discussions spanning hundreds of kilobytes (please let me know if I omitted a Frequent Argument). Famous mathematicians, like Bjorn Poonen, have made arguments here, that are then quickly ignored because changes that support one side are usually interpreted as POV even if it is a change that reflects mainstream math. (PS. Trovatore: the arguments do belong in wiki's Talk section) MvH —Preceding undated comment added 23:18, 9 February 2014 (UTC)

Actually, no, they don't, not unless they're directed to improving the article. And we don't base our articles on how mathematics "should" be, so arguments about that are not going to lead to improvements. I have a lot I could say on this particular subject, even have said it in the past, but I've come to realize that this was largely an error. --Trovatore (talk) 23:28, 9 February 2014 (UTC)

The goal here is to save time, to compress large amounts of prior arguments into a shorter text. — Preceding unsigned comment added by MvH (talk • contribs) 23:49, 9 February 2014‎

The correct way to "save time" is not to have the off-topic arguments on this page. Please sign your comments with four tildes. --Trovatore (talk) 00:04, 10 February 2014 (UTC)

Trovatore, before entering a big debate, let me reiterate my view that we can agree on a main page even if we can not agree on the issue "1 or undefined". I don't propose a 1-sided article, and I do agree with the "reliable sources with due weight". A big problem is that current views are not so easy to obtain in publications. People do not publish papers about 0^0 because you can't get publish things in good journals that are already known. My view is that things like empty product is 1 rule, and its applications, are now much more common than in older works. But I am not sure how to prove or disprove this view (citing papers would only be anecdotal evidence that can easily be rebutted by similar anecdotal evidence in the opposite direction).

On the main page, let me start with an example that may look fine to you, but looks biased to me. For example, the section that starts with: Any nonzero number raised by the exponent 0 is 1. To me, this section is obsolete because a few lines later in the section "negative exponents" it says again that b^0=1 if b is not 0. It looks as though the only purpose of this section is to reiterate the view that I don't like (the view that 0^0 is problematic). Surely we could delete this section, or merge it with the "negative exponents" section, without violating POV? It'll still tell the reader that 0^0 is problematic but this time, it won't say it twice within 10 lines from each other.

More troubling is the line: when 0^0 arises as a limit... This should be worded more precisely. What exactly is the word "it" in that sentence referring to, does it refer to 0^0, or the limit of f(x)^g(x)? If we can get rid of ambiguities, while keeping the same content, then that'd be a good thing. MvH (talk) 00:50, 10 February 2014 (UTC)MvH

I replaced "arises as" by "is viewed as". I think that is an improvement. Another problem is that there are multiple definitions of "indeterminate form" in the literature. Some are them are OK, while others are ambiguous, including the current one on wikipedia, which first says that it is an algebraic expression, but later on says that it is form of a limit. The section Indeterminate form#Evaluating indeterminate forms makes sense in the second view (evaluating a limit of that form) but does not make sense in the first view (evaluate an algebraic expression like 0/0, which is an impossible task). There are sources that do this correctly though. MvH (talk) 01:36, 10 February 2014 (UTC)MvH

I have no objection to "is viewed as". I don't see it as very different and I certainly am not interested in arguing the coin toss. However, see my last para below.

The subsections with positive, zero, negative exponents seem fairly natural to me. I suppose we could unify "zero" with "negative" into a single "nonpositive exponents" subsection but it looks sort of forced. (By the way, in my own "on the merits" view, I have no problem with 0^0 being interpreted as 1 when the exponent is a natural number, and if we were going that way, then it might be OK to leave the "undefined" discussion out of a "natural number exponents" section. That's off-topic, as I've explained, and I mention it in passing only to clarify where I'm coming from.).

On your final point, I would say "it" refers to 0^0. More precise than "arises as a limit" would be something like "is obtained by substituting 0 for a real-valued expression in the exponent". The point is not so much whether it comes from a limit, as whether the notation is being used for real-to-real as opposed to real-to-natural exponentiation. However I'm not proposing that as text for the article because it's likely unsourceable and is in any case probably too technically worded — I'm just proposing it as background to keep in mind. --Trovatore (talk) 02:03, 10 February 2014 (UTC)

On the phrase "When 0⁰ arises as a limit of the form...", this feels linguistically sloppy. What about "When the limiting form 0⁰ arises as a limit of the form..."? —Quondum 02:31, 10 February 2014 (UTC)

An older way of putting it 'When the form 0^0 arises in the context of continuity' expressed it better I thought. I tried saying that in the integer case it was usually taken as 1 but see the other section for in general but that was reverted on te basis that the business should all be dealt with at one place. Dmcq (talk) 09:07, 10 February 2014 (UTC)

I concur with you: the point is the form 0⁰, since 0⁰ is not what is meant. The "older way" is less clumsy than mine. To me this seems unrelated to the other discussion here; it is a matter of correct use of the language for what is being expressed. —Quondum 00:49, 11 February 2014 (UTC)

Trovatore, merging "zero exponent" and "negative exponents" is a relatively minor issue. My worry is though that even if we found the perfect formulation, the edit would still be reverted. In any case, the more problematic issue is the definition of the phrase "indeterminate form". Somehow, the two expressions "lim f(x)^g(x)" and "(lim f(x))^(lim g(x))" (i.e. 0^0) are both referred to as "indeterminate form", and this imprecise definition ends up equating "lim f(x)^g(x)" to "(lim f(x))^(lim g(x))". But these two expressions are very different (the value of the first can not be determined from the value of the second) and so they should definitely not be equated to each other.

The edit "arises" --> "is viewed" was a minor attempt to avoid equating "lim f(x)^g(x)" to "(lim f(x))^(lim g(x))", but it was quickly reverted. The same problem, mixing up different things, appears on the indeterminate form page as well, but there are sources that do it correctly.

Another way to say it is like this: if f(x) and g(x) converge to 0, this does not imply that lim f(x)^g(x) is 0^0. The page indeterminate form says that we can evaluate the indeterminate form, but that only makes sense if that refers to lim f(x)^g(x) (what would be the point of evaluating 0^0? It would say nothing about lim f(x)^g(x). Unless of course 0 really just means f(x), and the other 0 means g(x), but I sure hope that it doesn't!). MvH (talk) 03:08, 10 February 2014 (UTC)MvH

We've been through this a dozen times and we perfectly well understand that a limit does not have to equal the value at a point. What you are missing is how indeterminants arise and why they are different from functions written as functions rather than operations. You are also missing the practicalities of real life engineers and the importance of continuity in maths and physics as well as everyday life. They do not just arise as simple exercises in a book. Your edit only improved things in your own mind, it would not make anything clearer to anyone else that I can think of. The point is that in most cases a person would only try and evaluate the indeterminate because the value at a point was unknown, they wouldn't have though of doing a limit beforehand but they know something is continuous. Making it known means they don't bother evaluating the limit and they get a result with a discontinuity without ever a thought such a thing might happen. So 0/0 most definitely does arise. Dmcq (talk) 08:47, 10 February 2014 (UTC)

BTW I haven't found any earlier use of Bjorn Poonen's name in Talk:Exponentiation archives so I don't know why you brought him up. As far as I can see he does an undergrad course in differential equation but all his own work is in discrete mathematics, I'm not sure why he would show any interest in all this. I've come to the conclusion looking around it was probably some personal communication. Dmcq (talk) 09:36, 10 February 2014 (UTC)

I think you gave a much better argument than what has been given so far. The argument "undefined is good for practical reasons" has merit, whereas the argument "undefined is good for mathematical reasons" is impossible to swallow. Practical reasons can favor "undefined" in the calculus of limits, and favor "1" in discrete math and computer science, but in any case, practical arguments do have merit, especially when presented as such. MvH (talk) 15:21, 10 February 2014 (UTC)MvH

Many thanks to MvH for this fine efford. I do share your worry that even if we found the perfect formulation, the edit would still be reverted. If Dcmq really understood that a limit does not have to equal the value at a point then he wouldn't think that lim_{(x,y)→(0⁺,0)} x^y should be equal to 0⁰. The limit is indeterminate while the value is not. Bo Jacoby (talk) 14:23, 16 February 2014 (UTC).

Well I think your response at this point illustrates pretty clearly for MvH why I said "We've been through this a dozen times". Dmcq (talk) 17:16, 16 February 2014 (UTC)

Yes Dmcq, and you still make no sense. And we still don't know if you still believe that 0+i0≠0. Bo Jacoby (talk) 18:20, 16 February 2014 (UTC).

That aside, I recently had a discussion with someone who seems to believe strongly that 0⁰ = 1 is obvious nonsense. We should get the two of you together. MvH, nice paraphrase of the for argument. I find it sad that the notation for exponentiation with other than integer exponents is suggestive of a fundamental operation rather than of a function, since this appears to lead even mathematicians into zones of fuzzy assumptions. I would love to see a treatise by Bourbaki addressing the handling of the topic, but the contention relates to a matter of convention and context, not fundamental maths, and we do not have the necessary references other than to outline the contention. As has been made clear by Travatore and others, WP content is intended to be a reference of what is actually out there, not what simply makes sense to the editors. —Quondum 19:48, 16 February 2014 (UTC)

That user was actually being very reasonable, he or she pointed out that we have a page, indeterminate form, that can be interpreted (due to its ambiguous nature) as saying that 0^0 is undefined, while at the same time, we have a page exponential function that implies 0^0 to be 1. That user was correct to point out the contradiction between those two pages. Of course, everyone here is already aware of that, no need to go over it again. But leaving this implicit contradiction out there is indeed likely to keep causing complaints. The simplest solution is to fix that contradiction, but we can not do that; it would be considered OR and controversial. MvH (talk) 17:17, 21 February 2014 (UTC)MvH

Reasonable, perhaps. Correct, no. Any lack of clarity should be addressed in the article Indeterminate form, not elsewhere. I find that article very clear on this, though. —Quondum 02:25, 22 February 2014 (UTC)

What is actually out there is

1. consensus that f(0)=a₀ when f(x)=Σ_i a_i xⁱ
2. consensus that the empty product is = 1.

Bo Jacoby (talk) 10:50, 17 February 2014 (UTC).

I don't think anyone is disputing having those in the article. Dmcq (talk) 13:03, 17 February 2014 (UTC)

As the undefiners have a history of reverting my contributions I must leave it to the undefiners to clean up the mess themselves. Quondum's suggestion: "When 0⁰ arises as a limit" is no good because 0⁰ never arises as a limit. x^y is not continuous around x=y=0. Bo Jacoby (talk) 11:17, 25 February 2014 (UTC).

I know the current text is a hard fought compromise, and overall it is a good representation of the various points of view, but Benson's quote really ought to have an extra footnote. He writes that defining 0^0 is a matter of convenience, which is a correct statement that nevertheless promotes a poor understanding of mathematical practice; something that we should not endorse so prominently. Convenience is the sole justification for every definition in mathematics. Why does the modern definition of prime no longer include 1? Convenience. Saying that "0^0 = 1" is convenient means exactly the same thing as saying that it is a good definition. Benson is correct when he writes that defining 0^0 is a matter of convenience, but he is misleading the reader by pretending that 0^0 is special in that sense: defining 0^1 or 2^4 or 3^3 is a matter of convenience too. MvH (talk) 14:20, 26 April 2014 (UTC)MvH

While I cannot comment on the relative notability of Benson and Knuth, I agree with what you're saying. In addition, I find it striking that an alternative resolution of the conundrum is essentially absent from the article. This idea is exemplified by IEEE's choice to completely separate it into two separate functions, which is very much the route of "good definition". I would hope that this approach is adequately reflected in the literature, in which case it should be added to the article. —Quondum 15:22, 26 April 2014 (UTC)

The IEEE powr function is rarely used and is too limited to serve as a definition of x^n. For example, powr(-1, 2) won't compute (-1)^2 = 1. MvH (talk) 17:29, 26 April 2014 (UTC)MvH

You completely misunderstand me. According to the article, IEEE defines pown and powr as two separate functions. What I am saying is that there are sound mathematical reasons for defining two distinct functions, and I mention the IEEE choice only because it suggests that there exists a notable documented mathematical reasoning behind this approach. —Quondum 18:03, 26 April 2014 (UTC)

We already have two definitions. One is for integer exponents, which satisfies rules like (a^b)^c = a^(bc) for every a and every integers b,c. Another is for non-integer exponents (where that rule no longer holds for every a). I would not advocate adding another definition modeled after powr (I don't think powr is used much). Actually, we even have two separate definitions for integer exponents, one when the base is a real or complex number, and one for general rings (which for example defines M^n for square matrices). MvH (talk) 00:40, 27 April 2014 (UTC)MvH

Well, then I don't know where we're missing each other; you seem to be agreeing with me, but yet you did not seem to realize when you posted above that keeping the definitions (any ring as base with integer exponent vs. positive real base with a wide range exponent types) separate makes both Knuth's and Benson's debates on defining 0⁰ irrelevant. In the first case it unconditionally makes sense to define this as the ring's identity, and in the second case 0 is excluded as a base. —Quondum 02:17, 27 April 2014 (UTC)

What I was trying to say is that if we do not wish to mislead the reader, then a footnote needs to be attached to the quote from Benson. Even though he is correct when he says that the only argument for defining 0^0 is convenience, he is misleading the reader by failing to point out that the same is true for every definition in mathematics. MvH (talk) 16:01, 27 April 2014 (UTC)MvH

That this has the potential to mislead the reader, I definitely agree. However, I feel more strongly about it than adding a footnote: your argument is a good reason to remove the quote from Benson altogether (not to merely to add a footnote) and to simply include him as another one of the authors in the refnote [22] attached to the point that mentions the problematic nature of defining a single value. We also should not editorialize, which limits what we can put into footnotes.

I would also like to see a sourced third bullet here that points out the resolution through separation of definitions. I consider this obvious enough that it may be difficult to source (every mathematician worth their salt will automatically apply the context-appropriate definition of exponentiation, but may not write about the process of choosing it), but recognition of the role that notation plays in the confusion and of this resolution seem to elude most (on WP, at least). —Quondum 16:45, 27 April 2014 (UTC)

About removing that quote: It is one of the most common arguments for "undefined" (the most common argument is: "0^x = 0 for all x"). Even though these arguments are wrong, they convince many people that defining a single value for 0^0 is problematic, so I doubt we can get a consensus for completely deleting one of their main arguments. That's why I was aiming for a more modest goal, to add some clarification in a footnote. MvH (talk) 17:47, 27 April 2014 (UTC)MvH

In the correct context (specifically, when restricted to the "second case" that I referred to), that is closely related to a valid argument. As such, the quote is presented out of context. We should then at least restore the context, which would perhaps demand more than a footnote. I think that the context of all these arguments is a confused attempt at conflating the two cases. ;-) —Quondum 18:11, 27 April 2014 (UTC)

Quondum, for integer exponents, I do not understand the need for separate definitions. There are many theorems in which 0^0 can occur inside a formula. In all of those theorems, the value is 1 (if a theorem existed where 0^0 needs to be 0, it would cause a foundational crisis whether we define 0^0 or not, because of the existence of theorems that imply 0^0 = 1). MvH (talk) 19:47, 27 April 2014 (UTC)MvH

Are you saying that when the exponent is an integer, why have separate definitions (actually, separate functions)? What about the flip side: when the exponent may be a noninteger, or the base may be nonpositive or nonreal, why have separate definitions? Because only one of the definitions works at all. The same goes for when the exponent happens be an integer, but it is more subtle: generally only one of the definitions works at all, e.g. if the base can be zero or negative. If we were to replace every occurrence of the form x^y with either Π^y x (the product of y copies of x, defined as the empty product where y = 0, and defined through division for negative y), or x↑y ≝ exp(y log x). In every theorem where we require 0⁰ = 1, we mean Π⁰ 0. So, with this separation of notation, what contention remains? With this we cleanly get Π⁰ 0 = 1 and 0↑0 undefined (or NaN), exactly as required by theorems, analysis etc. I think that this is sufficient motivation for separate definitions, and indeed notations, for the two cases. There is only a limited set of circumstances where the distinction of which function we're dealing with is unimportant: when the inputs are restricted to x ∈ R⁺ and y ∈ Z. —Quondum 21:18, 27 April 2014 (UTC)

The point is that the most convenient definition is also the best definition. The most convenient definition is that in the x^y notation, 0^0 is always defined, and in cases where you do not want that, you can use the notation exp(y log(x)) instead. MvH (talk) 01:19, 28 April 2014 (UTC)MvH

There is a difference between a good definition and a convenient notation. You seem to be speaking about the latter. I consider this secondary to the failure to properly treat the topic of the distinction between the two functions in the article. Knuth and Benson both evidently failed to understand this, if I interpret the quotations correctly. —Quondum 01:44, 28 April 2014 (UTC)

People in general aren't going to use $\exp(y\log(x))$ instead of $x^{y}$ any more than they use $x\oplus y$ instead of $x+y$ when they have an operation analogous to addition. Just get over it. Dmcq (talk) 07:56, 28 April 2014 (UTC)

Understood and agreed. But differentiating notations where the writer wants to be very clear about the distinction between the functions has utility, but that is up to the writer. —Quondum 13:26, 28 April 2014 (UTC)

Quondum, shouldn't the definition of a term match the way the term is actually used? If you search for every instance of x^n inside a theorem, in wikipedia or in publications, you'll see that 0^0=1 is the right value in all of them. There is no theorem in which 0^0 = 1 is wrong (if such a theorem existed, combining it with the BT would result in a foundational crisis) (the often claimed "0^x = 0 for all x" is not a theorem). MvH (talk) 15:26, 28 April 2014 (UTC)MvH

And that is basically what the text says for those cases. However you are going beyond that and sticking in Knuth's ideas into the section about continuity which is not about those cases. And you did it without even a citation. I explained above why this causes trouble and you seemed to accept it then but now you've come back with your original idea of just pushing Knuth. Dmcq (talk) 15:50, 28 April 2014 (UTC)

Would you stop edit warring to get your stuff in please and wait for a consensus here first. You stuff has no citation and can be removed straightaway per WP:V. Dmcq (talk) 16:01, 28 April 2014 (UTC)

MvH, no, just because every case where x^y is defined for x = 0 corresponds to the case of Π^y x does not allow one to conclude that 0⁰ = 1 always, which seems to be what you're doing. And on the edit, while I agree that the out-of-context quote of Benson is misleading, I don't agree with the text of what you're adding, so I tend to concur with Dmcq here. Let's first work out what will fix this particular problem in the article. —Quondum 16:20, 28 April 2014 (UTC)

It really would be better if we had a good way of distinguishing the two main cases with the amount of trouble this has caused. I quite liked your $\prod ^{n}x$ but of course that's not what's out there. I see there is a claim in the last comment that you agreed with what is there. Dmcq (talk) 16:52, 28 April 2014 (UTC)

I'm not sure I follow you about what I agreed to. I agreed that the bald quote of Benson's claim is potentially misleading, but not to a digression in logic. I would like to see its impact toned down to reflect the point of the bullet, i.e. that defining 0⁰ once and for all is problematic. The quote, on the other hand, gives the impression that Benson believes that there is some potentially universal concept of 0⁰, and it is misleading about convenience being used to decide about this universal concept; it would have been better to state that whether it is defined is determined by what it is (i.e. which function it is intended to denote), which can usually be determined trivially. As I suggested before, I'd prefer to see the quote removed as not being adequately representative of the point being made.

If there is some consensus, it would be possible to highlight that there are two main cases via article structure etc., and hopefully that the confusion only arises as a consequence of an amalgamation of them – distinguishing notation such as mine could be used purely for explanatory purposes. —Quondum 17:33, 28 April 2014 (UTC)

Sorry I mean the last edit comment says in effect that you agree with the footnotes MvH put in. Dmcq (talk) 18:55, 28 April 2014 (UTC)

I said above "I don't agree with the text of what you're adding", which should be clear enough. I agreed to the principle of a footnote only, probably for reasons that were not fully clear. I don't object to the statement made in the article ("... is problematic"), only to this quotation of Benson in support of this point; my meaning may have been misinterpreted. On the flip side, if the quote is to stay, I might support a qualification (e.g. footnote) saying that there are criteria other than "convenience" that lead to case-specific definitions. Would you lean towards removing the quote, leaving no footnote or adding a footnote that in effect draws Benson's quote into question? —Quondum 19:42, 28 April 2014 (UTC)

My least favourite option is an uncited footnote which changes the meaning of the main text. I think the main text should be fixed or the cite removed if there's a problem with it. Dmcq (talk) 20:23, 28 April 2014 (UTC)

Dmcq, you don't like the footnotes I added; they can likely be improved. First, the sentence with the word "problematic". Math has binary logic, true or false. A set of statements is either contradictory, or it is not. There is no intermediate truth-value. So there is no mathematical interpretation for that sentence that ends in the word "problematic"; the sentence does not say that assigning 0^0 is contradictory, nor does it say that it is not contradictory, instead, it tries to say something in between, but there is no in between. Such a sentence is unacceptable in math, it has no meaning. You have to pick one or the other, either you say that there is a contradiction or you don't say that there is a contradiction. But it is not OK in math to try to say something in between (yes, I know there are logics with more than 2 truth-values but that's not related to this topic). MvH (talk) 17:40, 28 April 2014 (UTC)MvH

In summary: in math, one should not write a sentence that to the reader is likely to imply C, except if you really mean to claim C, but in that case, one should claim C without ambiguity. That's true in general (here C = "defining leads to contradictions"). MvH (talk) 17:52, 28 April 2014 (UTC)MvH

If there is a problem with how a cited source is summarized then how it is summarized should be changed to something that does summarize the source better. Just leaving summary that gives a wrong idea and sticking in your own thoughts about it in a footnote is leaving in one problem and putting in another, it doubles the problems rather than solving them. As to binary Wikipedia is not a maths textbook, it summarizes what people say on a topic and tries to do so with appropriate weight for the various views. And no I don't like the footnotes. Dmcq (talk) 18:47, 28 April 2014 (UTC)

The problem is not how the source is summarized. It accurately states a common belief that is not based on evidence. It tells the reader that defining 0^0 is enough to make math contradictory. What is so wrong about a simple footnote reassuring the reader that no actual contradiction has been found? (at least, not since Russell's paradox, which lead to a major overhaul) MvH (talk) 01:21, 29 April 2014 (UTC)MvH

Putting Knuth's argument into context

In the discussion about the value of 0⁰, indeterminate forms and the complex domain are presented as examples where 0⁰ cannot be defined as 1. Latter on, some opinions on the subject are presented expressed by notable mathematicians. In Knuth's treatment of the issue it is mentioned that it distinguishes between 0⁰ as a value and 0⁰ as an indeterminate form. Nothing is said about what Knuth thinks abouth 0⁰ in the complex domain. To the eyes of the casual reader this leaves Knuth's treatment of the issue as incomplete. Is this a correct impression? I tried to make this evident by adding that "... he does not address the issue of 0⁰ never being 1 in the complex domain.". This was reverted an oversimplification. I think this issue must be addressed because, as it is, Knuth's view is confusing. It may not be confusing for the academic community, where his article was addressed, but to someone who is no expert in indeterminate forms and the complex domain it certainly is. Nxavar (talk) 09:16, 21 February 2014 (UTC)

Hmm, well, I do think you have a point, in some sense. It's not at all clear whether Knuth intended to exclude the possibility of a complex-to-complex exponential function that would not assign a value to (0+0i)⁰⁺⁰ⁱ.

But there are a couple of problems with your addition as you wrote it. One is that I don't find it to be very clearly stated; it has a confusing double negative and, reading it out of the context of your comments, I wouldn't really be very sure what you were trying to say. Actually, even with your comments, I'm not sure quite what you mean to say.

That could be cleaned up, though. The more serious problem is that we don't have a source for the claim that he didn't address the issue. Usually, it's cleaner and more "encyclopedic" to report what people said, not what they didn't say. At some point Knuth did say something like "in this much stronger sense, 0^0 is less defined than...", can't remember how it finishes; that might be a better way to get at your point. --Trovatore (talk) 09:47, 21 February 2014 (UTC)

I think the best one can say from this is that Knuth did give a counterpoint to his "it has to be 1", which softens it. His very use of language tells us that he is arguing value to us, not mathematical necessity. It is evident (to me) that he was not addressing anything but the real-to-integer and real-to-real cases, but if this is mentioned at all, it should be stated that these are what he spoke of, not what he did not speak of, as T says. After all, there are many cases that he did not include aside from complex. I feel that the more important point is that Knuth was clearly not being categorical. —Quondum 02:09, 22 February 2014 (UTC)

I recently went back into studying infinitesimal calculus. What I understand now is that 0⁰ cannot be 1 when limits are involved. This explains why it's undefined in the context of calculus and arithmetic analysis. In that sense, the terminology "indeterminate form" in Knuth's writing means that he implicitly refers to the complex domain as well. Nxavar (talk) 11:46, 29 April 2014 (UTC)

A Question About Terms

If b^n=c, then what word indicates c?

    The base is b, the exponent is n, and the ________ is c.

It seems that product is a poor choice for the blank. Suppose that c=1 and n=0. In what sense, then, is c a product? Power, too, seems like a poor choice, for we are already in the habit of saying things like b raised to the the nth power and b to the power (of n). So a power can be either the exponent, n, or the result, c, of the operation b^n. A frustrated newcomer to mathematics should be forgiven for suspecting (i) that there's a gap in the glossary and (ii) that mathematicians are trying to confuse students by using the word "power" to refer both to c and to n.

64.107.153.120 (talk) 18:42, 29 April 2014 (UTC)

The frustration and confusion due to the use of the same word in maths for different concepts, often in related contexts, is pretty much unavoidable. Mathematics is full of this, and it is not intentional. As if this weren't enough, notation also gets used in distinct but related contexts – the notation bⁿ being a prime example. For your question "In what sense, then, is c a product?", we would call it an empty product. —Quondum 21:25, 29 April 2014 (UTC)

Out-of-place quotation

In the following text from Exponentiation#History of differing points of view

• Some argue that the best value for 0⁰ depends on context, and hence that defining it once and for all is problematic. According to Benson (1999), "The choice whether to define 0⁰ is based on convenience, not on correctness."

I see that the Benson quote was originally introduced by CBM in 2007. at the time, it was clearly intended to illustrate nothing more than that there was no definitive single answer, which made sense. Now the text has been refactored and the quote now appears intended to support of the statement that precedes it. As such, I propose deleting the quote as being out of place.

Going further, this section creates a false impression: that the debate is and historically has been essentially about only whether 0⁰ is best treated as undefined or is best defined as 1. But I will not address that now. —Quondum 21:58, 28 April 2014 (UTC)

OK, there seem to be two issues here: The Benson quote per se, and the footnotes added by MvH.

I'm somewhat undecided on the quote per se. I think it has some value, maybe not a lot. We cannot leave the impression that there is a consensus among mathematicians that 0⁰=1 is the correct choice, because that is not true. We could instead use the quotes from Carrier, Krook, and Pearson; Gonzalez; Meyerson that are referenced in the "continuous exponents section". Those are more unequivocal statements that 0⁰ is undefined; unfortunately, they omit the question of context.

However the footnotes, at least as written, have to go. For the "no theorem becomes false", it's not wholly clear exactly what it is supposed to be saying, but as a counterexample, how about this?

Given any term formed using only the operations of addition, multiplication, division, and exponentiation, the real-valued function defined by the term (when its variables are allowed to range over the reals) is continuous at every point of its implied domain.

As for the second one, it's an unsourced mini-lecture on the nature of mathematical definitions, linked to an article that is relevant only by analogy, with much argument to be made in the middle.

MvH, I do not see the "consensus" you claim for these two notes. I plan to remove them unless you can show it to me, or unless a better plan is proposed. --Trovatore (talk) 01:31, 29 April 2014 (UTC)

Is there no consensus for clarity? I can not be sure what exactly is meant by the sentence that ends with the word "problematic", different readers will surely interpret it different ways. But for the reader that interprets it as "contradiction", it would be important to know that no contradiction has been found. If the text was meant to convey that defining 0^0 once and for all leads to a contradiction, then the text should explicitly say it (authors that define 0^0 as unconditionally 1 would certainly want to know if their position could contradicts theorem in math!) If it was not meant to say that, it should not imply it either. The footnotes (or the text itself) can likely be improved, if it were not for the fact that any edit in this section is automatically reverted. For now, my only plan is to get consensus that we should have statements whose meanings are clear, and if they are not, they can be clarified by edits or footnotes. MvH (talk) 02:52, 29 April 2014 (UTC)MvH

I am certainly not against clarity, but I do not think that those notes really add clarity. Well, in any case, the Carrier-et-al quotes from an earlier section are certainly clear. One course of action is simply to state unequivocally that some mathematicians regard 0⁰ as undefined, and move (or copy) those refs from the earlier section.

Is that preferable? It is in one way; it definitely stays further from OR. But it is a bit of a pity not to bring in context dependence somehow. --Trovatore (talk) 03:12, 29 April 2014 (UTC)

The section currently creates the impression that the topic is contentious, whereas I expect that it is so only to a small minority of mathematicians. It would be clearest if we gave due weight to the majority view, which I suspect is that when the function exp(y log x) is intended, whenever x ≤ 0 it is undefined (unless an extension is specifically defined in the context), and conversely, etc. Simply put, due weight must be given to the majority perspective of mathematicians, which demands a rewrite of the section. —Quondum 03:28, 29 April 2014 (UTC)

Your counterexample to 'no theorem becomes false' is exactly the basis for the usual handling of indeterminates so it certainly has real world implications. Possibly the text portrays this problem as too much as a conflict whereas people can quite happily use exponentiation in both senses in a single sentence. If the exponent is i you know 0^0 will just be 1 and if it is x you know it is undefined - and even though treating as an indeterminate may give a limit value there and that very probably is 1 it quite possibly acts rather nastily in the neighbourhood. Telling people 0^0 is 1 end of story is just setting them up for a fall is how I feel about it, but iif the sources did give that with much greater weight I would go with it as the mainline in Wikipedia. Fortunately there is no such great weight. There are just the ways it is actually used. There is the problem though of people who want expressions to have a value irrespective of context and to give a value whenever possible with no ifs ands or buts. Can the section be written to quieten this subject down and make it clearer and less worrisome? Dmcq (talk) 10:54, 29 April 2014 (UTC)

I know there are people who share your worries. But can you really find a theorem that says that x^y is undefined whenever exp(y log x) is undefined? I do not think that a lot of mathematicians would agree with that. About the other "contradictions", what about ln(3), which is outside the radius of convergence of the well known series for ln(1+x), does that make ln(3) undefined? Finding a formula that doesn't tell you the value of ln(3) might cause worries but does not prove that anything is wrong.

In abstract algebra, in every ring with identity, x^0 is defined as 1 for all x. This also appears on the main page, and in textbooks. Do you think the authors in abstract algebra believe their definition can cause a contradiction? (or that they don't know that the reals form a ring with identity?) Beyond calculus and basic analysis, the belief that 0^0=1 can cause contradictions is not common. We can mention this belief, you are certainly not the only one that thinks this, but we should not be telling the reader that this is based on an actual contradiction when none has been found (what else is the reader supposed to conclude from "problematic" and "not based on correctness").

For comparison, how would you feel about a line in the evolution article that states: evolution is "problematic" (and not based on "correctness"). Wouldn't you like to at least add a footnote that this view is based on misunderstandings, and that no real evidence against evolution has been found? Creationists can find many citations that say that evolution is problematic. But when you read the arguments, what you find is misunderstandings. Likewise, the "problem" with 0^0 is based on the misunderstanding that f(0) must be obtainable from the limit of f(x) as x -> 0. This misunderstanding is common, until students learn that this is only correct when f is continuous at the origin, which we know x^y not to be, but by that point, the belief "problematic" has already settled. MvH (talk) 13:44, 29 April 2014 (UTC)MvH

Have you a citation for a book on abstract algebra that says that for a ring with identity every element to the power zero is the identity? Dmcq (talk) 15:11, 29 April 2014 (UTC)

This usually appears in undergraduate set theory, a prerequisite for graduate abstract algebra. I've taught both courses recently, and the set theory textbook explicitly says 0^0 = 1 (note that set theory is the foundation of modern math, and that a contradiction is not a minor issue!). I'm at home right now, so I can't look up the page number until tomorrow, but to give one more example, the OEIS (online encyclopedia of integer sequences) also defines 0^0 = 1, and this is used by many researchers in math. But to get back to the topic of the offending footnote, my question is still: do you have an example of an accepted mathematical theorem for which 0^0 = 1 is contradictory/incorrect? A reference would be great, but an example is also OK. It seems to me that all you have is are citations that suggests the existence of such a problem, but no actual example, and that as soon as one starts looking for an example, one only finds misunderstandings such as the belief that every function must be continuous on its domain. MvH (talk) 16:02, 29 April 2014 (UTC)MvH

Give me day or so to suggest a new wording for the subsection. This back-and-forth is not going anywhere; starting from a new text should make things simpler. As to whether most of us will be happier with it ... we'll have to just see. —Quondum 16:17, 29 April 2014 (UTC)

I think I should have a look at fixing up the abstract algebra section as it is a bit messy. I don't believe the $x^{0}$ bit there makes sense but I'll wait around for the citation. Probably it should just be in the group part. Dmcq (talk) 16:57, 29 April 2014 (UTC)

Quondum, what about replacing my controversy-causing footnote: "However, no currently accepted theorem becomes invalid if we defined 0^0=1 once and for all" by something that expresses the same dissent in a less controversial way. Instead of my version (that 0^0 doesn't cause any contradictions) it can be toned down to say that we have not found an agreed-upon theorem that becomes invalid under 0^0. There are people who are concerned about contradictions, as we can clearly see in this controversy, but it is not agreed-upon that contradictions actually occur. I'm sure there is a good way to say that, one that is better than simply implying that the other side is "problematic" and lacks "correctness". MvH (talk) 17:10, 29 April 2014 (UTC)MvH

What does this even mean? Of course you don't get "contradictions" from definitions. Of course definitions don't make theorems "invalid". Definitions are just definitions; they don't change the collection of theorems at all, provided you rewrite the theorems correctly to take them into account. You could define 2+2 to be 17 without causing any contradictions or making any theorems invalid, though it would be an awful lot of work to put in the silly conditions. There's also no approved list of "agreed-upon theorems". So put all that together, and I think you can see why I don't think the notes add the hoped-for clarity. --Trovatore (talk) 18:44, 29 April 2014 (UTC)

Dmcq, why do you think that a theorem like the binomial theorem implies 0^0 = 1 (which, under the usual rules of math, is already enough to force this to be accepted). The only reason the theorem can imply 0^0 = 1 is because it already used it in the proof. The same thing is used in other proofs too, in subtle ways that are not so easy to find; there is no easy revert here. MvH (talk) 17:14, 29 April 2014 (UTC)MvH

We've got to base what is there on citations with due weight. Can't you see how space has been wasted in these talk pages with people trying to stick in their own ideas about what's common or useful or obvious without you doing it too? Dmcq (talk) 17:32, 29 April 2014 (UTC)

I worried for a moment that someone here wanted to undefine 0^0 in the abstract algebra section as well (but perhaps I misunderstood that), which would open up more cans of worms (that would undefine x^0 for non-zero x as well, for example, take two non-zero square matrices whose product is 0).

Lets take a few steps back; looking at the article, it is a good article overall, it does represent views from literature, and contains useful stuff. But I think the same can also be done without implying that my side is "problematic" and lacks "correctness" (which you can imply with quotes but not with evidence). There is no need for major changes or battles. MvH (talk) 18:23, 29 April 2014 (UTC)MvH

Just produce a citation saying what you want to say or what you stuck in is going to get thrown out. Dmcq (talk) 22:38, 29 April 2014 (UTC)

In that case I'll delete Benson's quote, a mathematical nobody. Giving his personal opinion the same weight as Knuth (a founding father of his field) violates the undue weight rule. Also, it is obvious from the context what citation my footnote is referring to: all it does is clarify what is, and is not, in the cited source. It is allowed to say what is, but also, what is not, in the cited source. MvH (talk) 11:41, 30 April 2014 (UTC)MvH

When there are different definitions

Earlier in the debate, a separation of definitions was mentioned to address confusion. Suppose that we have two formulas/definitions that assign values to the same function F. Formula/definition #1 has a certain domain, say D1, and formula/definition #2 has domain D2. If a point P lies in the intersection of D1 and D2, it is required that the formulas/definitions #1 and #2 have to produce the same result for P. There is no requirement that D1 be a subset of D2, or vice versa. If P lies in the intersection, we have two definitions to choose from, and it doesn't matter which one you use, and if P lies in only one of these sets, then you have only one choice. It is not unusual to have several formulas/definitions for the same function F, and this is acceptable if and only if they agree on the intersections of their domains. For the case F(x,y) = x^y, we have several definitions, and none of them has a domain that is a subset of the domain of another one. Say D1 is the domain of the definition with integer exponents, and D2 is the domain of exp(y log(x)). If P is in only one of the domains (e.g. 0^2 is in D1 and e^(1+i) is in D2) then we still have a definition for the value of F(P). If P is in the intersection of D1 and D2, then we can choose from two formulas/definitions, but that is OK because they'll give the same result.

It may seem confusing to have several definitions/formulas, but it is OK because for any point P=(x,y) that lies in the intersection of two domains, the values of the definitions/formulas agree. The point is this: (1) I suggest that we do not use the word "generalization" when the domain of definition #1 is not a subset of the domain of definition #2. (2) Please do not suggest that there is tension between the definitions if they agree perfectly on the intersections of their domains. MvH (talk) 15:13, 30 April 2014 (UTC)MvH

It is fine to have a function F for which a given definition defines its values on a subset of its domain, and another definition that defines its values on another subset. What is crucial is that we be referring to the same function, i.e., that regardless of what the definitions for portions of its domain are, that the definitions will not disagree where they both have something to say about F. Note that you are using the term "domain" incorrectly: a function has a domain, but a definition does not. As soon as the domains are not identical, we are talking about distinct functions.

There is a distinction to be drawn between when a function is defined so that its domain excludes a given point, and when a particular definition does not apply to an element of the domain. This is precisely what we are dealing with in the case of 0⁰. In analysis, this case is deliberately defined as being excluded from the domain. Hence, we are not dealing with the same function if we choose to define 0⁰.

We could naturally define two functions with partially overlapping domains and which agree on the intersection of their domains, and then define another function as being defined over the union of the domains. Unfortunately, in this case this is not a useful definition generally, and in particular, it is a bad definition for the purposes of analysis. I wonder whether you could even source such a definition.

So, on your points: (1) Makes sense, but you'd have to be more specific. (2) I've addressed: a function has only one domain. —Quondum 16:17, 30 April 2014 (UTC)

Indeed, I was using the word "domain" incorrectly, what I meant was "instances where the definition can be applied" but that's not a domain or even a set. Strictly speaking, the "function" x -> x^4 is not a function until one specifies a domain, but the collection of choices one can make for that domain is a category, not a set. Any ring or group could be chosen as a domain, and strictly speaking they all give different functions, all having the same notation. So in the general setting (not restricting to one particular domain such as the complex numbers) then x^4 is not really a function, it is just notation. For many people, the real number 0 is the same as the integer 0, but from a category theory point of view it is not necessary to identify them. MvH (talk) 17:39, 30 April 2014 (UTC)MvH

There should be a section about $0^{0}$ in calculatores, just as there is now ...in software

Kjetil B Halvorsen 14:44, 6 May 2014 (UTC) — Preceding unsigned comment added by Kjetil1001 (talk • contribs)

We'd need reliable sources about it rather than people just testing them. I don't think anyone is that interested in them as far as that is concerned. In fact some of the software stuff probably shouldn't be here but there is reasonable stuff like how powers are treated in IEEE that is reliably sourced. Enough people are interested in it too I think to keep it alive and it isn't big. As for calculators there's too many and they've no standards. Dmcq (talk) 16:22, 6 May 2014 (UTC)

A proposed revision

Here follows a proposal for replacing the subsection of the same name. The significant change is that it gives weight to the majority, rather than ignoring the majority of authors as is currently the case. This could be shortened as the rest of the article deals with the topic accordingly. —Quondum 04:56, 30 April 2014 (UTC)

Unfortunately, I think it is not going to be so easy to source the "three functions" part. I do think it's a pretty good description of actual practice (with a reservation that case (i) is clearer for natural-number exponents, counting 0 as a natural number, than it is for integer exponents). But there is hardly ever any reason to codify this practice, so finding it in a reliable secondary source may be a challenge. --Trovatore (talk) 08:26, 30 April 2014 (UTC)

I've put my response after the proposal. Dmcq (talk) 12:33, 30 April 2014 (UTC)

History of differing points of view

The debate over the definition of 0⁰ in the case of a continuously variable exponent has been going on at least since the early 19th century. At that time, most mathematicians agreed that 0⁰ = 1, until In 1821 Cauchy listed 0⁰ along with expressions like 0/0 in a table of indeterminate forms. In the 1830s Libri published an unconvincing argument for 0⁰ = 1, and Möbius sided with him, erroneously claiming that $\scriptstyle \lim _{t\to 0^{+}}f(t)^{g(t)}\;=\;1$ whenever $\scriptstyle \lim _{t\to 0^{+}}f(t)\;=\;\lim _{t\to 0^{+}}g(t)\;=\;0$. A commentator who signed his name simply as "S" provided the counterexample of (e^−1/t)^t, and this quieted the debate for some time. More historical details can be found in Knuth (1992).

More recent authors interpret the situation above in different ways:

• Most authors simply distinguish between two distinct types of function despite being denoted similarly, never intending more than one of these for a given use. These distinct uses may occur in the same expression, often without comment, but with the reader nevertheless being expected to distinguish them. Further variants generally are associated with an explanation of exactly what is intended. These are:

  (i) The exponentiation bⁿ, with b being relatively unconstrained (e.g. being real, complex, etc.) and n any integer, defined in terms of repeated multiplication and division. It is generally considered as being defined for all b and integer n, with the exception of b = 0 when n < 0. In this case, b⁰, where it occurs, is nearly universally taken as being 1 for all b, including when b = 0.

  (ii) The exponential function exp(x), denoted e^x with an explicit e.

  (iii) The exponentiation b^x with b restricted to a positive real number and x relatively unconstrained (e.g. being real or complex numbers), typically defined in terms of the exponential function as b^x = exp(x log b). It should be noted that its is generally considered undefined for zero and negative b, unlike the first case above.

  (iv) A further variant, explicitly written b^1/n is sometimes intended to mean "the nth principle root of b" and may be defined for negative b when n is odd.

• A minority of authors typically writing for the popular press have treated the above functions as a single function. This has led to difficulty in reconciling the different intended uses, resulting in confusion about how to define 0⁰.

  (a) Some attempt to resolve the apparent conflict between the benefits of defining 0⁰ = 1, clearly useful and often used, and the benefits of leaving 0⁰ undefined, clearly useful in other contexts. According to Benson (1999), "The choice whether to define 0⁰ is based on convenience, not on correctness."

  (b) Others argue that 0⁰ should be defined as 1. Knuth (1992) contends strongly that 0⁰ "has to be 1", drawing a distinction between the value 0⁰, which should equal 1 as advocated by Libri, and the limiting form 0⁰ (an abbreviation for a limit of $\scriptstyle f(x)^{g(x)}$ where $\scriptstyle f(x),g(x)\to 0$), which is necessarily an indeterminate form as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."

(refs)

(end of proposal)

I think the 'for the popular press' bit can be left out - Knuth isn't the popular press! It would be better if we could find a list like this somewhere. The cases (ii) and (iii) are really the same as far as this is concerned. Case (iv) $b^{\frac {1}{n}}$ is an extension of case (i) whereas $e^{\frac {2\pi i}{n}}$ is really a case of (ii). So I would say there were the two main usages - the discrete and the continuous cases as described in the text. Perhaps fractional exponents in the form of the cube root of minus one could be used to illustrate the difference between the two usages. Dmcq (talk) 08:33, 30 April 2014 (UTC)

Trovatore, Dmcq: good comments. I disagree that (ii) and (iii) are really the same, but agree with the "as far as this is concerned": (ii) is not an example of exponentiation, but merely uses the same notation; it is a completely separate function, and does not belong. Removing it improves this no matter how you look at it. I was silly to include (iv): it has to be treated as one of the numerous "special extensions", and is really a shorthand for something else (roots).

I realize that it may be difficult to source the codification of this practice. I feel that this falls under the category of "allowed inclusions": those that are so obvious that they do not have to be sourced, though hopefully a source will pop up somewhere. The alternative of omitting the description of what is widespread practice creates a misleading impression, one which I hope to correct here.

On (i), should we omit negative integer exponents in this description, treating them as an extension? Inclusion of at least a mention of this extension here is fairly natural, since with it then the two cases (i) and (iii) act as clean starting points for two very powerful but distinct generalizations, both particularly significant in complex analysis but extending to matrices etc. —Quondum 13:50, 30 April 2014 (UTC)

Trovatore, I'm reading Cauchy ( https://archive.org/stream/coursdanalysede00caucgoog#page/n93/mode/2up ) but so far I have not yet found something that looks controversial from my point of view. He explains that with singular expressions (e.g. 0^0) you have to watch out for any infinitely small deviations in the variables. My question is: Does Cauchy support the view that 0^0 is best left undefined? If so, where did he write that? MvH (talk) 13:02, 1 May 2014 (UTC)MvH

It looks like the reference to Cauchy in the main page is highly misleading. It says that Cauchy put 0^0 among the "indeterminate forms" but the dictionary says that "indeterminate" means "undefined". So as currently formulated, our page says that 0^0 is undefined according to Cauchy. But the actual word he uses is "singular expression" which is uncontroversial. Unless one can find a reference that Cauchy supports the view that 0^0 is indeterminate/undefined, this ought be corrected. MvH (talk) 13:11, 1 May 2014 (UTC)MvH

On the page 69 you point at he uses the expression 0^0 and says its value could be anything from zero to infinity. I really don't see how you align that with him being happy with its value being 1. It's pretty certain he would have said that he was happy with it being 1 if he actually was given the context in which he wrote it being that others had asserted that. Dmcq (talk) 13:40, 1 May 2014 (UTC)

I think "indeterminate" cannot be construed as having the same meaning as "undefined". But I think that the historical debate probably concerned only the limit and not defining the value; in this sense the article may be representing the picture. —Quondum 14:20, 1 May 2014 (UTC)

Dmcq, a little while ago you demanded references, but it seems that you did not make that request in good faith because when I inserted a reference (OEIS and Sloane's book are highly cited!) you simply deleted it. The values (0..infinity) for 0^0 arise when you use infinitely small deviations (see page 68). Cauchy does not explicitly say that 0^0 is undefined, that's just your interpretation. Contrary to what the main page says, he does not use the word "indeterminate" or "undefined" MvH (talk) 14:48, 1 May 2014 (UTC)MvH

What you put in was "For more examples/references from combinatorics see the On-Line_Encyclopedia of Integer Sequences, e.g. http://oeis.org/A000312". We don't need discussions in the references and the reference wasn't a halfway decent one for what was said in the text. Here is a citation that actually says the sort of thing the text says and a reader can check it reasonably easily.

• Joshi, K D (1989). Foundations of Discrete Mathematics. New Age International. p. 80. ISBN 81-224-0120-1.

If you could provide citations for the other places you said instead of a personal attack that would make life much easier thanks. I wasn't looking for something copy edited, just something fairly reasonable. Dmcq (talk) 15:51, 1 May 2014 (UTC)

Just realized that cite wasn't right either. We want one saying something like the number of zero length strings or tuples of 0 possible characters or elements is 1. Had a quick look with google an didn't find one but I'm sure there must be a good one around. It is the sort of thing that's needed for the principle of inclusion and exclusion. Dmcq (talk) 16:32, 1 May 2014 (UTC)

Dmcq, my apologies for that. Thank you for finding references (finding them is time consuming). For Cauchy's reference, page 69 does not imply your interpretation of page 69. You can clearly see a y^x in front of that 0^0. The meaning is perfectly clear: y^x (near x,y = 0,0) evaluates to (0..infinity). Not the number 0^0 itself (what would that even mean?). The main page misrepresents what Cauchy wrote here. MvH (talk) 16:45, 1 May 2014 (UTC)MvH. RE: personal attack. I want to emphasize that my apology is sincere. Even when we are in conflict, I know that you and others here have done lots of great work for wikipedia, a valuable service to all. MvH (talk) 18:03, 1 May 2014 (UTC)MvH

It is not perfectly clear otherwise it would be clear and it isn't clear. It is only clear in your mind because you are following Knuth who distinguishes the cases and want to believe that Cauchy would have agreed with you. Knuth does not say it is clear Cauchy was okay with his idea, are you saying Knuth who really would have liked Cauchy to agree with him was wrong in his interpretation of what Cauchy was saying? Dmcq (talk) 17:59, 1 May 2014 (UTC)

Hang on, I interpret Cauchy to be saying that you can get any number [0,∞] from a suitable limit and thus that the two-variable limit does not exist, and nothing else. And I think that this is exactly what MvH is saying. I feel the article misrepresents this by mentioning it in the context of whether the value of 0⁰ is (or should be) defined. I don't see that Knuth had any different interpretation of what Cauchy wrote. —Quondum 18:23, 1 May 2014 (UTC)

That is what Cauchy said, but he was arguing against Libri who was saying that 0⁰ was 1. As Knuth says "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side". What MvH was trying to make out was that Cauchy would be happy with 0⁰ being defined as 1 as he was only talking about limits. That is quite wrong and it goes against the standard way indeterminants are handled in calculus which depends on the value not being defined. For instance just looking at the very first book in google when I stick in "indeterminate calculus" we get

http://books.google.com/books?id=rTHCAAAAQBAJ&pg=PT122

where it starts the section on Indeterminate forms with "Indeterminate forms like 0/0 have no definite value, however when a limit is an indeterminate l'Hôpital's rule can often be used to evaluate it. KEY POINTS Indeterminate forms include 0^0..." Dmcq (talk) 23:55, 1 May 2014 (UTC)

I still don't see quite how you draw the conclusion that Cauchy was saying that 0⁰ is necessarily undefined, nor do I see your point with the Google search. That Cauchy was arguing against Libri only leads me to conclude that he was (quite successfully) finding fault with Libri's logic, which is to say that Cauchy showed that to conclude that 0⁰ necessarily equals 1 is incorrect. This is not the same as concluding that defining it as 1 breaks any algebraic rules, in the same way as defining 0/0 = 1 would. I'm not suggesting that Cauchy would be happy defining 0⁰ as 1 for continuously varying exponents, only that we must not try to read between the lines and infer that he said what he didn't. —Quondum 04:50, 2 May 2014 (UTC)

The book says "indeterminate forms have no definite value, when a limit is an indeterminate form...", it does not say they only have no definite value if they are limits or that some indeterminate form may have a definite value but you still have to evaluate its limit as a limit. Cauchy used the rule in that Calculus book. It is necessary that 0^0 not be defined at the limit point otherwise the rule wouldn't be applied. In fact I haven't yet seen a decent reformulation of the process to take care of 0^0 being defined, at best one should rewrite the exponentiation as function calls as in $\exp(y\log x)$ first. Dmcq (talk) 07:40, 2 May 2014 (UTC)

No matter how you slice it, at the moment we have no evidence that Cauchy placed 0^0 in a table of indeterminate forms; on the pages we looked at he did not use the word indeterminate/undefined. MvH (talk) 12:15, 2 May 2014 (UTC)MvH This is not merely a matter of semantics. There is no dispute that the function is discontinuous. The disputed step is: "discontinuous implies indeterminate/undefined". Without that step, we have no controversy. The question is who introduced this step? MvH (talk) 13:33, 2 May 2014 (UTC)MvH

Go back to page 45 of that text by Cauchy and read what he says at the beginning of the section about singular values. It is practically identical to that Calculus book. He says if a case arises where the value at a point cannot immediately give the value then we seek the value to which it converges. That would exclude 0^0 if he thought 1 was okay as it would immediately give the value. Dmcq (talk) 13:41, 2 May 2014 (UTC)

You are assuming that Cauchy also meant the converse of what he actually wrote. He writes that if a function is not defined at a point, then compute limits (singular values of f). If A then B. You interpret that as "if B then A". MvH (talk) 15:06, 2 May 2014 (UTC)MvH

RE: page 45 Cauchy. I typed page 45 so editors can copy/paste it into Google translate:

Lorsque, pour un systeme de valeurs attribuees aux variables qu'elle renferme, une fonction d'une ou de plusieurs variables n'admet qu'une seule valeur, cette valeur unique se deduit ordinairement de la definition meme de la function. S'il se presente un cas particulier dans lequel la definition donnee ne puisse plus fournir immediatement la valeur de la fonction que l'on considere, on cherche la limite ou les limites vers lesquelles cette fonction converge, tandis que les variables s'approchent indefiniment des valeurs particulieres qui leur sont assignees; et, s'il existe une ou plusieurs limites de cette espece, elles sont regardees comme autant de valeurs de la fonction dans l'hypthese admise. Nous nommerons valeurs singulieres de la fonction proposee celles qui se trouvent determinees comme on vient de le dire. Telles sont, par example, celles qu'on obtient en attribuant aux variables des valeurs infinies, et souvent assi celles qui correspondent a des solutions de continuite. La recherche des valeurs singulieres des fonctions est une des questions les plus imporantes el les plus delicates de l'analyse: elle offre plus ou moins de difficultes, suivant la nature des fonctions et le numbre des variables qu'elles renferment.

The output from google translate is not perfect, some lines only make sense if you compare with the original. A few key lines are "In case the definition given could not immediately provide the value of the function that we consider, we look at limit or limits" (google translate has "boundary" instead of "limit"). And "we will call singular values ​​of the function those proposed are determined as we just said".

The first line tells us to look at limits if the function is not already defined there (which is the case in one camp, and not in the other camp). The second line tells us that [0..infinity) are not values of 0^0 but they are "singular values of the function x^y". MvH (talk) 14:52, 2 May 2014 (UTC)MvH To interpret that first line in the historical context, keep in mind that at the time Cauchy wrote this, 0^0 was already defined (undefining it would have been a break in tradition, it is a big leap to assume that he wanted to do that if he didn't explicitly say so). MvH (talk) 15:50, 2 May 2014 (UTC)MvH

So you are saying you think Knuth was wrong in saying Cauchy had listed 0⁰ as an undefined form? And that all the people between him and Knuth also misunderstood him? Dmcq (talk) 17:19, 2 May 2014 (UTC)

Anyway I've found a translation by a professional in [2]. People then considered functions as being defined by analytic expressions and that they should be continuous and differentiable. Dmcq (talk) 17:38, 2 May 2014 (UTC)

By the way Knuth provides no citation for his saying most people thought 0⁰ was 1 before Cauchy. Dmcq (talk) 17:58, 2 May 2014 (UTC)

RE Knuth wrong on Cauchy? Figuring out history is a time consuming task, it's tempting to copy history from others without checking it. I don't know if Knuth did that, but this does suggest that we can't automatically trust his version of history. The sentence "listed it in a table of indeterminate forms" is ambiguous, it can be interpreted in several ways, and we don't know which (if any) of these interpretation Cauchy would agree with. We could also say: "he listed it in a table of discontinuous functions and listed their singular values". The latter version doesn't require one to read "B -> A" where he wrote "A -> B". I don't know how math was written historically, but in the modern age you would never conclude that the author meant the converse too, unless clearly indicated. MvH (talk) 18:14, 2 May 2014 (UTC)MvH

Dmcq, you write "People then considered functions as being defined by analytic expressions and that they should be continuous and differentiable". That is a good way to explain how this became so controversial. I have always wondered how it was possible for so many to go from "defined" to "undefined" (assuming that that's what happened, which I'm not completely sure of anymore). If people only accepted continuous and differentiable functions, it may have come as a shock (the kind of thing that causes drastic change) when Cauchy pointed out that x^y has an infinite set of limits (singular values) at the origin. MvH (talk) 18:24, 2 May 2014 (UTC)MvH

I'm afraid I still think this proposal is original synthesis. It's too bad, because it is a pretty good summary of actual practice. But my strong suspicion is that most mathematicians don't explicitly think of it this way, mostly because they don't bother to think about it at all. They do distinguish between contexts, in practice, but they don't bother to codify the practice, because there's no need.

In my ideal world, people would recognize the distinction between the real number 0 and the natural number 0 (and also that it's usually correct and convenient to elide it), and would observe that the arguments for 0^0=1 (with one exception, the "analytic f and g" thingie) are all implicitly using the natural-number exponent, and that for the case of the real-number exponents, there just really aren't any good reasons to define it. But it isn't the common mathematical practice so we can't report it as though it is, just as we can't report that there's a consensus for 0^0=1, because there isn't. I think the tripartite codification falls into the same crack. It would be lovely if we could find one of these solutions (distinction between real 0 and natural 0, or tripartite codification) in some high-quality source, but even then we could only report that as what that source says, not as standard mathematical practice, because it isn't.

So I think the whole thing can't really be tidied up much. Definitely the argumentative footnotes should be removed though. Not sure about the Benson quote that provoked them; at least, we should think about whether that quote really says as much as the passage quoting it seems to suggest. --Trovatore (talk) 22:17, 3 May 2014 (UTC)

I think that what is there is rather problematic in that it creates the incorrect impression that the majority practice does not exist; one is thrown into unnecessary alternatives. So it would be worth seeing what we can change to mitigate this impression; what is said should be toned down in some way because it is nonencyclopaedic to create a misleading impression. The description above may be inaccurate in the sense that exponentiation (both with natural/integer exponents and other exponents) is mostly dealt with as though it is a shorthand rather than a formal function, which make things even fuzzier. This might partly underlie lack of codification. Perhaps we should qualify mention of the debate with "There has been some debate..." rather than emphasizing the enduring nature of the debate? —Quondum 02:36, 4 May 2014 (UTC)

another book [3] The Trouble With Zero by AJ Corcoran, refers to Wikipedia unfortunately but has a bit of his own thought. Dmcq (talk) 11:29, 4 May 2014 (UTC)

Reading through a few pages makes me conclude that Corcoran is an irretrievably fuzzy thinker: 'This is another case of "undefined" in mathematics because of lack of clear understanding of what zero is' (p. 44) is really only saying that Corcoran's own ideas of zero are different, and he uses all sorts of heuristics based on atrocious statements that seem to have no link to any axioms. Examples: 'Neither zero nor infinity are numbers and do not actually exist on the number plane ...' (p. 42), 'That is, zero has no reciprocal other than itself ...' (p. 43). I can't even figure out what is meant by some of the statements around the '"nothing" { } view of the quantity' (p. 45). Whatever the "bit of his own thought" might be, it has little to do with mathematical rigour. So, not much use for anything, I'd say. —Quondum 12:28, 4 May 2014 (UTC)

Yep it ain't good. It is obvious there isn't any general concern about this since so few say anything about it except to say what it is in their field which basically is that it is 1 outside of calculus books and undefined in calculus books. Dmcq (talk) 23:24, 4 May 2014 (UTC)

I will rework the proposal a bit over time based on the comments above. For now I'm removing the most salient OR, but please do not regard it as a proposal until I have managed to look at it again. —Quondum 16:54, 6 May 2014 (UTC)

?

i think that this arcticle should be a featured article. because it's richer that the featured article in herbew--ᔕGᕼᗩIEᖇ ᗰOᕼᗩᗰEᗪ (talk) 14:47, 3 September 2014 (UTC)

Extension to all reals

Article currently reads:

No natural extension to all real b and n exists, but when the base b is a positive real number, bⁿ can be defined for all real and even complex exponents n via the exponential function as e^{n log b}.

I am not sure what this is supposed to mean. The definition bⁿ = e^{n log b} is valid for all non-zero b, not just "positive real" b. It is a "natural extension". I would write simply:

For all real and complex n and b (other than b=0), bⁿ is naturally defined via the exponential function as e^{n log b}.

Is the problem the multivaluedness? That is true even for positive b. After all, 1^(1/2) equals both 1 and -1. --Macrakis (talk) 21:35, 28 August 2014 (UTC)

I think a "natural extension" to the reals would mean a real-valued function, not a complex-valued one. Of course you can take complex exponentiation (say on a certain branch) and restrict it to the reals, but that's a restriction of the extension to the complex numbers rather than a natural extension to the reals. --Trovatore (talk) 22:03, 28 August 2014 (UTC)

OK, if we want to restrict that passage to a real-to-real function, that's fine, but then we should not talk about complex n. And then the next sentence should presumably be something like "Indeed, the exponential definition allows us to define bⁿ for all real or complex n and b (b ≠ 0).", right? --Macrakis (talk) 22:24, 28 August 2014 (UTC)

Yes the problem is multivaluedness. A multivalued function is not a function (mathematics) and gives all sorts of problems. The natural extension of exponentiation for reals gives that $1^{\frac {1}{2}}$ is 1, not −1. A possible value of the square root of 1 is −1, that is something quite different, it is also one of the possible values when the base 1 is considered to be a complex number rather than a real. Dmcq (talk) 22:33, 28 August 2014 (UTC)

A cleaner way of looking at it is that the [real function or the principle value of the] log function over positive real b is unambiguously defined, but there is no such function without this constraint on b. Would it make sense to define exactly what log function is meant in the lead, or rather to leave it implicit as it currently is? —Quondum 00:03, 29 August 2014 (UTC)

It's not an extension to the reals. In the context of the real numbers, (−1)^0.5 doesn't have two values — it has no values. You're talking about the complex context; that's a different thing. --Trovatore (talk) 00:33, 29 August 2014 (UTC)

I am? What in "log function over positive real b" allows the value b = −1? —Quondum 06:38, 29 August 2014 (UTC)

Oh, sorry, I confused your comments with Dmcq's. And I'm not sure I interpreted those correctly anyway. But what stuck out was "the problem is multivaluedness", because I don't think that is in fact the problem. --Trovatore (talk) 18:27, 29 August 2014 (UTC)

;-) Well, I think we are agreed on the conclusion (that there is no natural extension), if not on the explanation, which is quite involved. And I agree with Dmcq's comment below that the lead should not be expanded – but it didn't hurt to pose the question. —Quondum 20:38, 29 August 2014 (UTC)

I think what has been stuck into the lead is quite enough and is okay. We don't have to spell every single thing out. It is the lead not the article. Dmcq (talk) 13:31, 29 August 2014 (UTC)

The example of using i=(-1)^(1/2)=(1/-1)^(1/2)=1/{(-1)^(1/2)}=-i is wrong. The correct expression is i=(-1)^(1/2)=(1/-1)^(1/2)=±1/{(-1)^(1/2)}=±i. -i is a fictitious root from taking the square root. The correct result is i=(-1)^(1/2)=(1/-1)^(1/2)=-1/{(-1)^(1/2)}=i. This section should be revised. 70.53.228.108 (talk) 16:38, 21 November 2014 (UTC)Cucaracha

Your have misread the example. In the article, there is a ≠ instead of = in the middle of the formula. Thus the article says essentially the same thing as you. D.Lazard (talk) 17:43, 21 November 2014 (UTC)

Imaginary Exponents Section Needs Clarification

I'm trying to wrap my head around complex exponents, but the description in the "Imaginary Exponents with Base e" is confusing. It states "Consider the right triangle (0, 1, 1 + ix/n)". I'm not familiar with a way of repesenting a triangle with 3 numbers, except if it's simply to state the length of each side as a Pythagorean triple (e.g. 3:4:5) or to indicate the 3 angles, as in a 30-60-90 triangle. The given values would not represent a right triangle using either of these methods. The quoted text includes a link to the "right triangle" Wikipedia article, but it sheds no light on what the three values might represent.

I think the meaning of the 3 number set should be explained or linked to or perhaps there should be 3 sets of Cartesian coordinates listed instead, as I believe this is a more familiar notation to most people.

Todd in Houston — Preceding unsigned comment added by 75.108.198.254 (talk) 17:34, 11 October 2014 (UTC)

I have edited the section for explaining what is the triangle. However, the "explanations" that follow are very difficult to understand, even for a professional mathematician, and seem to be incorrect ("almost equal" replaced by "equal"). I have thus tagged the section as "too technical" and the text as "clarification needed". D.Lazard (talk) 18:38, 11 October 2014 (UTC)

Are you tagging the entire sentence for clarification or just part of it? {{Clarify span}} can be used to identify the exact text that needs clarification. Is this clause coherent: "the triangle is almost a circular sector"? --50.53.58.19 (talk) 20:32, 11 October 2014 (UTC)

It is the whole paragraph that needs clarification, but more specifically the deduction implied by "so", which is not trivial nor elementary. I have edited the tag. D.Lazard (talk) 20:52, 11 October 2014 (UTC)

I took a shot at reworking the argument. It's still pretty handwavey, tho. And I'm not sure where the triangle similarity came in. Easy Secrets (talk) 08:16, 22 November 2014 (UTC)

Really I was looking for clarification to "Consider the right triangle (0, 1, 1 + ix/n)".

However, I read this section again this morning and suddenly realized what the triangle represented by the 3 numbers was, probably because I'd been ruminating on the later part of the article that mentions "Using exponentiation with complex exponents may reduce problems in trigonometry to algebra." Anyway, the piece of the puzzle that was missing for me is that each number in that set is a complex number, and therefore is representative of a pair of coordinates on the complex plane. I think the statement could be changed to either "Consider the right triangle (0, 1, 1 + ix/n) on the complex plane" or "Consider the right triangle (0 + 0i, 1 + 0i, 1 + ix/n)" and that would eliminate the ambiguity.

By the way, I'm brand new to interacting with Wikipedia, but I'll take a shot at suggesting that edit myself.

Thanks! Oddacorn (talk) 14:27, 12 October 2014 (UTC)

it was clarified yesterday already, just re-read it: "Consider the right triangle in the complex plane, which has 0, 1, 1 + ix/n as vertexes." —Quondum 14:51, 12 October 2014 (UTC)

What is the value of floor(0)?

(A) Many textbooks contain formulas that imply 0^0 = 1.

(B) However, many of the same textbooks leave 0^0 undefined.

(B1) Some of these books say nothing about 0^0. (B2) Some of them write that 0^0 is undefined.

(C) Some textbooks define 0^0 it as 1.

Options (B1) and (B2) are common in calculus/analysis textbooks, while (C) is common in discrete math, graduate algebra, set theory.

I think textbooks that do (A)+(B) contradict themselves, but I have to admit that (I) such books are common, and (II) wikipedia has to report what common sources say, even if these sources say things that are self-contradictory.

We have to report that:

(A) there are common formulas where 0^0 must be evaluated as 1.

(B) there are sources where 0^0 is not defined.

(C) there are sources where 0^0 is defined as 1.

The 0^0 section is reasonable as long as it says (A)+(B)+(C), which at the moment, it does.

The key argument in support of (B) is that indeterminate forms should not be assigned a value. To test of this is a valid argument, we can apply it to floor(0), which is an indeterminate form because the left and right limits of floor(x), with x going to 0, do not match.

If 0^0 is undefined, then by the same logic, floor(0) should be undefined as well, a conclusion that few people would be happy with. But in the end, this argument, nor the other excellent arguments that support "1", have much bearing on wikipedia, because it must report what common sources say, and "undefined", is indeed surprisingly common. Lets make sure that each of (A)+(B)+(C) is represented properly. If so, the page itself should be uncontroversial despite the long-lasting disagreements about the merits of each argument. MvH Feb 7, 2014.

I agree that all reputable positions about 0^0 should be mentioned.

However, floor has nothing to do with this: it is explicitly defined to have a discontinuity at every integer, with floor(x)=x where x is integral, even though the left limit is of course x-1. --Macrakis (talk) 14:20, 23 December 2014 (UTC)