Talk:Zero to the power of zero

Recent edits

The following comment was left by MvH on my talk page, concerning this edit; I repost it here with permission:

You reverted my edits, saying "the information that the different arguments depend on context is crucial" but that is not a neutral point of view. People fall in several groups: "always 1" or "depends on context" or "undefined". Picking the middle option seems reasonable, but that doesn't mean that it is neutral. For those who agree with Knuth, the value is simply 1 in all contexts. Authors can define any expression in any way they want, so rather than saying that the value depends on the context, it is more accurate to say that the value depends on the author. MvH (talk) 17:44, 26 September 2018 (UTC)

What the lead of the article currently says is that there are at least two common things people do when they see 0^0: declare it to be 1, or declare it to have no particular value. This is true. It also says that there are justifications for both. This is true. And finally it says that the justifications depend on the context in which the expression 0^0 appears. This is also true. Moreover, beyond being true, this is a correct summary of what our article actually says: that there are context-specific justifications for both answers. About Knuth in particular, I hope you do not assert that he believes that the bivariate limit $\lim _{(x,y)\to (0,0)}x^{y}$ in $\mathbb {R} _{\geq 0}^{2}$ is equal to 1 -- his arguments are limited to certain contexts.

I do agree with you about "agreed-upon" being better than "obvious" and I will restore that change now; I apologize for removing it.

Of course, further comments (from anyone) are invited. --JBL (talk) 10:26, 28 September 2018 (UTC)

I have now gone back and re-read Knuth's paper (it's been a while). Here is what he has to say on the topic:

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

--JBL (talk) 10:37, 28 September 2018 (UTC)

I do not mind the formulation of this paragraph at all, but I think "expression with no agreed-upon value" could be moderated to "expression with no generally agreed-upon meaning". It is obvious to me that people agree on defining this expression to have a specific value (for usefulness, or even involving limits), depending in which context it turns out. There are also purists agreeing to nail the expression as "undefined" (which is no value to me), because there is no value they consider generally appropriate. Purgy (talk) 11:21, 28 September 2018 (UTC)

I think that's a good suggestion. --JBL (talk) 11:26, 28 September 2018 (UTC)

Agree. MvH (talk) 13:39, 29 September 2018 (UTC)

The article is quite good at the moment, it presents both sides without choosing sides. One thing I don't like about the phrase "context" in this debate is that it allows for imprecise statements such as "B is true depending on the context". Instead of "... depending on the context" we should write "if A then B" where A and B are well-defined statements. Imprecision is at the core of the debate. For some people, the $0$'s they see in $0^{0}$ are just that: $0$'s. But others interpret those $0$'s not as numbers, but as functions that converge to $0$. For the first group, the limit argument is not a valid argument, for the second group, it is valid. If you interpret these $0$'s as not as numbers, but as functions that converge to $0$, then it would be best to simply say that rather than saying less precise things such as "depends on context". MvH (talk) 14:27, 28 September 2018 (UTC)

That is not quite right either. As it says at Indeterminate form "if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form". Talking about limits the way Knuth does is waving hands rather than dealing with that properly, you have to just get rid of the concept of an indeterminate form to do what he talks about. Dmcq (talk) 19:51, 28 September 2018 (UTC)

You won't find that definition in textbooks because it would make floor(0) an indeterminate form. In textbooks, "indeterminate form" simply means member of this table. MvH (talk) 22:27, 28 September 2018 (UTC)

Floor(0) is a function call. If we did all the arithmetic using add(x,y),subtract(x,y), multiply((x,y),divide(x,y),power(x,y) we wouldn't have to worry about indeterminate forms - provided we didn't automatically apply simplifications knowing the properties of these functions. Indeterminate forms allow us to not worry about using normal arithmetic rules. If we used an operator for floor and automatically applied rules combining it with other things then yes floor(0) would have to be counted as an indeterminate form or in fact something rather strange as it is defined with a value at 0, especially if it occurred at all often in calculus. But we don't. Indeterminate forms are not an arbitrary collection like you seem to be making out, the table is a result not a starting point. Pure maths people might be happy tuning '+' into 'add' but this is simply not a starter for applied maths. Also distinguishing between floor(x) and ceil(x-1) could be rather difficult for physical processes! Dmcq (talk) 08:38, 29 September 2018 (UTC)

Note that the place you pointed at said "The following table lists the most common indeterminate forms and ...." Dmcq (talk) 08:46, 29 September 2018 (UTC)

The only way to resolve the controversy is to use different notations for different things. We teach students that 0/0 is a math error, but in calculus the same expression could also denote what you call an indeterminate form. Why not just denote that indeterminate form as "0/0". The ambiguity would disappear and the debate would be resolved. The two expressions 0/0 and "0/0" are very different, 0/0 is always an error, whereas "0/0" means: use l'Hopitals rule in order to prevent an error. MvH (talk) 17:50, 29 September 2018 (UTC)

Besides the fact that putting around quote marks in calculus would make engineering students reject mathematics that simply doesn't work. Try writing a program to do what you are trying to think of and you'll find that you were simply waving your hands around. If power(0,0)) evaluates to 1 then the value will be defined and there is no question about the value at the point, there is no requirement to find a limit even though one has a discontinuity. Dmcq (talk) 23:05, 29 September 2018 (UTC)

A few quotes is a small price to pay for mathematical consistency. Clear and consistent standards make life easier for programmers. They lead to less handwaving, not more. Consistency is a good thing. pow(0,0) does not need to know what the context is. Instead, pow(0,0) should simply follow the IEEE standard and thus return 1 MvH (talk) 00:15, 30 September 2018 (UTC)

My suggestion may be belittled as in no way satisfying "If A then B", but, imho "depending on the context" IS the "program" that "inserts" scare quotes for the vast majority of readers in the appropriate place, at least in the context of not absolutely formalized texts. Purgy (talk) 06:07, 30 September 2018 (UTC)

Yes that is fine and is what happens in effect but MvH says "Consistency is a good thing. pow(0,0) does not need to know what the context is." which most definitely is the direct opposite. The whole business of doing algebraic simplification and using indeterminate forms for the limits doesn't work with that. Dmcq (talk) 09:48, 30 September 2018 (UTC)

The IEEE standard clearly states that pow(0,0) is 1. It does {\em not} say that pow() needs to be aware of context. Computer scientists tend to be careful with notation. I do think we can agree on some things though. The page says "mathematical expression with no agreed-upon value". Regardless what we write, the "no agreed-upon" is accurate either way. MvH (talk) 20:27, 30 September 2018 (UTC)

Sorry I was assuming you were using pow(0,0) as a way of writing 0^0 rather than the IEEE function. pow() is a mixture of pown and powr to my way of thinking. Dmcq (talk) 20:36, 30 September 2018 (UTC)

Yes, the IEEE 754-2008 pow() basically comes from C. But it is cleaner to have separate functions (depending on the domain), when possible. That's why pown and powr have been defined. Vincent Lefèvre (talk) 00:08, 1 October 2018 (UTC)

Asserting things will be cleaner is not the same as making things actually work. Anyway the article seems to describe the actual situation as described in reliable sources reasonably well whatever about what you'd like. Dmcq (talk) 09:41, 30 September 2018 (UTC)

As far as this discussion is about the article, I agree 100% with Purgy. The justifications in the article depend on the mathematical context in which the symbols 0^0 are to be understood, and the lead correctly reflects this. As far as it is about what is really true, this doesn't seem like the right venue. --JBL (talk) 13:45, 30 September 2018 (UTC)

It is true that words/names/expressions can have multiple meanings. But unless stated otherwise, we should assume that the most obvious meaning is the intended meaning. If the most obvious meaning is not the intended meaning, then one should indicate that. I don't understand why this is so controversial. MvH (talk) 20:42, 30 September 2018 (UTC)

Not sure what you're referring to. IF pow then I see no reason why you introduced that. Dmcq (talk) 21:13, 30 September 2018 (UTC)

Here is an example. Suppose f,g both converge to 0, and that f^g converges to 2. A student is asked to compute the limit of f^g. In step 1, the student replaces the limit of f^g with 0^0. In step 2, the student replaces 0^0 by 1. The final answer is wrong. This implies that at least one step was wrong, but which step(s)? If we allow notation with ambiguous meanings, then we can prove everything, including 2 = 1. But if 0^0 is unambiguous, then we can say for certain that step 1 was wrong. So, in the context of limits, if our notations are consistent then we can't prove that step 2 was wrong because there was already an error in step 1. This is why we should want consistent notation, it makes it possible to pinpoint the error. Now some may also decide that step 2 was wrong as well, but logic does not compel us to make that conclusion. MvH (talk) 21:23, 30 September 2018 (UTC)

Sounds to me like you are again trying to put the cart before the horse. For straightforward engineering and science if you stick in the limit of the variable into an expression then the value you get is the limit value if it evaluates okay. If the value of 0^0 is 1 then they would get the wrong result as far as science and engineering is concerned as they'd get something discontinuous rather than continuous. If we want to be able to easily apply algebraic simplifications this is what happens. As far as a pure mathematician is concerned x/x may be 1 except when x is 0 but if we're extending results to limit points it is 1 and there's no real difference for 0^0 except that the result 1 is much more common in other areas so defining it as 1 saves a lot of bother. Extending it to everything though is just causing trouble in applied mathematics. Dmcq (talk) 21:49, 30 September 2018 (UTC)

I think what you said about'a student' about covers it. The algebraic simplification bit is important. If we just have students given f^g then that is top level and obvious. I think perhaps Knuth was thinking like that too, though I wonder if he ever did have to tech anything like that rather than combinatorics. Dmcq (talk) 21:55, 30 September 2018 (UTC)

Just to make sure I understand your response, are you saying that step 2 is OK in pure math but not in science/engineering/applied math? MvH (talk) 23:18, 30 September 2018 (UTC)

Something like that, pure maths nowadays seems to be mainly concerned with algebra and combinatorics and logic even in things like topology. I'm saying I agree with the tenor of the article. Of course even there it could be left undefined in general but that would be less convenient in general for them. Dmcq (talk) 10:48, 1 October 2018 (UTC)

I tried to gently imply this in my last comment, but let me now be more direct: this conversation has long since stopped relating to the article in any identifiable way. If someone would like to propose some actual edits, then by all means do so; but otherwise please see WP:NOTFORUM. The current article handles these issues very well, and with reasonable sourcing. --JBL (talk) 23:58, 30 September 2018 (UTC)

Previous discussions

There have been numerous discussions on this topic at Talk:Exponentiation, Talk:Empty product, and Talk:Indeterminate form (and their archives), as well as the Stack Exchange discussion.--126.204.173.85 (talk) 18:52, 22 November 2018 (UTC)

Interesting Equation for 0 to 2

$((0^{0})+((0^{0})/(0^{0})))^{(}(0^{0})/(0^{0}))^{0}=2$ which Google reports as being equal to 2.

How in the world can you ever get 2 by multiplying and dividing (adding/subtracting) 0?

What would a proof for 0^0=0 look like? I think it is very self evident as to perform "power" is to multiply in repetition, which multiplication is just addition in repetition.. So the basic conclusion to take 0 and bring it to the 0th power would be to multiply 0 times 0 zero times which is to add 0 plus 0 zero times..

I find it most remarkable that all computers seem to handle 0^0=1 almost as if it was more convenient to respond that way than handle the fault similar to divide by 0. — Preceding unsigned comment added by 35.136.117.228 (talk) 16:27, 7 March 2019 (UTC)

This is WP:NOTAFORUM, but I'll answer anyway. 0^0 is indeterminate, as described in the article. For any value other than zero, $x^{0}=1$. power~enwiki (π, ν) 18:22, 7 March 2019 (UTC)

Nothing interesting actually: Google just regards 0^0 as equal to 1 (probably for reasons given in the article), and if one replaces 0^0 by 1 in the above expression, one simply gets 2 using basic mathematics. Vincent Lefèvre (talk) 23:07, 7 March 2019 (UTC)

Just for boring "fun", considering the IP's LaTeX fluency, it should be perhaps

$\left(0^{0}+{\tfrac {0^{0}}{0^{0}}}\right)^{\left({\frac {0^{0}}{0^{0}}}\right)^{0}}=2,$

and there is really, really nothing deserving interest. Could the thread be deleted, finally? Purgy (talk) 07:39, 8 March 2019 (UTC)

Neighborhood

@Jasper Deng and Vincent Lefèvre: With respect to this: the "neighborhood" condition is just wrong for the argument being made here, because different things go wrong for $x<0$. The point of this sentence is that you cannot define a continuous function on $\{(x,y)\mid x>0\}\cup \{(0,0)\}$ to agree with $x^{y}$ when $x>0$ and also to be continuous. I am happy to discuss ways to write this to be clearer, but including "neighborhood" is really not what's needed. --JBL (talk) 09:23, 11 April 2019 (UTC)

I included it for lack of a better word. Basically, the statement as-is was false, because one need not restrict to a subset on which there are multiple limiting values. I think it'll be better to just state that we cannot do it for the right half-plane. --Jasper Deng (talk) 10:13, 11 April 2019 (UTC)

Agreed. The new version is OK for me. Vincent Lefèvre (talk) 12:43, 11 April 2019 (UTC)

Excellent, thanks both! --JBL (talk) 14:58, 11 April 2019 (UTC)

Documenting programming language conventions

How should we go about removing the many "[citation needed]" notes in the "Treatment on computers section? Do we have to have a specific statement from the documentation (as is given for e.g. Lua and Perl), or else remove it as uncited? Can we use a secondary source that documents it (e.g. some programmer's blog or StackExchange answer stating their observations)? Can we link to implementation details (e.g. this line of Python's source code)? None of these seem especially satisfactory... -- Perey (talk) 09:07, 4 July 2019 (UTC)

IMHO, the behavior should officially be documented. Information just base on tests is not sufficient, as it could actually depend on the platform or some library, be undefined behavior, or whatever. I think that in the absence of documented behavior, the behavior deduced from the implementation could be mentioned here, but it should be clear that this is so, and not a documented behavior. Vincent Lefèvre (talk) 11:23, 4 July 2019 (UTC)

I agree with Vincent Lefèvre. --JBL (talk) 13:24, 4 July 2019 (UTC)

Info about not destinguishing this from 0^0 limit representation

Obviously, there is a difference, because this is just a notation (much better representation would be (-> 0)^(-> 0)). Also we cannot define indeterminate form (as it can be -1 or 1), while we can define zero in the power of zero to be 10, 5, e or 1. Or anything. But we cannot define zero in power of zero to be two values (functions can only have one result). The history of this should only be in this article: https://en.wikipedia.org/wiki/Indeterminate_form#Indeterminate_form_00

We should add smth. like this in very beginning: This article is about numeric value. For the limiting form, see... ZBalling (talk) 19:22, 24 August 2019 (UTC)

If only it was that simple. Perhaps you would care to explain to a computer the rule at the start of the article on indeterminate forms "In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form". Unfortunately defining 0^0 would mean that 0^0 would give enough information to evaluate it and there is no straightforward way I know of getting around that. In practice I distinguish the type of the power as being either an integer or a real but lots of people have problems with the idea of a strong distinction between integers and reals in mathematics. Dmcq (talk) 22:58, 24 August 2019 (UTC)

"by replacing these functions by their limits" it is not direct replacement. It only looks like it. I mean there is a difference between -> 5 and = 5 and the second is because https://en.wikipedia.org/wiki/Continuous_function (rather complicated thing IMO) and all that difference between limit of sequence and limit of function (Heine definition). Sin x / x is also classic (we can define value in x = 0, so...)

"defining 0^0 would mean that 0^0 would give enough information to evaluate it". No, I do not thing so, even if we define it and forget about all continuaty, the function will still only have right-hand limit. But it is really rather late here)) or maybe https://en.wikipedia.org/wiki/Classification_of_discontinuities#Removable_discontinuity (this one is not removable, smth. like that) ZBalling (talk) 00:43, 25 August 2019 (UTC)

Yeah. It is (non)removable singularity. Even if we define a 0^0 as one, this will not affect limit in any way. Like at all. One dot never affects limit (in this particulr case, that is for sure) ZBalling (talk) 01:04, 25 August 2019 (UTC)

And BTW, https://www.wolframalpha.com/input/?i=limit+x-%3E0%2B+x%5Ex cool, is not it? So the limit does exist. ZBalling (talk) 01:18, 25 August 2019 (UTC)

Oh, my. Now I understand how wrong I was, because it is never = 5. I mean what does limit x-> 0 sinx/x = 1 means? It means that y->1 when x->0. Not equels 1! It is never one actually. ZBalling (talk) 02:04, 25 August 2019 (UTC)

This conversation is nearly incomprehensible, but: the very first sentence of the section "Continuous exponents" is "Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form." To me, this seems like appropriate placement of the link to indeterminate form in the body. (It is also linked in the second sentence of the section "History of differing points of view".) Am I right that the original question is whether the words "indeterminate form" should also appear somewhere in the lead section? --JBL (talk) 11:40, 25 August 2019 (UTC)

Note that in practice, there are two or three power functions: one over ℝ×ℤ (i.e. with integral exponents) and one over ℝ×ℝ or ℝ⁺×ℝ (i.e. with real exponents, possibly with a restriction on the first argument). 0⁰ is an "indeterminate form" only in the latter case (real exponents). Vincent Lefèvre (talk) 12:20, 25 August 2019 (UTC)

Yes, we should mention that this article is not about indeterminate form. It is about x^x where x = 0, not where x -> 0. And I know about "three power functions" trick ZBalling (talk) 16:33, 30 August 2019 (UTC)

Ok. I do not think that I agree with you: part of this article clearly is about the indeterminate form. I do think that it would be worthwhile to have a link to indeterminate form in the lead, but not as a "this is something this article is not about"-type link. --JBL (talk) 18:14, 30 August 2019 (UTC)

Have you read this. Very logical. And just what I am talking about. https://en.wikipedia.org/wiki/Talk:Characterizations_of_the_exponential_function#Error_in_series_for_exponent ZBalling (talk) 16:33, 2 September 2019 (UTC)

You have lapsed back into incomprehensibility. The present article discusses 0^0 occurring in expressions for polynomials very clearly. It also discusses 0^0 occurring in other contexts. --JBL (talk) 16:40, 2 September 2019 (UTC)

The subject of the article is not intermidiate form; the history indeed has some people mixing it with limit representation that is incorrect and that is why we should place a placeholder. 2A00:1370:812C:B802:90A9:870:DB2A:829A (talk) 17:16, 11 May 2020 (UTC)

set theory

I have removed set theory from the claim

In algebra, combinatorics, or set theory, the generally agreed upon value is 0⁰ = 1,….

I understand where the claim is coming from. It's true that if you consider 0 to be a cardinal or ordinal number, the set-theoretic definitions for 0⁰ yield 1.

The problem is that the statement as given conflated these with the real number zero, which in any of the standard versions of the implementation of mathematics in set theory is a distinct object (or maybe more to the point, the exponential function in question is distinct from the ones defined on the cardinals and ordinals).

It's important to remember that the origins of set theory are in real analysis. The behavior of the reals is not at all an afterthought for set theory; it's actually the main preoccupation of a large segment of set theorists. --Trovatore (talk) 22:45, 7 February 2020 (UTC)

It is not clear where it comes from. It could be from the power set, but in this case, this is related to combinatorics anyway. Vincent Lefèvre (talk) 22:59, 7 February 2020 (UTC)

Not immediately clear to me what it has to do with the powerset, but that's not important right now. I just think it's problematic to claim it's the agreed value in "set theory" without further specification. --Trovatore (talk) 23:10, 7 February 2020 (UTC)

Sorry, I mean the set of functions, which is mentioned in Power set. The number of functions can be expressed as |X|^|Y|, and applied to the number of functions from the empty set to the empty set, one gets 0⁰ = 1. But that's actually combinatorics. Vincent Lefèvre (talk) 23:40, 7 February 2020 (UTC)

Yes, that's the viewpoint I had in mind when referencing exponentiation of cardinal numbers, as in my initial comment. I agree that "combinatorics" covers this case. --Trovatore (talk) 03:05, 8 February 2020 (UTC)

I wanted to add my disagreement with Trovatore's initial comment and reversion. "0" without qualification in set theory is an ordinal, and you're not constructing real numbers without defining 0 first. Ordinals form the backbone of the universe of sets, cardinal and ordinal numbers are fundamental to set theory, real numbers are not, and the fact that real analysis prompted the birth of set theory is not relevant to the statement "0^0 = 1 in set theory", which is entirely accurate. Both ordinal and cardinal math in set theory define 0^0 = 1. Any set theory textbook will define these values prominently, while you'll have a pretty hard time finding information on the calculus of indeterminate forms in a set theory textbook. Just because you can define functions within set theory where 0^0 has any value you like doesn't mean 0^0 has no agreed-upon set theoretical value: ordinals are of the most fundamental context in set theory. I strongly (but respectfully) dissent. (I actually came to the talk page because I was surprised to see that set theory was not included in the lede, and clearly I am not the only one.) TricksterWolf (talk) 19:13, 10 October 2021 (UTC)

The lede is just a summary. If set theory is not considered later, then it does not have its place in the lede. — Vincent Lefèvre (talk) 22:31, 10 October 2021 (UTC)

A set-theoretic interpretation is explicitly mentioned in the section Zero to the power of zero#Discrete exponents, with the same emphasis given as in the combinatorial interpretation. --JBL (talk) 23:31, 10 October 2021 (UTC)

OK, but that's the power set I mentioned above. Nothing new. — Vincent Lefèvre (talk) 00:38, 11 October 2021 (UTC)

Ah! I think I see where you're confused. The cardinality of the power set of the empty set represents 2^0, not 0^0 (the 2 in the target represent "member is in subset" and "member is not in subset"). The fact that 2^0 = 1 is already agreed-upon and not what is being discussed. The empty function (which is referenced in the article) has cardinality zero in almost any treatment, and is neither conceptually speaking nor in any obvious construction the power set of any set. TricksterWolf (talk) 21:55, 11 October 2021 (UTC)

I think it's super misleading to put "set theory" here. It makes it sound as though the set-theoretic viewpoint is an argument for the value of 1 even in the real-to-real or complex-to-complex context, which it isn't at all. In set theory, the natural number (equivalently ordinal number or cardinal number) 0 is a completely different object from the real number 0. --Trovatore (talk) 22:40, 11 October 2021 (UTC)

Only if you don't read the immediately preceding sentence, which establishes that the context in which it occurs is relevant. --JBL (talk) 01:00, 12 October 2021 (UTC)

But it isn't true that 0^0=1 "in the context of set theory". It's true if you take 0 to be an ordinal, yes. But set theory is much more than ordinals.

I have to disagree with TricksterWolf that reals are not "fundamental to set theory". Reals are extremely fundamental to set theory. They're the first objects that appear past the hereditarily finite. Set theorists think about the reals all the freakin' time. And the real number 0 is completely different in character from the ordinal 0. --Trovatore (talk) 04:14, 12 October 2021 (UTC)

I think I see your point with that elucidation: you're suggesting that 0^0 conjures "real arithmetic" for most readers (it doesn't automatically do that for me), so they may mistakenly think the set theoretical definition is an application to real numbers(?)—but I still contend that this would be a weird interpretation, especially given that algebra and analysis are both already explicitly provided as examples. Maybe "discrete mathematics" would be less confusing than set theory, but that isn't the same perspective IMO. TricksterWolf (talk) 01:07, 12 October 2021 (UTC)

@TricksterWolf: I'm not talking about the power set of the empty set, but about this section, where |X| = |Y| = 0, giving 0^0, that is the number of functions from the empty set to the empty set. — Vincent Lefèvre (talk) 00:16, 12 October 2021 (UTC)

That doesn't have anything to do with the power set, though. That's just the definition of cardinal exponentiation into which you're plugging 0 and 0. The definition is provided there to explain why |P(S)| = 2^|S|, not 0^|S|. The power set of a set isn't related to 0^0 in any way that I can see, even when it's the power set of the empty set. TricksterWolf (talk) 01:14, 12 October 2021 (UTC)

I think that's covered by "combinatorics". --Trovatore (talk) 04:14, 12 October 2021 (UTC)

Depends on context vs. no agreed-upon value

I was wondering what people thought about saying in the first sentence that "0⁰ depends on context" vs. saying that "0⁰ has no agreed-upon value". To me the latter suggests that there are factions, some arguing for one value and some arguing against it. But is it closer to the truth to say that it is not so much a controversy with different people thinking different things, but rather that most experts believe both that there are contexts (e.g., in discrete math) in which 0⁰ = 1 and that there are other contexts (e.g., limits) in which it is better to consider it indeterminate? (I had tried to simplify the lead to reflect this, but it seems that at least one editor disagrees, so I thought I'd ask here to find out what the consensus is.) Ebony Jackson (talk) 19:40, 7 November 2020 (UTC)

I think your formulation is probably grosso modo accurate, but the problem is that the question is rarely discussed at that level of generality in good sources. What we have is lots of sources that say 0⁰=1 and lots of others that define exponentiation in a way that leaves it out. Weaving them together in a way that distinguishes whether the exponent is discrete has a flavor of WP:SYNTH, even if it gives the correct result. --Trovatore (talk) 20:03, 7 November 2020 (UTC)

@Ebony Jackson: Thanks for raising the issue here. The current version says that there is no universally agreed-upon value, and then immediately goes on to explain that there are two common reasonable interpretations that arise in different contexts. I agree with you that it would be problematic only to say the first part without further specifying, but it's not like the statement about context is hard to find, it's literally the second sentence in the article. Similarly, with respect to simplifying, the current lead is one short paragraph, only four sentences long, and not unduly jargon-y or technical -- I guess I just have a hard time seeing it being an impediment to readers. --JBL (talk) 17:47, 8 November 2020 (UTC)

OK, thank you both. I see your points. Ebony Jackson (talk) 22:11, 10 November 2020 (UTC)

0^0 is undefined.

If "n" is "any number or variable" and we all know that any number or variable is equal to itself, then n must be equal to n right. So n=n. We also know that any number or variable subtracted to itself is equal to zero (0), then n-n=0. Lastly, we also know that any number or variable divided by itself is equal to one (1), then n/n=1. So, we will derive the equation n/n=1; this equation is also (n)(n^-1)=1; n^(1-1)=1; n^0=1. This means any number or variable to the power of zero (0) is equal to one (1). But this "n" could not substitute to zero (0) value because there has no zero (0) divided by zero (0); 0/0. You cannot say that there has "one" zero in a zero, 0^0=1. Therefore, zero (0) to the power of zero (0), 0^0 is undefined.— Preceding unsigned comment added by Lights Beatlemania (talk • contribs)

Maybe you should read the whole article -- you might learn something! --JBL (talk) 11:43, 7 April 2021 (UTC)

0^0 = 1 isn't saying there's one zero in a zero, that'd be 0/0 = 1 (which you point out). 0^0 is saying that when you multiply no 0's at all, you get 1 (the identity of multiplication). This is in part why x^0 = 1 in general; 0! is 1; the empty product is 1; etc. TricksterWolf (talk) 01:34, 12 October 2021 (UTC)

This doesn't make sense to me

I don't agree.

In algebra and combinatorics, the generally agreed upon value is 0^0 = 1

Should 0^0 = 0 (its power to itself) - Itself.

— Preceding unsigned comment added by 81.106.29.7 (talk) 18:31, 14 April 2021 (UTC) 

@81.106.29.7: per WP:TALK this talk page isn't supposed to be for discussing the underlying math, except to the extent that it's connected with discussion about what should appear in the article. You might consider asking a question at the mathematics reference desk. --Trovatore (talk) 20:09, 14 April 2021 (UTC)

The article's last section implies Mathematica makes 0^0=1. When I check the ref link (36), I see otherwise, and using WolframAlpha, it is also undefined.Billymac00 (talk) 18:22, 25 April 2021 (UTC)

There is no such rule "power to itself". Power function is defined so that any number in power 0 is 1 (a^n is 1 * a*a*a*a (a repeated n times, so 0^0 is just 1)). 0 is no exception. 2A00:1FA0:84A:C216:6CF5:7552:9B80:28EE (talk) 12:02, 3 November 2021 (UTC)

Current version biased & needs revision

Because of interviews that we were conducting for Algebra teachers in our math department, I became curious about what Wikipedia says and just read this page for the first time. As a mathematician, I like the diverse examples and different situations, but I couldn't help but feel that the page is biased towards encouraging the reader to take 0^0=1, presenting it as if it were the consensus in the profession, when it is not. Within very specific areas of specialty, it may be practice to define it to be 1 for some of the reasons made, but that is not very clear by this article.

My experience is that the broader mathematical community leaves it undefined and refers to it as indeterminant because there is more than one value that it could hold while still being consistent with the definitions and properties of operations; and only context & usefulness can dictate from one situation to the next what we might want to define it as.

The article already presents situations in which 0^0=1 makes sense. It also discusses situations involving limits of functions in which 0^0 represents f^g for continuous f & g. But it states rather definitively that 0^0 must be 1, because x^0 needs to be continuous. x^0 representing 1 in polynomials is a notational definition of its own. What is ignored is that exponents and their properties arise and are defined from the natural numbers by taking n^m to represent having m copies of n multiplied together. x^y is then developed & defined from this by continuity arguments. This itself leads to 0^0 being indeterminant, because 0^n would always be 0, being n copies of 0. This then leads to 0^x being 0 for all rational number x, because all roots of 0 are 0. This then leads to 0^x being 0 for all real numbers x by continuity. So just as one might argue that 0^0 must be 1 to make x^0 continuous; in the same way 0^0 must be 0 to make 0^x continuous.

Furthermore, 0^0=0 satisfies the exponent properties, as 0^0=1 does. So either would work. So, as far as real number operations go, it truly is indeterminant.

Now, I know a lot of work has gone into this article, so I have not changed the article at all, but hope that those of you committed to it would do two things: 1st, balance out the arguments in favor of indeterminant by acknowledging & explaining that 0 is also a consistent value that it could represent, which is one reason for why it is indeterminant; and 2nd, make it clearer that the math community as a whole does NOT define it to be 1, but that within specific contexts and specialties, it is often defined as such because it makes certain results easy to state. Thank you. Jkbelnap (talk) 05:14, 13 July 2022 (UTC).

The article does a very good job of laying out the conventional choices of various contexts, with a reasonable attempt to provide supporting sources. Your arguments (which appear just to be your own, and lack supporting sources) about one particular context are misguided. In particular, This itself leads to 0^0 being indeterminant, because 0^n would always be 0, being n copies of 0 misses the point: it punts the question to "what does it mean to multiply 0 copies of something together?" And the answer to that is "In any context where the sentence makes sense, multiplying 0 copies of something together gives 1, even if that something is 0." Your discussion of the expression 0^x is also confused: the section in question concerns polynomial functions, which 0^x is not, so its continuity (or not) is irrelevant; it is also certainly not true that 0^x = 0 for all rational or real numbers x ! --JBL (talk) 19:00, 13 July 2022 (UTC)

The article is indeed biased. It's biased towards correctness. You write that 0^x is 0 for all rational numbers x. That's not correct. Clearly x = -1 is a counter example. MvH (talk) 18:12, 27 February 2024 (UTC)

"Justification"

I propose to change to "offered the "justification" (n should be 0 to avoid 0/0 and this shows one cannot set 0^0 to 2, for example, only to 1 or 0)." Why did not you check the paper? Valery Zapolodov (talk) 11:29, 19 October 2022 (UTC)

Your suggested addition is incomprehensible. I do not understand your final question or its relevance. JBL (talk) 18:44, 19 October 2022 (UTC)

See my latest reverted edit. Valery Zapolodov (talk) 13:34, 21 October 2022 (UTC)

Note that in its annotation, Mascheroni wrote $a-a$ instead of just 0. So perhaps $a-a$ means something a bit more than a plain 0 (perhaps a formal term that would evaluate to 0). So any additional info about the justification would need to be based on a secondary source. There could be a note saying to be careful with this justification (pointing the problem of its interpretation, but not trying to interpret it), though. — Vincent Lefèvre (talk) 16:46, 21 October 2022 (UTC)

Is not the point he got to 0^0 = 0^0/0^0? Then 0^0 can be only 1 or 0 (0 makes it all undefined since 0/0 is undefined). Valery Zapolodov (talk) 01:48, 2 November 2022 (UTC)

To interpret what Mascheroni wrote, you need to know why he wrote $a-a$ instead of just 0, which is simpler. — Vincent Lefèvre (talk) 01:57, 2 November 2022 (UTC)

a - a is 0. What? Valery Zapolodov (talk) 02:11, 2 November 2022 (UTC)

@Valery Zapolodov: I have looked at the paper, and I too think that it is not our role here to try to guess what Mascheroni meant. So, although I understand your desire to make sense of it, I think it is best not to try to add your own reasoning about it. I'm sure that you can find better ways to use your knowledge to improve Wikipedia. Best wishes, Ebony Jackson (talk) 02:44, 2 November 2022 (UTC)