# Talk:Exponentiation

Exponentiation has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.
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## Promotion of notational-history section

In a recent edit, User:Dan6233 moved the "history of notation" section up to the first section after the lead.

My first reaction was that this puts too much weight on notation. Notation is much much much much less important than the math. Radically less important. Exponentially less important.

However, when I look at the article as it currently stands, I don't find it too objectionable. It's a reasonably short section, and it is traditional to deal with history near the top. As long as it doesn't grow, it seems sort of reasonable. But I'm concerned that it might grow.

Here's my thought, not really fleshed out: What if we turned it into a history section, rather than history of notation specifically? We could treat the history of the concept and of the notations sort of in parallel. If done well, that might yield a section that could grow sort of organically, without overemphasizing notation.

Thoughts? --Trovatore (talk) 00:00, 23 September 2015 (UTC)

All sounds reasonable to me. I agree with it being of less importance for the people coming here but we can afford to order to some degree by largest importance/size first. Just naming it History sounds good and leaves a bit of room for expansion. Dmcq (talk) 14:10, 23 September 2015 (UTC)

I am not a mathematician , and I found the history of notation very helpful. Thanks for keeping it up front.

May I suggest that after the sentence "Thus they would write polynomials, for example, as ax + bxx + cx3 + d," you provide in parens the modern notation for polynomial equations? That would allow the nonexpert reader, who does not have that information handy in the local memory bank, to see the difference between the two. The sentence would read, "Thus they would write . . . (instead of . . . , the accepted notation today).

Thanks. KC 16:50, 29 February 2016 (UTC) — Preceding unsigned comment added by Boydstra (talkcontribs)

## ii

At the bottom of the section named Computing complex powers, there is a sentence that says: "[...] there is an infinity of values which are possible candidates for the value of ii, one for each integer k". I would like to be clarified on this: does that include negative integers, too? Mvpo666 (talk) 22:10, 15 September 2016 (UTC)

Yes. Dmcq (talk) 23:33, 15 September 2016 (UTC)

## On 0^0 being correct

In the article we have

"According to Benson (1999), 'The choice whether to define ${\displaystyle 0^{0}}$ is based on convenience, not on correctness.'"

Just what constitutes being "correct"? Correct based on what? Exponentiation starts with ${\displaystyle x^{n}=x*x*x*...nfactors}$ with x real and n a positive integer. I think we all consider this "correct". For positive x we use various means to generalize to negative exponents, rational exponents, and then real exponents. We also use exp(ix) = cos x + i sin x to generalize ${\displaystyle x^{n}}$ for x and n complex numbers with the proviso x != 0. Now, just what does it mean to have 3.7 factors, or sqrt(2) factors, or i factors for example? What makes these generalizations "correct"? We generalize ${\displaystyle x^{n}}$ the way we do because it is useful, convenient, satisfies the exponent laws, and "makes sense". In this light, how is ${\displaystyle 0^{0}=1}$ incorrect? It is useful, convenient, follows the exponent laws, and make sense according to so many things. So I ask again, "correct" based on what?

We can use the exponent laws to narrow the possible choices to 0 or 1. The value 0 is useless. And as the article says, the value 1 is immensely useful. That makes it correct as much as any of the above generalizations.

Shouldn't we add something like the above? Perhaps a more concise version?

Betaneptune (talk) 18:18, 6 October 2016 (UTC)

The word correctness is used only in a quotation, and not elsewhere in the article. Nevertheless, in mathematics, a definition cannot be incorrect, unless it is self contradictory or induces a contradiction in mathematics. Generally, definitions are chosen for making statements as simple as possible. This is for this reason that 1 is not a prime number (otherwise, the uniqueness of factorization in the fundamental theorem of arithmetic would be difficult to express). Here, this is the same: ${\displaystyle x^{y}}$ is defined for satisfying the exponent laws and being continuous; there is only one function of x and y with y > 0, that satisfy both conditions (if continuity is dropped, there are many such functions). The problem, for ${\displaystyle 0^{0}}$ is that continuity is impossible, as ${\displaystyle 0^{y}\to 0}$ and ${\displaystyle x^{0}\to 1.}$ This is for this reason that generally one defines ${\displaystyle 0^{0}=1,}$ when only integer exponents are considered, and one considers ${\displaystyle 0^{0}}$ as undefined or undetermined, when exponents may varies continuously. Typically, in computer programming, if the type of the exponent is "integer", ${\displaystyle 0^{0}=0.0^{0}=1,}$ sand if the type of the exponent is "real" or "float" the result is NaN (Not a Number): ${\displaystyle 0^{0.0}=0.0^{0.0}=NaN.}$ Nevertheless, any definition for ${\displaystyle 0^{0}}$ is correct, even ${\displaystyle 0^{0}=\pi .}$ But some definitions are more useful in some context, and the most useful definition depends on the context. I believed that this was clearly explained in the article. Maybe you have not read it with enough care. D.Lazard (talk) 21:02, 6 October 2016 (UTC)
(1) Then the article wasn't clear. I didn't see anything about making 0^0 anything other than 1 or undefined. I see no advantage to it being undefined. And why does x^y have to be continuous everywhere? What horrors ensue if it's discontinuous at (0,0)?
(2) Now consider 1^oo. Clearly this is 1. But when you consider 1^oo as a type of indeterminate form you can have lim(h->0) (1+h)^(1/h)=e. Would you then say that 1^oo equals e? Of course not. So why should limits mean anything for 0^0? (This is separate from the continuity argument.) You yourself said that definitions are chosen to make things simpler, as in 1 not being a prime number. Why not do the same for 0^0? Having 0^0 = 1 simplifies a lot of things.
(3) Every math book I've ever seen that gives e^x in summation format says it is good for _all_ x. This can only be true if 0^0 = 1. So you have a choice: 0^0 = 1, or you can undefine 0^0, which means you have to qualify the summation formula to be valid only for x != 0, and add a line saying e^x = 1 for x = 0. You can't have it both ways. Do you really want e^x have a two-line definition? And the same for other series starting with a constant term?
(4) As for the article, I think one could comment that the quote is ambiguous, because it is not clear what "correct" means. Perhaps the quote was taken out of context.
Betaneptune (talk) 04:05, 7 October 2016 (UTC)
The definition 00=1 is assumed in all cases where 00 occurs. The polynomial ${\displaystyle \sum _{k=0}^{n}a_{k}x^{k}}$ is supposed to be equal to ${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}}$, even for x=0. The discontinuity of yx around x=y=0 cannot be helped by undefining 00. However the undefiners insist. Bo Jacoby (talk) 03:05, 7 October 2016 (UTC).
I totally agree. Thank you. Betaneptune (talk) 04:05, 7 October 2016 (UTC)
This talk page is not for litigating what the conventions should be. In the article, we simply report what they are, in their existing multiplicity. Your remark above, and Bo's, are off-topic. --Trovatore (talk) 04:35, 7 October 2016 (UTC)
As Trovatore says Wikipedia is an encyclopaedia which describes what is out there. It is not a place for us to put in our own point of view and remove any other as in WP:NPOV. And by the way ${\displaystyle 1^{\infty }}$ is another indeterminate form. For many purposes it would be clearly 1 but in other cases where we have ${\displaystyle x^{y}}$ and x and y are reals that happen to be one and infinity it would have no a priori value. What D.Lazard says above is a good description of the general situation. And by the way defining indeterminate forms to have definite constant values does give problems in calculus. I know calculus isn't being taught so much these days but it is still a very important part of mathematics. Dmcq (talk) 08:11, 7 October 2016 (UTC)
You just said that this is not a place to put our own point of view. Then you go on to add your own! I added my "POV" to explain why "correctness" in that quote is ambiguous, which is my primary point. One could simply add to the article: "But his criteria for what constitutes "correctness" is not given." That's not a point of view; it's a fact. If not, tell me what constitutes "correctness" in his quote and how it's obvious. Correct according to what?
I added my "POV" to support my proposal to add that one sentence. And [in response to your comment] you missed my point about form vs. value. As values, 0^0, 0/0, 1^oo are 1, "undefined", and 1, respectively. As indeterminate forms they are symbols that represent different types of limits. Not the same! Here's another example: 0/0. I think we'd both agree that this is undefined. Why? Because there is no unique solution to 0*x = 0. But we have lim(x->0) sin x / x = 1. This doesn't mean that 0/0 = 1. (!) So the limit and the value are different. With 1^oo, again the limit of an expression in that form and the value may be different. And so it is with 0^0. 0^0 = 1, but a limit in that form will vary. And the value of a function at a given point need not equal the limit at that point unless the function is continuous. But that's a property of the function, not the fixed number, whose symbol is used to represent the type of limit.
As Thomas says in his calculus book on p.654 (4th edition):
"Remark 2. Many people feel that the expression 1^oo cannot represent an indeterminate form. They say that one to any finite power is one, hence one to the power infinity is also one. But note, in the example above, that the base 1 + h is a variable which is not exactly equal to 1 except in the limit as h -> 0. There is a difference, in other words, between 1 ^ (1/h) and (1+h) ^ (1/h), even though both expressions formally become 1^oo as h -> 0."
So there is a difference between 1^oo the form, and 1^oo the value. And so it is with 0^0. And this is why 0^0 = 1 causes no problems in calculus.
And you say calculus isn't being taught much anymore? What? I beg to differ with that.
And D.Lazard saying 0^0 = Pi is correct? (1) It doesn't satisfy the exponent laws, which I think is a quite reasonable requirement. (2) There is no reason for it to be anything other than 1.
On D.Lazard's point about the implementation of 0^0 in computers, that is a decision made by the programmer -- not some ultimate truth or arbiter of "correctness".
Sorry if I went too far giving arguments for 0^0 = 1 in my original comment. I was just trying to demonstrate that 0^0 = 1 is a generalization or extension of the starting definition of x^n just like generalizing the exponent in x^n from integers, to rational numbers, to real numbers, and to complex numbers -- and just as valid.
Seems to me that after all this we can summarize it thusly: (1) I think we all agree (except for the remark of 0^0 = Pi) that the exponent laws must be satisfied. (2) There is disagreement on whether we need continuity at 0^0. (3) There is confusion between fixed values and limiting forms. OK, I'm guessing that Benson insists on continuity, but I don't see why that is unassailably "correct".
Anyway, aside from all the 0^0 = 1 bit, I don't propose to definitively state that 0^0 = 1 (even though it is!); I propose to add "But his criteria for what constitutes "correctness" is not given" as a comment on the quote by Benson. And I have presented my case.
Betaneptune (talk) 16:13, 7 October 2016 (UTC)
Saying 'But his criteria for what constitutes "correctness" is not given' would be us commenting on what a person said rather than an outside source doing it. He said what he said and thought it was good enough, I think we should just accept that. Dmcq (talk) 17:05, 7 October 2016 (UTC)
Betaneptune: The 0^0 section is a compromise. There is tension between what textbooks say about 0^0, and what they use+imply about 0^0. As you point out in your comment (3), textbooks give numerous formulas that assume+imply that 0^0 is 1. In other words, consistency requires that 0^0 is defined as 1. But many books don't explicitly say that 0^0 is 1. The 0^0 section is a compromise between two goals (a) consistency, and (b) reporting only what textbooks say (and not what they imply). But the two goals (a)+(b) are not compatible, which means that there's no easy way to fix the 0^0 section that wouldn't violate at least one of (a) or (b). MvH (talk) 02:46, 19 November 2016 (UTC)
I think the section is fixed. It reports the situation fairly well. Reading it again I thought it used to also mention that 0^0 was sometimes taken to be 0 in the complex domain, I wonder what happened to that. Dmcq (talk) 12:08, 19 November 2016 (UTC)
I agree that it reports the situation well. But given the tension between (a) and (b), we can't do better than a compromise. The line that Betaneptune quoted "The choice whether to define ${\displaystyle 0^{0}}$ is based on convenience, not on correctness" is a compromise. We can't simply delete it because of goal (b). But at the same time, no mathematician should ever write a line like that, because even though it is technically correct, it is also misleading (every definition in math is based on convenience; singling out 0^0 is misleading). MvH (talk) 16:33, 19 November 2016 (UTC)
0^0 is singled out because there are a lot of situations where it makes sense to define it as 1 but it is not immediately apparent that it should be so and there is no logical reason arising from some some simple axioms to do so and there's good reasons to not define it like for 0/0. Dmcq (talk) 17:13, 19 November 2016 (UTC)
Regarding arguments for/against "0^0 = 1 in all cases", the point is that there are valid arguments (e.g. "a lot of textbooks don't explicitly define 0^0"), and there are invalid arguments, such as "the definition 0^0 = 1 is based on convenience rather than correctness" (that argument is without merit because the same is true for every definition in math).
But it may not be possible to delete every invalid argument and still have a balanced presentation of commonly held views. The current page is a good compromise.
Regarding axioms, historically, the definition of x^n is not based on any axioms, instead, the definition is motivated by convenience. Initially this meant: compactness of notation. Historically, x^3y^4 is simply an abbreviation of xxxyyyy.
Initially, exponents were at least 3 because x^2 is not shorter than xx. But of course, extending the definition to more exponents was a natural thing to do when that became useful, and extensions that satisfy more axioms tend to be more useful. But still, the rules/axioms come after the definition, not before. The definition is, as every definition in mathematics, based on convenience. Axioms only play an indirect role, in the sense that definitions that satisfy more axioms are more convenient. MvH (talk) 21:40, 19 November 2016 (UTC)

The Undefiner's POV assume the Trovatore conjecture: ${\displaystyle 0\neq 0}$. Unfortumately the Trovatore conjecture is not main stream mathematics. Bo Jacoby (talk) 20:16, 7 October 2016 (UTC).

Bo Jacoby's conjecture is that Trovatore has conjectured something he has never written. Please stop such ridiculous assertions and stop to pretend that your personal point of view is main stream of mathematics. D.Lazard (talk) 21:35, 7 October 2016 (UTC)
I doubt that D.Lazard has read what Trovatore has written. Trovatore's POV is that ${\displaystyle 0^{0}}$ is defined when the exponent is integer zero, but not when the exponent is real zero. That is ${\displaystyle 0\neq 0}$. Bo Jacoby (talk) 06:34, 8 October 2016 (UTC).
Bo, I'm not interested in responding to how you characterize my remarks. But please don't do it here. It's off-topic on this page. --Trovatore (talk) 06:45, 8 October 2016 (UTC)
Trovatore, it is the topic of this talk page that once again a reader is confused that ${\displaystyle 0^{0}}$ is sometimes undefined. Why shouldn't I comment here? Bo Jacoby (talk) 07:23, 8 October 2016 (UTC).
If you have suggestions as to how to help readers understand why many sources leave 00 undefined, and if these suggestions are themselves sourceable, then yes, that is on-topic here. --Trovatore (talk) 07:37, 8 October 2016 (UTC)
Could you do it without breaking the 2nd and 4th pillars in WP:5P please. Dmcq (talk) 09:15, 8 October 2016 (UTC)
When the topic is to define the exponential, then the sources that do not define the exponential are off-topic. Bo Jacoby (talk) 06:39, 9 October 2016 (UTC).
Wikipedia:Silence_and_consensus. Bo Jacoby (talk) 05:54, 11 October 2016 (UTC).
No. --Trovatore (talk) 07:28, 11 October 2016 (UTC)
Wikipedia:Silence means nothing. Dmcq (talk) 09:37, 11 October 2016 (UTC)
The article needs to be cleaned from the off-topic undefining of 00. I leave it to you undefiners to contemplate that you got yet another user badly confused. Bo Jacoby (talk) 12:20, 14 October 2016 (UTC).
Perhaps Conservapedia would be more to your taste if you want an expurgated encyclopaedia, they like that sort of definiteness about the world. They might like that being put in there. Dmcq (talk) 15:04, 14 October 2016 (UTC)
The Trovatore conjecture fits nicely into the Conservapedia way of thinking. Bo Jacoby (talk) 08:42, 15 October 2016 (UTC).
I haven't been active on this article for a while, but it is clear that 0^0 is often considered undefined, so it is a violation of the pillars to assert that 0^0 is defined. The multiple-domain approach is WP:OR, but is accurate. As I see it, it's not saying that 0 ≠ 0.0, but that there are different definitions of the exponential function on different domains, and they may not have the same value at pairs which are in different domains. — Arthur Rubin (talk) 04:12, 10 January 2017 (UTC)

## math to nowrap

Sometime in the the past year, all the {{math}} templates were changed to {{nowrap}}, for no apparent reason. I just changed all the {{nowrap}} templates back to {{math}}. — Arthur Rubin (talk) 04:24, 10 January 2017 (UTC)

The status as of the beginning of 2016 was that some where {{math}}, and none were {{nowrap}}. An anon added a lot off {{math}}s on Feburary 2, 2016, which was promptly reverted. However, no consensus or attempt at consensus is evident in the edit comments or on this talk page. I think the status quo ante is better represented by use of {{math}} than by {{nowrap}}. — Arthur Rubin (talk) 04:39, 10 January 2017 (UTC)