Talk:Exponentiation/Archive 2015

Archive 2010

←

Archive 2013

Archive 2014

Archive 2015

Archive 2016

Archive 2017

→

Archive 2020

Domains

1. ℤ × ℕ (taking the convention that 0 ∈ ℕ), and, in general, R × ℕ for any ring R, or R × ℕ⁺ for any rng R.
2. (ℝ × ℤ) \ ({0} × -ℕ⁺), or, in general, (R × ℤ) \ ({0} × -ℕ⁺) for any field R. (Here, 0⁰ is usually 1, but is sometimes undefined.)
3. (ℝ⁺ × ℝ) ∪ ({0} × ℝ⁺). Here, 0⁰ is usually undefined, but sometimes 0 or 1.
4. (ℝ × ℚ₂ \ ({0} × ℚ₂^≤0); here, 0⁰ is usually undefined, but sometimes 1.
5. (ℂ × ℂ) \ ({0} × (ℂ \ ℝ⁺)) with two versions; one using the principle value of the logarithm, and one using the multivalued version of the logarithm. Sometimes 0^z is considered to be 0 if z has positive real part, rather than just being real, but that seems non-standard. 0⁰ is rarely defined in this version.

Generally, 1 is contained in 2; 2 is almost contained in 3 and in 4; and 1, 2, and 3 are usually compatible with 5A, 5A is contained in 5B, and 3 is almost contained in 5B. We still need references for these definitions and for the relationships, but if I, for example were to self-publish the relationships, that might be adequate. — Arthur Rubin (talk) 23:36, 22 December 2014 (UTC)

References

While one can define arbitrary functions that extend well-behaved exponentiation to a larger domain, but I would suggest that if it does not retain certain fundamental properties, it can be regarded as "trying too hard". Without such a restriction, pretty much any function can be called exponentiation. Let's therefore restrict ourselves to at least the property that

$x^{b}x^{c}=x^{b+c}$

whenever the left is defined.

This leaves us with effectively three cases – first, 1 & 2 can be combined by extension of the definition to zero and negative integer exponents where the identity exists and division respectively are defined for the base; second, 3, except that the exponent can be extended to any real power-associative algebra, and third, 5 being the multi-valued case (and which can presumably be extended beyond complex numbers). Case 4 fails the criterion. The multivalued treatment is interesting, but not being a function could be eliminated in this discussion. So I'd say there are only two general cases of exponentiation as a function. And except in the restriction to domains that fall entirely in the intersection such as Bo gave in an earlier response to me (which at the time I could not have been bothered to waste the keystrokes mentioning because this seemed so obvious to me), these two cases cannot be combined into a single function without breaking the property. —Quondum 02:43, 23 December 2014 (UTC)

1 and 2 can be combined if the ring in 1 has no zero-divisors, by making the ring in 2 the quotient field. But the multiplication in all cases, except 5A (principle value), satisfies both

1. $(xy)^{a}=x^{a}y^{a}$, and
2. $x^{a+b}=x^{a}x^{b}$,

(where all are defined), and where 0⁰, if defined, is 0 or 1.

Actually, equation 2 hold even for 5A

For #4, it can be verified using the fact that if you define the parity of a/b to be the parity of a, then it satisfies all the usual properties of parity:

positive^any is positive

negative^even is positive

negative^odd is negative

even + even = odd + odd = even

even + odd = odd + even = odd

I think

$x^{ab}=\left(x^{a}\right)^{b}$

(where all are defined) holds for all except 5(principle value), but I'm not sure, and where we say that all values on the left are attainable on the right for the multi-valued version.

— Arthur Rubin (talk) 03:38, 23 December 2014 (UTC)

Your first identity does not hold in general when x is drawn from a noncommutative ring, so if this identity is required, it would imply a restriction on the domain (specifically the base).

The rational exponent case may be a sort of edge case that fits neither of the main categories; I consider it "trying too hard" (and is rather pathological, including about the limitation on where every expression is defined) to be of interest, but others may find it interesting (I'm not arguing against your logic, but have not verified it).

The final identity fails for noninteger exponents in many cases, mainly because the codomain of the first exponentiation is not contained in the the set for the base in the domain of the second exponentiation. One may be able to find many esoteric cases where it works, but the only one I feel has merit is when the base is positive real, and the exponent is real or a subset. I have not verified this, but I'm pretty sure that it fails in the multivalued complex case.

That all said, hunting down the definable cases is not really sensible here; my intent is really only to bring attention to two important classes of exponentiation that are incompatible (i.e. that some fundamental identities break when one tries to combine them into one function), and that consequently to treat all exponentiation (even on the reals) as a single function does not make sense, even when these agree on the intersection of their domains.

—Quondum 04:49, 23 December 2014 (UTC)

I accept your analysis about the multi-valued version, as it is difficult to say what "equality" means, and your note about my equation one restricting the domain for version 1 (and possibly 2) to be commutative.

In mathematics, unlike most other subjects, we can generally discuss what is true on talk pages, without it going in to the article. However, we need reliable sources to map these discussions into the article. I'm saying I believe reliable sources exist for each of the 5 or 6 domains of exponentiation, but not necessarily for relationships between them. We can discuss what should be in the article if we could find sources, but we need to find the sources before we place these in the article. Most of us are probably sufficiently expert for this analysis to be used to support the article, if it were published, even self-published. As has been pointed out (even by me), my Google-fu is not the best, so I don't know what has been said about the different domains of exponentiation.

I'm sure domain 1 is the one used in the article on polynomial rings, and 2 in the obvious generalization to negative exponents (although, I admit, I haven't actually seen it in the literature.) 3 and 5 are the definitions generally used in real and complex analysis (respectively) using variations of the formula $x^{y}=\exp(y\log x)$ and 4, although I admit I haven't seen it used except in the solution of math puzzles, is an attempt to unify the concept of the well-defined cube root over the reals to a consistent exponential function. A precise definition is

$\left[{x^{a/b}}\right]_{4}=\left[{(\operatorname {sgn} x)^{a}}\right]_{2}\left[{|x|^{a/b}}\right]_{3}$

for x ≠ 0 real, a and b integers, with b odd. — Arthur Rubin (talk)

This discussion is in the spirit of background discussion on the talk page, such as you suggest (and hence does not need sourcing)).

In contexts such as power series and polynomial rings, a definition in terms of repeated multiplication is natural, including the empty product in any ring with identity.

Just by way of completing the thought, the natural interpretation of $\left(x^{a}\right)^{b}$ in the C×C multivalued case (with multivalued input) seems to be as the union of the multivalued sets obtained from the multi-valued second exponentiation applied to each of the elements of the input set. Using this interpretation/definition, one sees that $x^{ab}$ is generally a proper subset of $\left(x^{a}\right)^{b}$. I would define equality as equality of sets in this context, so the rule is retained as $x^{ab}\subseteq \left(x^{a}\right)^{b}$.

I presume your subscripts in the rational equation mean the class/definition of function; an interesting combination/definition. I guess it is reasonably natural to use exponential notation in the context of real nth roots with n restricted to being odd (a curious restriction: omitting even roots!), so we can distinguish three nonequivalent definitions that satisfy the three equations. I have omitted the multivalued case in this count, since it must be treated quite differently.

In all this, it seems clear that one needs to track which definition (and hence which function) is being used. —Quondum 22:34, 23 December 2014 (UTC)

Guys, please, take it off-line. While comments on talk pages are not subject to a sourcing requirement per se, they do have to be related to how the article should (or should not) be changed. I do not think there is any reasonable prospect of putting this sort of speculative extension into the article itself.

(I don't think Arthur self-publishing it is remotely good enough. That might be OK in an article on some very specialized topic, but not on exponentiation. We need to wait for a convention to be adopted by a significant segment of the mathematical community.) --Trovatore (talk) 22:42, 23 December 2014 (UTC)

We need to find the definitions in the literature, but I think self-published comments by an expert should be sufficient to describe connections between the definitions. For the definitions themselves, importance needs to be established, as well as accuracy; but there is no question that the connection between different, actually used, definitions of exponentiation is important, so a self-published source could be used. It would be better if a single sourced used more than one of the definitions, though. — Arthur Rubin (talk) 23:51, 23 December 2014 (UTC)

I don't think there's any way this is going to fly. The definitions themselves are not "actually used" in a sufficiently notable way to be included in an article about as general a topic as exponentiation. If they were, one of us would already know about it.

Arthur, I really think you should drop this, as it relates to this article at least. This seems like recreational math more than anything else. I don't see any real mathematical progress coming out of a definition that distinguishes cases based on whether the exponent has an even denominator. But if it does happen, fine, report it once it's an accepted part of the theory surrounding exponentiation. --Trovatore (talk) 00:04, 24 December 2014 (UTC)

Actually, the article cube root makes note of the fact that the principle value of ∛-8 is -2 if "-8" is considered real, and as 1 + i √3 if "-8" is considered complex. As cube root is a specialization of nth root, which is a specialization of exponentiation, we have a link. I'm afraid it's not sourced, there, either, and may be WP:SYNTHESIS, but the individual facts can be sourced. — Arthur Rubin (talk) 23:42, 14 January 2015 (UTC)

To state that the nth root is a specialization of exponentiation, especially for negative values, is pushing it too far, IMHO. To do so sensibly and rigorously is also apparently a severe challenge. It is also not related to the original thought in this thread. I'm with Trovatore on this; this direction does not have encyclopaedic relevance. —Quondum 06:41, 15 January 2015 (UTC)

Gelfond–Schneider theorem

It should appear in this article somewhere:

It reads: If a is algebraic and not 0 or 1, and if b is algebraic and irrational, then a^b is transcendental.

We may need to clarify which definition we are using, though. — Arthur Rubin (talk) 21:12, 29 January 2015 (UTC)

 Done. Although the theorem is true for non-real a and b, I have inserted this as a subsection of the section on positive a and real b, because this is sufficient to understand the theorem. However, I have given the complete statement, with the required clarification. D.Lazard (talk) 17:07, 13 March 2015 (UTC)

The undefiners have a lot of editing to do

There is no controversy in, say, Polynomial#Definition that $\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+\cdots +a_{n}x^{n}$, implying that x⁰=1 for all values of x and not only for nonzero values of x. So the undefiners of 0⁰ have a lot of editing to do in wikipedia. Bo Jacoby (talk) 16:28, 7 November 2014 (UTC).

There's no problem in the "different domain" representation of exponentiation. That one uses the X × N ↦ X exponential function (for X an arbitrary ring), which is defined at (0, 0). — Arthur Rubin (talk) 01:40, 22 November 2014 (UTC)

IMO, it would have been good to have emphasized the "different domain" in the article (i.e. X × Z → X vs. R_>0 × X → X) definitions more in the article. I've campaigned for it, but ... —Quondum 02:12, 22 November 2014 (UTC)

The "different domain" approach is WP:OR. Bo Jacoby (talk) 09:22, 20 December 2014 (UTC).

Going around pushing what Donald Knuth said as if he is God is just plain stupid. His experience is in discrete maths and computing with hardly anything at all to do with continuous functions. He is not in the same league as Cauchy as far as this is concerned. The different domain approach is how it is actually done in practice. I have no source telling me that London in England is different from London in Ontario but we have separate articles on them. If I was ever to accept an argument from a person like Knuth it would be from William Kahan not him, he has an argument why returning 1 is probably a better choice in computing even for the real case which basically is the same as the argument in the article about analytic functions, but he never says it actually is 1 and nowadays in the IEEE standard there are separate functions pown and powr for integers and reals where powr(0, 0) returns undefined, as well as the original Unix pow function which is defined if it is defined by either pown or powr. Dmcq (talk) 10:27, 20 December 2014 (UTC)

Please note (the main text is misleading on this point) that Cauchy did not write that 0^0 is undefined. All he did is to point out the nowadays uncontroversial fact that when f(x) and g(x) both converge to 0, that this is not enough information to determine lim f(x)^g(x). While this led some to conclude "undefined", don't include Cauchy in the "undefined" camp; Cauchy did not write that 0^0 is undefined. MvH (talk) 01:43, 22 March 2015 (UTC)MvH

The different domain approach is in practice done with exponentiation exactly as it is done with other compositions such as addition. Addition of real numbers is another domain than addition of integers, but the same symbol (+) and the same name (addition) is used, and nobody need to distinguish between the two kinds of addition. 0+a=a and 0+0=0 no matter whether 0 is considered an integer or a real number. In computing, however, where reals are approximated rather than represented exactly, the discontinuity of the function 0^x for x=0 makes it reasonable to warn that powr(0,0) is dangerous because 0 may be an approximation to some nonzero number x. 0^x depend critically on whether x=0 or x>0. Bo Jacoby (talk) 17:41, 20 December 2014 (UTC).

I have not reviewed the edit, but am only reacting to Bo's statement here. The real/integer example is not appropriate: Z×Z can be embedded in R×R in your example. No such embedding is possible for exponentiation. —Quondum 19:21, 20 December 2014 (UTC)

Just trying to force stuff in that is not supported by the sources is disruptive. That FAQ about 0^0 doesn't even mention complex numbers and yet you stuck it in as supporting the value of 1 for them despite the books either saying it is undefined or arbitrarily setting it to 0. Why on earth that should be included on a level with books and peer reviewed articles I on't know, it just gives reasons for one side and does not even discuss the other so it doesn't really present an argument just a biased view. Dmcq (talk) 19:40, 20 December 2014 (UTC)

(edit conflict)Comments. Apparently, an edit war is starting. IMO, Dmcq's version is the best one, as Bo Jacoby suffers of several logical weaknesses: firstly, the headings are confusing, as a limit is a numerical value. Therefore, the reader, that has not read this talk page, cannot understand what is the difference between these two headings. Moreover, the references to "branches" in the complex case involves the notion of continuity (required to distinguish the branches), and thus the notion of limit (required for defining continuity). Thus the paragraph on the complex domain is misplaced in Bo's version. As Dmcq version does not suffer of similar weaknesses, I'll restore it.

To Bo: When writing The "different domain" approach is WP:OR, you are wrong: IEEE754 is not OR, and is a reliable source. It defines different exponentiations depending on the nature (type) of the exponent. Your example of the addition is a bad one, as the result does not depends on the nature of the arguments. But this is not true for square root and exponentiation: formulas may be true or not depending of the nature of the variables appearing in them. For example $|x|={\sqrt {x^{2}}}$ is a true formula if x is a real number, and is wrong if x is viewed as a complex number, whichever definition of the square root is chosen in this case. This is the fact that the exponentiation has a different definition in the case of integer and irrational exponents, which makes necessary to take care of the nature of the 0 in the exponent. Your above sentence about computing is almost nonsense: in computing, the reals are represented approximatively by floating points number, while integers are represented by machine integers. Therefore, to call pown on 0 obtained as the result of an approximation, one needs to converting it first to an integer. Every programmer should know that type conversions need to be careful. D.Lazard (talk) 19:53, 20 December 2014 (UTC)

I just looked up up the editor of the maths FAQ on 0^0 and he is a professor of computing and researches constraint programming and search engines and suchlike. He is not an analyst nor is he in practical engineering or anything like that associated with continuous functions. There seems to be some sort of domain difference between the people saying it is indeterminate and the ones saying it is defined and they just don't read or do anything of real concern for the other side. We can't have the article saying one side is correct in all cases. Dmcq (talk) 20:03, 20 December 2014 (UTC)

The debate only involves low-level undergraduate math (continuous functions). Certainly anyone trusted with the FAQ of sci.math (which at the time was filled with high level mathematicians) is more than qualified to discuss such a basic topic. As for powr(0,0), this is besides the point, powr(-1, 2) = NaN and few people are seriously proposing that (-1)^2 should be undefined in the main definition of x^n. So powr has really no bearing on this issue. MvH (talk) 03:15, 22 March 2015 (UTC)MvH

I would also like to try yet again to correct the misapprehension evident in them saying about sending back NaN for powr(0,0) because they might really have meant a number near 0. That is a secondary reason, but even for powr(0,0) NaN is wanted as a marker that the value is not defined at that point. People do not want an arbitrary value back. For instance if their function is practically equal to (e^(-1/x))^x near 0 the limit at 0 is 1/e and they do not want a discontinuity at 0 where it suddenly becomes 1 - they want either nothing or 1/e. The engineering solution is to just use the straightforward function and be warned about problems. Experience will have shown no discontinuity at 0 so why should they have put any thought into it? They are not given an exam question about fining the limit at 0. There is no question about being 'in the context of limits'. If 0^0 is defined as 1 then the function is defined at the point and is discontinuous. The people just working with discrete numbers only seem to think of this as book questions with the limit point known where they have to do something strange. That isn't the order, one finds the problem first and then finds the limit. Defining the value replaces finding the problem with not finding the problem but still having one. Dmcq (talk) 20:33, 20 December 2014 (UTC)

To Arthur Rubin. I agree, there is no problem, and there is no need to present a problem. a⁰=1 without exceptions. Presenting the different domain definitions as different exponentiations would have been WP:OR. They have the same notation exactly because there is no problem.

To Quondum. Consider the sets A=ℕ×ℕ and B=ℝ₊×ℂ. Addition is defined on A, and on B, and the results of the two additions are equal on A∩B. 2+2=4 no matter which definition you choose. Likewise exponentiation is defined on A, and on B, and the results of the two exponentiations are equal on A∩B. 2²=4 no matter which definition you choose. In this respect exponentiation is like addition.

To Dmcq. You need to distinguish London in England from London in Ontario, but you do not need to distinguish 0⁰ from 0⁰.

The function f₁ defined by f₁(x)=(e^-1/x)^x is undefined for x=0.

The function f₂ defined by f₂(x)=(e^-1/x)^x for x≠0 and f₂(0)=e^-1 is continuous for x=0.

The function f₃ defined by f₃(x)=(e^-1/x)^x for x≠0 and f₃(0)=0⁰ for x=0 is discontinuous for x=0.

This problem is not solved by undefining 0⁰.

To D.Lazard. If you prefer the old version, then OK. What I want is that a⁰=1 be introduced as basis for the recursive definition aⁿ⁺¹=a⋅aⁿ. Any such attempt on my part was immediately reverted by the undefiners. I did not fight edit wars. IEEE is about computing and not about mathematics. In computing, integers are not reals. In mathematics, all integers are reals. The formula $|x|^{2}=x^{2}$ is a true formula if x is a real number, and is still true if the same x is viewed as a complex number. It is not true if the complex number x is not real. The fact that the exponentiation has a different definition in the case of integer and irrational exponents does not make it necessary to take care of the nature of the 0 in the exponent. One definition works, and no other definition objects. "Undefined" is not a definition. You wouldn't write something you consider undefined. If you do write x⁰ then you mean 1, even for x=0.

Bo Jacoby (talk) 16:17, 21 December 2014 (UTC).

Bo, I really dislike the recursive definition. It is not necessary and it is also not historical. When it was first used, x^3 was short for xxx. And x^3 y^4 is short for xxxyyyy. So x^n simply stands for the product of n copies of x. Rather than defining x^(n+1) as x times x^n, the more basic thing to do is to say that x^(n+1) is the product of n+1 copies of x. Things like negative n come later, fractional n later still. The very first definition (which makes sense for non-negative integers n) of x^n should simply be: the product of n copies of x. That's as short and simple as it gets, and historically the most accurate. MvH (talk) 03:24, 22 March 2015 (UTC)MvH

@Bo Jacoby: You are wrong, both as to my views, and as to the mathematics involved. It's true that Z is a subset of R, which is not always true in computation (although the 32-bit integers are a subset of the 64-bit reals), but there are at least three domains of exponentiation, and two of them clearly produce different results for (-2)^(1/3). That 0⁰ is defined for some domains and undefined in others is a minor refinement. Unfortunately, I have not found a reliable source for the fact that there are different domains, but there are reliable sources for each definition, and undefined can be part of a definition. Your view has some weight, and probably should be mentioned in the article, but it is not generally accepted. As for formulas, $\left({x^{3}}\right)^{1/3}=x$ is true in the reals, but false in the complex numbers, when x is a negative real.

As for expert comments, the question of "what is an expert" comes to mind. Knuth is an expert in computer science and pedagogy of computer science, but not necessarily in mathematics or even pedagogy of mathematics. — Arthur Rubin (talk) 05:49, 22 December 2014 (UTC)

Thanks Arthur Rubin. A 32-bit integer is not the same thing as a 64-bit real even if they represent the same number. So programmers do need to distingish between real 365 and integer 365. Mathematicians don't. I agree that there are three definitions for odd roots of negative number, e.g. (-8)^1/3=-2 and (-8)^1/3=1+i√3 and (-8)^1/3={1+i√3,-2,1-i√3}. But that is not what I am talking about. I am talking about the harm done by undefining 0⁰. As to Knuth, did you read Concrete Mathematics? Bo Jacoby (talk) 10:19, 22 December 2014 (UTC).

Concrete Mathematics says explicitly that "the goal is for each reader to become as familiar with discrete operations as a student of calculus is familiar with continuous operations". Like Knuth's note where he says 0^0 must be 1 the book is about discrete maths which is what Knuth is interested in. And yet his work there is taken by you as being definitive for all mathematics. You simply have not understood what I said above about engineering you have treated the problem as a set question in an exam where the limit is to be found. What equivalent harm can you point to by having 0^0 being defined only for discrete powers? In fact what harm at all? More to the point as far as Wikipedia is concerned WP:WEIGHT is the appropriate policy and with the current evidence that does not justify changing the article to say 0^0 is always 1. Dmcq (talk) 11:06, 22 December 2014 (UTC)

To Dmcq. Knuth knew about Cauchy and Cauchy did not know about Knuth. What you said about engineering seemed to be that undefining 0⁰ solves some problem, and I responded that it does not. The warning comes from the discontinuity of x^y at x=y=0, not from undefining 0⁰. The harm done by having 0⁰ being defined only for discrete powers is :

1. 0=0. Readers and mathematicians need elsewhere not distinguish between real zero and integer zero. Zero is both integer and real.
2. If $f$ is defined by $f(x)=\sum _{k=0}^{n}a_{k}x^{k}$ then $f(0)=a_{0}$ , not undefined.

Some sources do define 0⁰ and others do not, but no source supports your view that 0⁰ is sometimes defined and sometimes undefined.

The WP:WEIGHT says: "Generally, the views of tiny minorities should not be included at all, except perhaps in a "see also" to an article about those specific views. For example, the article on the Earth does not directly mention modern support for the Flat Earth concept, the view of a distinct minority; to do so would give undue weight to it." Nor should your view, that 0⁰ is sometimes undefined, be given undue weight. Bo Jacoby (talk) 14:21, 22 December 2014 (UTC).

You didn't even come close to addressing the problem, you turned it into an exam question about finding the limit at a given point without considering the phase before then. Indeterminate forms come to light because a function is evaluated and the form is seen whilst trying to evaluate a function, it is only then that a person might use L'Hôpital's rule to get a value as given by continuity. This is why having powr(0,0) returning NaN is the right thing to do for continuous functions for instance. The problem is not in the context of explicit limits, it is well before then in the context of continuous functions and only implicitly in the context of limits. We do not need to choose between authors on Wikipedia, we give both sides if there are two widely held points of view. 0^0 is not something like that the earth is flat. Dmcq (talk) 15:41, 22 December 2014 (UTC)

Unlike Dmcq, I don't see any harm in defining 0⁰ = 1, as long as it recognized the exponentiation function is not continuous at (0, 0). I agree with Dmcq, as opposed to Bo Jacoby, that it isn't usually done in the context of working in R⁺ × R or C × C but is almost always defined to be 0 in the context of R × Z^≥0 (where "R" is a ring (mathematics) or (R × Z) \ ({0} × Z^<0) (where R is a field (mathematics)). — Arthur Rubin (talk) 16:13, 22 December 2014 (UTC)

I was just pointing out the usual way of recognizing discontinuity in standard operations is by saying they are indeterminate rather than by having a function that gives a value and then having to wrap it in something else. After all one can define 0/0=1 if one wants to, it is a matter of convenience that one doesn't, if it were defined as 1 then one would have to write something like if(x≠0 or y≠0, x/y, raise error()). Doing different things in the discrete and continuous cases is the straightforward thing to do and they come from different definitions that make the differences natural. The number of cases where we want to do anything else is quite small. Dmcq (talk) 17:12, 22 December 2014 (UTC)

As User:Dmcq says, the way Wikipedia operates when there is a disagreement among reliable sources and authorities is to report on all notable positions, not to try to decide which is the best. It is also important to distinguish between different fields. Talking about NaNs is very useful in the context of floating-point calculation, but meaningless in mathematics. --Macrakis (talk) 18:15, 22 December 2014 (UTC)

My two cents worth here: I agree with this point. WP exists to report the current facts for a given topic, and that requires presenting the major authoritative interpretations of those facts, even (or especially) when they differ. Since there currently is no consensus that 0⁰ is defined among the authoritative mathematical sources, then we should say so. Whether or not 0⁰ = 1 works is not relevant to this article; what the experts have to say about it is. — Loadmaster (talk) 18:04, 29 December 2014 (UTC)

I see the assertions In mathematics, all integers are reals and It's true that Z is a subset of R. I disagree. These sentences are only standard abuses of language. The correct sentence is the set of integers is canonically isomorphic (as an ordered ring) to a subset of the reals. This allows to identify each integer with its image in the reals. Said otherwise, an integer may not be "an equivalence class of Cauchy sequences of rational numbers, which are themselves equivalence of classes of pairs of integers". In usual traditional mathematics, there is generally no harm to identify canonically isomorphic objects. However, there are some cases, such as this one, where this identification of the integers with a subclasses of the reals may generate some errors in mathematical reasoning. These cases occur very frequently in computer science, and this is essentially for this reason that type theory has been introduced. This does not changes mathematics, but introduces more rigor, when traditional mathematics become insufficiently accurate.

Bo, you are wrong when saying that IEEE is about computing and not about mathematics. IEE74, is a norm established for making mathematically correct the computer operations on numbers. Its authors, leaded by Bill Kahan are both mathematicians and computer scientists, and their work involve more mathematical rigor than usual in elementary mathematics. D.Lazard (talk) 18:59, 22 December 2014 (UTC)

Well I agree reals are not the same as integers. However IEEE754 was devised more to enable correct reasoning about the computer arithmetic and easy handling of some corner cases rather than corresponding exactly with how the reals are usually handled. If anything I'd characterize it as being like non-standard arithmetic with finite infinitesimals ;-) Once certainly doesn't have a separate negative zero in normal arithmetic though it can actually be quite convenient. Dmcq (talk) 20:29, 22 December 2014 (UTC)

D.Lazard, If "the set of integers is canonically isomorphic (as an ordered ring)" to some other set, then this other set is indistinguishable from the original set of integers. There is no point in denying that ℤ⊂ℝ. As you put it: "there is generally no harm to identify canonically isomorphic objects". Then you say: "However, there are some cases, such as this one, where this identification of the integers with a subclasses of the reals may generate some errors in mathematical reasoning". That is not true. There are no such cases.

I do not question the qualifications of IEE74 or of Bill Kahan, but computing with 64-bit floating point numbers is not mathematically correct, due to round off errors, and computing with 32-bit integers may produce overflow. In computing, powr(x,y) and pown(i,j) are different procedures, but in mathematics, exponentiation x^y covers both cases. The complexities of approximate computing need not and should not interfere with mathematics.

I hope that you are not suggesting that type theory be introduced in our article on exponentiation.

To Dmcq: The different domain approach, that 0≠0, is quite like that the earth is flat. Bo Jacoby (talk) 21:21, 22 December 2014 (UTC).

As far as I'm concerned, canonical embeddings can be ignored, so that we can say ℤ⊂ℝ. We're not dealing with foundations of mathematics here, so the specific representations of the domains are not important, nor are we dealing with the question of whether ℤ is an elementarily definable subset of ℝ. (It isn't.) However, we can define the exponential function on the domains mentioned in the next section. — Arthur Rubin (talk) 23:36, 22 December 2014 (UTC)

In non-standard arithmetic you can have a zero with a negative non-standard part and a zero with a positive non-standard part. They are still considered equal though as the standard parts are equal. This is the same as for the floating point zeroes, the positive zero is equal to the negative zero when the usual equality compare is done. It is also IEEE 754, D.Lazard just mistyped that. The difference between powr and pown has nothing to do with finite precision, IEEE floating point functions are defined for infinite precision and then rounded to the particular format. William Kahan is now emeritus professor of mathematics and of electrical engineering and computer sciences, basically he has received loads of awards and is very well qualified to talk about analysis and engineering. Actually his original proposal included having an option flag to support having an unsigned zero and infinity - the projective infinity - but it was abandoned later.

Donald Knuth never showed how to make what he was saying workable, he is only really interested in discrete mathematics, and people teaching calculus will just say 0^0 is an indeterminate, they don't say it is really 1 but is a limiting form that has to be dealt with specially in the context of limits. I don't know why you are going on and on about this. The literature supports two different views. The two different views are described. That's basically the end of the story as far as Wikipedia is concerned. Dmcq (talk) 00:28, 23 December 2014 (UTC)

To Dmcq: What did Donald Knuth say that he did not make workable? I explained many times why I am going on and on about this, so you should know by now. It is simply because you are wrong. The literature supports two different views, but no single source supports both views (just as no source supports both round earth and flat earth views). Neither should Wikipedia support both views. Apparently these "people teaching calculus" believe that all functions are continuous, since they think that $\lim _{(x,y)\rightarrow (0,0)}x^{y}=0^{0}$. Our Wikipedia article is confusing because you (and your fellow undefiners) insist that the formula x⁰ = 1 applies for nonzero x only. Bo Jacoby (talk) 04:50, 23 December 2014 (UTC).

I have tried to explain to you but you just don't seem to be able to see what I'm talking about. You jump straight to sticking in a limit operation without saying why a person would stick it in. You assume the final form without doing the working out. Donald Knuth has never tutored in calculus that I know of and has never tried to get some students past this. It is hand waving with a big hole in the centre. The article does not say that x⁰ = 1 applies for nonzero x only, it says that is generally agreed to be 1 since the power 0 is not part of the range of a continuous variable but the discrete number 0. Dmcq (talk) 12:02, 23 December 2014 (UTC)

I'm confused as to what is being suggested in this thread with regard to editing the article. I see a lot of mathematical debate about defining 0⁰, but do not see that this is about any change to the article. In particular, from an encyclopaedic perspective, we cannot proceed to define it in any particular way. What we have is a pretty reasonable reporting of the current state. —Quondum 14:32, 23 December 2014 (UTC)

To Quondum. The definition x⁰=1 as the basis for the recursive definition of exponentiation with positive integer exponents x¹⁺ⁿ=x⋅xⁿ was reverted by those who insist that x=0 must be an exception. That is what all this is about. Bo Jacoby (talk) 19:19, 25 December 2014 (UTC).

To Dmcq. Your example above was this: "For instance if their function is practically equal to (e^(-1/x))^x near 0 the limit at 0 is 1/e and they do not want a discontinuity at 0 where it suddenly becomes 1 - they want either nothing or 1/e. The engineering solution is to just use the straightforward function and be warned about problems". I answer that (e^(-1/x))^x near 0 has nothing to do with 0^0. The limit of (e^(-1/x))^x is not equal to 0^0, even if the limits of both e^(-1/x) and x are 0. Explain if this is not correctly understood. Bo Jacoby (talk) 19:19, 25 December 2014 (UTC).

I did explain it there quite adequately I thought. I explained in the past a couple of times too but you still go on about it. Try saying what you said just using the word continuous as far as possible not limit and forgetting that you knew that there was a problem at 0. Continuous functions are always in the context of limits at all points they are defined. That's why continuous exponents is the right heading for the section rather than in the context of limits. The simple reason the article isn't like how you want it is because current reliable sources aren't in any way united on the subject and Wikipedia tries to reflect what's out there. Personally I'd be a lot more convinced your view was workable if I saw a calculus course which showed how to deal with indeterminate forms without assuming that calculating the function would give an error at the relevant points. I'm perfectly aware one can have a value at a point which is not equal to a limit there for instance with the signum function and it would probably save a lot of bother if the continuous power was written as $e^{y\log x}$ but as the say ' if wishes were fishes...'. Dmcq (talk) 15:27, 26 December 2014 (UTC)

To Dmcq. I go on about it because you are mistaken. You request me to forget that I know there is a problem at 0. If I forget that, then I may believe that $\lim _{(x,y)\rightarrow (0,0)}x^{y}=0^{0}$, which is not the case, because exponentiation is not continuous everywhere. Wikipedia should reflect the consensus that if $f(x)=\sum _{k=0}^{n}a_{k}x^{k}$ then $f(0)=a_{0}$. This means that 0⁰=1. Denying this gives you a lot of editing to do. Bo Jacoby (talk) 19:49, 28 December 2014 (UTC).

The 'forget about 0' was supposed to be a guide towards the problem but you just won't go there. I've told you that I understand about the difference between the limit and the value and I wasn't talking about that. I'm sorry I've tried again and again but I seem to be unable to explain the problem to you, you insist on knowing about the problem at 0 and sticking in a limit and won't go to before then with just the formula and continuity. I will not try again - I believe any further attempt would be a waste of my time. The article does say that sums should be done like you say but the sources do not have the same consensus about the conclusion you draw. I don't have to do any editing - I just have to point to the sources in which it is pretty clear there is no such consensus and the Wikipedia policy about showing the major opinions on a subject. Dmcq (talk) 23:43, 28 December 2014 (UTC)

To Dmcq. You want me to "go to before then with just the formula and continuity". A formula involving x^y and continuity may define a limit for x=y=0 which is not 0⁰. So continuity has nothing to do with 0⁰. Do you agree? The sources need not explain that "if $f(x)=\sum _{k=0}^{n}a_{k}x^{k}$ then $f(0)=a_{0}$" means that $0^{0}=1$. Just insert $n=0$ and $a_{0}=1$ and $x=0$ to show that $f(0)=1\cdot 0^{0}$ and $f(0)=1$. Do you agree? Bo Jacoby (talk) 13:55, 29 December 2014 (UTC).

This is Wikipdia. It is up to us to follow the sources as per WP:WEIGHT. Dmcq (talk) 20:20, 29 December 2014 (UTC)

To Dmcq. You are repeating yourself. My answer was and is: The consensus that x⁰=1 without exception is expressed in Polynomial#Definition, as I wrote above, and in countless uses in the literature. Your point of view leads to 0≠0. Read what WP:WEIGHT has to say about flat earth believers. Bo Jacoby (talk) 08:37, 30 December 2014 (UTC).

Knock it off, Bo Jacoby. You have presented the same argument a number of timrs, and nobody agrees. The mathematical literature confirms that in some contexts, 0⁰ is 1, and in other contexts is undefined. Actually, to be precise, some literature supports each view. It is wrong to assert it is 1 in all contexts, or even most significantly presented contexts. — Arthur Rubin (talk) 10:07, 30 December 2014 (UTC)

Arthur, when you claim it is wrong to assert 1 in all contexts, you imply that there are contexts where 1 is inconsistent. But such contexts do not actually exist (if they did exist, then the binomial theorem would be inconsistent too, but such situations have never been found) (discontinuities haven't been considered contradictions in over a century). MvH (talk) 03:23, 13 March 2015 (UTC)MvH

MvH, I'd suggest that you knock it off too. It is inconsistent with how it is defined in some contexts, and to assert otherwise in this article would be "wrong", by which I mean contrary to the standards required by WP. —Quondum 14:38, 13 March 2015 (UTC)

There's no need to attack me like this when it was perfectly clear what I meant with "inconsistent" (namely: leading to mathematical contradictions). Arthur implied that defining 0^0 once and for all is inconsistent in that sense. It is important to add here that the textbooks that leave 0^0 undefined never rigorously prove that this undefined is necessary to avoid contradictions. I'd like WP to point out that the textbooks omit that part. I don't see how that would be against WP standards. MvH (talk) 04:09, 14 March 2015 (UTC)MvH

Okay, let's back up a little: the consistent interpretation of what Arthur meant by "wrong" is not that it is algebraically inconsistent, but effectively that it would be incorrect to make a false assertion (that it is defined in all contexts) in the article. Are you suggesting that is not against WP standards to analyze what is omitted from textbooks? —Quondum 07:17, 14 March 2015 (UTC)

To Arthur Rubin. Exactly how many people agree with me, or with Dmcq, nobody knows, but you are free to compromise your credibility by pretending that you do. It seems that many people, including Dmcq, agree that $f(0)=a_{0}$ when $f(x)=\sum _{k=0}^{n}a_{k}x^{k}$. It also seems that 0=0 is widely accepted as a mathematical truth although some undefiners deny it. Their point of view, that 0≠0, is revolutionary to scientific practice, like that of the flat earth believers. My two questions to Dmcq, 13:55, 29 December 2014 (UTC), remain unanswered. Bo Jacoby (talk) 13:03, 31 December 2014 (UTC).

Bo, I too say drop it, lest you get yourself labelled as a troll. It is apparent to everyone but you that you are missing the point being made to you. Your missing the point at a mathematical level I can understand, but your continued excessive belabouring of a point that you will clearly not get consensus for on this talk page is becoming inexcusable. You are fortunate that you still seem to be regarded highly enough to merit a reply. —Quondum 13:49, 31 December 2014 (UTC)

Bo has been fruitlessly arguing this point for over eight years now; don't stop him now, it would ruin the streak! — Steven G. Johnson (talk) 02:09, 16 January 2015 (UTC)

Yes, and in these years Steven has insisted that 0≠0. Bo Jacoby (talk) 15:57, 14 March 2015 (UTC).

Caret

Please replace all superscripts here to carets. — Preceding unsigned comment added by PiotrGrochowski000 (talk • contribs) 16:23, 6 April 2015 (UTC)

You've got to be kidding. Actually, looking at your edit history, you really appear to believe that superscript notation has been superseded by caret notation. Please acquaint yourself with the MoS before continuing with your personal reformation of notation on WP. —Quondum 17:14, 6 April 2015 (UTC)

Editing detail

MvH, my edit is almost exactly a revert, but is not meant in this light. I'm not objecting to the content, nor to the removal of references to indeterminate forms, but what I removed is exactly and unambiguously implied by the rest of the sentence. Thus, the sentence now says the same thing exactly, only more elegantly. —Quondum 02:18, 28 April 2015 (UTC)

Reference to Cauchy's work

Quondum, I noticed you reverted my edit with the quote "ou aurait trouvé que cette dernière devient indéterminé", but this doesn't refer to 0^0, instead, it refers to a limit of that form. The idea that 0^0 itself is indéterminé did not appear in Cauchy's cited work, the main page is wrong on this issue. MvH (talk) 18:37, 23 April 2015 (UTC)MvH

MvH, perhaps you are referring to this revert by Dmcq? —Quondum 18:48, 23 April 2015 (UTC)

Oops, sorry for mixing up your names. MvH (talk) 23:12, 23 April 2015 (UTC)MvH

Have we got a source giving your interpretation of what is said? Dmcq (talk) 21:59, 23 April 2015 (UTC)

Yes, Augustin-Louis Cauchy, Cours d'Analyse de l'École Royale Polytechnique (1821). In his Oeuvres Complètes, series 2, volume 3. MvH (talk) 23:12, 23 April 2015 (UTC)MvH

I think probably you are reading too much into that the complete two word phrase 'indeterminate form' was not used till later. Cauchy seems to me to talk about the singular value being completely determine din some cases and in other cases that it was indefinite and one would have to evaluate the limit to find the value. It's not how one thinks about the business nowadays which is that the limit need not equal the value at the point. Dmcq (talk) 22:43, 23 April 2015 (UTC)

The two issues (1) the value of 0^0, if any, and (2) the issues with limits of f(x)^g(x), are entirely independent. Regardless of what one decides for (1), either way one has to be careful with limits of f(x)^g(x). There is no hint in the Cauchy's cited work about issue (1), it is clearly about (2). Yet, the main article strongly suggests that Cauchy thinks 0^0 is undefined (or indéterminé). Someone made the step from (2) to (1) but this didn't happen in Cauchy's cited work. MvH (talk) 23:27, 23 April 2015 (UTC)MvH

Nicely explained. I fail to see the "step from (2) to (1)" in the article, i.e. that "the main article strongly suggests that Cauchy thinks 0^0 is undefined". —Quondum 00:13, 24 April 2015 (UTC)

The phrase "indeterminate form" has different meanings, depending on who you ask and depending on when you ask. The dictionary says that "indeterminate" means "not defined" so that is certainly one of the meanings that is being conveyed, adding to the overall confusion. MvH (talk) 02:49, 24 April 2015 (UTC)MvH

In WP, the phrase indeterminate form should only be interpreted as defined in that article (sense (2)), meaning of indeterminate limit behaviour, and never as an undefined expression (sense (1)). Thus, one may refer to "the indeterminate form 0⁰", and to "the expression 0⁰", and one would be referring to distinct concepts. I do not see that this article does anything wrong in this regard. —Quondum 04:07, 24 April 2015 (UTC)

If (2) was the intended meaning, then why is the issue "indeterminate form" even brought up in section "Zero to the power of zero"? If (2) was the only intended meaning, then it would have no bearing on this issue. The key step in the argument for "undefined" is this: Demonstrate (2), then conclude (1). We make that key step in section "Continuous exponents", then attribute it to Cauchy in the next section "History of differing points of view". It is true that someone must have been the first to make this leap of logic, that (2) somehow implies (1), but I could not find this in Cauchy. MvH (talk) 10:52, 24 April 2015 (UTC)MvH

Why did you ignore what I wrote below? Dmcq (talk) 11:27, 24 April 2015 (UTC)

I don't think Cauchy's text backs you on a split between (1) and (2). At the start of the section about singular values he says it is a value given by the limit when the value of the function is not immediately determined by the value of the variables there. You are reading the modern way into Cauchy's text. He considered the value of 0/0 0^0 etc to not be immediately determined. Splitting (1) and (2) is a more modern idea. Dmcq (talk) 09:59, 24 April 2015 (UTC)

II found this (with the help of Google translate): "In case the definition given could not immediately provide the value of the function that we consider, we look at limit or limits". You assume that he also meant the converse of this statement. If Cauchy disagreed with the (common at the time) view 0^0=1, then where did he say that, or imply that? (in math, a statement does not imply its converse). MvH (talk) 21:10, 24 April 2015 (UTC)MvH

I think all that we should read into Cauchy is that he felt that 0⁰ was not to be thought of as self-evidently equal to anything specific. However, I don't see where in the article we are attributing anything other than that to him. Perhaps you can point to the wording that you are objecting to? —Quondum 21:21, 24 April 2015 (UTC)

If he thought that 0^0 was 1 then the definition would immediately provide the value of the function. There would be no singular value. There is no need to look at any converse. Dmcq (talk) 21:47, 24 April 2015 (UTC)

Quondum, I don't know what Cauchy felt. Dmcq. I don't know what Cauchy thought. All we know is what he wrote. MvH (talk) 23:29, 24 April 2015 (UTC)MvH

We don't need to know what he thought about the value of 0⁰; all we're using what he said for is that the limiting form is indeterminate. This is sufficient for us to use to state that he thought the limiting form was indeterminate, and also that there is no limiting value for x^y as x, y → 0. This is important in the context of continuity. It is also somewhat useful in a context of determining whether to define the expression 0⁰ in the context where we would like to define a function such that it exhibits topological continuity at every point of its domain (it might be good to put it this explicitly in the article). —Quondum 00:14, 25 April 2015 (UTC)

Quondum, I agree that there is no limiting value for x^y as x,y -> 0, that this is important in the context of continuity, and that it is somewhat useful in the context of 0^0. The phrase "the limiting form was indeterminate" is ambiguous because this might be a statement about limits, or might be a statement about 0^0 itself, or both. MvH (talk) 12:18, 25 April 2015 (UTC)MvH

There are dozens of proofs (and disproofs) for P=NP. Wikipedia doesn't present any of them as though it was a valid proof.

On the surface, the 0^0 section is a balanced section as it contains support for both viewpoints. The problem is that the argument on the main page that 0^0 is undefined reads like a mathematical proof. That's a problem, because we know that (a) there is a gap in the proof, (b) we know exactly where the gap is, (c) this gap has been pointed out in the literature [failure to draw a distinction between two separate meanings, see Knuth], and (d) the main page still presents the proof as though it was valid, without even a footnote at the gap.

The first line "When the form..." in the proof is somewhat ambiguous, then come limits, which is fine, and then comes the conclusion "So 0^0 is an indeterminate form". Looking up the word "indeterminate" in the dictionary tells the reader that this line means that 0^0 is undefined. The proof is defended by saying that "indeterminate form" means one thing, when the plain dictionary meaning says something else, it is a classic bait and switch.

At this point, readers will think that we've already proved "undefined". The history section tells the reader some people (Cauchy, Benson) agree with this and others (Knuth, Mobius) do not, and gives two explanations for this discrepancy ("Mobius made an erroneous claim", and Knuth's "drawing a distinction"). MvH (talk) 12:21, 25 April 2015 (UTC)MvH

I think we might be interpreting the wording in the article differently.

• "... the argument on the main page that 0^0 is undefined ..." – The article should not make any assertion that it is undefined and should aid the reader in not thinking that it does so; it should only point out the implications of any given choice of definition. However, I still fail to see how you can possibly read this into what the article says.
• I've made an edit to reduce ambiguity: "So 0⁰ is an indeterminate form" I've changed to "Thus, a limiting expression of the form 0⁰ is an indeterminate form." Does this help reduce the potential for confusion with the defined value of the expression 0⁰?

I think it is necessary to make it clear in the article when we are talking about a limit of an expression, and when we are talking about the value of a function at that point. I interpret "the form 0⁰" as exclusively the former, whereas you may tend to the latter. —Quondum 15:50, 25 April 2015 (UTC)

The implication of assigning a value to 0^0 is it assigns a function value at a non-removable discontinuity. Another issue that needs clarification is this: The main reason 0/0 is undefined has nothing to do with limit arguments (0/0 was already undefined long before limits were considered). MvH (talk) 12:40, 27 April 2015 (UTC)MvH

MvH, That you should even think that something this obvious would need any clarification suggests perhaps that some of the ambiguity you see derives from a failure by you to distinguish adequately between an expression and a limiting form, e.g. between 0/0 as an expression and the form 0/0. It is unfortunate that we use the notation "the form 0/0". I bet that if the notation was different, none of this would have needed discussing. —Quondum 16:18, 27 April 2015 (UTC)

Limit arguments play a prominent role in the argument to undefine 0^0. If you agree that limits are not the main reason that 0/0 is undefined, then should limit arguments matter for 0^0? MvH (talk) 18:06, 27 April 2015 (UTC)MvH

For the form "0/0" one can get away with some ambiguity because 0/0 doesn't make sense as a number, and isn't used in formulas that produce numbers, so if one sees "0/0" one can pretty much guess that this refers to a limit requiring l'Hopitals rule (computer algebra system probably won't guess this and simply give an error message, but that's a perfectly valid response). This is different for 0^0, which does make sense as a number, one that is required for commonly accepted formulas to be true. Here, if a computer algebra system returns anything other than 1, it means that well established formulas either can't be implemented, or can give unexpected results. MvH (talk) 18:27, 27 April 2015 (UTC)MvH

In the context of evaluating the value of a continuous function occurrences of 0/0 or 0^0 are indeterminate forms. They do not have values. Even if 0^0 is 1 in other contexts it is not 1 in this context. This is before any mention of limits. The thing that is relevant is the continuity of the function. There is no particular need to distinguish between expressions and forms. In the context of limits as opposed to continuous functions there may be discontinuities and at discontinuities a function is not equal to a limit or no limit may even exist. In calculus we are basically dealing with continuous functions for this reason together with it being used in algebraic manipulations, 0^0 is not defined even though we do also deal with discontinuous functions in calculus. Dmcq (talk) 17:03, 27 April 2015 (UTC)

When you talk about different contexts, it is as though you view 0 as a number in some contexts but as a function in other contexts. This is simply not rigorous. The only thing that has been established is that x^y has a non-removable discontinuity at the origin. This neither proves nor disproves that 0^0 has a value. MvH (talk) 18:06, 27 April 2015 (UTC)MvH

I'm talking about 0 as a number, nothing else. I don't see where you get the idea I'm talking about it as anything else. You evaluate a function. You get the number zero to the power zero. You are in the context of calculus. The answer is unknown, not determined. Zero to the power zero is an indeterminate form. It is not 1. Zero to the power zero is not given a particular value in calculus. Doing so would give problems just like not doing so makes life difficult elsewhere in maths. If you were dealing with the exponential function where 0^0 is 1 then the value would be 1. We could say things like determine the limit near that point but there would be no good reason for saying it was an indeterminate form as we wouldn't normally be interested in the limit as the function has a defined value there. Dmcq (talk) 19:11, 27 April 2015 (UTC)

Your claim "doing so would give problems" has not been rigorously defined/demonstrated. All that was demonstrated is that the function x^y has a non-removable discontinuity at the origin. Why not simply write exactly that, in unambiguous terms? MvH (talk) 20:27, 27 April 2015 (UTC)MvH

I get the feeling you're a bit like Alice in Wonderland trying to believe two opposite things and reconcile them. Yes one can say 0^0 is always 1 and that there is something called the indeterminate form 0^0 which is different from etc. but it removes the basic ideas of continuity and simplifying expressions which drove the whole business. I have not seen any halfway reasonable way to square that circle and I would put a windy road symbol beside it. Dmcq (talk) 19:43, 27 April 2015 (UTC)

In evaluating the limit of f(x)^g(x) (when f(x) and g(x) go to 0), where is the error: step 1 student replaces f(x)^g(x) by 0^0, or step 2, student replaces 0^0 by 1? I hope that we can agree on at least one thing, that step 1 is wrong. MvH (talk) 20:38, 27 April 2015 (UTC)MvH

I disagree with Dmcq to some extent: leaving 0⁰ undefined is merely a question of convenience in the application in limits, not an algebraic necessity not particularly an issue that needs more warning than many similar gotchas in mathematics. But in response to MvH, the expression 0⁰ is ill-defined, because the expression b^x is ill-defined. This has nothing to do with limits or continuity. If you want to preserve the algebraic identities, you have to distinguish which of (at least) two distinct functions you are dealing with. Once you have specified which function you mean, the argument becomes simpler: in one case defining 0⁰ ≝ 1 is the only sensible choice; in the other, it makes sense to leave 0^x undefined for all x. Because the two functions are distinct, there is no confusion about having 0⁰ = 1 in one context (e.g. a power series) while leaving it undefined in another (namely, when we take b^x ≝ exp(x ln b). This does not preclude defining the expression in the latter case, but here Dmcq's argument has full validity and consistency as an argument of convenience. I see nothing to argue about; this is the algebraically rigorous approach; trying to meld the two functions is a logical nightmare. —Quondum 21:54, 27 April 2015 (UTC)

I'm glad you deleted that bit and that's why I emphasize continuity rather than limits. Basically as I said above of course one could do limit calculations and yet have 0^0 always equal 1, but in calculus one does not assume that as it could introduce discontinuities when doing algebraic manipulations on an expression. Introducing possibly removable discontinuities and then maybe removing them does not make for a very satisfactory scheme of work, it is error prone and confusing and sounds like fudging. Calculus books treat indeterminate forms as singularities which might be removable. We should just follow the sources. Dmcq (talk) 00:36, 28 April 2015 (UTC)

Yeah, sorry, I never meant the deleted bit; I somehow confused two answers and noticed it after posting. —Quondum 01:34, 28 April 2015 (UTC)

I think the article pushes limits too much and ignores the context. There was an edit a while ago changing continuous exponents to in the context of limits for the title of the section and I changed it back,. Saying in the context of limits does not explain why anyone does this or show what's happening. The context is the important thing, as I tried before to explain to somebody just evaluating limits is an exercise for students without a context and somehow people seem to have got the idea that all one has to worry about is limits. Limits are used because one is in the context of continuity. One does not notice 0^0 in the context of limits one spots it when one evaluates a continuous function and require the function to be extended in a continuous manner to a point. Otherwise a student may think they can just stick 1 in instead of 0^0 because they don't spot that a limit is required. 0^0 is an indeterminate form without mention of limits - it is an indeterminate form in the context of continuity. Limits are used after one spots the indeterminate form. Dmcq (talk) 01:09, 26 April 2015 (UTC)

The statement "0^0 is an indeterminate form without mention of limits" means one thing to one person, and another thing to another person. The phrase indeterminate form is inherently ambiguous, no matter how we may try to define it, the meaning "undefined" will be the default interpretation because most textbooks don't define this term, and if we look it up in a dictionary we find: "undefined". Try to add other examples that meet the definition in the indeterminate form page, and see how quickly your edit will be reverted! This tells me that there isn't a widely agreed upon formal definition, there's just a list. There is simply no way to use this phrase without strongly implying the plain dictionary meaning (the other members of the list do satisfy the dictionary meaning). As long as we use this term, one can imply one thing while proving something else. The only way to exit the infinite loop in the debate is by using unambiguous terminology. MvH (talk) 18:20, 26 April 2015 (UTC)MvH

Lets look at an example, the first line: "it must be handled as an indeterminate form.". In my view the word "must" is way too strong given the fact that different people will interpret this sentence differently. First of all, "it" refers to 0^0 or to limit? But with the inherit ambiguity in "indeterminate form" there's really no way to fix this sentence other than simply deleting it (that would improve the article). MvH (talk) 18:27, 26 April 2015 (UTC)MvH

In the context of continuity requirements. And if you agree with Bo's proof about $e^{y\ln x}$ below then you'd have to say that $0\times -\infty$ is 0 even though it is also an indeterminate form. The rest of the indeterminates can also be given values going along that path. Dmcq (talk) 20:29, 26 April 2015 (UTC)

? My hope was to simply remove ambiguity, not to make matters worse. We have to distinguish the limits from the expressions themselves. Even for the other indeterminate forms, the limits and the expressions themselves are very different things. If f(x) and g(x) converge to 0, and you type limit(f(x)/g(x), x=0) in a computer algebra system, then this may, or may not, return a number. But if you type 0/0 it'll never return a number. These are completely different things. Please agree to distinguish them more clearly. MvH (talk) 03:19, 27 April 2015 (UTC)MvH

The notations for the seven indeterminate forms (e.g. 0⁰) are standard. The best we can can do to make it clearer is to prefix it with a phrase such as "form" or "limiting form", and by contrast we can use the prefix "the expression" or "the value of". I agree that we need to avoid ambiguity. —Quondum 05:17, 27 April 2015 (UTC)

The page Classification of discontinuities gives unambiguous terminology to describe what is proved. MvH (talk) 10:58, 27 April 2015 (UTC)MvH

No, not quite. There is no such thing as a discontinuity at a point not in the domain. So your latest wording is problematic; saying that x^y has a discontinuity at (0,0) assumes that (0,0) is in the domain. --Trovatore (talk) 22:07, 27 April 2015 (UTC)

It is correct on the main page though, "... has a discontinuity at the origin that is not removable, ... no matter how one chooses to define 0^0". So the assumption "define" is in that sentence. It spells out the issue without ambiguity: if it is defined, then you have a non-removable discontinuity. MvH (talk) 00:48, 28 April 2015 (UTC)MvH

I'll add an extra "if then" to ensure it is read as intended: If 0^0 is defined, then we have this discontinuity. Keep in mind, that's the only thing that's actually proven. There is no proof that anything worse than a discontinuity occurs. MvH (talk) 02:02, 28 April 2015 (UTC)MvH

Fair enough; to me the two are so intertwined I barely separate them. However, continuity is more intuitive and is the "purpose" of limits, which are merely a tool. However, one that is phrased more naturally in terms of limits is when infinity is involved. If we were dealing with the (affinely or projectively) extended real line, continuity would handle it all. Do you think we should reword "When the form 0⁰ arises as a limit of f(t)^g(t), it must be handled as an indeterminate form" in terms of continuity, e.g. "When the form 0⁰ arises in the context of an everywhere-continuous function b^x, it must be handled as an indeterminate form."? I don't think I've got it right, but perhaps there's something better. —Quondum 01:56, 26 April 2015 (UTC)

If a function f is not continuous then lim f(x)=f(lim x) is not necessarily true. So 0⁰ never arises as a limit of f(t)^g(t). Bo Jacoby (talk) 07:01, 26 April 2015 (UTC).

If one is working in the context of continuity and sticks in a value of x into the function and finds one comes across 0⁰ then that is an indeterminate form. That is what the textbooks say and that is what we should say. If a student evaluated that as 1 they would not get a mark for their work. Dmcq (talk) 10:10, 26 April 2015 (UTC)

Working with a discontinuous function is not working in the context of continuity. Bo Jacoby (talk) 12:41, 26 April 2015 (UTC).

Bo, you seem to be thinking I'm talking about something I'm not talking about. I was not referring to any discontinuity in f, but rather to discontinuity in the two-variable exponentiation function, i.e. to the function ρ : (b, x) ↦ b^x being discontinuous at (0, 0), because no limit exists at this point. And the limiting form 0⁰ is essentially a shorthand to describe a family of limits of the form f(t)^g(t) (with constraints on f and g). I can't quite work out what it is that you are trying to say. —Quondum 14:07, 26 April 2015 (UTC)

Setting lim(y^x)=(lim y)^{lim x} is incorrect because the exponentiation function is not necessarily continuous. Then substituting lim y=0 and lim x=0 gives correctly (lim y)^{lim x}=0⁰ . Now you get suspicious! You believe that 0⁰ is dangerous, although it is perfectly well defined: 0⁰=1 . You believe that undefining 0⁰ or calling it names will save you, but that only introduces inconsistencies like 0≠0 when you say that 0⁰ is defined for integer values of 0 but not for real values of 0 . The substitution 0⁰=1 is safe but the substitution lim(y^x)= (lim y)^{lim x} is dangerous. I am trying to tell you that your point of view is mathematically inconsistent. Bo Jacoby (talk) 15:35, 26 April 2015 (UTC).

You illustrate well the problems of just saying limits rather than continuity. In calculus $x^{y}$ is treated as equivalent to the continuous function $e^{y\ln x}$ and 0⁰ is treated as an indeterminate form. Dmcq (talk) 17:55, 26 April 2015 (UTC)

What is your preference, should 0^2 be 0 or undefined in calculus? MvH (talk) 14:41, 28 April 2015 (UTC)MvH

The limit there from positive x is 0 rather than indeterminate. That point is normally included in the domain rather than bothering with thinking of it as a limit but sometimes only x>0 is considered. Dmcq (talk) 15:39, 28 April 2015 (UTC)

The function e^{y ln x} is not continuous. In calculus a polynomial f(x)=∑_k a_k x^k has the value f(0)=a₀, proving that 0⁰=1 . Undefining and name-calling solves no problem. Bo Jacoby (talk) 19:11, 26 April 2015 (UTC).

Find a calculus book that says 0^0 is 1 before going on about it in relation to the section about continuous exponents. Dmcq (talk) 20:09, 26 April 2015 (UTC)

Sources that leave 0^0 undefined are easy to find, that's true and uncontroversial. Plenty of people believe that 0^0 "must be undefined", is "problematic", "leads to errors" etc. But the main text should not pretend that there is actually a source that contains a rigorous proof that 0^0 must be undefined. All that's actually proven is that defining 0^0 leads to a discontinuity, we can't claim more than that. MvH (talk) 12:55, 28 April 2015 (UTC)MvH

Calculus is a low-level math course (for instance, a rigorous definition of the real numbers is usually omitted). At this level, people may be uncomfortable with discontinuities (and jump to all sorts of conclusions) because they may not have seen a lot of examples yet. Suppose the limit of f(x)^g(x) is e. Suppose that in step 1, the student replaces f(x)^g(x) by 0^0, and in step 2, the student replaces 0^0 by 1. There is no way to prove that the student's wrong answer was due to what he did in step 2 because there is already an error in step 1. Benson's claim is not based on evidence, he simply didn't understand this subtlety, but Knuth did. Going back further in history, Cauchy/Libri/Mobius, this debate took place at a time when there was no rigorous definition of real numbers, continuous functions, etc., so confusion and disagreement are likely outcomes. MvH (talk) 12:55, 28 April 2015 (UTC)MvH

Benson did not understand the subtlety. Jeezz. Could you stop treating Knuth's writings like Mao's little red book. Knuth never taught analysis and he is human like the rest of us. There is nothing wrong with writing 0/0 or 0^0 in calculus - it just denotes a singularity which is an indeterminate form not a value. There is no problem with it being both marking a singularity and being defined as the value 1 because it isn't 1 in calculus. If you want to write a calculus book that does anything else be my guest and perhaps we can use it as a source because Knuth certainly didn't. Dmcq (talk) 14:01, 28 April 2015 (UTC)

I owe Benson an apology. First of all, I had no right to criticize when I was too lazy to look up the quote to his book. But even without that, my phrase "simply didn't understand" was not justified either way. Hereby my apologies to him. And lesson learned, it appears that even in math, one can't rely on quotes without looking them up. MvH (talk) 01:42, 30 April 2015 (UTC)MvH

So you now owe him an apology because you think he agrees with you? But for people who don't agree with you you have the edit comment "Benson's quote was taken out of context (creationist tactics))". You really do think of this as a war zone don't you with you advancing a cause. The policy on Wikipedia is to say the major points of view. It is true that 0^0 is defined as 1 in most contexts but as Benson himself said there are textbooks that do not define it. And they do that for exactly the same sort of reasons. If you start defining 0^0 then in calculus certain things become unnecessarily awkward. As he puts it "The choice whether to define 0⁰ is based on convenience, not on correctness." That was what was relevant there. Dmcq (talk) 08:48, 30 April 2015 (UTC)

No, I already owed him an apology before that, and I was wrong for not doing that sooner. MvH (talk) 12:16, 30 April 2015 (UTC)MvH

Bo, e^{y ln x} is continuous on its entire domain, which is x > 0, y ∈ R. Please think about what you're saying. —Quondum 20:13, 26 April 2015 (UTC)

The function is discontinuous in the closure of the domain. Bo Jacoby (talk) 06:51, 27 April 2015 (UTC).

Repeated discussion with Dmcq

In calculus it is treated as not having a defined value. In that way the normal algebraic operations can be applied without sticking in conditions all over the place just the same as one can multiply the top and bottom of a fraction by an expression that might be zero at some point. When evaluating the result 0^0 is treated as an indeterminate form just the same as 0/0 is treated as an indeterminate form is whatever about your J language saying 0/0 is always 0. Find a calculus book that agrees with you first before going on about 0^0 always being 1. Dmcq (talk) 08:27, 27 April 2015 (UTC)

MvH, I totally agree that "We have to distinguish the limits from the expressions themselves". Bo Jacoby (talk) 21:02, 27 April 2015 (UTC).

Dmcq, as I explained above, an nth degree polynomial $f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}$ is − in many math books and even here in wikipedia − written $f(x)=\sum _{i=0}^{n}a_{i}x^{i}$. This very common notation assumes that $x^{0}=1$ for all values of $x$ , including of course $x=0$ . So it goes without saying that $0^{0}=1$. Bo Jacoby (talk) 21:02, 27 April 2015 (UTC).

Dmcq, Amongst "the normal algebraic operations" is interchange of computation and limits: $\lim f(x)=f(\lim x)$, but that substitution is unsafe unless the computation $f$ is continuous. You are misguiding your students when teaching them that $\lim(x^{y})=(\lim x)^{\lim y}$. Bo Jacoby (talk) 21:02, 27 April 2015 (UTC).

Dmcq, you wrote: "you'd have to say that $0\times -\infty$ is 0". That is not correct. We do not have to say that. Bo Jacoby (talk) 21:02, 27 April 2015 (UTC).

I was referring to your saying $e^{y\ln x}$ was 1 when x and y are zero. Dmcq (talk) 09:11, 28 April 2015 (UTC)

Dmcq, I did not say that. I said that (0,0) is in the closure of the domain of $f$ where $f(x,y)=e^{y\ln x}$, and that $f$ is discontinuous in (0,0). That discontinuity does not disappear by undefining − nor appear by defining − 0⁰. I have said this before. Please try to understand. Bo Jacoby (talk) 14:44, 28 April 2015 (UTC).

In calculus $f(x,y)=e^{y\ln x}$ has a singularity there, it doesn't have a defined value rather than being discontinuous. You define that $f(x,y)=e^{y\ln x}=1$ there. Dmcq (talk) 15:02, 28 April 2015 (UTC)

Dmcq, no, I did not define $f(0,0)$, but $0^{0}=1$. Nor do I define $f(-2,2)$ , but $(-2)^{2}=4$. The function $f(x,y)=e^{y\ln x}$ generalizes exponentiation with positive base and integer exponent to positive base and arbitrary exponent. Bo Jacoby (talk) 23:23, 28 April 2015 (UTC).

To editor Bo Jacoby: You say 0⁰ [...] is perfectly well defined: 0⁰=1. This is true that this is a perfectly correct definition, but, in mathematics, most definitions depends on some arbitrary choices, and different choices lead to different definitions, which are all perfectly correct in some context, but may be nonsenses in other contexts. This is true for 0⁰. This expression never occurs directly, but always results from substitutions in other expressions. If 0⁰ results from a substitution in x⁰ (as in the constant term of a polynomial), the choice 0⁰ = 1 makes the function x⁰ continuous, and is thus the best choice. If 0⁰ results of a substitution in 0^y, there is no good choice, except if one knows that y is never negative. In this case 0⁰ = 0 is an acceptable choice, but is dangerous because of the constraint on y, which may be difficult to verify and is easy to forget. If the substitution occurs in f(x)^g(x) there is no acceptable choice, as any choice is meaningless, if it does not preserve some continuity. This is the reason for which mathematicians have introduced the term "indefinite form". This not implies that something is undefined, but means that the substitution is not sufficient for getting the desired value, and some other computation is needed (such as a limit computation). Thus there is no value for 0⁰ that is convenient in every context. Your insistence for having a unique convention for the value of 0⁰ is not only unsourced, but there is no hope to reach a consensus in its favor among mathematicians. D.Lazard (talk) 16:16, 28 April 2015 (UTC)

The phrase "if .. results from a substitution" is not well-defined. How would you define if 2^3 comes from a substitution or not? Also, there is no theorem to support the idea that if f(x) is not continuous at x=0 then f(x) must not be defined at x=0. MvH (talk) 16:49, 28 April 2015 (UTC)MvH

I'll repeat the example, let me know what you think: Suppose the limit of f(x)^g(x) is e. Suppose that in step 1, the student replaces f(x)^g(x) by 0^0, and in step 2, the student replaces 0^0 by 1. There is no way to prove that the student's wrong answer was due to what he did in step 2 because there is already an error in step 1. All the "proofs" that 0^0 can't be defined are like this example, they're not rigorous and only apply to contexts where the student has already made an error. In contexts where no error has yet been made, evaluating 0^0 to 1 won't cause an error either. MvH (talk) 17:07, 28 April 2015 (UTC)MvH

In what way would replacing it with 0^0 be wrong? In finding a limit we're not trying to find a value for 0^0. All we do in calculus is not give a value to 0^0 but call it an indeterminate form. Dmcq (talk) 17:19, 28 April 2015 (UTC)

Replacing f(x)^g(x) by 0^0 is wrong because the limit can be obtained from the former but not from the latter. For a similar example, take the limit of sin(3x)/x with x->0. The limit can be computed from sin(3x)/x with l'Hopitals rule, but can't be computed from the expression 0/0. So if you replace sin(3x)/x by 0/0, it doesn't matter what you do with 0/0 afterwards because anything you do with the expression 0/0 is unlikely to lead to the correct answer. Likewise, defining 0^0=1 will not cause a wrong answer unless the student already made an error. To phrase this in computer terms, following the IEEE standard 0^0=1 does not cause computer programs to return a false result in any context (excluding programs that have other errors). MvH (talk) 17:31, 28 April 2015 (UTC)MvH

It seems to me that this is drifting back into a discussion of the merits of the various conventions. Please keep in mind that that discussion, interesting as it might be, is off-topic here. --Trovatore (talk) 20:24, 28 April 2015 (UTC)

There is no requirement to be able to compute something from 0^0. I haven't the foggiest where you got that idea from. The value of x^2 at x=3 is 9, that does not mean the answer 9 is somehow wrong because we are not able to calculate the slope 6 from it. There are three different power functions in IEEE, each for a different usage. Two of them give 1 for that and the third gives NaN. Dmcq (talk)

The error in this calculation

$\lim _{(x,y)\rightarrow (0,0)}x^{y}=(\lim _{x\rightarrow 0}x)^{\lim _{y\rightarrow 0}y}=0^{0}=1$

is in the last equality sign according to Dmcq and D.Lazard, but in the first equality sign according to MvH and myself. Bo Jacoby (talk) 21:20, 29 April 2015 (UTC).

Any response by me to what you said is irrelevant to changing the article. You need a source before you change the article to say what you want. Dmcq (talk) 22:58, 29 April 2015 (UTC)

Dmcq, as you already know: page 162 in Graham, Knuth, Patashnik, "Concrete Mathematics", second edition (1995): "Some textbooks leave the quantity 0⁰ undefined, because the functions x⁰ and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x⁰ = 1 , for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The theorem is too important to be arbitrarily restricted. By contrast, the function 0^x is quite unimportant. (See Donald E. Knuth, "Two notes on notation," American Mathematical Monthly 99 (1992), 403-422, for further discussion.)" Bo Jacoby (talk) 05:14, 30 April 2015 (UTC).

That's Knuth's opinion. It has not caught on to the extent you seem to think it has. --Trovatore (talk) 05:51, 30 April 2015 (UTC)

And here for example is one of those textbooks that Knuth says is mistaken. One note in a magazine does not automatically override multiple textbooks by multiple authors that have gone through multiple editions. See WP:WEIGHT. "Note that 0⁰ and 0⁻³ are undefined"Berresford, Geoffrey; Rockett, Andrew (2015). Brief Applied Calculus (7th ed.). Cengage Learning. p. 22. ISBN 9781305085329.. Actually I'd be a bit more circumspect than them but the point stands. Dmcq (talk) 09:26, 30 April 2015 (UTC)

Dmcq, you merely requested a source, which I did provide. Bo Jacoby (talk) 20:13, 30 April 2015 (UTC).

Nobody here is denying that there are a lot of calculus books that leave 0^0 undefined. Regarding WP:WEIGHT, it is wrong to rank calculus books higher than the IEEE standard, the sci.math faq, and a founding father of computer science, etc. Also, a text that simply gives an opinion (whether it is 1 or undefined) without supporting arguments, should have much less weight than a text that is supported by arguments. MvH (talk) 12:50, 30 April 2015 (UTC)MvH

No it shouldn't.

Look, this is a question of convention. There are reasons people adopt conventions, but when you talk about things like what has been "proved", that's all completely beside the point. The proofs are all trivial; there's no dispute about the mathematics, except in some very broad sense of the word. We report what the conventions are and what people have said about them. We don't automatically promote one proposed convention just because its proponents have made a lot of noise about it, whereas the ones who use the traditional convention have just kept on using it without bothering much about it. --Trovatore (talk) 14:13, 30 April 2015 (UTC)

Not entirely besides the point. There are dozens of "proofs" for P=NP (and also dozens that P is not NP), yet, none of them are presented as fact on wikipedia. Clearly, it does matter if something is opinion or backed by a verified proof. There are significant differences among mathematicians about how formulas are interpreted. We usually doesn't notice this, except here, where "0^0 is unambiguously 1" is obviously true to some mathematicians, and obviously false to others. It may be true (I'm currently trying to find this out with the help of D.Lazard, but this page is not the right place for it) that there is more than one internally-consistent way to interpret mathematical formulas, that agree on most things except this. Presenting one side as "fact" and the other side as "mere convenience" is going to be controversial. Calculus are low-level math courses, they deserve appropriate weight but they don't trump everything else. We'll probably always disagree on this issue. It might not matter, I'm not planning further edits. MvH (talk) 15:33, 30 April 2015 (UTC)MvH

OK, I saw that on Lazard's talk page, and I will permit myself a brief off-topic note here that you really should listen to Carl, who is a professional mathematical logician (as BTW I have also been in a previous life stage of my career). --Trovatore (talk) 16:13, 30 April 2015 (UTC)

I was conflicted about ignoring Carl because it probably came across as rude. Trying to understand how another mathematician interprets math is remarkably difficult, adding one more to this discussion would make it harder still. There is another angle to this issue that Lazard mentioned, how to interpret math comes strongly in focus when you're trying to put it into a computer algebra system. There is a lot of tension between having powerful expressive notation and having unambiguous notation. MvH (talk) 17:12, 30 April 2015 (UTC)MvH

Rude or not, the point is that Carl knows what he's talking about and you could learn from him. His response on that page expressed one of the key points clearly and concisely. --Trovatore (talk) 17:17, 30 April 2015 (UTC)

To editor CBM:, OK, then I'll respond. Carl wrote in complex analysis, $x^{y}$ is defined as an abbreviation for $\exp(y\log x)$ for all complex numbers. He also wrote: A similar phenomenon happens, for example, if we compute $2^{-1}$ in $\mathbb {Q}$, $\mathbb {Z} _{5}$, $\mathbb {Z} _{7}$, and $\mathbb {Z} _{8}$, and find it has values 1/2, 3, 4, and "undefined", respectively.

Reply: In this case, I wouldn't choose "undefined" as the default value for the number $2^{-1}$. I'd only choose that if we know that the characteristic is even. Likewise, there is no majority-support for letting "undefined" be the default value for 0^2, which it would be if we insist that $\exp(y\log x)$ trumps everything else. This issue is already mentioned on the main page, and I'd prefer not to redo that debate (I think Bo would pull out his hair...). MvH (talk) 18:39, 30 April 2015 (UTC)MvH

Not here, please. This discussion is off-topic on this page. --Trovatore (talk) 18:47, 30 April 2015 (UTC)

IEEE and Maple

And whilst the IEEE function pow(0,0) gives 1 the function powr(0,0) gives NaN. So they haven't gone and said it is 1 in all circumstances. Dmcq (talk) 14:30, 30 April 2015 (UTC)

Dmcq, in the mathematics software section, it says that Maple evaluates 0^0 to 1. But this isn't the full story. The way Maple acknowledges the other circumstances is like this: it evaluates 0.0^0.0 to Float(undefined) (in Maple 0 = exact zero and 0.0 = approximate zero). Should I add that in Maple's line in the software section? MvH (talk) 19:39, 30 April 2015 (UTC)MvH

I did a quick search, and I wasn't able to find where the Maple docs refer to 0.0 as "approximate zero". Is this your own interpretation? From what I was able to glean from the docs, admittedly without spending much time on it, it seems to me that it would be interpreted as "floating point zero". It's true that floating-point numbers in computing generally represent approximate quantities, but that is not the same as saying that Maple considers that value to be an approximate zero. (However, if they do refer to it that way, which is certainly not impossible, please do point me to where.) --Trovatore (talk) 20:02, 30 April 2015 (UTC)

Proposal: After the line "...and evaluates 0^0 to 1." on the main page, add a short line like this: "It evaluates 0.0^0.0 to Float(undefined)." I'll wait a while to see if people agree/disagree. MvH (talk) 22:08, 30 April 2015 (UTC)MvH

I think the bit about Maple is fine. We can't however do that for the article in general. I think it describes what happens in practice fairly well but we don't have sources saying that and even if we did it would just be yet another point of view. Reminds of this from xkcd [1] ;-) And while we're disclosing I'm afraid I'm not a professional mathematician, I'm what I like to call a computer toolsmith and as to logic I'm currently reading through Thomas Jech's 'The Axiom of Choice' but finding it rather heavy going. Dmcq (talk) 22:25, 30 April 2015 (UTC)

I also support adding the note about Maple. Not sure of the best wording. I have some concerns that "Float(undefined)" is a little obscure to non-Maple users and that it could be read as something like "evaluates to Float, which is undefined". Maybe something like "evaluates to an undefined floating-point value"? --Trovatore (talk) 00:04, 1 May 2015 (UTC)

Maple documentation says The quantity Float(undefined) represents a non-numeric object in the floating-point system. This value can be returned by a function or operation if the input operands are not in the domain of the function or operand. In other words "Float(indefined)" is a synonymous of NaN (IEEE754) or "indeterminate form".

The complete story about 0⁰ in Maple is the following. Firstly, 0.0 and .0 are both displayed as "0." (quotes are here and in the following to distinguish the punctuation mark from the decimal mark). Then "0 ^ 0" returns "1" and "0. ^ 0" returns "1." (in other words, if the exponent is the integer 0, then the result is the float "1." or the integer "1", depending on the type of the base); both "0 ^ 0." and "0. ^ 0." return "float(undefined)" (that is the result is "float(undefined)" if the exponent is the floating point "0.", independently of the type of the base). D.Lazard (talk) 08:37, 1 May 2015 (UTC)

Gosh. That's exactly what the IEEE functions pown and powr do and what I'd hope for thinking of 0.0 as approximate, but I must admit I'm surprised they've actually done it! Here is a full list of what powr does from the standard.

For the powr function (derived by considering only exp(y × log(x))):

powr (x, ±0) is 1 for finite x > 0

powr (±0, y) is +∞ and signals the divideByZero exception for finite y < 0

powr (±0, −∞) is +∞

powr (±0, y) is +0 for y > 0

powr (+1, y) is 1 for finite y

powr (x, y) signals the invalid operation exception for x < 0

powr (±0, ±0) signals the invalid operation exception

powr (+∞, ±0) signals the invalid operation exception

powr (+1, ±∞) signals the invalid operation exception

powr (x, qNaN) is qNaN for x ≥ 0

powr (qNaN, y) is qNaN.

Invalid operation means it will return NaN unless some flags are set. IEEE has signed zeroes which is a strange idea for mathematicians but can be quite useful sometimes thinking of the 0.0 as an approximation or limit. Dmcq (talk) 10:24, 1 May 2015 (UTC)

A short formulation would be this: "In Maple, 0^0 is 1 while 0.0^0.0 is undefined". MvH (talk) 13:52, 1 May 2015 (UTC)MvH

This short formulation would be incomplete (e.g. it omits 0^0.0=Float(undef) and 0.0⁰=1.0). Maple is clearly type-sensitive. It might make sense to characterize this behaviour in a broader perspective: I'm guessing that when the exponent has the type Float, the base x is implicitly promoted to Float and with value x≤0 gives undefined Float (powr behaviour), whereas when the exponent is of type Integer, the result is only undefined for a base x=0 and exponent y<0, and the type of the result is that same as that of the base (overloaded pown behaviour). This would allow determination of all the 0⁰ cases for these two types, and well as the behaviour for all other values. —Quondum 18:27, 1 May 2015 (UTC)

To editor D.Lazard:, I would have preferred a much shorter Maple entry, where the text on the page is very short but where we use links to other pages (e.g. float) for the people that want to read more. Wikipedia pages sometimes tend to grow too long and need occasional trimming (which can often be done without deleting real content, by using links). MvH (talk) 13:23, 2 May 2015 (UTC)MvH

To make this explicit, I repeat the line I proposed, this time with a link added: "In Maple, 0^0 is 1 while 0.0^0.0 is undefined." This is sufficient; a detailed explanation is only 1 click away. MvH (talk) 14:15, 2 May 2015 (UTC)MvH

I too am in favor of a concise statement here. The only thing I don't like about your proposed addition is the piped link to floating point from the text 0.0^0.0 . Wikilinks are not explanatory notes; in general, a reader should be able to predict where a link points from the text of the link (see least surprise principle for the general notion in UI design). I would expand it out just slightly to give a more direct anchor for the link, ideally without using pipes. Something like this, maybe: In Maple, 0^0 is 1, whereas 0.0^0.0 is undefined (where 0.0 denotes a floating point representation of zero). --Trovatore (talk) 19:58, 2 May 2015 (UTC)

Agreed. Lets wait a day or so, if nobody objects, then lets write it that way. MvH (talk) 16:54, 3 May 2015 (UTC)MvH

I do not use Maple so please tell me, does Maple, like Trovatore, deny that 0=0. ? What does if 0=0. do ? Bo Jacoby (talk) 17:16, 1 May 2015 (UTC).

This depends on how you define equality. In Maple, 0 and 0. are stored differently. They are thus not syntactically equal. Moreover, they behave differently when used in some functions (see above). However, the equality test evalb(0=0.) (for "eval to a boolean") tests the semantic equality, and return true. This is very similar with what appears when one defines formally the reals: A real number is commonly defined as a class of equivalence of Cauchy sequences of rational numbers. An integer is not such a class, but there is canonical injection of the integers into the real numbers, which allows to "identify" each integer with the corresponding class of Cauchy sequences. This "identification" is essentially a semantic equality. D.Lazard (talk) 18:14, 1 May 2015 (UTC)

More pragmatically, the equality operator is probably overloaded as two independent operations integer=integer and float=float, and integer is automatically promoted to float if the parameter types do not match. This implicit promotion of types is typical in computer languages of where the mathematical equivalent is a canonical embedding between types that are considered distinct by the language. This says the same thing as D.Lazard in the language of computer science. —Quondum 18:40, 1 May 2015 (UTC)

I would not actually call it "semantic equality". A more cautious way to phrase it would be to say that, for many purposes, it is convenient to treat the canonical embedding as though it were inclusion. But that's quite off-topic here; anyone who wants to discuss it, I'm willing to do so, up to a point, on some user talk page. --Trovatore (talk) 23:47, 1 May 2015 (UTC)

The difference between 0 and 0.0 in maple is not similar to the issue of embedding Z into R (integers into reals). Z and R are rings, but floating point numbers do not form a ring, they don't satisfy the rules of arithmetic (there is no finite ring of characteristic 0). If a,b are floating point numbers, then a=b will evaluate as "true" if a,b match within the error-tolerance. Hence, if you ask Maple if 0 is an element of the set {0.0} then it'll say "false" but if you ask Maple if 0 = 0.0 then it'll say "true" because they match within the error-tolerance. MvH (talk) 02:27, 2 May 2015 (UTC)MvH

Again, this seems to be your own interpretation, and in any case is off-topic. --Trovatore (talk) 03:58, 2 May 2015 (UTC)

Thanks to D.Lazard for the answer, which in short is: Maple does not agree with Trovatore that 0 ≠ 0.0 .

Obviously Maple and IEEE do not agree with the elementary calculus books that 0⁰ is undefined.

Does Maple agree with Dmcq that $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\neq \sum _{i=0}^{3}a_{i}x^{i}$, (the right hand side being undefined at x = 0 according to Dmcq)? Does the answer depend on whether i is declared integer or float? Bo Jacoby (talk) 03:00, 2 May 2015 (UTC).

Bo, as usual, you're completely mischaracterizing everything, and it isn't worth discussing it with you. --Trovatore (talk) 03:58, 2 May 2015 (UTC)

There's no point in addressing me directly - I look at this page and anyone can respond. That isn't what I said. As to the question it looks like it depends on whether i is an integer or a float. In how you wrote it i would be an integer so the two sides would match. Dmcq (talk) 14:32, 2 May 2015 (UTC)

Trovatore, then don't. By the way I agree with you that the computer representations of numbers, float and integer, is off topic. They need not appear in our elementary article on exponentiation. Nor does the issue of embedding ℤ into ℝ. Bo Jacoby (talk) 09:23, 2 May 2015 (UTC).

There's just the one article not an elementary or advanced one. We don't need something like Introduction to quantum mechanics I hope! Dmcq (talk) 14:40, 2 May 2015 (UTC)

For the record, I did not in fact say that the computer representation of numbers was off-topic. --Trovatore (talk) 18:12, 2 May 2015 (UTC)

I hope that our article can be elementary in the sense that it does not talk about computer implementations of float and int, nor about theoretical constructions of ℤ and ℝ. It was unclear to me what Dmcq meant by referring to a calculus book that said that 0⁰ is as undefined as 0⁻³. It is unclear to me if Quondum accepts that e^{y ln x} has a discontinuity in the closure of the domain, and that continuity in the domain is insufficient. It is perfectly clear to me that Trovatore thinks that 0≠0.0, and perfectly clear also that this point of view is incompatible with mathematics. Bo Jacoby (talk) 17:11, 2 May 2015 (UTC).

Trovatore's point of view is incompatible with what Bo Jacoby calls "mathematics", but Trovatore's point of view is compatible with that of almost all mathematicians. This logically implies that Bo Jacoby calls mathematics his personal point of view and nothing else. Pursuing this kind of logical deduction, one could conclude that Bo Jacoby considers that he is the only one who knows what is mathematics. D.Lazard (talk) 17:41, 2 May 2015 (UTC)

Bo has chosen to ping me, so I'll respond: any argument about the closure of the domain of exp(y ln x) is completely extraneous. —Quondum 20:09, 2 May 2015 (UTC)

D.Lazard, do you really want to teach our readers that 0≠0.0? Quondum, (0,0) is in the closure of the domain of e^{y ln x} but not in the domain. So please explain what you find extraneous. Bo Jacoby (talk) 22:25, 2 May 2015 (UTC).

Bo, you introduced the closure of the domain apropos nothing, only to make an irrelevant statement. No-one else has said anything about it. Its relevance here is nil. How much clearer do I need to make it? This is the epitome of wikt:extraneous. —Quondum 22:46, 2 May 2015 (UTC)

No one is proposing to say in the article that 0≠0.0, nor would anyone, including me, put it that way, except in very particular situations. I do not know why it so offends your personal mathematical aesthetic to distinguish between the natural number 0 and the real number 0.0, but I do know that it is not necessary to argue it on this page. As to the second point, there is no such thing as a discontinuity at a point not in the domain. --Trovatore (talk) 22:52, 2 May 2015 (UTC)

I used to write 0.5−0.5=0 meaning the same thing as 0.5−0.5=0.0 , that's why I am offended by Trovatore's 0≠0.0 . Our article distinguishes between discrete exponent and continuous exponent. To me 0 is neither discrete nor continuous, it is simply a number. There is such thing as a discontinuity at a point not in the domain. A discontinuity at a point in the closure of the domain can make the function discontinuous. Undefining such function at the point of discontinuity does not make it continuous. This is crucial to our discussion about undefining 0⁰. Undefining 0⁰ does not make e^{y ln x} continuous at (0,0), such as Quondum seems to think. Bo Jacoby (talk) 06:31, 3 May 2015 (UTC).

Bo Jacoby, you are trolling. Please stop it. You have no cause to keep throwing such false assertions at individuals. —Quondum 14:50, 3 May 2015 (UTC)

Quondum , you have so far not been able to make sense to me. I am seriously trying to find out what you mean. That is not trolling. Do you really mean that e^{y ln x} is continuous at (0,0)? Bo Jacoby (talk) 15:42, 3 May 2015 (UTC).

No. —Quondum 16:57, 3 May 2015 (UTC)

Thanks! Do you think that this discontinuity depend on whether 0⁰ is defined or not? Bo Jacoby (talk) 17:31, 3 May 2015 (UTC).

Please take this discussion somewhere else as it is off-topic. Bo, your use of language is nonstandard; by standard terminology there is no such thing as a function being discontinuous at a point not in the domain. However I concede that this is a linguistic point and that I do understand what you mean. But it doesn't matter because nothing in the article is going to depend on it, just as nothing in the article is going to depend on whether we think that 0⁰ ought to be defined. Your repeated attempts to re-litigate the merits of the various conventions not licit here. --Trovatore (talk) 18:28, 3 May 2015 (UTC)

Trovatore, I respectfully disagree. The article should make sense, which it presently does not. Bo Jacoby (talk) 19:24, 3 May 2015 (UTC).

Different people read math differently (not that this is easy to understand, but I can't conclude otherwise). Arguments that are solid for one side don't make sense for the other side, and vice versa, leading to an infinite loop. I'm going to attempt to end this infinite loop with a positive note: Compare our article with that of wolfram (which says that f(a,0)=1 for all a, and f(0,a)=0 for a>0, are mutually contradictory statements!). Regardless how you feel about our article, I hope you agree that it is much better than wolfram's! MvH (talk) 15:36, 4 May 2015 (UTC)MvH

Thanks MvH. Our article is bad, and it is a poor consolation that Wolfram's is still worse. I have a hard time in taking the undefiners seriously, and I am not always succesful. Best luck to you. Bo Jacoby (talk) 18:12, 4 May 2015 (UTC).

Usage of "indeterminate form"

There are been several lengthy discussions, here, about the value of 0⁰. Surprisingly, although the word "indeterminate" appeared frequently, the term indeterminate form appeared only rarely, as well as the link to the article about it (indeterminate form is not even linked in the section about 0⁰ in the article). In my opinion, this lack of references is one of the main reasons of the misunderstands that lead to these endless discussions. In fact, when considering continuity and limits, the following theorem is widely used, although rarely explicitly stated:

Let f be a function, which is built up by composition from continuous functions whose domains and ranges are included in the projectively extended real line. If f may be evaluated at a without encountering any indeterminate form, then f is defined and continuous at a, or may be prolongated by continuity to this value.

A similar theorem exists for functions defined on the extended real number line, which requires the use of a signed zero. These theorems are important, as, in many cases, they allow to prove the continuity of a function without considering the definition of limit. It is analogous to the derivation rules that allow to prove differentiability and compute the derivative without considering the definition of the derivative through limits. Therefore, considering that 0⁰ has a specified value when dealing with continuity would prevent to use this theorem. It was that I had unclearly in mind when I wrote, in a previous post, that 0⁰ is not considered by itself, but appears only in function evaluation.

My opinion is that both this article and Indeterminate form deserve to be edited to take these theorems into account (presently, I have not found any place where it is said that the absence of any indeterminate form proves the existence of a limit). D.Lazard (talk) 10:13, 5 May 2015 (UTC)

The phrase indeterminate form is already mentioned six times on the main page (not counting references). Please don't make it more prominent than it already is. It is inherently ambiguous because it has two separate meanings that continue to be confused. One meaning is its definition in indeterminate form and the other is its plain dictionary meaning. This ambiguity leads to intractable circular debates because it allows you to prove one thing but then conclude something else. (PS. I added two more links to indeterminate form) MvH (talk) 13:01, 5 May 2015 (UTC)MvH

I think the theorem is interesting for the page indeterminate form because it is basically says that the list on that page completely covers the functions {+, /, *, -, ^}. As it is right now, the definition on the page is a bit vague (it appears from the examples that the terminology only applies to these functions {+, /, *, -, ^}, but the definition doesn't explicitly say that, and there don't appear to be references that are any clearer on this). Your probably do need to find a reference for the theorem before you add it to the indeterminate form page, because otherwise chances are that someone will revert your work. MvH (talk) 13:48, 5 May 2015 (UTC)MvH

If the there is a formal definition of an indeterminate form (and there must be, if there is a theorem that relies in it), let's start by getting that definition formalized on the page Indeterminate form. Giving the theorems mentioned here on that page would be a next step. —Quondum 14:23, 5 May 2015 (UTC)

Let's put it this way: There is no standard formal abstract definition of "indeterminate form". I wouldn't be surprised if someone, somewhere, has ventured to give it a definition. But in typical usage, it's just a closed-end list of (usually) seven expressions, with no provision for expansion. --Trovatore (talk) 15:30, 5 May 2015 (UTC)

Quondum, the theorem Lazard mentioned can be formulated with or without a formal definition, the only thing it needs is just the list. The theorem allows every continuous function from the projectively extended real line to itself, as well as the bivariate functions {+, /, *, -, ^}. Since there's nothing to be proven for the continuous functions, the only functions that matter for the theorem are the bivariate functions {+, /, *, -, ^}. So what the theorem is saying is that every discontinuity (from the projectively extended line to itself) that can arise by combining {+, /, *, -, ^} must come from a member of the list given in indeterminate form. So to formulate the theorem, one only needs the list, no additional formal definition is needed. The main thing Lazard would need is a reference (otherwise chances are that Trovatore will revert his edit...). MvH (talk) 16:00, 5 May 2015 (UTC)MvH

Sure, accepted. The theorem can be formalized better though; it needs to define what its terms mean. —Quondum 19:01, 5 May 2015 (UTC)

The theorem itself can also be interpreted as a definition of indeterminate forms: Let C be the Projectively extended real line. If one extends the functions {+, /, *, -, ^} to functions from CxC to C as much as possible, then the list of essential discontinuities is precisely the list of indeterminate forms. MvH (talk) 17:15, 5 May 2015 (UTC)MvH

As a theorem, the inclusion of exponentiation and the associated indeterminate forms are redundant. This is because they are covered by the theorem already using the four elementary operations: exponentiation is merely "built up by composition from continuous functions ...". At least if, as D.Lazard points out, it has been extended beyond the domain upon which it is defined as a composition of functions. The argument that such an extension breaks the theorem seems good to me. —Quondum 19:01, 5 May 2015 (UTC)

I'm starting to get some doubts about Lazard's theorem (I'm not saying it is wrong, I may be reading it wrong; a reference or more precise formulation could be helpful). The function 2^(1/x) is discontinuous at x=0, with left-limit 0 and right-limit infinity. Yet, this didn't go through an indeterminate form (1/0 is not an arithmetic operation but it's also not an indeterminate form). MvH (talk) 02:11, 6 May 2015 (UTC)MvH

It goes through the discontinuous function x ↦ 1/x, though on the projectively extended real numbers I guess that would be still be considered continuous at x = 0, with value ∞ = 1/0. However, the point ∞ is not in the domain of exp, and hence the premise of the theorem is not satisfied. So this is not a counterexample. —Quondum 19:45, 6 May 2015 (UTC)

I mixed up domains. The function 1/x is continuous on the projectively extended real numbers, but not on the extended real number line, but for exp(x) it is the other way around. So by combining them I get an example that doesn't apply to either of those two domains. MvH (talk) 21:51, 6 May 2015 (UTC)MvH

Zero to the Power of Zero: Discrete Exponents

(Removed off-topic exposition).

Sorry, Danchristensen, but this is not the place for this. Per WP:TALK, this is not the place to discuss what the definition should be, but only what we should do with the article. --Trovatore (talk) 05:31, 1 May 2015 (UTC)

Shouldn't readers know that there is a solid rationale for having 0^0 undefined on N? — Preceding unsigned comment added by Danchristensen (talk • contribs) 05:56, 1 May 2015 (UTC)

I think the collapse templates {{cot}} and {{cob}} around something like that are better than removing them. For what should be in the article we really need citations rather than peoples own reasons no matter how good, and generally we don't put in anything like a proof unless it is notable or very short. Cited references provide that sort of thing. So a sentence like 'Some mathematicans think that even in the discrete case the value should not be defined' followed by a couple of citations would be fine. Dmcq (talk) 07:59, 1 May 2015 (UTC)

Trovatore, I can't argue with the rules, but it does seem a bit harsh. Dan, I read your argument. Keep in mind that the current section on integer exponents already doesn't say that 0^0 is 1. Are you proposing to add things to that section? MvH (talk) 14:33, 1 May 2015 (UTC)MvH

How about:

Using a simple table-building exercise, it can be shown that there are infinitely many binary functions ^ on N that satisfy

n^2 = n.n

n^3 = n.n.n

n^4 = n.n.n.n

and so on.

They differ only on the value of 0^0, however. So, a representation of exponentiation as a partial function on N with only 0^0 being undefined may be justified.

--Danchristensen (talk) 14:25, 1 May 2015 (UTC)

Dan, this is what wikipedians want to know: Among all the definitions out there, which one is the most dominant/common/prominent in the literature? Which one is used the most? Every potential answer to these questions (maybe even the questions themselves?) is highly controversial. MvH (talk) 14:43, 1 May 2015 (UTC)MvH

Yes, I suppose you have to keep out all the kooks and cranks. It just seemed to me like such a simple, straightforward argument that no reference was really required. Sorry to bother.

--Danchristensen (talk) 14:53, 1 May 2015 (UTC)

Dan, if we write a recursive definition for n^m, then what should be the starting point? You started with m=2, the main page starts with m=1 while other sources start with m=0. That choice, start with m=0 or with m>0, that's the entire controversy in one tiny nutshell. In a typical calculus textbook, the definition starts with m=1 while the proofs start with m=0. PS. Your choice to start with m=2 would not only leave 0^0 ambiguous, but 0^1 as well (in non-reduced rings). PS2. I think if we continue this discussion here, we'll get in trouble with WP:TALK, but feel free to continue it on my talk page. MvH (talk) 15:23, 1 May 2015 (UTC)MvH

I almost wish the argument hadn't been removed; I haven't seen one, except over a rng, where, there being no "1", x⁰ cannot be defined if x is not a zero divisor (or 0), and the only reasonable definition of 0⁰ would be 0. — Arthur Rubin (talk) 16:51, 5 May 2015 (UTC)

But, what is important here (on Wikipedia) is whether any source fails to set 0⁰ = 1 in the context of R × Z^≥0, for any ring R. — Arthur Rubin (talk) 16:56, 5 May 2015 (UTC)

Yes, I like this direction (though we might get resistance from those in WP who wish to treat 0⁰ in a context-free way). The implication is that if we fail to find notable sources that consider it to be undefined in the discrete exponent context (by which I mean exponentiation defined via repeated multiplication and division), the disputed statement must be removed. —Quondum 18:43, 5 May 2015 (UTC)

Arthur, it's not the case that 0^0 is automatically a problem in a rng. For example, x^2 y^3 is simply notation for xxyyy. Likewise, x^1 y^3 is notation for xyyy and x^0 y^3 is simply notation for yyy. There's no need to prove x^0 y^3 = yyy, it holds in all situations, simply because x^0 y^3 is notation for yyy. This identity holds for every x,y in every ring (even in rings without identity). If you plug in x=0, it follows that 0^0 y^3 is well-defined for every y in every ring, even in rings without identity. It's simply notation for yyy. This doesn't imply that 0^0 itself is defined in rings without identity, because in rings without identity there is no good interpretation for an empty product. MvH (talk) 21:53, 5 May 2015 (UTC)MvH

Notes

References

Although not precisely what was asked for, this might have some bearing on the issue. I know of at least one algorithm in the finite setting that requires 0⁰ = 0 to work properly. The rub is that this is in the context of a finite field GF(q) and involves the interpretation of the expression x^q-1. In this field exponents are taken modulo q - 1, so x^q-1 ≡ x⁰ = 1 for all x ≠ 0 and 0 otherwise. Bill Cherowitzo (talk) 19:02, 5 May 2015 (UTC)

Which algorithm are you referring to? Note that x^(q-1)-1 = 0 and x^q-x = 0 are well known equations for Fq^* and Fq. It shouldn't come as a surprise that x=0 doesn't satisfy the equation for Fq^*. MvH (talk) 21:42, 5 May 2015 (UTC)MvH

I can see that xⁿ ≡ x^{n mod (q−1)} (mod q) for all n and x, except that when x ≡ 0 (mod q) and n ≤ 0 it is not necessarily defined. This creates an awkward hole in a very regular structure, which would be neatly filled if xⁿ ≡ 0 for x = 0 and all n ≤ 0. To gain some insight, one could look at the operations as applied to the elements of equivalence classes, and require that equivalence always holds when the operation applied to the elements of the class is defined. As integers, we would require that because q⁰ = 1, and q ≡ 0 (mod q), we must have 0⁰ ≡ 1 (mod q). I'm afraid that I do not see how we can reasonably salvage the mod-(q−1) pattern on exponents when n = 0 without requiring in integers that q⁰ = 0 (or undefined), which is exceedingly unpalatable. So pretty as the completion of this pattern on exponents modulo (q−1) might be, the hole in it for a zero (and negative) base must remain, and it does not stand as a counterexample to 0⁰ = 1; indeed this line of reasoning shows that necessarily 0⁰ = 1 if it is defined. —Quondum 10:36, 6 May 2015 (UTC)

It is often assumed that if a is congruent to b modulo m, that a^n will then be congruent to b^n modulo m. But this assumption is true if and only if 0^0 is 1. On a computer, if you choose anything other than 1 for 0^0, you're simply asking for bugs. MvH (talk) 16:13, 6 May 2015 (UTC)MvH

The pattern does not hold for any exponent when x = 0. One must unconditionally exclude a zero base from the "algorithm". Looking at it in the context of a projectively extended finite field, the argument becomes clearer: exponents mod q−1 simply does not apply when the base is zero.

I have a feeling that there might be an isomorphism argument on decomposable rings that provides a challenging case, though. Via isomorphism, one could argue that the ring of 2×2 matrices with only the top-left entry nonzero, the identity of the ring is with 1 in that position, and hence 0⁰ must equal that. Yet considered as embedded in the full matrix ring, the ring identity is different. Thus the operation 0⁰ has to "know" what the identity element of the ring is, not entirely satisfactory. —Quondum 20:12, 6 May 2015 (UTC)

Upon closer examination I must retract my statement about an algorithm, I had misremembered something. To be clear about what I was trying to say ... in any finite field 0^q-1 = 0 is clear and provable. This would lend credence to the definition 0^[0] = 0 where the brackets indicate the equivalence class mod q - 1. In rebuttal it was claimed that this would force [0]⁰ = 1 in the integers (this time the equivalence class is mod q). I don't find the argument convincing since I can't see how you can get from one of these statements to the other without passing through the problematic [0]^[0], which must be a poster child for an expression that is not well defined. Yet another place where a "natural" argument favors the value of 0 — consider the sequence of curves with affine equations y = xⁿ (integral n > 0), they all pass through the origin; so then should the "limiting curve" y = x⁰. I am not trying to favor any one definition over another, only pointing out that context matters and no universal definition will ever be satisfactory. Bill Cherowitzo (talk) 23:58, 6 May 2015 (UTC)

The expression [0]^[0] appears to be precisely what you are saying should be [0]? All I'm saying is that when the base is zero, the equivalence class mod (q−1) on the exponent does not apply, and any way you look at it and regardless of what (unital) ring the base might come from, and would step on the toes of too many directly related results. Taking the argument further, you'd get that 0⁻¹ = 0^[−1] = 0^q−2 = 0. Moving to the projectively extended finite field, you'd also end up with a flat-out contradiction, because 0⁻¹ = ∞, a well-defined value. A system that does not embed a field into its projectively extended field is decidedly problematic.

Surely the power example is little different from the same argument on sequence 0ⁿ? And based on little more than "intuitive expectation"? And for which we know that simple arguments like this are no help.

I think that this has strayed a bit off-topic. For the purposes of this page, I think that it is clear that we do not yet have any compelling 0⁰ ≠ 1 examples. —Quondum 00:47, 7 May 2015 (UTC)

Obvious domain restriction

In Exponentiation § Zero to the power of zero §§ Continuous exponents, we have two bullets. Are we not missing the obvious third point of the domain restriction arising from the definition? Specifically, in this context (of a real base and real exponent), the exponentiation function is defined via R_>0 × R → R : (b, x) ↦ limr(∈Q)→x b^r or via R_>0 × R → R : (b, x) ↦ exp(x ln b). This definition implies a domain constraint b > 0, which says that zero is excluded from the domain for all exponent values, and so even 0¹ is undefined for this definition. Any attempt to extend the domain is contrived, and inherently based on limits. Should we add a bullet to this effect? —Quondum 03:50, 13 May 2015 (UTC)

Sounds reasonable, I think that is fairly common in calculus textbooks. For instance Courant, Richard; John, Fritz (1989). Introduction to Calculus and Analysis, Volume 1. Springer-Verlag. p. 152. ISBN 9781461389576.. Dmcq (talk) 08:18, 13 May 2015 (UTC)

The case of real base and real exponent is already defined in Exponentiation#Limits of rational exponents, so the exp(y ln x) formula is not the primary definition for real number exponentiation. Its usefulness is simply because it is an easier formula than the one in Exponentiation#Limits of rational exponents. But just because one happens to give a formula that simplifies things in certain situations, this doesn't imply that the original definition in Exponentiation#Limits of rational exponents is no longer valid (if that were the case, then why even put it in the article?) MvH (talk) 12:29, 13 May 2015 (UTC)MvH

Do I need to point out that the same argument applies to the definition in terms of a rational limit? The limit does not exist for x ≤ 0. —Quondum 14:34, 13 May 2015 (UTC)

That's true, but that section doesn't say or imply that 0^2 is no longer defined. It just says: if b>0 then we can define... So that section only enlarges the domain, it doesn't remove points from the domain. If we raise the status of exp(y ln x) from "useful formula" to "the primary definition" then other sections (rational exponents, limits of rational exponents) become obsolete, and moreover, it removes points from the domain. And while exp(y ln x) is nice and short, I don't think this definition is used much in math/computing/etc. Definitions where 0^2 is defined are much more widely used, so there shouldn't be too much emphasis on the exp(y ln x)-based definition and the issues it raises. Try to find a formula in a physics book that contains f(x)^g(x) where f,g both vary continuously. I think that's very uncommon, such formulas are generally written as exp(...) instead of (...)^(...). On the other hand, expressions like x^2 are very common, and it would be bizarre to have, as primary definition, something where that would be undefined when x=0. MvH (talk) 15:25, 13 May 2015 (UTC)MvH

There's more than one major point of view on the subject as I think the reams and reams of talk before this indicate quite clearly. Dmcq (talk) 15:19, 13 May 2015 (UTC)

Quondum, I think the text as it is at the moment is fine. It doesn't say that the formula undefines 0^w, all it says is that the formula doesn't define 0^w. So whether 0^w (or 0^0) is defined or not is left to the reader which I think is a good compromise, there's no need to further elaborate on that. MvH (talk) 15:47, 13 May 2015 (UTC)MvH

It leaves the reader confused about the definition being made, which I don't think is fine. In one breath it says it says "The extension of exponentiation to real powers of positive real numbers can be done either by extending the rational powers to reals by continuity, or more usually ...", and then next it defines it as the limit over the rationals; these are not equivalent: the hallmark of a bad definition. The former (the extension) is actually very clunky when translated to symbols. I also is not clear how far the domain of the rationals was extended before extending it to the reals (there are some arbitrary choices to be made, including whether to define 0⁰, and whether to include negative bases for rational exponents with odd denominator. Have you ever seen the "extension to reals" inheriting all these points of the domain? —Quondum 16:45, 13 May 2015 (UTC)

I don't see a problem in that section. It says this:

$b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} )$

This definition assumes that b^r is already defined for positive real numbers b, and arbitrary rational numbers r. Under that assumption, it shows how you can define b^x for positive real numbers b and arbitrary real numbers x. This is precisely the definition I remember seeing in high school, so as far as I know it is (was?) the standard way to extend exponentiation to real exponents. MvH (talk) 02:03, 14 May 2015 (UTC)MvH

To rephrase this, every continuous function from Q to R can be extended uniquely to a continuous function from R to R. Fix a constant positive b in R, then r -> b^r is a continuous function from Q to R. Hence it extends to a continuous function x -> b^x from R to R. MvH (talk) 02:14, 14 May 2015 (UTC)MvH

If we start with that domain for the rational exponent function (real b > 0), the resulting function is defined only for b > 0. Then there is no argument: 0^x in the resulting real-exponent function is undefined for all x (contrary to what you say earlier in this thread). A problem only arises if you leave off the restriction b > 0.

So, back to my original point at the start of this thread. I've revised it to include the rational limit definition. —Quondum 04:50, 14 May 2015 (UTC)

I still don't understand the issue here. Suppose you define multiplication for positive integers. A while later you define multiplication for negative integers. Is the product of positive integers then suddenly no longer defined? Likewise, first the article defines b^n for every b (including 0) and every n in {1,2,3,...}. A little while later it says: if b>0 and x is a real number, then b^x := something (notice the "if" in that sentence). It never says that the original definition b^n (for n = 1,2,3...) is no longer valid. Thus, that definition still applies, and 0^2 is still defined. The original definition continues to hold unless explicitly stated otherwise. So defining b^x for positive b has no impact on 0^n, whatever its definition was, it stays that way. MvH (talk) 12:42, 14 May 2015 (UTC)MvH

To put it another way, unless explicitly stated otherwise, this formula exp(y ln x) is not a complete definition. It gives values for the function x^y on only a subset of its domain. MvH (talk) 12:51, 14 May 2015 (UTC)MvH

When a maths textbook says 'definition' then that's all there is to that definition. Any conjectures about other values are like wondering what somebody in a film did other than in the film. They don't exist outside of the film. If a definition adds to a previous definition it is defining an extension. If it replaces a previous definition it is defining a replacement and what was there before is obliterated, no more, gone in the definition. In for example the citation I give above it says 'This obviates the more clumsy 'elementary' definition and justification of these process by passage to the limit from rational exponents'. Such a definition removes everything completely as it is not an extension. If they had wanted anything else from before included in the definition they would have said so. Dmcq (talk) 13:01, 14 May 2015 (UTC)

I agree with Dmcq here, but I'll add the following: if you explicitly do bring along all the old points, you end up with a lot of problems, including that we do not have a clear definition of the original domain (in this article), and the fact that when b < 0, it does not fit MvH's description of a continuous function on Q: it is somewhat pathologically discontinuous, and hence there is no continuous extension. If we artificially limit the original function to b ≥ 0 just to avoid this, the extension is still discontinuous at the whole line b = 0 (only one-sided limits exist), and cannot be described as "the unique continuous extension from Q to R". And even ignoring all this, we are still left with 0⁰ being undefined or whatever we chose to define it as originally for exponents in Q. There are lots of problems to be seen, if you just look for them. One has to be very contrived to sidestep them. It breaks the identity b^x = exp(x ln y), and although though this is minor, in the context of real exponents, there is no particular benefit in defining 0^x for the function that is defined for real exponents, unlike the immense benefit from defining b⁰ = 1 for the exponentiation function that is defined for integer exponents. —Quondum 14:38, 14 May 2015 (UTC)

Dmcq, Quondum: What if a textbook gives multiple definitions? (as our page does). How do you interpret that: (A) only the last one hold (unless explicitly stated otherwise) or (B) they all hold (unless explicitly stated otherwise)? MvH (talk) 14:46, 14 May 2015 (UTC)MvH

WP:NOTTEXTBOOK Wikipedia isn't a textbook. A textbook will make it pretty clear what it is talking about or it isn't a very good textbook. Dmcq (talk) 14:53, 14 May 2015 (UTC)

I'm not trying to cause a disagreement here, this is a genuine question: The various definitions in our page come from various textbooks. If they are in conflict, then it is not obvious which one to take. If they are not in conflict, but have domains D1, D2 (with neither one a subset of the other) then it is not a priori clear whether just one of the definitions applies, or both (i.e. domain is D1 union D2). Combining material from various sources, it's not so obvious to determine what the overall intent of the definitions is. Is one definition meant to replace another, or is it meant in the sense of D1 union D2? How would we know? MvH (talk) 17:30, 14 May 2015 (UTC)MvH

(Again I agree with Dmcq, though the same goes for an encyclopaedia: it must be clear what it is talking about, no ambiguity "left to the reader".) Put what is being said into symbols. We have a partial function φ : R × Q ↛ R, which is the rational exponent case. From this we are defining a new partial function ψ : R × R ↛ R (and no, the original partial function remains the unchanged original φ). The fact is that ψ is a new function; there is no way in which you can pretend that φ is having its definition extended to new points without starting with a partial definition of φ : R × R ↛ R in anticipation of completing the definition by filling where you can by continuity. And that all said, let's say we do define ψ as being defined equal to φ wherever φ is defined, plus any new points that can be filled in by continuity. Are you seriously proposing that it makes sense to have ψ(–1, 2/3) = (–1)^2/3 = 1? Especially since if so, we lose some very useful identities on this function, as pointed out at the start of Exponentiation § Real exponents? —Quondum 16:45, 14 May 2015 (UTC)

This poses an additional difficulty, what if (-1)^(2/3) is a third root of unity in one textbook and 1 in another? How would wikipedias settle this? Our page says: "choosing a branch of log z" but as soon as one makes a choice, one should expect that some of the useful identities will no longer hold [Example: regardless which branch of log z you pick, (z^2)^w will differ from z^(2w) for some complex numbers z,w] MvH (talk) 17:30, 14 May 2015 (UTC)MvH

If one takes the proper approach I don't see that it does. The proper approach is to first identify which exponentiation function is being used. Defining two classes of exponentiation functions (those with only integer exponents, and those with only positive real bases) provides a useful set of well-behaved functions. Any exponentiation function that defined over a domain including both bases that are not positive reals and exponents that are not integers generally needs special care and breaks identities, becomes multivalued or whatever. Here be dragons. Over complex valued bases and noninteger exponents, no sane textbook is going to try to try to make universal claims about specific cases. —Quondum 18:23, 14 May 2015 (UTC)

Regarding the bullet point you were proposing for 0^w, would this be for integer w, real w, or for complex w (say, with positive real part)? One would have some freedom here, different books may make different choices. One could still argue that it isn't really necessary to do this, for instance, if one sees sum a_n x^n then it's clear from the context which definition (which exponentiation function) is intended. PS. With the issues involved in choosing a branch cut, wouldn't one expect that this definition is hardly ever used? MvH (talk) 19:11, 14 May 2015 (UTC)MvH

I thought it was clear: the exponentiation function on real base, real exponent. It would be a brief first point that both the definitions for the continuous exponent generally exclude it (depending on source; Dmcq gave one), essentially saying the definition as the limit over R_>0 × Q for positive bases or the exp version exclude (and cannot naturally accommodate) b ≤ 0 in the domain. I've noticed a flaw in my statement about the limit on the rationals: although the limit in the plane R² is nonexistant for b = 0, the limit on only the exponent (not the base) does exist for b = 0, x > 0. It is can thus be considered semicontinuous here: continuous in the exponent, discontinuous in the base. It is still (doubly) discontinuous at (0, 0), including in the exponent. —Quondum 20:20, 14 May 2015 (UTC)

Thousands separator

I've been reformatting large numbers by replacing the comma as a thousands separator in large values, using the {{val}} default of a thin space. I now realize that the article seems to have had a preponderance of comma-separated numbers. It is easily changed to commas with the fmt=commas parameter. In this article, should we standardize on comma or on space as the separator? —Quondum 20:30, 14 May 2015 (UTC)

"log" and "ln"

@Dmcq: It may be true that "ln" is not normally used for the (natural) logarithm of complex numbers; however using "log" for complex numbers and "ln" for real numbers in the same formula is confusing. — Arthur Rubin (talk) 18:09, 22 August 2015 (UTC)

If they have to be the same then they should be both be log like it was before someone stuck ln in. Dmcq (talk) 19:59, 22 August 2015 (UTC)

Makes sense to me. I think "log" is probably the preferable notation in mathematics articles anyway. (For people on the fence, here's a potential tiebreaker: "ln" is hard to read in sans-serif type; it looks like "In"). --Trovatore (talk) 20:05, 22 August 2015 (UTC)

Perhaps this needs to be continued at MOS:MATH? I haven't looked at the literature lately, and I don't have a real opinion as to which "looks better". Using both "log" and "ln" for the natural logarithm just looks wrong. — Arthur Rubin (talk) 21:15, 22 August 2015 (UTC)

I am generally not very concerned about cross-article consistency. Within this article, I would propend for "log". Shall I make it a formal RFC? --Trovatore (talk) 21:59, 22 August 2015 (UTC)

I already changed the article to say log instead of ln in the complex number section at the time of my first reply here. What had happened originally was that someone changed a number of log's to ln in that section and I'd changed it so only the ones referring to real numbers were ln - which gave the mix Arthur say. as in for instance ${\textstyle \log(w)=\ln(r)+i\theta }$. Brackets were put in at the same time and I left them there. Dmcq (talk) 22:40, 22 August 2015 (UTC)

My suggestion was that we drop "ln" from this article altogether (except for a mention explaining that it is sometimes used for natural log). But I don't care to press the point if there is no active problem. --Trovatore (talk) 19:48, 30 August 2015 (UTC)

I suggest using log everywhere in the article, but also adding a short "Alternative notations" note (or some such similar title) explaining the common use of log to mean common log or log base 10 (especially among engineers and scientific calculators) and ln to mean natural log or log base e. That way we do not confuse anyone but still cover all the common uses. — Loadmaster (talk) 18:04, 31 August 2015 (UTC)

Exponent and power are NOT synonyms

At the beginning of the article, it reads ... exponent (or power). That is wrong! The following explains the correct terminology:

Base^Exponent = Power

Examples of the correct use this terminology:

4, 8 and 16 are POWERS of 2.

32 is the POWER of 2 when the EXPONENT is 5.

10, 100 and 1,000 are POWERS of 10.

George Rodney Maruri Game (talk) 03:37, 9 September 2015 (UTC)

But power is also used in a manner synonymous with exponent. For example, $x^{n}$="x raised to the power n". Sławomir
Biały 10:27, 9 September 2015 (UTC)

Mmmm... "Wording" ≠ Terminology. The wording (how we read aloud a power) is not completely compatible with the preferred labels for the components of a power as an expression. That is unfortunate. The article contains an explanation about how to read powers, though. Most textbooks call the exponent "exponent" despite the confusing wording. Should a clarifying note be added to the article? I am an engineer and I think a mathematician would be more authoritative concerning this issue.

George Rodney Maruri Game (talk) 16:00, 10 September 2015 (UTC)

It doesn't really have to do with whether the thing is read aloud. One can find quite a large number of print sources that employ the phrase "raised to the power" in this sense. However, I would agree with a rephrasing, as long as it is consistent with established practices. I'll have a go at it. Sławomir
Biały 16:17, 10 September 2015 (UTC)

That definitely looks like an improvement to me. The old wording, "the exponent (or power)", was indeed a bit strange, but the new wording seems unobjectionable.

I stuck my head out the window and started countin' phone poles
Goin' by at the rate of four to the seventh power
And I put two an' two together an' added twelve an' carried five
An' come up with twenty-two thousand telephone poles an hour
— CW McCall

Agree that is an improvement. Thanks. I had a look but wasn't able to think of how to phrase it better myself. Dmcq (talk) 00:12, 11 September 2015 (UTC)

Hmmm...

1. 'raise b to n'.
2. 'b raised to the power of n'.
3. 'b (raised) to the power n'.
4. 'b to the n'.
5. 'b, raised n'.
6. 'b exponentiated by n'.
7. 'b powered by n' (like 'b multiplied by n').

So:

1. To raise 2 to 3 one adds 1.
2. 'b raised to the power of n' takes a lot of time, to speak and, worse, as often by many, to listen to.
3. 'b (raised) to the power n' is wrong as n is a number and a exponent and not a power, unless in itself, in which case it is confusing, considering:
4. 'b to the n' sounds like an interval (a range like 'from 3 to 10') ending on a special n, a part of a 'lim_b → n (...)' and a derivation. It is short but unexpressive (or too expressive and thus confusing).
5. 'b, raised n' states (merely) what is written (used in German (without the comma, thus 'hoch' is like 'plus')) and is the shortest solution (of those given here), but less expressive than:
6. 'b exponentiated by n': even one syllable more than solution 2, thus:
7. 'b powered by n' (like 'b multiplied by n'): the context prevents b from being plugged into n, it is expressive as b is turned into a power of and by n.

Now only 7000 million people have to follow this argument (easy part) and drop some of their habit (difficult part) – and one of them has to go and edit the entry, for who am I ... --Empro2 (talk) 23:44, 22 September 2015 (UTC)