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I see that Knuth's paper (Donald E. Knuth, Two notes on notation, currently reference 15) is available here on arXiv.com. I presume that this is not a copyright violation, so should we not link to it? — Quondum 16:26, 27 July 2013 (UTC)
Some of Libri’s papers are still well remembered, but [32] and [33] are not. I found no mention of them in Science Citation Index, after searching through all years of that index available in our library (1955 to date). However, the paper [33] did produce several ripples in mathematical waters when it originally appeared, because it stirred up a controversy about whether is defined. Most mathematicians agreed that , but Cauchy [5, page 70] had listed together with other expressions like and in a table of undefined forms. Libri’s justification for the equation was far from convincing, and a commentator who signed his name simply “S” rose to the attack [45]. August Möbius [36] defended Libri, by presenting his former professor’s reason for believing that (basically a proof that ). Möbius also went further and presented a supposed proof that whenever . Of course “S” then asked [3] whether Möbius knew about functions such as and . (And paper [36] was quietly omitted from the historical record when the collected works of Möbius were ultimately published.) The debate stopped there, apparently with the conclusion that should be undefined.
But no, no, ten thousand times no! Anybody who wants the binomial theorem
to hold for at least one nonnegative integer must believe that , for we can plug in and to get on the left and on the right.
The number of mappings from the empty set to the empty set is . It has to be .
On the other hand, Cauchy had good reason to consider as an undefined limiting form, in the sense that the limiting value of is not known a priori when and approach independently. In this much stronger sense, the value of is less defined than, say, the value of . Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side.
(−2)1/3 does have different defined values depending on the domains. If you let r denote the number 21/3, then
As we have noted above,
— Arthur Rubin (talk) 18:06, 30 December 2013 (UTC)
Carl undid my edit. Carl apparently does not accept that powers with nonnegative integer exponent can be defined by xn=1 when n=0 and xn=x⋅xn-1 when n>0. Even hardcore undefiners like Trovatore agree that 00=1 when the exponent is integer. But the undefiners are now a minority in this discussion.
There is no point in undefining (−1)2 or 00 even if these expressions are not evaluated by the formula xy=exp(y⋅log(x)). They are calculated by other means and they are very widely used. But stop teaching innoscent children that (−1)1/3=−1, because (−1)1/3=(1+i31/2)/2 is more useful and more widely used. In J it reads
_1^2 1 0^0 1 _1^%3 0.5j0.866025
Bo Jacoby ( talk) 07:54, 31 December 2013 (UTC).
It seems as you agree with my above suggestion : Talk:Exponentiation#merging_three_definitions. Right? Bo Jacoby ( talk) 10:51, 2 January 2014 (UTC).
Naming the different exponentiation functions differently is OR. It is not what the mathematical community does. The functions are all called xy. There is consensus in the literature that 22=4, that 02=0, that (−2)2=4, that 2−1=1/2, that eiπ=−1, and that 21/2=√2. The function is generalized to still bigger domains without renaming. Nor do we rename the plus-sign when addition is generalized from non-negative integers through integers, rationals, reals, complex numbers, vectors, matrices, et cetera. I wonder if The Undefiners, (Dmcq, Trovatore, Carl, Quondum), still think that 0≠0.0 and that 0≠0+i0 ? Bo Jacoby ( talk) 09:09, 3 January 2014 (UTC).
The Undefiners prevent our article from improving. Dmcq claims that 0+i0≠0 which is OR. Trovatore claims that 0.0≠0, which is OR. Quondum claims that the different definitions of exponentiation should lead to different function names, which is OR. Carl reverts that xn (for non-negative integer n) can be defined by x0 = 1 and xn+1 = x⋅xn. Dmcq is correct that silliness is a better word for describing the positions of The Undefiners.
Many authors do not define 00. That doesn't undefine 00 once it is defined, and it is quite unimportant, and it is a nuisance.
Dmcq thinks that not giving 0/0 a value does help. It is not advisable to divide by zero because x/y is not continuous around x=y=0, no matter whether 0/0 is defined or not. Dcmq could write more carefully:
and differentiation gives
"no telltale warning because of assigning a value to 0/0". Assigning a value to 0/0 does not remove the warning against dividing by zero. The discontinuity of x/y around x=y=0 is still there. Not giving 0/0 a value does not help.
Dmcq believes that limx→0 f(x) must be equal to f(0), such that the only way to avoid a discontinuity is to undefine f(0). This belief is a mistake. Undefining does not necessarity remove discontinuities. xy is discontinuous around x=y=0, even if you undefine 00.
We need two different functions if one is to be given a value. No, we don't. We just have to accept the discontinuity.
Some sources say it should be 1. Other sources say it is an indeterminate form and give no value. We don't extrapolate and say indeterminate forms have values. If a function f(x) is discontinuous for x=0, then the value f(0) is not defined by continuity as limx→0 f(x). Some sources call f(0) an indeterminate form when it is not defined as limx→0 f(x) . But f(0) may be defined by other means.
I want our article to be understandable. x0 is defined to be equal to 1 in the elementary section. Later it turns out that xy is discontinuous around x=y=0. This is not helped by undefining 00. Bo Jacoby ( talk) 08:17, 4 January 2014 (UTC).
If you are an undefiner no more, then I am not refering to you, and so you have no reason to be offended. If you are still an Undefiner, then take pride in being capitalized. You choose your words and I choose mine. But you are changing the subject. The question is: Do you by now understand that undefining serves no purpose and solves no problem? Bo Jacoby ( talk) 17:58, 4 January 2014 (UTC).
Not I, but the readers of our article, suffer censorship from The Four Undefiners: Dmcq, Trovatore, Quondum, and Carl. This fine suggestion, put forward by Javalenok and supported by Mark van Hoeij, and 128.186.104.253, and myself, could not be implemented. Bo Jacoby ( talk) 08:04, 5 January 2014 (UTC).
Stating that
is original research, which is prohibited in wikipedia. (ps. It is also doublethink). The lead of our article on complex number says:
The complex number 0 + i0 has zero imaginary part and so it is a real number. As the real part is also zero makes is clear that 0 + i0 is the real number zero. So it is correct that 0 + i0 = 0. There is no doubt about it. Denying it leads to contradiction. How would you explain it to the uninitiated wikipedia reader? How would you rewrite complex number ?
That
is not true. Undefining 00 solves no problem whatsoever.
You write:
From the fact "A", that some authors define 00 = 1, and the fact "B", that other authors do not define 00, The Undefiners conclude "therefore C", that 00 is defined in some contexts and undefined in other contexts. Note the rule in Wikipedia:OR#Synthesis_of_published_material_that_advances_a_position:
So The Undefiners' position is original research.
You write:
That is true and it is a nuisance, but it is a different problem from that of defining 00. Right now I prefer to concentrate on: exponentiation with arbitrary base and nonnegative integer exponent, exponentiation with nonzero base and negative integer exponent, and exponentiation with positive base and arbitrary exponent. Bo Jacoby ( talk) 19:18, 5 January 2014 (UTC).
The notation was used way before the 20th century, and nobody doubts that f(0)=a0, so it goes without saying that 0k=0 for k>0 and 0k=1 for k=0. Donald Knuth didn't invent 00=1. In the 18th century Leonhard Euler wrote ex with the same exponential notation as x2, so it is not original synthesis on my part. Bo Jacoby ( talk) 23:03, 5 January 2014 (UTC).
Do you really doubt that f(0) is ment to be equal to a0 when f(x) is defined by ∑k ak xk ?
It doesn't matter which exponential function was intended as long as the competing definitions produce the same result. 32 = 1⋅3⋅3 = 9 and 32 = exp(2⋅log(3)) = 9.
An analogous problem is well handled in Addition#Addition of natural and real numbers. Bo Jacoby ( talk) 07:07, 6 January 2014 (UTC).
Thinking that 00 is sometimes defined and sometimes undefined is not mathematics, it is doublethink. Bo Jacoby ( talk) 08:33, 6 January 2014 (UTC).
Show your good will by answering my question: Do you doubt that f(0) is ment to be equal to a0 when f(x) is defined by ∑k ak xk ? Bo Jacoby ( talk) 17:55, 6 January 2014 (UTC).
Do The Undefiners doubt that e0 = 1 where ex is defined by the power series ? Bo Jacoby ( talk) 08:02, 7 January 2014 (UTC).
As The Undefiners are not willing or not able to explain their own position, I will give it a try. Everybody agree that and that . So . Both the numerator and the denominator are empty products, and so 00=0!=1. Recursive definitions are: 0!=1 and n!=n⋅(n−1)! for n=1,2,3,... , and x0=1 and xn=xn−1⋅x for n=1,2,3,... . Carl is sensitive against 00, but Trovatore and Dcqm agrees that 00=1 for integer exponent 0. But, because some authors do not define 00, it is important to The Undefiners to claim that 00 is sometimes not defined. I don't follow this argument. Of course some authors do not define 00. Most authors write about something else. Dmcq thinks that the discontinuity of xy around x=y=0 is more spectacular when 00 is undefined, than when 00 is defined to be 1. Trovatore promotes the idea that 00 is undefined when the exponent 0 is real, but defined when it is integer. This idea is given the formulation: 0.0≠0 , real zero is not the same thing as integer zero. I don't follow this argument. Trovatore and Carl thinks that the set of integers, ℤ, is disjoint to the set of reals, ℝ, while I consider ℤ to be a subset of ℝ, such that operations on ℝ generalize operations on ℤ. Trovatore has problems with 1+0.5=1.5, because 1 is an integer and 0.5 is a (non-integer) real. I don't have the same problem because integer 1 and real 1 to me is the same number: 1=1.0. I find support for this interpretation here and elsewhere. The Undefiners polluted our article only by undefining 00 , but they should also undefine (−1)2 which also cannot be defined by xy = exp(y⋅log(x)). There is no real logarithm to −1. The fact that some authors do not define 00 should not force our article to become unreadable. Bo Jacoby ( talk) 22:36, 7 January 2014 (UTC).
What would Dcmq want The Undefiners to be called? The Doublethinkers? Bo Jacoby ( talk) 08:15, 8 January 2014 (UTC).
Is it also besides the point that the article is incomprehensible? Exponentiation#Arbitrary_integer_exponents says: "The case of 00 is controversial". In fact it is not at all controversial for integer exponents. Also the reader gets the warning: "Any nonzero number raised by the exponent 0 is 1". In fact x0 is always interpreted as 1. Nobody writes 00 if it is supposed to be undefined. Why not omit this nonsense? Bo Jacoby ( talk) 08:15, 8 January 2014 (UTC).
Also, a book that comes out and says directly how to the seeming issue with 00 in power series: Complex Analysis for Mathematics and Engineering, John Mathews, Russell Howell, Jones & Bartlett, 2010, p. 151.
This could be useful for the article at some point. — Carl ( CBM · talk) 20:13, 8 January 2014 (UTC)
Oh, thank you for allowing me to hold my view. But how can you or any other mathematician think that the above identity does not imply that 00 =1 ? Must wikipedia convey the impression that mathematicians are insane? Bo Jacoby ( talk) 10:44, 9 January 2014 (UTC).
I am not the only one to see that the stipulated identity defines 00. The position of The Undefiners (including Mathews and Howell) is manifestly insane. So the obvious answer to my rhetorical question is that wikipedia should stop conveying the impression that mathematicians are insane. The Undefiners must leave in shame. Sensible suggestions on this talk page were turned down by The Undefiners, and my very innocent edit was undone by Carl. This must stop. Bo Jacoby ( talk) 05:36, 10 January 2014 (UTC).
It is inconsistent that the article claims that 00 is more controversial than, say, (−3)2. Neither are defined by: xy=exp(log(x)⋅y), and both are defined by the recursive definitions: x0 = 1, and xn+1 = xn⋅x, and xn−1 = xn/x for nonzero x. If only one definition apply, then that is the one to be used, such as 31/2=exp(log(3)/2). If both definitions apply, then the result is the same: 32 = 9 by both definitions. So yes, there are multiple functions called exponentiation, but that is not a problem, and that is not the point. Bo Jacoby ( talk) 19:05, 8 January 2014 (UTC).
The two definitions for xn are the analytic one: xn = exp(log(x)⋅n), and the recursive one: xn = 1 for n=0 and xn = xn−1⋅x for n>0. These definitions produce identical results whereever they both apply. The analytic definition applies to positive x and general n. The recursive definition applies to general x and nonnegative integer n, and to nonzero x and negative integer n. "Undefined" is not appropriate when something is defined. Bo Jacoby ( talk) 06:46, 9 January 2014 (UTC).
Making something undefined is not a definition. I want to postpone the inconsistent definitions of (-1)1/3 as the discussion is sufficiently complicated by now. The 00 stuff is almost kept together in one place, except the warnings against 00 while treating integer exponents. I did not say that there is one true exponential function. The exponential notation may refer to different functions, as explained above. You have not told why you disagree that "xn can be defined by x0 = 1 and xn = xn−1⋅x for positive integer n". You undid my contribution. Bo Jacoby ( talk) 20:30, 8 January 2014 (UTC).
You mean that it is sometimes correct and sometimes not correct? This is doublethink. Bo Jacoby ( talk) 21:52, 8 January 2014 (UTC).
I need managable sections. You are welcome to rename the headings for your convenience. But how can making something undefined be a definition? What do you mean? Bo Jacoby ( talk) 06:20, 9 January 2014 (UTC).
We must distinguish between function f and function value f(x). Specifying the domain is part of the definition of a function. Removing x from the domain of f makes f(x) undefined. This redefinition of f is not a definition of f(x). Making f(x) undefined can not be a definition of f(x). Bo Jacoby ( talk) 05:07, 10 January 2014 (UTC).
Dmcq changed "Non-negative" to "Positive" and "" to "". Any reader need to know why. Is it incorrect to say that: "Formally, powers with non-negative integer exponents may be defined by the initial condition and the recurrence relation "? Does the uninitiated reader benefit from making an exception from the definition? Bo Jacoby ( talk) 18:25, 10 January 2014 (UTC).
So you found a book that left 00 undefined. The question still remains: Does the uninitiated reader benefit from making 00 an exception from the definition? Bo Jacoby ( talk) 21:46, 10 January 2014 (UTC).
I know what wikipedia is. But now I also know why your source does not define b0. Your source is about general algebraic rings, not only rings with an identity. So the ring of even numbers, 2ℤ, is an example. In this ring b0 is not defined, because 1 is not an even number, 1∉2ℤ. You could and should have discovered this yourself. Our article is about the rings with identity, ℤ or ℝ or ℂ. One is a number. Now that this is clear, would you please either restore the definition b0=1, or carefully explain why not. Bo Jacoby ( talk) 23:16, 10 January 2014 (UTC).
Does your source tell why it leaves 00 undefined? Bo Jacoby ( talk) 06:40, 11 January 2014 (UTC).
Shouldn't the links to sources leaving 00 undefined be collected here after the sentence "not all sources define 00" ? Bo Jacoby ( talk) 09:10, 11 January 2014 (UTC).
Why restrict b0=1 to nonzero b at this stage? An empty product doesn't depend on the value of a factor that isn't there.
Shouldn't "00=1 or undefined" be treated in the article rather than on the talk page? Bo Jacoby ( talk) 16:45, 11 January 2014 (UTC).
OK, I went there. Why not wait until Exponentiation#Zero to the power of_zero with mentioning "nonzero b" ? Even Trovatore agrees that b0=1 is not restricted to nonzero b when the exponent is integer (which it is). Bo Jacoby ( talk) 04:07, 12 January 2014 (UTC).
I am discussing Dmcq's edit right here. Please answer my questions. Bo Jacoby ( talk) 14:16, 12 January 2014 (UTC).
I moved that section "combinatorial interpretation" back to its original location. The article should certainly include that topic, but since this article is on an elementary subject, and that interpretation is slightly more advanced, it should not be placed so highly in the article. Compare WP:MTAA section 4.1. — Carl ( CBM · talk) 13:11, 12 January 2014 (UTC)
We probably should be revising the integer section and saying a bit more about 0^0=1 there, but just sticking it at the start when that is not generally agreed is simply WP:OR. How about just trying to fix the article up rather than trying to push a point of view? Dmcq ( talk) 10:58, 10 January 2014 (UTC)
I agree with Carl that it's better to keep all the 0^0 stuff together in one place. Assuming that x^0 means 1 in the context of a power series is actually the same thing as defining 0^0=1. (See #Answering_Trovatore). Bo Jacoby ( talk) 03:59, 12 January 2014 (UTC).
How can the reader decide whether a problem is "in the context of a power series" or not? When and why and how does 00 fade away from being 1 to being undefined? Bo Jacoby ( talk) 14:24, 12 January 2014 (UTC).
I don't understand what you mean. Bo Jacoby ( talk) 21:18, 12 January 2014 (UTC).
Dmcq undid my edit. Carl's source actually did define 0^0 . See above. Bo Jacoby ( talk) 11:02, 10 January 2014 (UTC).
I is a diplomatic challenge to negotiate with people who seriously deny that 0+i0=0. Step back and relaxe. You have still unanswered questions. Improvements are not made by your automatic undoing my contributions. I welcome your stated willingness to discuss. Bo Jacoby ( talk) 11:36, 10 January 2014 (UTC).
My edit was finished at 1047 o'clock and your straightforward section just above was made at 1058, so I didn't ignore it. I asked you what you guys want to be called instead of Undefiners, and you did not answer. I look forward to see your arguments on the subject matter, but not your comments on my histrionics. Your reversions are so quick that you haven't taken time to study my contribution in calm and detail. Bo Jacoby ( talk) 12:09, 10 January 2014 (UTC).
Although there is not much progress in this discussion, I do see some light. Trovatore seems to have stopped denying that 0.0=0, and Dmcq seems to have stopped denying that 0+i0=0, and Carl seems to have stopped denying that ℕ⊂ℤ⊂ℝ⊂ℂ, and to realize that the equation is no different from defining z0=1. So some progress has been made, and The Undefiners no longer constitute quite the same mathematical madhouse as they used to. Dcmq does not like being called an undefiner, and I will stop calling Dcmq an undefiner as soon as Dmcq stops being an undefiner. But because some sources do not define 00 it is important to Dcmq to claim that 00 is not always defined. Dmcq did find a source which does not even define b0 for nonzero b. but for unknown reasons that did not urge Dmcq to claim that b0 is sometimes not defined for nonzero b. Trovatore knows that the definition by = exp(log(b)⋅y) does not cover the case b=0, but neither does it cover the case b=−1, which is no problem to Trovatore. The Undefiners' attitude of infallibility made collaboration difficult and unpleasant. I did my duty. I regret to the wikipedia readers that I was not successful in removing the nonsense from our article. The simplicity and clarity of Donald Knuth's approach should be used in the article, and the obscurity of undefining should be concentrated in the appropriate subsection. Bo Jacoby ( talk) 15:17, 14 January 2014 (UTC).
Wow, how did we pass the seventh anniversary of this thread without celebrating? Congratulations to everyone (especially the inimitable Mr. Jacoby) for their stamina. — Steven G. Johnson ( talk) 21:06, 15 January 2014 (UTC)
Thanks to Mr. Johnson for the congratulation! Isheden's suggestion needs proofreading:
Shorthand for this is formally
and informally:
Be bold and make your edit. Undoing a contribution is not progress, and undefining a definition is not progress either. Bo Jacoby ( talk) 22:01, 15 January 2014 (UTC).
The definition 00 = 1 is always assumed in the usual expression for a polynomial or a series, where when . The definition b0 = 1 need not be restricted to nonzero b (or to positive b). But do not assume that . The function bx = elog(b)⋅x is defined for b>0, but it is not continuous around b=x=0, so don't even try to define 00 by continuity. Bo Jacoby ( talk) 23:37, 15 January 2014 (UTC).
Hi, following part didn't seem right to me, but I hesitated to edit.
The following identity holds for arbitrary integers m and n, provided that m and n are both positive when b is zero:
First, why write b like a variable if b is always zero. Why say arbitrary integers then say positive integers? What good is 0^(m+n), why not just say 0^m is zero when m>0? Why is there a dot after the equation? — Preceding unsigned comment added by 85.108.132.131 ( talk) 06:16, 10 January 2014 (UTC)
In many other Wikipedia articles Template:Round in circles and Template:FAQ are used to respond to long-running and often-repeated arguments that go nowhere.
The arguments over 00 seem to fall into this category. Arguments have been going on for over seven years now, with no really new arguments being raised nor any substantially different sources being cited as far as I can tell. Wouldn't it be more productive to create a FAQ entry and simply refer to that when similar arguments are raised in the future?
— Steven G. Johnson ( talk) 21:47, 15 January 2014 (UTC)
What is the point of this section if not to justify the continuity of the exponential function through the continuity of the logarithm through its own definition? The fact is one textbook prefers to define the logarithm as a continuous function first, and later introduce the exponential function and the justification for the existence of real exponents on said definition:
Toolnut ( talk) 07:25, 7 February 2014 (UTC)
But the section containing this subsection is titled "Real exponents" and talks at length about the justification for the extension to real exponents from rational exponents. Of note, "Since any irrational number can be approximated by a rational number ...": if you use the logarithm definition, that kind of justification becomes unnecessary; therefore, it is a different approach well worth noting. So you appear to be saying the reader should look for the logarithm definition elsewhere and not become aware of this finer point here, where it has been brought up. The separate article on logarithms actually begins by defining a logarithm as the inverse of the exponential function, not independently of it; all the more reason to make this fine point here. Toolnut ( talk) 10:32, 7 February 2014 (UTC)
In response to the request for a FAQ. I started by making a summary of frequently made arguments for "1". The goal of this section is to save time (i.e. the goal is not to do the debate all over again!). Instead, the goal is to give a summary of arguments that have been made dozens of times, over hundreds of kilobytes, and compress it down to this one section.
Identities in basic arithmetic should not be undefined simply because they do not fit in a later formula such as exp(y log(x)). If we were to allow that, then we can undefine not only 0^0 and 0^2, but pretty much anything else as well.
0^0 = 1 is basic arithmetic, it follows from fundamental rules such as the empty product rule, and cardinal number arithmetic, both of which are far more fundamental than the exp(y log(x)) formula, or the technical issue regarding limits that is used to undefine 0^0.
The limit issue used to undefine 0^0 ignores the fact that the same argument also undefines f(0) for any function that is discontinuous at x=0, such as the floor function. It is not logically consistent to use this argument to say that 0^0 is undefined but floor(0) is not. Other common arguments for "undefined" are equally inconsistent (the most common one says that "0^x = 0 for all x", blissfully ignoring x=-1).
Efforts to portray 0^0 as undefined lead to bizarre wordings in the text, for example, right now we have a line "regarding b^0 as an empty product" which is a strange thing to say because b^0 is an empty product, just like an empty bag has 0 applies.
To justify this bizarre wording you need to hold a circular "definition" of 0; to believe that 0 is not a counting number (0 is the number of apples in an empty bag, or in more sophisticated words, it is the cardinality of the empty set) but instead, one somehow views 0 as a function that converges to 0, ignoring the circular nature of this view of numbers.
Efforts for "undefined" also leads to conclusions that we would normally not make. For example, if a book says f(x) = g(x) for all |x|<1, but does not say that f(0) = g(0), then in normal circumstances, the conclusion f(0) = g(0) would not be Original Research. Yet, if we apply this to 1/(1-x) = sum x^n then it suddenly becomes Original Research? To keep 0^0 undefined, one must ignore the fact that 0^0 = 1 is used throughout mathematics, and that even the calculus books that say "undefined" require 0^0 to be evaluated as 1 in their formulas, and that they give plenty of formulas that say 0^0=1 after a simple substitution. You have to pick one:
(a) simple substitutions are OR, (b) simple substitutions are OK, or (c) simple substitutions are OK but if I don't like the outcome, then they are OR.
Now you might say, if calculus textbooks (through the binomial theorem, or 1/(1-x), or power series notation in general) imply 0^0=1, they still do not mean to say 0^0=1 (the argument goes something like: the x^0 in power series is not unconditionally 1, it is just a nice convention that makes formulas shorter). The problem with that argument is that these books do not actually say that x^0=1 is a convention. In any decent math journal, if you imply a statement, it means that you claim it is true. There are two possibilities: either the calculus textbook authors do not follow the standards held by math journals, or, more likely, they do not know that their own writings imply 0^0=1. If our 0^0 section does not point out that many common formulas imply the value 1, then we allow textbook authors to remain ignorant about the implications of the formulas they write.
In the table of indeterminate forms, the "0/0" does not refer to an arithmetic operation. If it did, that would already be an error. Instead, it is an abbreviation of a lim f(x)/g(x) where f(x),g(x) both converge to 0 and l'Hopitals rule is needed to proceed. Likewise, "0^0" in the table of indeterminate forms does not refer to an arithmetic operation 0 to the power 0, instead, it is an abbreviation for a limit that should be evaluated with a transformation followed by l'Hopital.
The statement that "0^0" is an indeterminate form has no implications for the arithmetic operation 0^0. The indeterminate form "0^0" is an abbreviation of a statement about functions, while 0^0 = ... is statement about numbers. Confusing "abbreviation of limits of functions" with "an arithmetic operation" works in favor of the wrong conclusion, so it is best to clear up this confusion.
The sci.math FAQ says that 0^0 = 1 is now a consensus. Of course, you can find webpages that say something else. To determine consensus, status counts too (Benson vs. Knuth? Clearly these are not equals).
Note that the consensus can change, even in math: 1 used to be a prime number, but not anymore. Closely related is that the empty product rule is now standard in mathematics, but it was not always so. That's why old references should have little weight compared with modern conventions (if old texts counted equally, then "Pluto is a planet" would still be the consensus). Computer algebra systems have to deal with people from the 21'st century, that's one way to get a sampling of the current consensus.
Of course, computers should follow math rules, instead of the other way around. However, "following math rules" is only possible if the rules are consistent! The option "undefined" breaks many rules, and "1" does not. MvH —Preceding undated comment added 18:48, 9 February 2014 (UTC)
Benson's claim "the best value for 0^0 depends on context" is an urban myth. Yes, people do believe that there exist contexts where 0 is a better value than 1, but nobody has actually seen such a context.
Looking at limits of 0^x and x^0 it seemed so plausible to Benson that both 0 and 1 are useful as values of 0^0 in some contexts, that he wrote this without actually checking it. In doing so, he underestimated how consistent math is when you apply general principles consistently. Wikipedia repeats Benson's claim, again without checking it, presenting it as a valid argument instead of an urban myth.
Of all the arguments for "undefined", the only one with actual merit is an educational argument: If the calculator returns 0^0 = undefined instead of 1, this might help a calculus student to avoid a mistake on a test about limits. The value of "undefined" is that it can function as a warning flag when a student works on limits.
However, the educational argument for "1" is stronger than that for "undefined". After calculus, you are unlikely to work with functions of the form f(x)^g(x) because whenever the exponent varies continuously, you would be advised to rewrite the expression as exp(...) at which point "undefined" no longer serves a purpose. Moreover, students taught "undefined" still need to know that 0^0 is 1 in order to interpret numerous formulas. In addition, 0! and 0^0 give an excellent teachable moment. Students start out being very uncomfortable with empty sets/lists/products/sums (and the concept of zero in general). But by using these concepts you end up with shorter and cleaner theorems/proofs/algorithms due to having to consider fewer cases. Dijkstra argued this years ago, but I still see this on a regular basis when I give programming assignments to students; inevitably they start out writing unnecessarily long programs for simple tasks, almost all have to learn these things. MvH ( talk) 23:23, 1 March 2014 (UTC)MvH
The inclusion of trigonometric functions in this article seems to belong in Exponential function rather than here. I propose moving all mention of trigonometric functions there. — Quondum 03:07, 5 March 2014 (UTC)
FAQ stands for Frequently Asked Questions. What you wrote has nothing to do with a FAQ, it is just your own thoughts on a topic that has bedeviled this article's talk pages. What a FAQ should be for something like this is
The section should not simply be "my own thoughts", the goal is to make a summary of arguments that have been written here dozens of times, in discussions spanning hundreds of kilobytes (please let me know if I omitted a Frequent Argument). Famous mathematicians, like Bjorn Poonen, have made arguments here, that are then quickly ignored because changes that support one side are usually interpreted as POV even if it is a change that reflects mainstream math. (PS. Trovatore: the arguments do belong in wiki's Talk section) MvH —Preceding undated comment added 23:18, 9 February 2014 (UTC)
Trovatore, before entering a big debate, let me reiterate my view that we can agree on a main page even if we can not agree on the issue "1 or undefined". I don't propose a 1-sided article, and I do agree with the "reliable sources with due weight". A big problem is that current views are not so easy to obtain in publications. People do not publish papers about 0^0 because you can't get publish things in good journals that are already known. My view is that things like empty product is 1 rule, and its applications, are now much more common than in older works. But I am not sure how to prove or disprove this view (citing papers would only be anecdotal evidence that can easily be rebutted by similar anecdotal evidence in the opposite direction).
On the main page, let me start with an example that may look fine to you, but looks biased to me. For example, the section that starts with: Any nonzero number raised by the exponent 0 is 1. To me, this section is obsolete because a few lines later in the section "negative exponents" it says again that b^0=1 if b is not 0. It looks as though the only purpose of this section is to reiterate the view that I don't like (the view that 0^0 is problematic). Surely we could delete this section, or merge it with the "negative exponents" section, without violating POV? It'll still tell the reader that 0^0 is problematic but this time, it won't say it twice within 10 lines from each other.
More troubling is the line: when 0^0 arises as a limit... This should be worded more precisely. What exactly is the word "it" in that sentence referring to, does it refer to 0^0, or the limit of f(x)^g(x)? If we can get rid of ambiguities, while keeping the same content, then that'd be a good thing. MvH ( talk) 00:50, 10 February 2014 (UTC)MvH
Trovatore, merging "zero exponent" and "negative exponents" is a relatively minor issue. My worry is though that even if we found the perfect formulation, the edit would still be reverted. In any case, the more problematic issue is the definition of the phrase "indeterminate form". Somehow, the two expressions "lim f(x)^g(x)" and "(lim f(x))^(lim g(x))" (i.e. 0^0) are both referred to as "indeterminate form", and this imprecise definition ends up equating "lim f(x)^g(x)" to "(lim f(x))^(lim g(x))". But these two expressions are very different (the value of the first can not be determined from the value of the second) and so they should definitely not be equated to each other.
The edit "arises" --> "is viewed" was a minor attempt to avoid equating "lim f(x)^g(x)" to "(lim f(x))^(lim g(x))", but it was quickly reverted. The same problem, mixing up different things, appears on the indeterminate form page as well, but there are sources that do it correctly.
Another way to say it is like this: if f(x) and g(x) converge to 0, this does not imply that lim f(x)^g(x) is 0^0. The page indeterminate form says that we can evaluate the indeterminate form, but that only makes sense if that refers to lim f(x)^g(x) (what would be the point of evaluating 0^0? It would say nothing about lim f(x)^g(x). Unless of course 0 really just means f(x), and the other 0 means g(x), but I sure hope that it doesn't!). MvH ( talk) 03:08, 10 February 2014 (UTC)MvH
Many thanks to MvH for this fine efford. I do share your worry that even if we found the perfect formulation, the edit would still be reverted. If Dcmq really understood that a limit does not have to equal the value at a point then he wouldn't think that lim(x,y)→(0+,0) xy should be equal to 00. The limit is indeterminate while the value is not. Bo Jacoby ( talk) 14:23, 16 February 2014 (UTC).
What is actually out there is
Bo Jacoby ( talk) 10:50, 17 February 2014 (UTC).
As the undefiners have a history of reverting my contributions I must leave it to the undefiners to clean up the mess themselves. Quondum's suggestion: "When 00 arises as a limit" is no good because 00 never arises as a limit. xy is not continuous around x=y=0. Bo Jacoby ( talk) 11:17, 25 February 2014 (UTC).
I know the current text is a hard fought compromise, and overall it is a good representation of the various points of view, but Benson's quote really ought to have an extra footnote. He writes that defining 0^0 is a matter of convenience, which is a correct statement that nevertheless promotes a poor understanding of mathematical practice; something that we should not endorse so prominently. Convenience is the sole justification for every definition in mathematics. Why does the modern definition of prime no longer include 1? Convenience. Saying that "0^0 = 1" is convenient means exactly the same thing as saying that it is a good definition. Benson is correct when he writes that defining 0^0 is a matter of convenience, but he is misleading the reader by pretending that 0^0 is special in that sense: defining 0^1 or 2^4 or 3^3 is a matter of convenience too. MvH ( talk) 14:20, 26 April 2014 (UTC)MvH
pown
and powr
as two separate functions. What I am saying is that there are sound mathematical reasons for defining two distinct functions, and I mention the IEEE choice only because it suggests that there exists a notable documented mathematical reasoning behind this approach. —
Quondum
18:03, 26 April 2014 (UTC)NaN
), exactly as required by theorems, analysis etc. I think that this is sufficient motivation for separate definitions, and indeed notations, for the two cases. There is only a limited set of circumstances where the distinction of which function we're dealing with is unimportant: when the inputs are restricted to x ∈ R+ and y ∈ Z. —
Quondum
21:18, 27 April 2014 (UTC)In the discussion about the value of 00, indeterminate forms and the complex domain are presented as examples where 00 cannot be defined as 1. Latter on, some opinions on the subject are presented expressed by notable mathematicians. In Knuth's treatment of the issue it is mentioned that it distinguishes between 00 as a value and 00 as an indeterminate form. Nothing is said about what Knuth thinks abouth 00 in the complex domain. To the eyes of the casual reader this leaves Knuth's treatment of the issue as incomplete. Is this a correct impression? I tried to make this evident by adding that "... he does not address the issue of 00 never being 1 in the complex domain.". This was reverted an oversimplification. I think this issue must be addressed because, as it is, Knuth's view is confusing. It may not be confusing for the academic community, where his article was addressed, but to someone who is no expert in indeterminate forms and the complex domain it certainly is. Nxavar ( talk) 09:16, 21 February 2014 (UTC)
If b^n=c, then what word indicates c?
The base is b, the exponent is n, and the ________ is c.
It seems that product is a poor choice for the blank. Suppose that c=1 and n=0. In what sense, then, is c a product? Power, too, seems like a poor choice, for we are already in the habit of saying things like b raised to the the nth power and b to the power (of n). So a power can be either the exponent, n, or the result, c, of the operation b^n. A frustrated newcomer to mathematics should be forgiven for suspecting (i) that there's a gap in the glossary and (ii) that mathematicians are trying to confuse students by using the word "power" to refer both to c and to n.
64.107.153.120 ( talk) 18:42, 29 April 2014 (UTC)
In the following text from Exponentiation#History of differing points of view
I see that the Benson quote was originally introduced by CBM in 2007. at the time, it was clearly intended to illustrate nothing more than that there was no definitive single answer, which made sense. Now the text has been refactored and the quote now appears intended to support of the statement that precedes it. As such, I propose deleting the quote as being out of place.
Going further, this section creates a false impression: that the debate is and historically has been essentially about only whether 00 is best treated as undefined or is best defined as 1. But I will not address that now. — Quondum 21:58, 28 April 2014 (UTC)
Earlier in the debate, a separation of definitions was mentioned to address confusion. Suppose that we have two formulas/definitions that assign values to the same function F. Formula/definition #1 has a certain domain, say D1, and formula/definition #2 has domain D2. If a point P lies in the intersection of D1 and D2, it is required that the formulas/definitions #1 and #2 have to produce the same result for P. There is no requirement that D1 be a subset of D2, or vice versa. If P lies in the intersection, we have two definitions to choose from, and it doesn't matter which one you use, and if P lies in only one of these sets, then you have only one choice. It is not unusual to have several formulas/definitions for the same function F, and this is acceptable if and only if they agree on the intersections of their domains. For the case F(x,y) = x^y, we have several definitions, and none of them has a domain that is a subset of the domain of another one. Say D1 is the domain of the definition with integer exponents, and D2 is the domain of exp(y log(x)). If P is in only one of the domains (e.g. 0^2 is in D1 and e^(1+i) is in D2) then we still have a definition for the value of F(P). If P is in the intersection of D1 and D2, then we can choose from two formulas/definitions, but that is OK because they'll give the same result.
It may seem confusing to have several definitions/formulas, but it is OK because for any point P=(x,y) that lies in the intersection of two domains, the values of the definitions/formulas agree. The point is this: (1) I suggest that we do not use the word "generalization" when the domain of definition #1 is not a subset of the domain of definition #2. (2) Please do not suggest that there is tension between the definitions if they agree perfectly on the intersections of their domains. MvH ( talk) 15:13, 30 April 2014 (UTC)MvH
Kjetil B Halvorsen 14:44, 6 May 2014 (UTC) — Preceding unsigned comment added by Kjetil1001 ( talk • contribs)
Here follows a proposal for replacing the subsection of the same name. The significant change is that it gives weight to the majority, rather than ignoring the majority of authors as is currently the case. This could be shortened as the rest of the article deals with the topic accordingly. — Quondum 04:56, 30 April 2014 (UTC)
The debate over the definition of 00 in the case of a continuously variable exponent has been going on at least since the early 19th century. At that time, most mathematicians agreed that 00 = 1, until In 1821
Cauchy
[1] listed 00 along with expressions like 0/0 in a table of indeterminate forms. In the 1830s Libri
[2]
[3] published an unconvincing argument for 00 = 1, and
Möbius
[4] sided with him, erroneously claiming that whenever . A commentator who signed his name simply as "S" provided the counterexample of (e−1/t)t, and this quieted the debate for some time. More historical details can be found in Knuth (1992).
[5]
More recent authors interpret the situation above in different ways:
RE: page 45 Cauchy. I typed page 45 so editors can copy/paste it into Google translate:
Lorsque, pour un systeme de valeurs attribuees aux variables qu'elle renferme, une fonction d'une ou de plusieurs variables n'admet qu'une seule valeur, cette valeur unique se deduit ordinairement de la definition meme de la function. S'il se presente un cas particulier dans lequel la definition donnee ne puisse plus fournir immediatement la valeur de la fonction que l'on considere, on cherche la limite ou les limites vers lesquelles cette fonction converge, tandis que les variables s'approchent indefiniment des valeurs particulieres qui leur sont assignees; et, s'il existe une ou plusieurs limites de cette espece, elles sont regardees comme autant de valeurs de la fonction dans l'hypthese admise. Nous nommerons valeurs singulieres de la fonction proposee celles qui se trouvent determinees comme on vient de le dire. Telles sont, par example, celles qu'on obtient en attribuant aux variables des valeurs infinies, et souvent assi celles qui correspondent a des solutions de continuite. La recherche des valeurs singulieres des fonctions est une des questions les plus imporantes el les plus delicates de l'analyse: elle offre plus ou moins de difficultes, suivant la nature des fonctions et le numbre des variables qu'elles renferment.
The output from google translate is not perfect, some lines only make sense if you compare with the original. A few key lines are "In case the definition given could not immediately provide the value of the function that we consider, we look at limit or limits" (google translate has "boundary" instead of "limit"). And "we will call singular values of the function those proposed are determined as we just said".
The first line tells us to look at limits if the function is not already defined there (which is the case in one camp, and not in the other camp). The second line tells us that [0..infinity) are not values of 0^0 but they are "singular values of the function x^y". MvH ( talk) 14:52, 2 May 2014 (UTC)MvH To interpret that first line in the historical context, keep in mind that at the time Cauchy wrote this, 0^0 was already defined (undefining it would have been a break in tradition, it is a big leap to assume that he wanted to do that if he didn't explicitly say so). MvH ( talk) 15:50, 2 May 2014 (UTC)MvH
i think that this arcticle should be a featured article. because it's richer that the featured article in herbew-- ᔕGᕼᗩIEᖇ ᗰOᕼᗩᗰEᗪ ( talk) 14:47, 3 September 2014 (UTC)
Article currently reads:
I am not sure what this is supposed to mean. The definition bn = en log b is valid for all non-zero b, not just "positive real" b. It is a "natural extension". I would write simply:
Is the problem the multivaluedness? That is true even for positive b. After all, 1(1/2) equals both 1 and -1. -- Macrakis ( talk) 21:35, 28 August 2014 (UTC)
I'm trying to wrap my head around complex exponents, but the description in the "Imaginary Exponents with Base e" is confusing. It states "Consider the right triangle (0, 1, 1 + ix/n)". I'm not familiar with a way of repesenting a triangle with 3 numbers, except if it's simply to state the length of each side as a Pythagorean triple (e.g. 3:4:5) or to indicate the 3 angles, as in a 30-60-90 triangle. The given values would not represent a right triangle using either of these methods. The quoted text includes a link to the "right triangle" Wikipedia article, but it sheds no light on what the three values might represent.
I think the meaning of the 3 number set should be explained or linked to or perhaps there should be 3 sets of Cartesian coordinates listed instead, as I believe this is a more familiar notation to most people.
Todd in Houston — Preceding unsigned comment added by 75.108.198.254 ( talk) 17:34, 11 October 2014 (UTC)
"the triangle is almost a circular sector"? -- 50.53.58.19 ( talk) 20:32, 11 October 2014 (UTC)
Really I was looking for clarification to "Consider the right triangle (0, 1, 1 + ix/n)".
However, I read this section again this morning and suddenly realized what the triangle represented by the 3 numbers was, probably because I'd been ruminating on the later part of the article that mentions "Using exponentiation with complex exponents may reduce problems in trigonometry to algebra." Anyway, the piece of the puzzle that was missing for me is that each number in that set is a complex number, and therefore is representative of a pair of coordinates on the complex plane. I think the statement could be changed to either "Consider the right triangle (0, 1, 1 + ix/n) on the complex plane" or "Consider the right triangle (0 + 0i, 1 + 0i, 1 + ix/n)" and that would eliminate the ambiguity.
By the way, I'm brand new to interacting with Wikipedia, but I'll take a shot at suggesting that edit myself.
Thanks! Oddacorn ( talk) 14:27, 12 October 2014 (UTC)
(A) Many textbooks contain formulas that imply 0^0 = 1.
(B) However, many of the same textbooks leave 0^0 undefined.
(B1) Some of these books say nothing about 0^0. (B2) Some of them write that 0^0 is undefined.
(C) Some textbooks define 0^0 it as 1.
Options (B1) and (B2) are common in calculus/analysis textbooks, while (C) is common in discrete math, graduate algebra, set theory.
I think textbooks that do (A)+(B) contradict themselves, but I have to admit that (I) such books are common, and (II) wikipedia has to report what common sources say, even if these sources say things that are self-contradictory.
We have to report that:
(A) there are common formulas where 0^0 must be evaluated as 1.
(B) there are sources where 0^0 is not defined.
(C) there are sources where 0^0 is defined as 1.
The 0^0 section is reasonable as long as it says (A)+(B)+(C), which at the moment, it does.
The key argument in support of (B) is that indeterminate forms should not be assigned a value. To test of this is a valid argument, we can apply it to floor(0), which is an indeterminate form because the left and right limits of floor(x), with x going to 0, do not match.
If 0^0 is undefined, then by the same logic, floor(0) should be undefined as well, a conclusion that few people would be happy with. But in the end, this argument, nor the other excellent arguments that support "1", have much bearing on wikipedia, because it must report what common sources say, and "undefined", is indeed surprisingly common. Lets make sure that each of (A)+(B)+(C) is represented properly. If so, the page itself should be uncontroversial despite the long-lasting disagreements about the merits of each argument. MvH Feb 7, 2014.
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I see that Knuth's paper (Donald E. Knuth, Two notes on notation, currently reference 15) is available here on arXiv.com. I presume that this is not a copyright violation, so should we not link to it? — Quondum 16:26, 27 July 2013 (UTC)
Some of Libri’s papers are still well remembered, but [32] and [33] are not. I found no mention of them in Science Citation Index, after searching through all years of that index available in our library (1955 to date). However, the paper [33] did produce several ripples in mathematical waters when it originally appeared, because it stirred up a controversy about whether is defined. Most mathematicians agreed that , but Cauchy [5, page 70] had listed together with other expressions like and in a table of undefined forms. Libri’s justification for the equation was far from convincing, and a commentator who signed his name simply “S” rose to the attack [45]. August Möbius [36] defended Libri, by presenting his former professor’s reason for believing that (basically a proof that ). Möbius also went further and presented a supposed proof that whenever . Of course “S” then asked [3] whether Möbius knew about functions such as and . (And paper [36] was quietly omitted from the historical record when the collected works of Möbius were ultimately published.) The debate stopped there, apparently with the conclusion that should be undefined.
But no, no, ten thousand times no! Anybody who wants the binomial theorem
to hold for at least one nonnegative integer must believe that , for we can plug in and to get on the left and on the right.
The number of mappings from the empty set to the empty set is . It has to be .
On the other hand, Cauchy had good reason to consider as an undefined limiting form, in the sense that the limiting value of is not known a priori when and approach independently. In this much stronger sense, the value of is less defined than, say, the value of . Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side.
(−2)1/3 does have different defined values depending on the domains. If you let r denote the number 21/3, then
As we have noted above,
— Arthur Rubin (talk) 18:06, 30 December 2013 (UTC)
Carl undid my edit. Carl apparently does not accept that powers with nonnegative integer exponent can be defined by xn=1 when n=0 and xn=x⋅xn-1 when n>0. Even hardcore undefiners like Trovatore agree that 00=1 when the exponent is integer. But the undefiners are now a minority in this discussion.
There is no point in undefining (−1)2 or 00 even if these expressions are not evaluated by the formula xy=exp(y⋅log(x)). They are calculated by other means and they are very widely used. But stop teaching innoscent children that (−1)1/3=−1, because (−1)1/3=(1+i31/2)/2 is more useful and more widely used. In J it reads
_1^2 1 0^0 1 _1^%3 0.5j0.866025
Bo Jacoby ( talk) 07:54, 31 December 2013 (UTC).
It seems as you agree with my above suggestion : Talk:Exponentiation#merging_three_definitions. Right? Bo Jacoby ( talk) 10:51, 2 January 2014 (UTC).
Naming the different exponentiation functions differently is OR. It is not what the mathematical community does. The functions are all called xy. There is consensus in the literature that 22=4, that 02=0, that (−2)2=4, that 2−1=1/2, that eiπ=−1, and that 21/2=√2. The function is generalized to still bigger domains without renaming. Nor do we rename the plus-sign when addition is generalized from non-negative integers through integers, rationals, reals, complex numbers, vectors, matrices, et cetera. I wonder if The Undefiners, (Dmcq, Trovatore, Carl, Quondum), still think that 0≠0.0 and that 0≠0+i0 ? Bo Jacoby ( talk) 09:09, 3 January 2014 (UTC).
The Undefiners prevent our article from improving. Dmcq claims that 0+i0≠0 which is OR. Trovatore claims that 0.0≠0, which is OR. Quondum claims that the different definitions of exponentiation should lead to different function names, which is OR. Carl reverts that xn (for non-negative integer n) can be defined by x0 = 1 and xn+1 = x⋅xn. Dmcq is correct that silliness is a better word for describing the positions of The Undefiners.
Many authors do not define 00. That doesn't undefine 00 once it is defined, and it is quite unimportant, and it is a nuisance.
Dmcq thinks that not giving 0/0 a value does help. It is not advisable to divide by zero because x/y is not continuous around x=y=0, no matter whether 0/0 is defined or not. Dcmq could write more carefully:
and differentiation gives
"no telltale warning because of assigning a value to 0/0". Assigning a value to 0/0 does not remove the warning against dividing by zero. The discontinuity of x/y around x=y=0 is still there. Not giving 0/0 a value does not help.
Dmcq believes that limx→0 f(x) must be equal to f(0), such that the only way to avoid a discontinuity is to undefine f(0). This belief is a mistake. Undefining does not necessarity remove discontinuities. xy is discontinuous around x=y=0, even if you undefine 00.
We need two different functions if one is to be given a value. No, we don't. We just have to accept the discontinuity.
Some sources say it should be 1. Other sources say it is an indeterminate form and give no value. We don't extrapolate and say indeterminate forms have values. If a function f(x) is discontinuous for x=0, then the value f(0) is not defined by continuity as limx→0 f(x). Some sources call f(0) an indeterminate form when it is not defined as limx→0 f(x) . But f(0) may be defined by other means.
I want our article to be understandable. x0 is defined to be equal to 1 in the elementary section. Later it turns out that xy is discontinuous around x=y=0. This is not helped by undefining 00. Bo Jacoby ( talk) 08:17, 4 January 2014 (UTC).
If you are an undefiner no more, then I am not refering to you, and so you have no reason to be offended. If you are still an Undefiner, then take pride in being capitalized. You choose your words and I choose mine. But you are changing the subject. The question is: Do you by now understand that undefining serves no purpose and solves no problem? Bo Jacoby ( talk) 17:58, 4 January 2014 (UTC).
Not I, but the readers of our article, suffer censorship from The Four Undefiners: Dmcq, Trovatore, Quondum, and Carl. This fine suggestion, put forward by Javalenok and supported by Mark van Hoeij, and 128.186.104.253, and myself, could not be implemented. Bo Jacoby ( talk) 08:04, 5 January 2014 (UTC).
Stating that
is original research, which is prohibited in wikipedia. (ps. It is also doublethink). The lead of our article on complex number says:
The complex number 0 + i0 has zero imaginary part and so it is a real number. As the real part is also zero makes is clear that 0 + i0 is the real number zero. So it is correct that 0 + i0 = 0. There is no doubt about it. Denying it leads to contradiction. How would you explain it to the uninitiated wikipedia reader? How would you rewrite complex number ?
That
is not true. Undefining 00 solves no problem whatsoever.
You write:
From the fact "A", that some authors define 00 = 1, and the fact "B", that other authors do not define 00, The Undefiners conclude "therefore C", that 00 is defined in some contexts and undefined in other contexts. Note the rule in Wikipedia:OR#Synthesis_of_published_material_that_advances_a_position:
So The Undefiners' position is original research.
You write:
That is true and it is a nuisance, but it is a different problem from that of defining 00. Right now I prefer to concentrate on: exponentiation with arbitrary base and nonnegative integer exponent, exponentiation with nonzero base and negative integer exponent, and exponentiation with positive base and arbitrary exponent. Bo Jacoby ( talk) 19:18, 5 January 2014 (UTC).
The notation was used way before the 20th century, and nobody doubts that f(0)=a0, so it goes without saying that 0k=0 for k>0 and 0k=1 for k=0. Donald Knuth didn't invent 00=1. In the 18th century Leonhard Euler wrote ex with the same exponential notation as x2, so it is not original synthesis on my part. Bo Jacoby ( talk) 23:03, 5 January 2014 (UTC).
Do you really doubt that f(0) is ment to be equal to a0 when f(x) is defined by ∑k ak xk ?
It doesn't matter which exponential function was intended as long as the competing definitions produce the same result. 32 = 1⋅3⋅3 = 9 and 32 = exp(2⋅log(3)) = 9.
An analogous problem is well handled in Addition#Addition of natural and real numbers. Bo Jacoby ( talk) 07:07, 6 January 2014 (UTC).
Thinking that 00 is sometimes defined and sometimes undefined is not mathematics, it is doublethink. Bo Jacoby ( talk) 08:33, 6 January 2014 (UTC).
Show your good will by answering my question: Do you doubt that f(0) is ment to be equal to a0 when f(x) is defined by ∑k ak xk ? Bo Jacoby ( talk) 17:55, 6 January 2014 (UTC).
Do The Undefiners doubt that e0 = 1 where ex is defined by the power series ? Bo Jacoby ( talk) 08:02, 7 January 2014 (UTC).
As The Undefiners are not willing or not able to explain their own position, I will give it a try. Everybody agree that and that . So . Both the numerator and the denominator are empty products, and so 00=0!=1. Recursive definitions are: 0!=1 and n!=n⋅(n−1)! for n=1,2,3,... , and x0=1 and xn=xn−1⋅x for n=1,2,3,... . Carl is sensitive against 00, but Trovatore and Dcqm agrees that 00=1 for integer exponent 0. But, because some authors do not define 00, it is important to The Undefiners to claim that 00 is sometimes not defined. I don't follow this argument. Of course some authors do not define 00. Most authors write about something else. Dmcq thinks that the discontinuity of xy around x=y=0 is more spectacular when 00 is undefined, than when 00 is defined to be 1. Trovatore promotes the idea that 00 is undefined when the exponent 0 is real, but defined when it is integer. This idea is given the formulation: 0.0≠0 , real zero is not the same thing as integer zero. I don't follow this argument. Trovatore and Carl thinks that the set of integers, ℤ, is disjoint to the set of reals, ℝ, while I consider ℤ to be a subset of ℝ, such that operations on ℝ generalize operations on ℤ. Trovatore has problems with 1+0.5=1.5, because 1 is an integer and 0.5 is a (non-integer) real. I don't have the same problem because integer 1 and real 1 to me is the same number: 1=1.0. I find support for this interpretation here and elsewhere. The Undefiners polluted our article only by undefining 00 , but they should also undefine (−1)2 which also cannot be defined by xy = exp(y⋅log(x)). There is no real logarithm to −1. The fact that some authors do not define 00 should not force our article to become unreadable. Bo Jacoby ( talk) 22:36, 7 January 2014 (UTC).
What would Dcmq want The Undefiners to be called? The Doublethinkers? Bo Jacoby ( talk) 08:15, 8 January 2014 (UTC).
Is it also besides the point that the article is incomprehensible? Exponentiation#Arbitrary_integer_exponents says: "The case of 00 is controversial". In fact it is not at all controversial for integer exponents. Also the reader gets the warning: "Any nonzero number raised by the exponent 0 is 1". In fact x0 is always interpreted as 1. Nobody writes 00 if it is supposed to be undefined. Why not omit this nonsense? Bo Jacoby ( talk) 08:15, 8 January 2014 (UTC).
Also, a book that comes out and says directly how to the seeming issue with 00 in power series: Complex Analysis for Mathematics and Engineering, John Mathews, Russell Howell, Jones & Bartlett, 2010, p. 151.
This could be useful for the article at some point. — Carl ( CBM · talk) 20:13, 8 January 2014 (UTC)
Oh, thank you for allowing me to hold my view. But how can you or any other mathematician think that the above identity does not imply that 00 =1 ? Must wikipedia convey the impression that mathematicians are insane? Bo Jacoby ( talk) 10:44, 9 January 2014 (UTC).
I am not the only one to see that the stipulated identity defines 00. The position of The Undefiners (including Mathews and Howell) is manifestly insane. So the obvious answer to my rhetorical question is that wikipedia should stop conveying the impression that mathematicians are insane. The Undefiners must leave in shame. Sensible suggestions on this talk page were turned down by The Undefiners, and my very innocent edit was undone by Carl. This must stop. Bo Jacoby ( talk) 05:36, 10 January 2014 (UTC).
It is inconsistent that the article claims that 00 is more controversial than, say, (−3)2. Neither are defined by: xy=exp(log(x)⋅y), and both are defined by the recursive definitions: x0 = 1, and xn+1 = xn⋅x, and xn−1 = xn/x for nonzero x. If only one definition apply, then that is the one to be used, such as 31/2=exp(log(3)/2). If both definitions apply, then the result is the same: 32 = 9 by both definitions. So yes, there are multiple functions called exponentiation, but that is not a problem, and that is not the point. Bo Jacoby ( talk) 19:05, 8 January 2014 (UTC).
The two definitions for xn are the analytic one: xn = exp(log(x)⋅n), and the recursive one: xn = 1 for n=0 and xn = xn−1⋅x for n>0. These definitions produce identical results whereever they both apply. The analytic definition applies to positive x and general n. The recursive definition applies to general x and nonnegative integer n, and to nonzero x and negative integer n. "Undefined" is not appropriate when something is defined. Bo Jacoby ( talk) 06:46, 9 January 2014 (UTC).
Making something undefined is not a definition. I want to postpone the inconsistent definitions of (-1)1/3 as the discussion is sufficiently complicated by now. The 00 stuff is almost kept together in one place, except the warnings against 00 while treating integer exponents. I did not say that there is one true exponential function. The exponential notation may refer to different functions, as explained above. You have not told why you disagree that "xn can be defined by x0 = 1 and xn = xn−1⋅x for positive integer n". You undid my contribution. Bo Jacoby ( talk) 20:30, 8 January 2014 (UTC).
You mean that it is sometimes correct and sometimes not correct? This is doublethink. Bo Jacoby ( talk) 21:52, 8 January 2014 (UTC).
I need managable sections. You are welcome to rename the headings for your convenience. But how can making something undefined be a definition? What do you mean? Bo Jacoby ( talk) 06:20, 9 January 2014 (UTC).
We must distinguish between function f and function value f(x). Specifying the domain is part of the definition of a function. Removing x from the domain of f makes f(x) undefined. This redefinition of f is not a definition of f(x). Making f(x) undefined can not be a definition of f(x). Bo Jacoby ( talk) 05:07, 10 January 2014 (UTC).
Dmcq changed "Non-negative" to "Positive" and "" to "". Any reader need to know why. Is it incorrect to say that: "Formally, powers with non-negative integer exponents may be defined by the initial condition and the recurrence relation "? Does the uninitiated reader benefit from making an exception from the definition? Bo Jacoby ( talk) 18:25, 10 January 2014 (UTC).
So you found a book that left 00 undefined. The question still remains: Does the uninitiated reader benefit from making 00 an exception from the definition? Bo Jacoby ( talk) 21:46, 10 January 2014 (UTC).
I know what wikipedia is. But now I also know why your source does not define b0. Your source is about general algebraic rings, not only rings with an identity. So the ring of even numbers, 2ℤ, is an example. In this ring b0 is not defined, because 1 is not an even number, 1∉2ℤ. You could and should have discovered this yourself. Our article is about the rings with identity, ℤ or ℝ or ℂ. One is a number. Now that this is clear, would you please either restore the definition b0=1, or carefully explain why not. Bo Jacoby ( talk) 23:16, 10 January 2014 (UTC).
Does your source tell why it leaves 00 undefined? Bo Jacoby ( talk) 06:40, 11 January 2014 (UTC).
Shouldn't the links to sources leaving 00 undefined be collected here after the sentence "not all sources define 00" ? Bo Jacoby ( talk) 09:10, 11 January 2014 (UTC).
Why restrict b0=1 to nonzero b at this stage? An empty product doesn't depend on the value of a factor that isn't there.
Shouldn't "00=1 or undefined" be treated in the article rather than on the talk page? Bo Jacoby ( talk) 16:45, 11 January 2014 (UTC).
OK, I went there. Why not wait until Exponentiation#Zero to the power of_zero with mentioning "nonzero b" ? Even Trovatore agrees that b0=1 is not restricted to nonzero b when the exponent is integer (which it is). Bo Jacoby ( talk) 04:07, 12 January 2014 (UTC).
I am discussing Dmcq's edit right here. Please answer my questions. Bo Jacoby ( talk) 14:16, 12 January 2014 (UTC).
I moved that section "combinatorial interpretation" back to its original location. The article should certainly include that topic, but since this article is on an elementary subject, and that interpretation is slightly more advanced, it should not be placed so highly in the article. Compare WP:MTAA section 4.1. — Carl ( CBM · talk) 13:11, 12 January 2014 (UTC)
We probably should be revising the integer section and saying a bit more about 0^0=1 there, but just sticking it at the start when that is not generally agreed is simply WP:OR. How about just trying to fix the article up rather than trying to push a point of view? Dmcq ( talk) 10:58, 10 January 2014 (UTC)
I agree with Carl that it's better to keep all the 0^0 stuff together in one place. Assuming that x^0 means 1 in the context of a power series is actually the same thing as defining 0^0=1. (See #Answering_Trovatore). Bo Jacoby ( talk) 03:59, 12 January 2014 (UTC).
How can the reader decide whether a problem is "in the context of a power series" or not? When and why and how does 00 fade away from being 1 to being undefined? Bo Jacoby ( talk) 14:24, 12 January 2014 (UTC).
I don't understand what you mean. Bo Jacoby ( talk) 21:18, 12 January 2014 (UTC).
Dmcq undid my edit. Carl's source actually did define 0^0 . See above. Bo Jacoby ( talk) 11:02, 10 January 2014 (UTC).
I is a diplomatic challenge to negotiate with people who seriously deny that 0+i0=0. Step back and relaxe. You have still unanswered questions. Improvements are not made by your automatic undoing my contributions. I welcome your stated willingness to discuss. Bo Jacoby ( talk) 11:36, 10 January 2014 (UTC).
My edit was finished at 1047 o'clock and your straightforward section just above was made at 1058, so I didn't ignore it. I asked you what you guys want to be called instead of Undefiners, and you did not answer. I look forward to see your arguments on the subject matter, but not your comments on my histrionics. Your reversions are so quick that you haven't taken time to study my contribution in calm and detail. Bo Jacoby ( talk) 12:09, 10 January 2014 (UTC).
Although there is not much progress in this discussion, I do see some light. Trovatore seems to have stopped denying that 0.0=0, and Dmcq seems to have stopped denying that 0+i0=0, and Carl seems to have stopped denying that ℕ⊂ℤ⊂ℝ⊂ℂ, and to realize that the equation is no different from defining z0=1. So some progress has been made, and The Undefiners no longer constitute quite the same mathematical madhouse as they used to. Dcmq does not like being called an undefiner, and I will stop calling Dcmq an undefiner as soon as Dmcq stops being an undefiner. But because some sources do not define 00 it is important to Dcmq to claim that 00 is not always defined. Dmcq did find a source which does not even define b0 for nonzero b. but for unknown reasons that did not urge Dmcq to claim that b0 is sometimes not defined for nonzero b. Trovatore knows that the definition by = exp(log(b)⋅y) does not cover the case b=0, but neither does it cover the case b=−1, which is no problem to Trovatore. The Undefiners' attitude of infallibility made collaboration difficult and unpleasant. I did my duty. I regret to the wikipedia readers that I was not successful in removing the nonsense from our article. The simplicity and clarity of Donald Knuth's approach should be used in the article, and the obscurity of undefining should be concentrated in the appropriate subsection. Bo Jacoby ( talk) 15:17, 14 January 2014 (UTC).
Wow, how did we pass the seventh anniversary of this thread without celebrating? Congratulations to everyone (especially the inimitable Mr. Jacoby) for their stamina. — Steven G. Johnson ( talk) 21:06, 15 January 2014 (UTC)
Thanks to Mr. Johnson for the congratulation! Isheden's suggestion needs proofreading:
Shorthand for this is formally
and informally:
Be bold and make your edit. Undoing a contribution is not progress, and undefining a definition is not progress either. Bo Jacoby ( talk) 22:01, 15 January 2014 (UTC).
The definition 00 = 1 is always assumed in the usual expression for a polynomial or a series, where when . The definition b0 = 1 need not be restricted to nonzero b (or to positive b). But do not assume that . The function bx = elog(b)⋅x is defined for b>0, but it is not continuous around b=x=0, so don't even try to define 00 by continuity. Bo Jacoby ( talk) 23:37, 15 January 2014 (UTC).
Hi, following part didn't seem right to me, but I hesitated to edit.
The following identity holds for arbitrary integers m and n, provided that m and n are both positive when b is zero:
First, why write b like a variable if b is always zero. Why say arbitrary integers then say positive integers? What good is 0^(m+n), why not just say 0^m is zero when m>0? Why is there a dot after the equation? — Preceding unsigned comment added by 85.108.132.131 ( talk) 06:16, 10 January 2014 (UTC)
In many other Wikipedia articles Template:Round in circles and Template:FAQ are used to respond to long-running and often-repeated arguments that go nowhere.
The arguments over 00 seem to fall into this category. Arguments have been going on for over seven years now, with no really new arguments being raised nor any substantially different sources being cited as far as I can tell. Wouldn't it be more productive to create a FAQ entry and simply refer to that when similar arguments are raised in the future?
— Steven G. Johnson ( talk) 21:47, 15 January 2014 (UTC)
What is the point of this section if not to justify the continuity of the exponential function through the continuity of the logarithm through its own definition? The fact is one textbook prefers to define the logarithm as a continuous function first, and later introduce the exponential function and the justification for the existence of real exponents on said definition:
Toolnut ( talk) 07:25, 7 February 2014 (UTC)
But the section containing this subsection is titled "Real exponents" and talks at length about the justification for the extension to real exponents from rational exponents. Of note, "Since any irrational number can be approximated by a rational number ...": if you use the logarithm definition, that kind of justification becomes unnecessary; therefore, it is a different approach well worth noting. So you appear to be saying the reader should look for the logarithm definition elsewhere and not become aware of this finer point here, where it has been brought up. The separate article on logarithms actually begins by defining a logarithm as the inverse of the exponential function, not independently of it; all the more reason to make this fine point here. Toolnut ( talk) 10:32, 7 February 2014 (UTC)
In response to the request for a FAQ. I started by making a summary of frequently made arguments for "1". The goal of this section is to save time (i.e. the goal is not to do the debate all over again!). Instead, the goal is to give a summary of arguments that have been made dozens of times, over hundreds of kilobytes, and compress it down to this one section.
Identities in basic arithmetic should not be undefined simply because they do not fit in a later formula such as exp(y log(x)). If we were to allow that, then we can undefine not only 0^0 and 0^2, but pretty much anything else as well.
0^0 = 1 is basic arithmetic, it follows from fundamental rules such as the empty product rule, and cardinal number arithmetic, both of which are far more fundamental than the exp(y log(x)) formula, or the technical issue regarding limits that is used to undefine 0^0.
The limit issue used to undefine 0^0 ignores the fact that the same argument also undefines f(0) for any function that is discontinuous at x=0, such as the floor function. It is not logically consistent to use this argument to say that 0^0 is undefined but floor(0) is not. Other common arguments for "undefined" are equally inconsistent (the most common one says that "0^x = 0 for all x", blissfully ignoring x=-1).
Efforts to portray 0^0 as undefined lead to bizarre wordings in the text, for example, right now we have a line "regarding b^0 as an empty product" which is a strange thing to say because b^0 is an empty product, just like an empty bag has 0 applies.
To justify this bizarre wording you need to hold a circular "definition" of 0; to believe that 0 is not a counting number (0 is the number of apples in an empty bag, or in more sophisticated words, it is the cardinality of the empty set) but instead, one somehow views 0 as a function that converges to 0, ignoring the circular nature of this view of numbers.
Efforts for "undefined" also leads to conclusions that we would normally not make. For example, if a book says f(x) = g(x) for all |x|<1, but does not say that f(0) = g(0), then in normal circumstances, the conclusion f(0) = g(0) would not be Original Research. Yet, if we apply this to 1/(1-x) = sum x^n then it suddenly becomes Original Research? To keep 0^0 undefined, one must ignore the fact that 0^0 = 1 is used throughout mathematics, and that even the calculus books that say "undefined" require 0^0 to be evaluated as 1 in their formulas, and that they give plenty of formulas that say 0^0=1 after a simple substitution. You have to pick one:
(a) simple substitutions are OR, (b) simple substitutions are OK, or (c) simple substitutions are OK but if I don't like the outcome, then they are OR.
Now you might say, if calculus textbooks (through the binomial theorem, or 1/(1-x), or power series notation in general) imply 0^0=1, they still do not mean to say 0^0=1 (the argument goes something like: the x^0 in power series is not unconditionally 1, it is just a nice convention that makes formulas shorter). The problem with that argument is that these books do not actually say that x^0=1 is a convention. In any decent math journal, if you imply a statement, it means that you claim it is true. There are two possibilities: either the calculus textbook authors do not follow the standards held by math journals, or, more likely, they do not know that their own writings imply 0^0=1. If our 0^0 section does not point out that many common formulas imply the value 1, then we allow textbook authors to remain ignorant about the implications of the formulas they write.
In the table of indeterminate forms, the "0/0" does not refer to an arithmetic operation. If it did, that would already be an error. Instead, it is an abbreviation of a lim f(x)/g(x) where f(x),g(x) both converge to 0 and l'Hopitals rule is needed to proceed. Likewise, "0^0" in the table of indeterminate forms does not refer to an arithmetic operation 0 to the power 0, instead, it is an abbreviation for a limit that should be evaluated with a transformation followed by l'Hopital.
The statement that "0^0" is an indeterminate form has no implications for the arithmetic operation 0^0. The indeterminate form "0^0" is an abbreviation of a statement about functions, while 0^0 = ... is statement about numbers. Confusing "abbreviation of limits of functions" with "an arithmetic operation" works in favor of the wrong conclusion, so it is best to clear up this confusion.
The sci.math FAQ says that 0^0 = 1 is now a consensus. Of course, you can find webpages that say something else. To determine consensus, status counts too (Benson vs. Knuth? Clearly these are not equals).
Note that the consensus can change, even in math: 1 used to be a prime number, but not anymore. Closely related is that the empty product rule is now standard in mathematics, but it was not always so. That's why old references should have little weight compared with modern conventions (if old texts counted equally, then "Pluto is a planet" would still be the consensus). Computer algebra systems have to deal with people from the 21'st century, that's one way to get a sampling of the current consensus.
Of course, computers should follow math rules, instead of the other way around. However, "following math rules" is only possible if the rules are consistent! The option "undefined" breaks many rules, and "1" does not. MvH —Preceding undated comment added 18:48, 9 February 2014 (UTC)
Benson's claim "the best value for 0^0 depends on context" is an urban myth. Yes, people do believe that there exist contexts where 0 is a better value than 1, but nobody has actually seen such a context.
Looking at limits of 0^x and x^0 it seemed so plausible to Benson that both 0 and 1 are useful as values of 0^0 in some contexts, that he wrote this without actually checking it. In doing so, he underestimated how consistent math is when you apply general principles consistently. Wikipedia repeats Benson's claim, again without checking it, presenting it as a valid argument instead of an urban myth.
Of all the arguments for "undefined", the only one with actual merit is an educational argument: If the calculator returns 0^0 = undefined instead of 1, this might help a calculus student to avoid a mistake on a test about limits. The value of "undefined" is that it can function as a warning flag when a student works on limits.
However, the educational argument for "1" is stronger than that for "undefined". After calculus, you are unlikely to work with functions of the form f(x)^g(x) because whenever the exponent varies continuously, you would be advised to rewrite the expression as exp(...) at which point "undefined" no longer serves a purpose. Moreover, students taught "undefined" still need to know that 0^0 is 1 in order to interpret numerous formulas. In addition, 0! and 0^0 give an excellent teachable moment. Students start out being very uncomfortable with empty sets/lists/products/sums (and the concept of zero in general). But by using these concepts you end up with shorter and cleaner theorems/proofs/algorithms due to having to consider fewer cases. Dijkstra argued this years ago, but I still see this on a regular basis when I give programming assignments to students; inevitably they start out writing unnecessarily long programs for simple tasks, almost all have to learn these things. MvH ( talk) 23:23, 1 March 2014 (UTC)MvH
The inclusion of trigonometric functions in this article seems to belong in Exponential function rather than here. I propose moving all mention of trigonometric functions there. — Quondum 03:07, 5 March 2014 (UTC)
FAQ stands for Frequently Asked Questions. What you wrote has nothing to do with a FAQ, it is just your own thoughts on a topic that has bedeviled this article's talk pages. What a FAQ should be for something like this is
The section should not simply be "my own thoughts", the goal is to make a summary of arguments that have been written here dozens of times, in discussions spanning hundreds of kilobytes (please let me know if I omitted a Frequent Argument). Famous mathematicians, like Bjorn Poonen, have made arguments here, that are then quickly ignored because changes that support one side are usually interpreted as POV even if it is a change that reflects mainstream math. (PS. Trovatore: the arguments do belong in wiki's Talk section) MvH —Preceding undated comment added 23:18, 9 February 2014 (UTC)
Trovatore, before entering a big debate, let me reiterate my view that we can agree on a main page even if we can not agree on the issue "1 or undefined". I don't propose a 1-sided article, and I do agree with the "reliable sources with due weight". A big problem is that current views are not so easy to obtain in publications. People do not publish papers about 0^0 because you can't get publish things in good journals that are already known. My view is that things like empty product is 1 rule, and its applications, are now much more common than in older works. But I am not sure how to prove or disprove this view (citing papers would only be anecdotal evidence that can easily be rebutted by similar anecdotal evidence in the opposite direction).
On the main page, let me start with an example that may look fine to you, but looks biased to me. For example, the section that starts with: Any nonzero number raised by the exponent 0 is 1. To me, this section is obsolete because a few lines later in the section "negative exponents" it says again that b^0=1 if b is not 0. It looks as though the only purpose of this section is to reiterate the view that I don't like (the view that 0^0 is problematic). Surely we could delete this section, or merge it with the "negative exponents" section, without violating POV? It'll still tell the reader that 0^0 is problematic but this time, it won't say it twice within 10 lines from each other.
More troubling is the line: when 0^0 arises as a limit... This should be worded more precisely. What exactly is the word "it" in that sentence referring to, does it refer to 0^0, or the limit of f(x)^g(x)? If we can get rid of ambiguities, while keeping the same content, then that'd be a good thing. MvH ( talk) 00:50, 10 February 2014 (UTC)MvH
Trovatore, merging "zero exponent" and "negative exponents" is a relatively minor issue. My worry is though that even if we found the perfect formulation, the edit would still be reverted. In any case, the more problematic issue is the definition of the phrase "indeterminate form". Somehow, the two expressions "lim f(x)^g(x)" and "(lim f(x))^(lim g(x))" (i.e. 0^0) are both referred to as "indeterminate form", and this imprecise definition ends up equating "lim f(x)^g(x)" to "(lim f(x))^(lim g(x))". But these two expressions are very different (the value of the first can not be determined from the value of the second) and so they should definitely not be equated to each other.
The edit "arises" --> "is viewed" was a minor attempt to avoid equating "lim f(x)^g(x)" to "(lim f(x))^(lim g(x))", but it was quickly reverted. The same problem, mixing up different things, appears on the indeterminate form page as well, but there are sources that do it correctly.
Another way to say it is like this: if f(x) and g(x) converge to 0, this does not imply that lim f(x)^g(x) is 0^0. The page indeterminate form says that we can evaluate the indeterminate form, but that only makes sense if that refers to lim f(x)^g(x) (what would be the point of evaluating 0^0? It would say nothing about lim f(x)^g(x). Unless of course 0 really just means f(x), and the other 0 means g(x), but I sure hope that it doesn't!). MvH ( talk) 03:08, 10 February 2014 (UTC)MvH
Many thanks to MvH for this fine efford. I do share your worry that even if we found the perfect formulation, the edit would still be reverted. If Dcmq really understood that a limit does not have to equal the value at a point then he wouldn't think that lim(x,y)→(0+,0) xy should be equal to 00. The limit is indeterminate while the value is not. Bo Jacoby ( talk) 14:23, 16 February 2014 (UTC).
What is actually out there is
Bo Jacoby ( talk) 10:50, 17 February 2014 (UTC).
As the undefiners have a history of reverting my contributions I must leave it to the undefiners to clean up the mess themselves. Quondum's suggestion: "When 00 arises as a limit" is no good because 00 never arises as a limit. xy is not continuous around x=y=0. Bo Jacoby ( talk) 11:17, 25 February 2014 (UTC).
I know the current text is a hard fought compromise, and overall it is a good representation of the various points of view, but Benson's quote really ought to have an extra footnote. He writes that defining 0^0 is a matter of convenience, which is a correct statement that nevertheless promotes a poor understanding of mathematical practice; something that we should not endorse so prominently. Convenience is the sole justification for every definition in mathematics. Why does the modern definition of prime no longer include 1? Convenience. Saying that "0^0 = 1" is convenient means exactly the same thing as saying that it is a good definition. Benson is correct when he writes that defining 0^0 is a matter of convenience, but he is misleading the reader by pretending that 0^0 is special in that sense: defining 0^1 or 2^4 or 3^3 is a matter of convenience too. MvH ( talk) 14:20, 26 April 2014 (UTC)MvH
pown
and powr
as two separate functions. What I am saying is that there are sound mathematical reasons for defining two distinct functions, and I mention the IEEE choice only because it suggests that there exists a notable documented mathematical reasoning behind this approach. —
Quondum
18:03, 26 April 2014 (UTC)NaN
), exactly as required by theorems, analysis etc. I think that this is sufficient motivation for separate definitions, and indeed notations, for the two cases. There is only a limited set of circumstances where the distinction of which function we're dealing with is unimportant: when the inputs are restricted to x ∈ R+ and y ∈ Z. —
Quondum
21:18, 27 April 2014 (UTC)In the discussion about the value of 00, indeterminate forms and the complex domain are presented as examples where 00 cannot be defined as 1. Latter on, some opinions on the subject are presented expressed by notable mathematicians. In Knuth's treatment of the issue it is mentioned that it distinguishes between 00 as a value and 00 as an indeterminate form. Nothing is said about what Knuth thinks abouth 00 in the complex domain. To the eyes of the casual reader this leaves Knuth's treatment of the issue as incomplete. Is this a correct impression? I tried to make this evident by adding that "... he does not address the issue of 00 never being 1 in the complex domain.". This was reverted an oversimplification. I think this issue must be addressed because, as it is, Knuth's view is confusing. It may not be confusing for the academic community, where his article was addressed, but to someone who is no expert in indeterminate forms and the complex domain it certainly is. Nxavar ( talk) 09:16, 21 February 2014 (UTC)
If b^n=c, then what word indicates c?
The base is b, the exponent is n, and the ________ is c.
It seems that product is a poor choice for the blank. Suppose that c=1 and n=0. In what sense, then, is c a product? Power, too, seems like a poor choice, for we are already in the habit of saying things like b raised to the the nth power and b to the power (of n). So a power can be either the exponent, n, or the result, c, of the operation b^n. A frustrated newcomer to mathematics should be forgiven for suspecting (i) that there's a gap in the glossary and (ii) that mathematicians are trying to confuse students by using the word "power" to refer both to c and to n.
64.107.153.120 ( talk) 18:42, 29 April 2014 (UTC)
In the following text from Exponentiation#History of differing points of view
I see that the Benson quote was originally introduced by CBM in 2007. at the time, it was clearly intended to illustrate nothing more than that there was no definitive single answer, which made sense. Now the text has been refactored and the quote now appears intended to support of the statement that precedes it. As such, I propose deleting the quote as being out of place.
Going further, this section creates a false impression: that the debate is and historically has been essentially about only whether 00 is best treated as undefined or is best defined as 1. But I will not address that now. — Quondum 21:58, 28 April 2014 (UTC)
Earlier in the debate, a separation of definitions was mentioned to address confusion. Suppose that we have two formulas/definitions that assign values to the same function F. Formula/definition #1 has a certain domain, say D1, and formula/definition #2 has domain D2. If a point P lies in the intersection of D1 and D2, it is required that the formulas/definitions #1 and #2 have to produce the same result for P. There is no requirement that D1 be a subset of D2, or vice versa. If P lies in the intersection, we have two definitions to choose from, and it doesn't matter which one you use, and if P lies in only one of these sets, then you have only one choice. It is not unusual to have several formulas/definitions for the same function F, and this is acceptable if and only if they agree on the intersections of their domains. For the case F(x,y) = x^y, we have several definitions, and none of them has a domain that is a subset of the domain of another one. Say D1 is the domain of the definition with integer exponents, and D2 is the domain of exp(y log(x)). If P is in only one of the domains (e.g. 0^2 is in D1 and e^(1+i) is in D2) then we still have a definition for the value of F(P). If P is in the intersection of D1 and D2, then we can choose from two formulas/definitions, but that is OK because they'll give the same result.
It may seem confusing to have several definitions/formulas, but it is OK because for any point P=(x,y) that lies in the intersection of two domains, the values of the definitions/formulas agree. The point is this: (1) I suggest that we do not use the word "generalization" when the domain of definition #1 is not a subset of the domain of definition #2. (2) Please do not suggest that there is tension between the definitions if they agree perfectly on the intersections of their domains. MvH ( talk) 15:13, 30 April 2014 (UTC)MvH
Kjetil B Halvorsen 14:44, 6 May 2014 (UTC) — Preceding unsigned comment added by Kjetil1001 ( talk • contribs)
Here follows a proposal for replacing the subsection of the same name. The significant change is that it gives weight to the majority, rather than ignoring the majority of authors as is currently the case. This could be shortened as the rest of the article deals with the topic accordingly. — Quondum 04:56, 30 April 2014 (UTC)
The debate over the definition of 00 in the case of a continuously variable exponent has been going on at least since the early 19th century. At that time, most mathematicians agreed that 00 = 1, until In 1821
Cauchy
[1] listed 00 along with expressions like 0/0 in a table of indeterminate forms. In the 1830s Libri
[2]
[3] published an unconvincing argument for 00 = 1, and
Möbius
[4] sided with him, erroneously claiming that whenever . A commentator who signed his name simply as "S" provided the counterexample of (e−1/t)t, and this quieted the debate for some time. More historical details can be found in Knuth (1992).
[5]
More recent authors interpret the situation above in different ways:
RE: page 45 Cauchy. I typed page 45 so editors can copy/paste it into Google translate:
Lorsque, pour un systeme de valeurs attribuees aux variables qu'elle renferme, une fonction d'une ou de plusieurs variables n'admet qu'une seule valeur, cette valeur unique se deduit ordinairement de la definition meme de la function. S'il se presente un cas particulier dans lequel la definition donnee ne puisse plus fournir immediatement la valeur de la fonction que l'on considere, on cherche la limite ou les limites vers lesquelles cette fonction converge, tandis que les variables s'approchent indefiniment des valeurs particulieres qui leur sont assignees; et, s'il existe une ou plusieurs limites de cette espece, elles sont regardees comme autant de valeurs de la fonction dans l'hypthese admise. Nous nommerons valeurs singulieres de la fonction proposee celles qui se trouvent determinees comme on vient de le dire. Telles sont, par example, celles qu'on obtient en attribuant aux variables des valeurs infinies, et souvent assi celles qui correspondent a des solutions de continuite. La recherche des valeurs singulieres des fonctions est une des questions les plus imporantes el les plus delicates de l'analyse: elle offre plus ou moins de difficultes, suivant la nature des fonctions et le numbre des variables qu'elles renferment.
The output from google translate is not perfect, some lines only make sense if you compare with the original. A few key lines are "In case the definition given could not immediately provide the value of the function that we consider, we look at limit or limits" (google translate has "boundary" instead of "limit"). And "we will call singular values of the function those proposed are determined as we just said".
The first line tells us to look at limits if the function is not already defined there (which is the case in one camp, and not in the other camp). The second line tells us that [0..infinity) are not values of 0^0 but they are "singular values of the function x^y". MvH ( talk) 14:52, 2 May 2014 (UTC)MvH To interpret that first line in the historical context, keep in mind that at the time Cauchy wrote this, 0^0 was already defined (undefining it would have been a break in tradition, it is a big leap to assume that he wanted to do that if he didn't explicitly say so). MvH ( talk) 15:50, 2 May 2014 (UTC)MvH
i think that this arcticle should be a featured article. because it's richer that the featured article in herbew-- ᔕGᕼᗩIEᖇ ᗰOᕼᗩᗰEᗪ ( talk) 14:47, 3 September 2014 (UTC)
Article currently reads:
I am not sure what this is supposed to mean. The definition bn = en log b is valid for all non-zero b, not just "positive real" b. It is a "natural extension". I would write simply:
Is the problem the multivaluedness? That is true even for positive b. After all, 1(1/2) equals both 1 and -1. -- Macrakis ( talk) 21:35, 28 August 2014 (UTC)
I'm trying to wrap my head around complex exponents, but the description in the "Imaginary Exponents with Base e" is confusing. It states "Consider the right triangle (0, 1, 1 + ix/n)". I'm not familiar with a way of repesenting a triangle with 3 numbers, except if it's simply to state the length of each side as a Pythagorean triple (e.g. 3:4:5) or to indicate the 3 angles, as in a 30-60-90 triangle. The given values would not represent a right triangle using either of these methods. The quoted text includes a link to the "right triangle" Wikipedia article, but it sheds no light on what the three values might represent.
I think the meaning of the 3 number set should be explained or linked to or perhaps there should be 3 sets of Cartesian coordinates listed instead, as I believe this is a more familiar notation to most people.
Todd in Houston — Preceding unsigned comment added by 75.108.198.254 ( talk) 17:34, 11 October 2014 (UTC)
"the triangle is almost a circular sector"? -- 50.53.58.19 ( talk) 20:32, 11 October 2014 (UTC)
Really I was looking for clarification to "Consider the right triangle (0, 1, 1 + ix/n)".
However, I read this section again this morning and suddenly realized what the triangle represented by the 3 numbers was, probably because I'd been ruminating on the later part of the article that mentions "Using exponentiation with complex exponents may reduce problems in trigonometry to algebra." Anyway, the piece of the puzzle that was missing for me is that each number in that set is a complex number, and therefore is representative of a pair of coordinates on the complex plane. I think the statement could be changed to either "Consider the right triangle (0, 1, 1 + ix/n) on the complex plane" or "Consider the right triangle (0 + 0i, 1 + 0i, 1 + ix/n)" and that would eliminate the ambiguity.
By the way, I'm brand new to interacting with Wikipedia, but I'll take a shot at suggesting that edit myself.
Thanks! Oddacorn ( talk) 14:27, 12 October 2014 (UTC)
(A) Many textbooks contain formulas that imply 0^0 = 1.
(B) However, many of the same textbooks leave 0^0 undefined.
(B1) Some of these books say nothing about 0^0. (B2) Some of them write that 0^0 is undefined.
(C) Some textbooks define 0^0 it as 1.
Options (B1) and (B2) are common in calculus/analysis textbooks, while (C) is common in discrete math, graduate algebra, set theory.
I think textbooks that do (A)+(B) contradict themselves, but I have to admit that (I) such books are common, and (II) wikipedia has to report what common sources say, even if these sources say things that are self-contradictory.
We have to report that:
(A) there are common formulas where 0^0 must be evaluated as 1.
(B) there are sources where 0^0 is not defined.
(C) there are sources where 0^0 is defined as 1.
The 0^0 section is reasonable as long as it says (A)+(B)+(C), which at the moment, it does.
The key argument in support of (B) is that indeterminate forms should not be assigned a value. To test of this is a valid argument, we can apply it to floor(0), which is an indeterminate form because the left and right limits of floor(x), with x going to 0, do not match.
If 0^0 is undefined, then by the same logic, floor(0) should be undefined as well, a conclusion that few people would be happy with. But in the end, this argument, nor the other excellent arguments that support "1", have much bearing on wikipedia, because it must report what common sources say, and "undefined", is indeed surprisingly common. Lets make sure that each of (A)+(B)+(C) is represented properly. If so, the page itself should be uncontroversial despite the long-lasting disagreements about the merits of each argument. MvH Feb 7, 2014.