Talk:Empty product

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This article incorporates material from PlanetMath, which is licensed under the GFDL.

On the old version of Netscape I'm using at this moment, the double "=" enclosing a paragraph title make it look bigger than the article's title. (I'm going to try Internet Explorer and Mozilla and see if that continues.) Michael Hardy 03:40 Feb 12, 2003 (UTC)

I'm not sure which version of the article you're talking about: with "=" or with "==". I've never heard anybody else reporting that "==" is larger than "="; if so, that's a significant bug in the software or in your browser. Wikipedia convention is to start with "==" and go from there; "=" is only for very rare occasions. (Indeed, we only really allow it because they use on the Polish Wikipedia.)

BTW, the headers are for sections, not for paragraphs. It's a good idea to break a large paragraph into smaller ones, especially given a typical reader's short attention span these days. I also shortened the titles of the sections for readability, but if I knocked out any important text in doing so, then you could always put it back into a paragraph. Finally, I rearranged the two main sections on the grounds that people need to have some sympathy for the empty product's being one before studying a possible exception.

-- Toby 20:21 Feb 12, 2003 (UTC)

I think it's really stretching things to consider set theory to be a part of discrete mathematics, although I'm not sure the latter term has a precise definition. Often "discrete" implies that something nearby is finite, and set theory deals primarily with very extreme kinds of infinity that most mathematicians never hear of. Michael Hardy 20:47 Feb 12, 2003 (UTC)

The set theory that we're talking about here is perfectly discrete. Furthermore, set theory deals with infinities only in discrete ways -- "discrete" doesn't imply finite; it's more the other way around, actually.

Also, why don't you think there's an "of" after "raised to the power"? The grammar doesn't make sense without it.

And I agree that the link to Almost everywhere doesn't work while that article is so incomplete, focusing only on measure theory. (I don't even know if there is a measure that includes our sense or not.) But it seems rather out of the way to stress the distinction in this article, so how about just leaving it unlinked for now?

Finally, you're absolutely right about the analytic functions.

-- Toby 22:00 Feb 12, 2003 (UTC)

I thought about this "discrete math" thing some more. Our page Discrete mathematics includes set theory in its list of topics -- correctly in my opinion -- but that's not really enough. Your argument in the 0⁰ section is focused on the combinatorics of finite sets. I think that I wanted a generic term, rather than a long list of applicable fields -- but that's better at the beginning of the article than in the spot that I've changed. So let me switch them; let me know what you think about it now. -- Toby 00:00 Feb 13, 2003 (UTC)

I'm not very happy with the 0^0 discussion: I think we should start out with making it clear that always and everywhere and without exceptions, in algebra, set theory, discrete mathematics or analysis, 0^0 is defined to be 1. Formulas break down left and right if you don't. (The binomial theorem could be mentioned as another example.) Then, after we have hammered that into the readers' brains, and repeated it three times, we might mention on the side that u^v is not continuous and in certain cases both u and v approach 0 but u^v does not approach 1. This is not an argument against the definition 0^0=1; it's just a defect of the function u^v. AxelBoldt 23:49 Feb 12, 2003 (UTC)

I added a paragraph to that section that I'd been thinking about. I don't know if it'll make you perfectly happy. I don't want to simply come out and say that 0⁰ = 1, period, because (as mentioned in my paragraph) a more nuanced approach is possible. I'd even argue that many calculus textbooks are secretly adopting such an approach, although not very well. (But then, since when are calculus textbooks well written?) You can probably find old posts by me on sci.math about this; although I was younger and more naïve in those days, I still agree with my ultimate conclusion there. -- Toby 00:15 Feb 13, 2003 (UTC)

The new paragraph is fine. Can I convince you to switch the first and second paragraph of the 0^0 section? First give the numerous contexts where 0^0=1 is the only possible definition, and the reasons, and then give the single exception where one may treat it as indeterminate, and the reason. If you want to make me perfectly happy that is... AxelBoldt 02:52 Feb 13, 2003 (UTC)

Toby, something else occurred to me the other day. If you want to call 0⁰ an "indeterminate form" on account of the discontinuity, then (-8)^1/3 should also be called an indeterminate form, for the same reason. AxelBoldt 03:37 Feb 16, 2003 (UTC)

I don't see any discontinuity there. (-8)^x is not defined (as a real number) for x near 1/3, so we are (if we fix -8) talking about an isolated point, where any function is continuous. (And of course there's no discontinuity when varying the -8). OTOH, if we pick a branch of the complex natural logarithm, then exponentiation is perfectly continuous on a (complex) neighbourhood there. So whether we're considering real or complex numbers, there is no discontinuity. (Really this discussion is primarily about indeterminate forms, but we don't have an article to discuss that yet.) -- Toby 01:23 Feb 26, 2003 (UTC)

I write out "one" as three letters, when I write "one example of this phenomenon", but if I'm referring to the mathematical object that is the number 1, then I write the digit rather than the three-letter word. Does anyone have opinions about the propriety or felicitousness or usefulness or comprehensibility of this usage? Michael Hardy 03:10 Feb 16, 2003 (UTC)

Ah, I see what you're thinking. -- Toby 01:24 Feb 26, 2003 (UTC)

sounds fine to me. the rule that i follow is write the numbers in letters from one to ten, and everything else is written in numerals, except if it starts a sentence (although I'd probably break that rule if it was a large number, for some spur-of-the-moment definition of large). i agree that if you're talking math, then it makes more sense to write it in numerals. Dze27 03:24 Feb 16, 2003 (UTC)

I found myself sticking "of" back in when I remembered that Michael is certain that this word is wrong ... so I just rewrote the phrases to avoid it! I hope that this keeps everybody happy? -- Toby Bartels 02:45, 13 Feb 2004 (UTC)

I noted that the "easy formula" sinned in dividing by zero. It might be something to delete since it doesn't really apply to this page anymore after one notices. --130.39.154.50 22:34, 28 Mar 2004 (UTC)

I wrote that a certain probability distribution "concentrates probability 1 at 0". Someone changed it to "concentrates with probability 1 at 0", saying that that word was missing. That is not correct. This is not a random variable; to say it does something "with probability 1" therefore makes no sense. Rather, it is the probability distribution of a random variable. I meant that it concentrates all of the probability at 0. Michael Hardy 19:47, 1 Apr 2004 (UTC)

[edit] Eequor's allegations of fallaciousness

On August 27th, Eequor deleted what in her summary she called a "fallacious argument", the thought-experiment of the calculator that only multiplies. I am appalled, and I put it back. A number of mathematicians have edited this page without complaining about that argument. I've also used it in teaching basic combinatorics in probability courses I've taught at several universities. One of those was MIT, where some of the student are exceedingly mentally acute, and no one complained. Could those concerned venture their opinions here? Michael Hardy 23:49, 26 Sep 2004 (UTC)

The most glaring error is the idea that a calculator which can only multiply must necessarily continue to usefully function if its value is cleared. It may easily be claimed that, in fact, the calculator is functioning exactly as designed if 3 is entered after clear is pressed and 0 is the result. It is probably buggy, and will never again produce any number other than 0, but it does what you said it should do. No further conclusion can be drawn from this, anyway, because exactly two numbers are being multiplied every time, not zero numbers.

Another problem: the argument supposes that the displayed value must be identical to the value stored in memory. This may sometimes be true in real life, but it is not necessary. For example, calculators must often round fractional values so they will fit on the display, but many calculators continue to store the least significant digits. The imaginary calculator might display 0 while storing 1 internally.

Additionally, the argument is inconsistent with itself. It supposes a calculator which "can only multiply", and then adds that the calculator also has a clear function. There is also no definition of what clear might do. Because of this, the result of pressing the clear key after multiplying 21 by 4 is undefined, and so no conclusions about the future behavior can be made.

One could define the clear key to mean "remove the current number and display 1", but this leads to a circular argument, additionally neglecting to demonstrate why the result cannot be 0. Defining the key as replacing the previous value with 0 gives an apparent reductio ad absurdum, but not of a sort that makes any conclusion about the definition of the empty product (it only shows the clear key must not produce 0 if it is desired that the calculator will continue to function). The key must be defined to produce no numerical value. Call this value nil, and let the calculator display nothing at all if its value is nil.

As commonly understood, a calculator displaying nothing accepts the next-input value as its new value (entering 3 when the value is nil produces 3). Since the calculator is claimed to only multiply, and because the only value which when multiplied by a number n produces exactly n is the multiplicative identity, 1, nil must be numerically equivalent to 1.

Now, consider the behavior of the calculator immediately after pressing clear. It is blank, and will remain blank indefinitely until a number is entered, yet when a number is entered the blankness will be numerically equivalent to 1. No numbers are multiplied until enter is pressed. Therefore the empty product is equal to 1. --[[User:Eequor|η

υωρ]] 01:11, 27 Sep 2004 (UTC)

I find the arguments above largely correct if construed literally, but completely lacking in merit, for reasons I would have thought were obvious. They're written by someone who is too literal-minded. Michael Hardy 19:56, 27 Sep 2004 (UTC)

No, you're simply incorrect. You needn't be rude when you're shown to be wrong. If this is the quality of education provided by MIT, I consider myself fortunate to have not been a student there. --[[User:Eequor|η

υωρ]] 00:47, 30 Sep 2004 (UTC)

I was not shown to be wrong. A somewhat hand-waving argument was given; you showed that there were holes in it if construed literally, but it's not-quite-literal meaning should have been clear. Another user has taken some trouble to rephrase it in view of your comments. What exactly is it you're calling rude? My statement that the fact that the meaning should have been obvious? Some people consider excessive literal-mindedness rude. Michael Hardy 01:46, 30 Sep 2004 (UTC)

Ad hominem is inherently rude.

"Ronald Reagan is a Republican, therefore the argument he just gave is wrong." That's ad hominem. But I don't think it's inherently rude, although fallacious, nor that Ronald Reagan would be offended by being called a Republican. Michael Hardy 20:26, 30 Sep 2004 (UTC)

Whether a person is "too literal-minded" is a matter of opinion.

OK, your removing a very good argument is literal-mindedness. OK? Michael Hardy 20:26, 30 Sep 2004 (UTC)

As written, your previous statement implies literal-mindedness should have been obvious, not the meaning of the thought experiment.

While its general meaning should be obvious to laypeople, and most people will probably accept it without question, the experiment has been presented poorly and makes its conclusion for the wrong reasons. This is an encyclopedia, not a grade school textbook. Articles should be made readable without sacrificing accuracy. Science does not accept arguments simply because they are "good enough", and neither should Wikipedia. --[[User:Eequor|η

υωρ]] 02:13, 30 Sep 2004 (UTC)

Eequor, you are rude and gratuitously belligerent. I have been polite in addressing you and have tried consistently to reconcile, but you persist in belligerence. Michael Hardy 20:26, 30 Sep 2004 (UTC)

Errr - I guess this is, at one further level of abstraction, about designing a finite state machine of a certain rather simple kind. So, that could be made more explicit. Not going to prove anything, one way or another. Charles Matthews 09:16, 27 Sep 2004 (UTC)

Maybe it would be nice to move paragraphs about "literal mindedness" etc. to somewhere else, and retain here just the essential relevant remarks concerning the original subject:

The calculator could indeed display nothing after pushing "CLEAR" and work in all other ways as required.
All agree that if a numerical value has to be assigned to the/an empty product, it cannot be anything else than 1.
(I quote) "This is an encyclopedia, not a grade school textbook. Articles should be made readable without sacrificing accuracy." (I defend the same thesis, but it does not always appear to be common concensus...)
(quote cont'd) "Science does not accept arguments simply because" + "of the (...) exceedingly mentally acute (...) no one complained". (NB: this was gluing together 2 quotes...)

I think it is about the same discussion on whether '0' is a 'natural', i.e. 'counting' number: do we start counting at 1 or at 0 (before actually starting...)? I remember discussion on this somewhere else, but could not find it now. I don't think there can be a categorical "one and only true" answer. It's a convention about terminology, which does not matter as long as everybody agrees on the mathematical facts. The worst that can happen is some possible confusion about the meaning of words, which can in turn cause errors for those who cite/use theorems from others without checking the author's conventions on what the words do mean. The best to avoid this is to take care to mention explicitely known ambiguities everywhere they may be relevant (is O in N? does "ring" imply "unital"?, ...) — MFH: Talk 16:17, 20 May 2005 (UTC)

[edit] back to the conceptual rationale

Whether or not zero is a natural number is a matter of convention. It could be either way. The empty product must be 1.

Here is another justification, based on computer programming. Suppose you have an array of numbers x(1) ... x(10) and you want to get the sum of them and the product of them. You would do it this way (pseudocode, not WikiCode):

sum := 0

prod := 1

for i := 1 to 10 do

sum := sum + x(i)

prod := prod * x(i)

The empty sum is 0. The empty product is 1. Someone may want to write this up for the article. I don't want to because I don't want to take any heat. Bubba73 02:44, 21 Jun 2005 (UTC)

It's already in the article. It's the section titled A conceptual rationale. Michael Hardy 22:51, 21 Jun 2005 (UTC)

The following program woud ~~be indeniably better~~ solve the given problem more efficiently and with less risk of ambiguities:

sum := x(1)

prod := x(1)

for i := 2 to 10 do

sum := sum + x(i)

prod := prod * x(i)

I maintain that this philosophical "problem" has its root in the question of counting: Do you start with the first member to count, or before counting the first member. — MFH: Talk 23:32, 21 Jun 2005 (UTC)

PS: indeed, depending on what are the objects to sum and multiply, O and 1 are not the same. E.g. a computer algebra program could give you an error if the x(i) were matrices and it would not allow the scalar 0 (resp 1) to be added to (resp multiplied by) a matrix. But I agree on the following:

The empty sum (resp. product) of objects in X should be the neutral element for addition (resp multiplication) in X.

Remains to know what is the neutral element for the (cartesian) product of sets. — MFH: Talk

ACtually, it is known what the neutral element for product of sets is, it's any terminal object (singleton). Because for any family of sets X, the product of X is isomorphic to the product of X union a singleton. Revolver 29 June 2005 16:57 (UTC)

Who ever said there's either a problem or a matter of philosophy? Your quotation marks around "problem" make it look as if someone other than you saw a problem here, or otherwise used that word. Michael Hardy 23:50, 21 Jun 2005 (UTC)

PS: "Indeniably"? Did you mean "undeniably"? Michael Hardy 23:50, 21 Jun 2005 (UTC)

Sorry, but you are wrong (not on "undeniably", though): The specified task is to add and multiply items x(i) starting with x(1). And, as I said, the other program is not correct since 0 and 1 are not necessarily of the same type than the x(i).

Indeed, I put "problem" in quotes, since it is not a true problem, just a matter of convention. Not even a convention about the contents (where all agree upon), but just on how to speak about... just a naming convention, in some sense. — MFH: Talk 00:00, 22 Jun 2005 (UTC)

"The following program woud be indeniably better:" I don't think so because what if you want to take the product of zero terms (which is what the topic is)? I should have put variables on the limits of the loop, then it gives the correct result when there are zero terms in the product. The one that is "indeniably better" gives the wrong result in the case of the empty product (unless x(1) happens to be 1). Bubba73 00:04, 22 Jun 2005 (UTC)

Please read above "the specified task is..." (or your "Suppose you have..."). And: The product of zero terms, is it an integer, a float, a matrix, or what? — MFH: Talk 00:09, 22 Jun 2005 (UTC)

An integer or a real number. (And if it is a matrix, then it has to be initialized to the identity matrix, and no other. It (the empty product) has to be the multiplicitive identity element in any case.) Since the topic of the article is "empty product", we want a routine that will return the correct value for that case. So

prod := 1

for i := y to z do

prod := prod * x(i)

where y and z can index any elements of the set. If z < y then it is the empty product, and this returns the correct result and the other one doesn't. Bubba73 01:37, 22 Jun 2005 (UTC)

Of course the empty product of matrices has to be the identity matrix of adequate rank. (I now put in boldface the universial concensus I cited just above.) In fact the type of the variables prod and sum should be declared initially (to be the same as the type of the x()'s), and the programming language should be intelligent enough to cast 0 resp. 1 into this type (or to interpret it as the corresponding neutral element). It is clear that in general if the x(i) are not scalars, they cannot be added to or multiplied with 0 resp. 1 if this has to be an integer or a real.

E.g. in the computer algebra system MuPAD you could initialize sum := 0*x(1) and prod := x(1)^0 to be the adequate neutral elements (this will e.g. return the 3x3 unit matrix if x(1) is such a matrix).

However, if you have no x(1), i.e. x is an empty list, then we cannot know (without type declaration) what the sum or the product of this list should be. It's just like the empty intersection: to know the value of the intersection of an empty family of subsets, you must specify where the (inexisting) factors of the intersection belong to - this universal superset will then be the value of the intersection. — MFH: Talk 17:20, 22 Jun 2005 (UTC)

PS: Anecdotically, I'm not sure if there are no languages (e.g. some BASIC dialects) in which "FOR i := 1 TO 0 DO..." would not execute the loop one first time with i=1 (and maybe some even more "intelligent" languages, a second time with i=0 - note b.t.w. that the common definition $\int_1^0f(x)dx=-\int_0^1f(x)dx$ suggests to put $\sum_{i=1}^0a_i =-a_1-a_0$ and not $\sum_{i=1}^0a_i = 0$ ). In fact the 'FOR' directive is somehow imprecise on that. The C-style "for( i=y; i<z; i++)" or a "i := y; WHILE (i < z) DO ...." construct are a bit better, but there is always some "temptation" to start out doing something for i=y, which is then (maybe) frustrated. Maybe the best would be Maple-style syntax, "FOR i in indices DO ...", where indices is the list or set of indices to sum over.

(PPS: ...R¹⁺⁰... : moved to section #associativity_of_Cartesian_product.)

Maybe, to avoid the discussion drifting away from the subject, it would be good to number the different given statements and to refer to the precise statement whenever someone wants to express he disagrees about something. (I tried to do this in my post just above the new subsection header, but it was somehow ignored and I did not have the idea of the more concrete numbering.) So let's go: (I intentionally number "explicitely" and not using the wiki #, for known reasons.)

(1) An empty product is equal to 1 (number).

(2) In (1), '1' stands for an abstract or concrete multiplicative identity.

(3) The calculator could display nothing (i.e. show an empty display) after pressing CLEAR and serve exactly the same practical use.

(4) Via the logarithm, the empty product is related to the empty sum,

under the assumption that the logarithm can be defined for the objects under consideration.

(5) Via (4), the empty product is equal to exp( 0 ), where 0 is the neutral element for addition.

(6) Using the common (power series) definition of the exponential, this implies and/or requires that 0⁰ = 1. (see also (2) and (4)).

(7) In their most elementary meaning, addition and multiplication are associative binary operators, i.e. semigroups.

(8) (Without associativity, it is not possible to define the product of more than 2 elements, neither a power greater than 2 of an element, without further specification.)

(9) In a semigroup, the product of an ordered, finite list of k=2 or more numbers is well defined in a natural way. If all elements of the list are equal, this is the k-th power of the element.

(10) It is a useful extension to (9) to generalize a product (and the power) of elements to the case k=1, by defining it to be equal to the given element.

(11) This (10) is a new definition, extending (and maybe logically but not formally implied by) the original definition of the binary operation and the previous notion of a product and a power.

(12) Addition and multiplication allow for an identity element, i.e. we have the structure of a monoid. In a monoid, it is useful to define the zeroeth power of any element, and also the product of an empty list of elements, to be equal to the (unique) identity element.

(13) These two definitions, the zeroeth power and the empty product, are related.

(14) Both of these definitions are "new" definitions, (maybe logically but) not formally implied by the original definition of the binary operation.

(15) Both of these definitions are the only reasonably possible choice for the considered expression.

(16) Both of these definitions are logically implied if one wants to extend formulas for the product or for powers to include empty products or zero powers, and also if one wishes "cancelling" formulas and "adding exponents" formulas to hold in presence of invertible elements.

(17) Also by definition and for the same reasons (14-15), the inverse of an invertible element and its powers are written as negative powers.

(18) This (17) is a new definition, extending (and maybe logically but not formally implied by) the previous definitions.

(19) The definition (17) further justifies a posteriori the definition of x⁰=1 for invertible elements x.

Feel free to add any new propositions, and to discuss the validity of the above, if possible without changing them in-place. — MFH: Talk 16:21, 23 Jun 2005 (UTC)

Wow! A thought experment that evokes such debate. Hehe! ok. It is NOT nessesary to have a clear function. You just use the property of inversion to return to the identity element. (1). Since One is the identity element, any finite number of identity elements can be multiplied. If you enter '0' into the calculator, it becomes only usefull for 0*n=0, otherwise it will retain its wide range of calculating powers.

I would modify (1) to read "Empty products are equal to the identity element (1)" which enhances its operational definition. Products are arithmatic.

(6) must except 0^0, as indeterminiate.

(20) Negitive powers are indeterminate for bases equal to either 0 or Inf. Artoftransformation 13:48, 25 December 2005 (UTC)--

[edit] Conceptual issues

I hate to jump in on this. If Michael actually thinks his above behavior is polite, that Eequor is the one being rude, and that he has actually made attempts to reconcile, then there is probably no compromise that can be offered, at least in his lights. Yet I think that the conceptual section needs serious modification; I find it unconvincing as written. If one has written something that many people do not understand, one is not entitled to presume that one has written clearly and everyone else is just dense. So perhaps someone else can clarify these issues.

First, the example of a cancelled fraction is curious. Numbers when cancelled do not simply disappear; they divide to 1. Thus the numerator of the fraction in the article is certainly not (NULL * NULL) = 1 or (NULL) = 1, but 1 * 1 = 1. Nothing strange here.

Second, the calculator example is unconvincing for all of the reasons described above. The example works *only if the clear button returns 1*. I think the article is attempting to say that the calculator *should* be programmed to work that way because it wouldn't work otherwise. That's fine, though it could be stated more explicitly. I have two problems with this: (1) I have no idea where the multiplication of the empty set has gone on. If the clear button returns 1, then where is the empty set? (2) Why does the behavior of the calculator generalize to anything else?

I suggest that both of these examples be deleted.

(1) I have no idea where the multiplication of the empty set has gone on. If the clear button returns 1, then where is the empty set?

Seems obvious to me: the empty set is the set of factors that have been entered after the "clear" button was pushed. Or one can say: zero is the number of factors that were entered after "clear" was pushed. So the product of that many factors is 1.

(2) Why does the behavior of the calculator generalize to anything else?

To multiply by no factors at all is to multiply by 1. Michael Hardy 00:19, 25 March 2006 (UTC)

Anonymous user: Since you have a user name, please don't leave these things unsigned.

Your example of cancelling with fractions misses the point. The fact that there's more than one way to look at something doesn't mean what is being looked at is something different.

Why do you speak of many people not understanding? And who are those people? If a non-mathematician does not understand an article about the Riemann hypothesis, does that mean it's badly written? Michael Hardy 00:29, 25 March 2006 (UTC)

I apologize for failing to sign my comments--I am fairly new here and simply forgot. I think the fraction example has been clarified and is much more intuitive now.

I think that a good conceptual example of the empty product needs to meet two criteria: (1) it should provide a clear, reasonably common instance of a situation in which the empty product arises; and (2) it should show that in that circumstance, it is useful for the empty product to be defined as 1. I think the original example did not meet criterion 1. Here's why: As far as I know, everybody learns that when you cancel, you cancel to 1. This, of course, is because 2/2=1 and 3/3=1. The original example seemed to suggest that it was equally valid to cancel to NULL (which implied in turn that 2/2=NULL=1). My immediate reaction was "This has nothing to do with the empty product and everything to do with simple arithmetic."

By adding the description of deletion, and by describing it as an *alternative* to cancellation in this instance, the article is less likely to lead people astray.

On a general note, I don't think that all parts of all math articles need to be understandable to the population as a whole. (That would probably involve deleting vast amounts of material....) I do think, though, that when an article involves material that people come across in basic math (such as 0^0 or fractions), that material *should* be clear to an everyday audience. Moreover, when a conceptual example is presented, it should be as clear as possible. This may include spelling out steps that are intuitive and painfully obvious to a mathematician (or to undergrads who are in the midst of a math course at MIT) but not obvious at all to the average Wikipedia reader.

--Trilateral chairman 04:00, 26 March 2006 (UTC)

[edit] technical justification

I suggest to simplify the section Empty product#A_more_technical_justification and remove everything unnecessarily general (log w.r.t. arbitrary base,....) in order to focus on the principal idea. (One could also just take the log of the product.) — MFH: Talk 16:28, 23 Jun 2005 (UTC)

I suggest deletion of that section because it only confuses a simple concept and only applies in limited circumstances. The properties of logarithms and exponentials are much more advanced than the concept of an empty product, and one usually tries to explain complex concepts in terms of simpler ones, not the other way around. --McKay 05:51, 14 December 2006 (UTC)

No objection being raised, I'm deleting it. McKay 03:20, 18 February 2007 (UTC)

[edit] 0⁰ is indeterminate?

I once read somewhere that 0⁰ is indeterminate, because 0⁰ = 0^1-1 = 0¹/0¹ = 0/0 which is indeterminate. And the proof that n⁰ = 1 to which I'm most familiar is n⁰ = n^1-1 = n¹/n¹ = n/n = 1 which of course doesn't apply if n = 0. --Army1987 11:26, 6 Nov 2004 (UTC)

The first argument is no good; the second one explains something. Charles Matthews 11:30, 6 Nov 2004 (UTC)

0^0 is indeterminate, but your first argument fails because the division of powers by subraction only holds for numbers NOT zero. Please, Please, Please read [[1]]. Having looked at this for more than 30 years, and have come apon an extrodinary explanation, I would refer all questions to there.

The sense in which 0⁰ is indeterminate is that if f(x) and g(x) both approach 0 as x approaches something then f(x)^g(x) may approach any positive number, depending on what functions f and g are. But for many purposes, including both formal power series and convergent power series, and many of the purposes of combinatorics and probability, one should take 0⁰ to be 1. Michael Hardy 19:16, 6 Nov 2004 (UTC)

.... However, we should figure out what the best way is to address this point in the article. To be continued ... Michael Hardy 19:18, 6 Nov 2004 (UTC)

I think we should be bold (or: honest) enough to admit explicitely that this is a ~~pure~~ matter of convention. We can then motivate it by the numerous cases where it is "the only good thing to do", and maybe should once again be honest enough to list afterwards some of the extremely rare cases where it is not convenient, and where it could be justified (by convenience) to use another convention, say 0^0 = 0 (I can't think of any other.)

(In some sense, like 0×∞=0 in measure theory, but not in 1st year calculus on limits.) — MFH: Talk 18:40, 12 May 2005 (UTC)

But it is not just a matter of convention. I think the article makes that clear. But maybe we should also add examples of combinatorial identities and other formulas that rely on this fact, in order to solidify this point further. Michael Hardy 00:49, 17 May 2005 (UTC)

OK, I delete "pure", but it's a matter of convention, and it can be useful in some situations to put "0⁰=0"; e.g. I had to consider spaces $N(r)=\{ x\in\mathbb C^{\mathbb N}\mid\lim|x_n|^{r_n}=0\}$ where r are sequences decreasing to zero, and in this context it was useful to take the (exceptional) convention "0⁰=0". Of course, I agree with the standard definition, but it's nevertheless somehow like the cited convention in measure theory. — MFH: Talk 14:36, 19 May 2005 (UTC)

In this discussion is worth noting that if y(x) is defined recursively as y=x^y then lim_x->0y(x) -> {0,1}. If an approximation of this limit is taken at an odd iteration it equals 0, and 1 for even iterations. This function is quite cool on its own merits, for example, it ceases to be multiply-valued on the interval [e^-e,1].

[edit] Thank you

Great article, Michael Hardy! I've always wondered why a number raised to the power of zero equals one, and my math teachers don't seem to know/get annoyed when I ask questions that I'm not required to ask to complete the homework assignment. By the way, what does the Wikipedia stress meter mean?

[edit] Cancelling numbers

About the cancelling numbers example: If both the numerator and denominator are divided by 6 there exists 1 in the numerator. I don't see a need for an empty set there. I'm definitely not trying to argue with the authors of this article but only telling the impression I got.

[edit] empty product of sets

I came to this page following the "see also" link on cartesian product, but I did not find much info on that on this page, more precisely, only the 1st phrase of the last section of the introduction:

More generally, given an operation of multiplication on some collection of objects, the empty product is the result of multiplying no objects together. It is generally defined to be the identity element with respect to the given operation, if such exists.

And I can't see what is the identity element for the cartesian product. Except if we're willing to think "modulo isomorphism", and then it is any singleton. E.g., { 0 }, but also, equivalently, { 12 + 5 X + 2005 X² } or anything else.

Now, this makes sense if we think of Rⁿ as n-dimensional vector space, but not if we see Rⁿ as mapping of n (= {0,...,n-1}, 0={}, see natural number) into R.

At least, until a mapping of empty domain could be identified with a singleton... which, admittedly, would be a possibility (would be somehow coherent with identification of a nullary function with a constant (but, in fact, any constant)) but for some reason, I would have identified such a function rather with the empty set, in view of the cardinality of its graph. (This issue should be discussed in function.)

So: R^{} = {0} or R^{} = {} ? — MFH: Talk 18:27, 12 May 2005 (UTC)

(Remark added) - in view of {0}=1 (see natural number), maybe R^{} = {0} = 1 would indeed be a good convention (empty product equal to 1) - "How neatless it all fits together", in Snoopy's words. — MFH: Talk 14:01, 20 May 2005 (UTC)

Well, in category theory, it is most definitely not just a convention that the product of the empty family in Set is (up to isomorphism) any terminal object. This is a well-known fact, and can be verified by closely following the precise definition of categorical product in this case. Revolver 29 June 2005 17:23 (UTC)

[edit] associativity of Cartesian product

In the section "Complex numbers", it is written

although the associativity of Cartesian product is nowhere stated.

I think associativity isn't really used here (and there are other flaws in this paragraph, as the logic implies that the imaginary line {0}×R would also be "equal" to R), but on the contrary, by definition the Cartesian product (of two sets, i.e. seen as binary operator) is the set of (ordered) couples (a,b), defined e.g. by (a,b)={a,{a,b}} and it is immediate to check that (A×B)×C is NOT equal to A×(B×C); even if A=B=C, they are completely different sets (and both are sets of 2-tuples). — MFH: Talk 18:58, 12 May 2005 (UTC)

No, Cartesian product is not strictly associative, but it is associative up to bijection (i.e. up to isomorphism in the category of sets), and this is what people mean when they say "cartesian product is associative". In fact, taking the skeleton of Set, this is exactly why multiplication of cardinal numbers is associative. Revolver 29 June 2005 17:12 (UTC)

I now noticed that this passus is "stolen" from PlanetMath. I object to it, because using exactly the same argument, you can deduce

$\mathbb R =\mathbb R^1 =\mathbb R^{0+1} =\mathbb R^0 \times\mathbb R^1 =\{0\}\times \mathbb R = i\,\mathbb R$

with the identification (x,y) = x + i y used in the original. — MFH: Talk 17:31, 22 Jun 2005 (UTC)

I've removed this problematic section, because it doesn't seem to be about empty products. Paul August ☎ 18:09, Jun 22, 2005 (UTC)

Well done. — MFH: Talk 14:41, 23 Jun 2005 (UTC)

[edit] Why I reverted the deletion of material by User:WAS 4.250

This user wrote:

this is nonsense if "1" is displayed typing in "5" results in"15"

The behavior of the hypothetical calculator in this thought experiment was described: If a number appears in the display and another number is entered, then the product of the two numbers appears in the display. That means if "2" is displayed, and you enter "5", what will appear in the display is 10.

How actual calculuators behave is not relevant.

But you are wrong about how actual calculators behave as well: If a normal calculator returns "2" as the answer to a problem, and you press the key that says "5", you will not see "25"; you will just see "5". That's also how the calculator in the thought experiment behaves; it is only after the "ENTER" key is pressed that you see the number that results from multiplying.

You CLEARLY have not carefully read the material that you removed. Michael Hardy 01:56, 27 March 2006 (UTC)

[edit] content in question

Conceptual justification

Imagine a calculator that can only multiply. It has an "ENTER" key and a "CLEAR" key. One would wish that, for example, if one presses "CLEAR", 7, 3, 4, then the display reads 84, because 7 × 3 × 4 = 84. More precisely, we specify:

A number is displayed just after pressing "CLEAR";
When a number is displayed and one enters another number, the product is displayed;
Pressing "CLEAR" and entering a number results in the display of that number.

Then the starting value after pressing "CLEAR" has to be 1. After one has pressed "clear" and done nothing else, the number of factors one has entered is zero. Therefore it makes sense to define the product zero numbers as 1.

[edit] Comments

If the point of this section is "Conceptual justification" then it should be deleted because one can program a computer or calculator to display anything whatsoever and the choice of displaying zero or one or blank or anything else depends on (assuming it meets user needs) prior convictions of the user, not limitations in what the computer programmer can program the device to do. If the user is, as the author, convinced that a one must be displayed for the following entry of seven to be seven and not zero then to sell it to him the programmer will dutifully have it display one. On the other hand, if the buyer insists it should initially display zero and entering seven will result in seven being displayed then that is what will happen. Oh, but it can only multiply and that's not multiplying. Well, "clear" doesn't multiply, so clearly it does some things other than multiply. And what is the enter key for if every time a number is pressed, that number is multiplied by the existing displayed value? You could make some kind of sense of this by deleting mention of the "enter" key, but you are still left with describing something that is in no way a calculator and only illustrates that a programmer could satisfy your mathematical beliefs by showing a "one" when you hit a clear key (or when you turn it on I presume). This subsection justifies nothing and is an insult to any programmer who ever programmed a human interface. Just say seven times one is seven and seven times zero is zero. When you bring a calculator into it questions of "state=off", "display=blank", and "enter key does what" enter the picture and distract rather than than justify. It is no justification at all, even if I have not convinced you of that. WAS 4.250 03:46, 27 March 2006 (UTC)

The comments above are irrelevant. This was a thought-experiment. It was obviously not intended to be taken literally as being about some actual electronic device. Michael Hardy 22:09, 27 March 2006 (UTC)

The change to the article you made is such that: If that is what I had seen when I first read the article, I would have said nothing. So I say nothing further here. WAS 4.250 01:18, 28 March 2006 (UTC)

I vote delete. Imho, this does not justify much: So many complicated assumptions etc. are necessary to make it work, and all it says is that if x*a=a for any a, then x=1. — MFH:Talk 18:20, 29 March 2006 (UTC)

I don't think the assumptions are complicated, nor are the many assumptions. It's a really simple thought experiment. Do you object to phrasing something like this in a way that can be understood by those not comfortable with algebra? (And BTW, please note the following differences in appearance in mathematical notation:

if x*a=a for any a, then x=1.

if x × a = a for any a, then x = 1.

We're not limited to plain ASCII, even when not using TeX.) Michael Hardy 22:32, 29 March 2006 (UTC)

I'm used from lessons on group theory to denote by * an "arbitrary" multiplication, and still advocate for the fact that the "empty product" is not only about real numbers ;-) — MFH:Talk 19:01, 21 October 2006 (UTC)

Agreed. The rule that the empty product is the multiplicative identity applies outside the real numbers. For instance, the empty product of size-n matrices is the identity matrix of size n. Of course there is no single answer to the question "What do you get if you multiply no matrices together?" since you need the size. --FOo 20:19, 21 October 2006 (UTC)

[edit] Convention or fact

I only recently learned that some respectable mathematicians regard things like this a conventions rather than facts. I was shocked. They are wrong. I think they simply have never given the issue a moment's thought. I'm a bit rushed now, but I'll post on this further later. We've had this argument a number of times on this page over the last couple of years. For now I'll just say: see the footnote on this in my paper "Combinatorics of Partial Derivatives" in the 2006 volume of the Electronic Journal of Combinatorics. More later...... Michael Hardy 21:50, 4 April 2006 (UTC)

[edit] conceptual rationale etc.

I still stumble on 2 paragraphs of this article:

the conceptual justification. Considering the hypotheseses, it is clear that what it demonstrates is precisely the following statement:

If the empty product is a (real) number, then it must be the number 1.

which is equivalent to its contraposition

The empty product cannot be a number different from 1.

I think there is no doubt about this. No author would define the empty product to be some number different from 1. Some authors decide to not define it at all. This is a priori excluded by the axiom "After pushing CLEAR, the calculator must display a number." It does not not mean that the calculator would not work exactly as expected, if after pushing CLEAR, the display would be empty.

the technical justification, with (product) = exp( sum (logarithms)). I think it's unnecessarily complicated, and ill-defined (since the logarithm is an unambiguous function only on strictly positive reals). Furthermore, much emphasis is put on "e" and the "natural logarithm", while one could exactly as well take 2 or 10 instead of e.

why is the following simple justification missing: aⁿ is the product of n factors of a; for n=0n we get a product of zero factors (i.e. an empty product), and it is equal to 1, without any doubt for any nonzero number a, but in fact the result does not depend at all on a (it is an empty product: there are no factors of a in it).

also it should be emphasized more that an empty product makes sense not only for real or complex numbers, but in particular in any monoid (leaving aside other more exotic products like the cartesian product of sets); and in that case the "result" (value of the empty product) is (must be) the identity element which may be not the number 1, but e.g. the identity matrix, the identity function, an "abstract" identity element e. — MFH:Talk 18:57, 21 October 2006 (UTC)

Yeah, there's some cleanup needed. The problem with this sort of article is that everyone rushes to add their own justification for what may to some people seem a counterintuitive fact. Thanks for pointing out that we were missing the empty Cartesian product; I've added it in now. EdC 22:18, 21 October 2006 (UTC)

[edit] zero to the zero

I know this is temerarious of me, because this is one of those issues that can result in endless unproductive discussions and is usually better just left alone. However I don't think the current 0⁰ discussion is acceptable as it stands, especially with the assertion about the "consistent point of view". Another "consistent point of view" is that there is a distinction between the natural number 0 and the real number 0.0, and that while 0⁰ is clearly 1 in natural-number exponentiation, 0.0^0.0 is naturally left undefined in real-number exponentiation (though 0.0⁰ is 1.0, which takes care of the binomial theorem problem). --Trovatore 22:56, 10 December 2006 (UTC)

Could you be more specific (i.e. quote) about which text you are objecting to? JRSpriggs 06:58, 11 December 2006 (UTC)

The identity 0=0.0 implies that 0⁰=0.0^0.0. The empty product a⁰ equals 1, no matter which factor a the empty product does not contain. Bo Jacoby 13:17, 11 December 2006 (UTC)

But it is not in fact clear that 0 (the natural number) and 0.0 (the real number) are identical. They are of equal magnitude, of course. What is not clear is that they are the same kind of thing. --Trovatore 17:26, 11 December 2006 (UTC)

So you say there is a difference between the image of the integers under the group homomorphism that maps (in your terminology) 1 to 1.0 and the integers? Kusma (討論) 17:31, 11 December 2006 (UTC)

I'm saying that's one reasonable way of looking at it, yes. I tend to see both the naturals and the reals as sui generis (or more precisely, I see the ordinals as sui generis, and the naturals as finite ordinals, not as a special set of reals). --Trovatore 17:33, 11 December 2006 (UTC)

The reals as sui generis? Wow. I see them as the topological closure of the set of computable numbers. (That is, they're a fiction invented to spare the blushes caused by Cantor's diagonal construction.)

As such, I'd find making 0.0^0.0 undefined delightfully retrograde. In every step of the construction of numbers, from the naturals on, we postulate a conservative extension that gives solutions to previously unsolvable formulae. You're proposing the exact opposite step.

Out of curiosity, where do you see the fateful undefinability creeping in? Is (1-1)^1-1 undefined? Is (½-½)^(½-½)? How about (π-π)^π-π?

<strikeout>As an aside, setting 0.0^0.0 = 1 gives 0^x as a nicely succinct Dirac delta function.</strikeout> EdC 19:17, 11 December 2006 (UTC)

On this machine I can't read the π symbol, so I'm not sure what you're asking. But basically I'd say the integers and rationals are fundamentally algebraic, whereas the reals are fundamentally geometric/topological. Operations on the reals are naturally defined by continuity, not by algebra (even 0.0+0.0 does not inherit its value directly from 0+0; rather, 0.0+0.0 is the equivalence class of a sequence of rationals obtained by taking two Cauchy sequences converging to zero, and adding them pointwise). --Trovatore 20:02, 11 December 2006 (UTC)

π is pi, which is non-algebraic (i.e. transcendental) but computable. Hmm... that said, exponentiation isn't well defined in the computables around 0⁰. That doesn't really matter, though. Aside from that, Cauchy sequences aren't to be identified with the reals; they're a way of working with the reals. What's the Cauchy-sequence definition of exponentiation?

Note that exponentiation on the rationals, x^y, is well-defined for half the plane and a bunch of lines (y integer) less the open ray x=0, y<0. Thus the computable definition of exponentiation is well-defined within that plane or when the exponential is known (by knowing x to be zero and y to be nonnegative, or y to be integral) to be defined. EdC 17:56, 12 December 2006 (UTC)

As to "Cauchy sequences aren't to be identified with reals" -- right, didn't I say the reals are sui generis? If they were identified with Cauchy sequences, they wouldn't be. But Cauchy sequences are the most natural of the set-theoretic representations I know. The Cauchy-sequence definition of exponentiation would be slightly complicated (because a rational to a rational already needs to be taken as a Cauchy sequence, or something like it). As I say below, the cleanest, but not most satisfying, definition of exponentiation on the reals is a^x = e^{x log a}, for a>0 (not defined for a≤0).

With respect to the computable reals -- to be honest I don't find the computable reals to be particularly interesting or enlightening as reals. They're really a stand-in for computable elements of Baire space or Cantor space, which are much cleaner to work with. --Trovatore 07:21, 13 December 2006 (UTC)

(responded below) --EdC 19:32, 13 December 2006 (UTC)

The set of integers is identified with a subset of the set of reals, and the integer 0 is identified with the real 0.0. If that identification is not made, we are bound for trouble. We cannot accept if a formula has one meaning for reals and another meaning for integers. The binomial expansion (1+0)¹=1¹·0⁰ + 1⁰·0¹ must be true for integer as well as for real computation. Unless 0⁰ = 1 we are in deep trouble. Bo Jacoby 18:00, 11 December 2006 (UTC)

Bo, how much do you know about the set-theoretic representations of things like ordinals, naturals, integers, rationals, reals, as developed by people like Zermelo and von Neumann? By these codings, the natural number 0 is simply the empty set, whereas the real number 0.0 is something much more complicated (exactly what it is depends on whether you go with Dedekind cuts or Cauchy sequences -- personally, I consider the latter a bit more natural, but the point can be made with Dedekind cuts as well). The real number 0.0 is a certain equivalence class of Cauchy sequences of rational numbers.

Now, I'm not ascribing any deep significance directly to that. It's a coding, after all. We have considerable freedom to code things as we find convenient. But, if you've done any software development, you'll have noticed this curious fact: If we have an option of two ways of representing data, and one of them works smoothly, and the other requires lots of hacks and special cases, it usually means that the one that works smoothly is the one that corresponds conceptually to the way we should have been thinking of the problem all along. Not an infallible dogma, but a very reliable rule of thumb.

Insisting on the identity of 0 and 0.0 works out to be a hack. It requires an awkward special case in our treatment of the reals. --Trovatore 20:35, 11 December 2006 (UTC)

I think I know sufficient of that. No matter how the field of reals is constructed it contains a subset isomorphic to the ring of integers, so that the real 0 maps to the integer 0. (Computer programmers must distinguish between real 0 and integer 0 having different computer representations, and that is a source of error. Mathematicians do not distinguish). There is no advantage at all in not defining 0⁰ = 1, so why create this unnecessary trouble? Bo Jacoby 23:10, 11 December 2006 (UTC)

Yes, of course it has a copy of the integers (or, more to the point, of the natural numbers). The question is whether we should consider that copy to be the natural numbers. I say we should not. There is a fundamental conceptual difference between natural numbers and "natural number reals", and while it rarely makes a difference, it occasionally does (witness the current discussion).

The natural number zero counts, whereas the real zero measures.

Zaphod spun around, wild-eyed.

"Ford," he said, "how many escape capsules are there?"

"None," said Ford.

Zaphod gibbered.

"Did you count them?" he yelled.

"Twice," said Ford.

--D. Adams, The Restaurant at the End of the Universe —The preceding unsigned comment was added by Trovatore (talk • contribs) 23:24, 11 December 2006 (UTC).

Counting is a type of measurement; it's the measurement of the size of finite sets. Also, measurement is invariably operationally defined in terms of counting, in a way which is very similar to the characterisation of the computables; you specify a precision you want a measurement to, and I select a suitable measuring stick and count how many times the stick (and/or fractions thereof) needs to be laid down. EdC 19:32, 13 December 2006 (UTC)

Conceptual differences doesn't count in mathematics. Isomorphic structures are considered identical. This formalistic point of view has been accepted since the times of Hilbert. Bo Jacoby 23:46, 11 December 2006 (UTC)

Conceptual difference do indeed count in mathematics. The idea that isomorphic structures serve equally well is not formalism, but rather structuralism, and you should learn the difference before pontificating on these things. I am a structuralist myself, but I am not saying here that the copy of the naturals contained in the reals would not serve equally well -- the point is a different one, which I may not have expressed very well: The operations are defined differently. Even addition -- see my remarks to EdC above on 0.0+0.0 . It turns out that we get an isomorphic embedding from the naturals into the reals for the operations of addition and multiplication, but arguably not for the operation of exponentiation, because the justifications for assigning a value of 1 to 0⁰, convincing in the naturals, are no longer convincing in the reals. --Trovatore 00:09, 12 December 2006 (UTC)

Yeah, actually I muddled this a bit. The direct way of expressing it is to say that the natural copy of the naturals within the reals is an isomorphic copy when only addition and multiplication are considered, but ceases to be an isomorphic copy when exponentiation between elements is added. Note that Bo's remarks talked about a subset of "the ring of integers". Rings do not have exponentiation. (That is, not between their elements -- of course you can naturally define exponentiation with a ring element as the base, and a natural number, ~~or even an integer~~, as the exponent.) --Trovatore 06:47, 12 December 2006 (UTC) "Integer" would work over a field; for rings, we'd better stick to "natural number" --Trovatore 06:49, 12 December 2006 (UTC)

The convincing justification for 0⁰=1 is simply that 0 and 1 are integers or naturals, even if they are reals too. Bo Jacoby 09:09, 12 December 2006 (UTC)

Let's keep the notation clear, shall we? If we're arguing about whether the real number zero is the same as the natural number zero, then for you to use the notation "0" to talk about the real number is begging the question.

Rephrasing your claim, it would be:

The convincing justification for 0.0^0.0=1.0 is simply that 0.0 and 1.0 are integers or naturals, even if they are reals too.

But that is not a convincing reason, and perhaps not even a true reason. Oh, to be sure, in most contexts if you ask if 1.0 is an integer, I'd say "yes". Because I assume that what you're really asking is whether it's an integer real, which is often a useful thing to know.

However, even if we identify the naturals with their images in the reals, it still does not get you where you need to be, because we could still have different notions of "real exponentiation" and "exponentiation with natural number exponent", and the justification for 0⁰=1 is convincing only for the latter.

This is not at all implausible or unique. Consider that we ordinarily identify the cardinal $\aleph_0$ with the ordinal

ω

. In the standard coding they are the very same object. And yet, $\aleph_0^{\aleph_0}$ is the cardinality of the continuum, whereas

ω ω

is a rather small countable ordinal. The difference is not in the objects, but in the kind of exponentiation that is understood (cardinal exponentiation in the former case, and ordinal exponentiation in the latter). --Trovatore 16:37, 12 December 2006 (UTC)

Just for the record: I can't agree with that. Aleph-null is defined by $\aleph_0 = |\{0, 1, 2, ...\}|$ , whereas the first infinite ordinal is defined by

0,1,2,...,ω

. They're not the same object under any coding I'd recognise. EdC 08:21, 14 December 2006 (UTC)

If I may say so, the exponential function presents problems which are not limited to 0⁰. For example, in the real numbers we would want $(-1)^{\frac{1}{3}}\!$ to be -1. But in the complex numbers we want it to be $\frac{1}{2}+ i \frac{\sqrt{3}}{2}\!$ . JRSpriggs 10:45, 12 December 2006 (UTC)

Yes. a^b is multivalued unless b is an integer (where a^b is singlevalued) or unless a=e (where a conventional choice for e^x is made). See Exponentiation#Multivalued_power. The escape from multivaluedness is to stick to a=e whenever b is not an integer. The current 0⁰ discussion is not actually a problem, but the discontinuity of a^b around a=0, b=0 creates confusion. Bo Jacoby 12:47, 12 December 2006 (UTC)

It seems to me that the central issue here is not the distinction between 0 and 0.0 but rather between the operations of natural number exponentiation and real exponentiation. That is, we have two functions natexp(x,y) and realexp(x,y) where the former is defined only when y is a natural number. It makes sense to set natexp(0,0) = 1 but not to set realexp(0,0) = 1. (I'm purposely not making a distinction between the natural numbers and their images in the reals here.) -- Fropuff 17:21, 12 December 2006 (UTC)

That seems to be the issue, yes. But as realexp(x,y) will be discontinuous at (0,0) no matter whether realexp(0,0) is defined, we can just as well decide to let realexp be an extension of natexp and therefore set realexp(0,0)=1. Kusma (討論) 18:05, 12 December 2006 (UTC)

Of course we can, but it isn't natural. The operations on the reals are naturally defined by continuity. As I said earlier, even when you define addition on the reals, you don't make a special case for when the numbers being added are integer reals, or rational reals; you use the definition given by continuity, and it happens to come out the same. --Trovatore 18:18, 12 December 2006 (UTC)

I agree with Kusma. Leaving 0.0^0.0 undefined serves no purpose and is supported by no argument but a subjective feeling of being natural. Bo Jacoby 22:18, 12 December 2006 (UTC)

What's lacking is a convincing argument to give it a definition. The operations on reals do not automatically inherit their values from the naturals or rationals; you have to provide a definition. To get the result you want, you'd have to put a special case into the definition of exponentiation on reals. You haven't given any good reason to do that. --Trovatore 22:29, 12 December 2006 (UTC)

It depends on how you introduce exponentiation. Let's say I start by defining it on the natural numbers (including the definition of 0⁰, then extend that definition to rational bases, then I add rational exponents by taking roots. Now I can extend by continuity to the reals. I do not think it is natural to undefine 0⁰ during this process of extending the definition of exponentiation, since real exponentiation where 0⁰ is undefined is not more natural or consistent than real exponentiation where 0⁰=1: both are discontinuous. If I decide that 0⁰ is undefined, I have needlessly decided to limit my arithmetic. Kusma (討論) 22:46, 12 December 2006 (UTC)

But the right way to extend it does not need to ask whether something "already had a definition". Look at the case of addition again. How do you define addition on the reals? If you use Cauchy sequences, you define a+b by looking at a representative Cauchy sequence for a and one for b, and adding them pointwise. Then you note that this is also a Cauchy sequence, and that its equivalence class does not depend on which representatives you chose. Therefore you define a+b to be the equivalence class of that sequence.

This does not require any special case for when a and b happen to be images of naturals/integers/rationals under the natural embedding.

The right definition for a^b will have the same properties (with modifications necessary, of course, for the fact that a rational to a rational power is not itself necessarily rational). I think fundamentally a^b, for a, b real, should be defined only for a greater than 0. A somewhat more liberal definition is possible when you move to the complex numbers.

You're not unnecessarily limiting arithmetic; you're taking note of the fact that reals are fundamentally different creatures from naturals or rationals. The naturals embody counting, and the rationals simple (linear) algebra; the reals are about continuity. --Trovatore 22:56, 12 December 2006 (UTC)

1. See exponentiation for the definitions.

2. An operation on reals that does not reproduce addition of integers should not be called addition. (You may according to Trovatore also define a+b by looking at a representative Cauchy sequence for a and one for b, and subtracting them pointwise. Then you note that this is also a Cauchy sequence, and that its equivalence class does not depend on which representatives you chose. Therefore you define a+b to be the equivalence class of that sequence.)

3. Whether "reals are fundamentally different creatures from naturals or rationals" has nothing to do with the case. Bo Jacoby 23:37, 12 December 2006 (UTC)

1. It's not a given that the current state of the WP articles is correct.

2. Of course it would be silly to define addition that way. It's not an automatic thing; some thought has to be put into definitions. The thought process involves considering what you're trying to represent conceptually.

3. Wrong. See point 2 above. --Trovatore 00:42, 13 December 2006 (UTC)

1. Surely. It is not an argument but merely an information to spare you from defining exponentiation here.

2. Surely. The silliness consists in not checking that an extended, real, definition includes the restricted, integer, definition as a special case. Just as the real addition 1.0+2.0=3.0 must match the integer addition 1+2=3, even if reals are fundamentally different creatures from integers, so must real exponentiation 0.0^0.0=1.0 match integer exponentiation 0⁰=1. That is why the suggestion that 0.0^0.0 is anything but 1, is 'silly'. Bo Jacoby 06:21, 13 December 2006 (UTC)

I'm not saying it's anything but 1 (or 1.0). I'm saying there is no such thing as 0.0^0.0. It's not something else; there just is no such thing at all. The point (0.0,0.0) is not in the domain of the real-to-real exponentiation function.

No, it does not have to match what happens on the natural numbers. It's a different concept, called by the same name for convenience because it does match in many cases. By the way, it's not just 0.0 that's a problem -- you can't raise −2.0 to a real power, either, though you can easily raise it to a natural number power.

I've said all I care to say on the reasons for this. You're obviously not going to be convinced, and that's OK. However the article is still unacceptable. There are plenty of authors who say that 0⁰ is undefined, though they rarely get into this level of detail, distinguishing between naturals and their corresponding reals. In fact I'm sure lots of references can be found defining a^x as e^{x log a} (by the way, this is the cleanest, though not the most intuitively satisfying, definition on the reals), and this is not defined for a≤ 0. --Trovatore 06:41, 13 December 2006 (UTC)

What exactly do you mean by e^{x log a}? Do you mean the power-series definition? In that case, how do you show that this coincides with rational exponentiation? EdC 19:32, 13 December 2006 (UTC)

Surely I understand that you accept integer exponentiation 0⁰=1 but that for unknown reasons you want to leave 0.0^0.0 undefined. This difference is sufficient for not calling the two operations by the same name or by the same notation. Surely you can raise −2.0 to a real power, see Exponentiation#Complex_powers_of_complex_numbers. Surely a^x=|a|^x·e^{(2n+1)·π·i·x} is defined as a multivalued function for a<0. Bo Jacoby 08:09, 13 December 2006 (UTC)

No, it's not a sufficient reason not to use the same notation. There is no such thing as a "multivalued function"; if it's multi-valued, it's not a function. But in any case I'm done arguing with you. The fact is that the article does not fairly represent the usages of many mainstream authors, and therefore must change, whatever you or I may think about it. --Trovatore 17:08, 13 December 2006 (UTC)

Well, if you want to move into the complex plane, a^x = e^{x log a} is an complex analytic function of a and x for all a ≠ 0. There is, of course, a branch cut running from 0 off to ∞ in some direction, but that's not a real problem. The point a = 0 is bad because it's a singular branch point. There is simply no sensible way to define 0⁰. If you want to be obstinate you could extended your definition to a = 0 as long as Re(x) ≠ 0 via

$0^x = \begin{cases}\infin &\mathrm{Re}(x) < 0\\ 0 & \mathrm{Re}(x) > 0\end{cases}$

But you can't find a sensible value for 0^iy for y real no matter how hard you try. And no matter what value you assign to 0⁰ the function won't be analytic there (or even continuous for that matter). So in analysis it is wrong to assign a value to 0⁰. -- Fropuff 17:15, 13 December 2006 (UTC)

Yes, thanks, well put. That's basically what I was talking about when I said a more liberal definition was possible when considering complex numbers. I have focused on the real category in this discussion -- if you want to stay inside the reals, you can't reasonably define (−2.0)^1.0, even though the answer comes out real once you add complex numbers as an intermediary. --Trovatore 17:19, 13 December 2006 (UTC)

But you don't have to assign a value to 0⁰, if you just agree to carry the definition through from the naturals. I don't understand how allowing real exponentiation to be a conservative extension of natural (integer, rational) exponentiation is going to introduce problems. EdC 19:36, 13 December 2006 (UTC)

That's an awkward special case. It makes real-to-real exponentiation a definition by cases when it doesn't have to be. It says, "If we can impute the value from the one on the naturals, do so; otherwise do something else". --Trovatore 19:52, 13 December 2006 (UTC)

Yes, but what's it a special case of? EdC 08:17, 14 December 2006 (UTC)

OK, let's see if I understand your position. You're arguing that exponentiation on the reals is a completely distinct operation from exponentiation on the rationals, and they just happen to share a name and some values? EdC 19:32, 13 December 2006 (UTC)

It's not an accident of course, no. The conceptual bases of the two operations overlap, but are not identical. --Trovatore 19:52, 13 December 2006 (UTC)

The article multivalued function offers a fair explanation. Trovatore is free to stop arguing with me at any time of his choice, and I am free to stop arguing with Trovatore at any time of my choice. Trovatore is correct in saying: "no matter what value you assign to 0⁰ the function won't be analytic there", but the function won't be analytic there even if one don't assign any value to 0⁰, so Trovatore's suggestion doesn't improve that situation at all. The value of 0⁰ cannot be defined by continuity, and so what? There are other ways. No mainstream author have argued like Trovatore that 0 ≠ 0.0 or that 0.0^0.0 does not mean exactly the same thing as 0⁰ Bo Jacoby 21:54, 13 December 2006 (UTC)

Well, there's at least one counterexample to that (I am a mainstream author). Whether there are others I haven't checked. However there is definitely mainstream literature that asserts that 0⁰ is undefined. Therefore the current state of the article is unacceptable. --Trovatore 22:02, 13 December 2006 (UTC)

But a mathematics article should be held to a higher standard than merely repeating what mainstream authors have said, especially where they contradict one another. Given that you still haven't told us enough to understand your position, it'll be hard to write an article that doesn't leave the lay reader hopelessly confused. EdC 08:17, 14 December 2006 (UTC)

I can't imagine what information you're missing, to understand my position. To summarize: That empty products are the multiplicative identity is convincing; that 0.0^0.0 is an empty product is not convincing. Your suggestion to allow 0.0^0.0 to inherit the value from 0⁰ is a definition by cases (requiring one definition when the values are integer reals, and another when they're not). Definitions by cases, especially of fundamental operations, tend to be unnatural and are best avoided when possible.

(To anticipate one possible objection: No, I am not proposing a definition by cases myself here. I am proposing a definition of two different, though related, operations.)

I think I've said this all before, but maybe the summary will help. Is there anything you still feel you're missing? --Trovatore 08:29, 14 December 2006 (UTC)

Yes. Your real exponential function: where is it defined, and how is it defined? (Please answer in terms of well-defined operations, e.g. Cauchy sequence addition as you mentioned above.) --EdC 09:04, 14 December 2006 (UTC)

I could do it in terms of Cauchy sequences but it'd be a bit messy -- basically given sequences (r_n) and (s_n) of rationals, I would construct a sequence of better and better rational approximations to the pointwise values ${r_n}^{s_n}$ . With a sufficiently fast notion of "better and better" that would result in a Cauchy sequence whose equivalence class depended only on the equivalence classes of the input sequences.

Terser, though less intuitive, would be a^x = e^{x log a}. You asked above what I meant by e^x; there are several ways of dealing with this. My preferred approach is to define the exponential function in terms of a differential equation, and log as its inverse. But you could equally well define log as an integral, and the exponential function as its inverse (that's the approach taken by Swokowski). --Trovatore 06:19, 15 December 2006 (UTC)

Oh, you also asked where it's defined. It's defined for a>0, and any real value of x. Really that's the only region we have any use to have it defined. The uses mentioned in the article -- power series, binomial theorem -- are instances of real-to-the-natural-power or natural-to-the-natural-power, not of real-to-the-real power. --Trovatore 07:19, 15 December 2006 (UTC)

That's begging the question, I'm afraid, or something close to it. You're choosing to work with the reals as Cauchy sequences, noticing that operations on Cauchy sequences only yield well-defined values when the operation is continuous around the limit, concluding that the operation on the reals is only well-defined where it is continuous around the limit, and justifying this to yourself by saying that you're only interested in those values where the operation is continuous around the limit. EdC 21:36, 16 December 2006 (UTC)

But that is all I'm interested in -- and all that it's ordinarily useful to be interested in -- in this particular context. If you find an application for a broader-based definition, more power to you -- just explain it briefly in the paper, and everything will be fine. Most of the time, though, strong typing is a guard against conceptual errors. --Trovatore 21:47, 16 December 2006 (UTC)

About your definition of e^x - is that constructive? EdC 21:36, 16 December 2006 (UTC)

Don't know. It could probably be reformulated that way. But frankly I'm not extremely interested in whether it's constructive or not; I'm not a constructivist. --Trovatore 21:47, 16 December 2006 (UTC)

This is what EdC considers to be the surface of x^y on the reals

I don't know why the 0⁰ stuff is on this page at all. It has hardly anything to do with the topic of the page, which is empty product. The value of an empty product is 1, even if the values that would be there except that there aren't any of them are 0. Exponentials, derivatives, etc, etc, are completely irrelevant. --McKay 05:40, 14 December 2006 (UTC)

I have to agree with you there. The whole section on 0⁰ should probably be moved to the exponentiation article. -- Fropuff 06:00, 14 December 2006 (UTC)

Quite possibly, except that n⁰ is best understood as an empty product. EdC 08:17, 14 December 2006 (UTC)

Trovatore suggests that a computation in integer arithmetic ceases to function when repeated in real number arithmetic. This loss of backwards compatibility is a high price to pay for no benefit at all. Thanks to EdC for the nice picture. Do you consider improving the exponentiation article? Bo Jacoby 15:39, 14 December 2006 (UTC).

I'll take a look, if I have time. EdC 21:36, 16 December 2006 (UTC)

The history of 0⁰ is told in Knuth, ``Two notes on notation". Libri and Möbius and Knuth himself argues that it should be =1 while Cauchy considered it undefined. Trovatore's proposal, that 0⁰ is defined for integers but not for reals, is not considered by Knuth. It is not mainstream mathematics. Bo Jacoby 02:49, 15 December 2006 (UTC)

I don't propose that the article should say 0⁰=1 but 0.0^0.0 is undefined, unless it can be directly sourced. I can find sources from which you can infer it (for example Swokowski, my high school calculus text, defines a^x only for a>0), but I agree that getting from there to what I'm saying requires too much analysis beyond what the source actually says. However it is definitely within the scope of mainstream mathematics to point out that, in the usages of many authors, 0⁰ is undefined. As for Knuth, he's a great computer scientist. He's not a mathematician. --Trovatore 06:13, 15 December 2006 (UTC)

Actually, now that I've looked at his WP bio, I take the last bit back -- he is a mathematician. That doesn't change the fact that not all mathematicians agree. --Trovatore 07:15, 15 December 2006 (UTC)

I'm pleased that Trovatore's proposal is withdrawn and that backwards compatibility between integer arithmetic and real arithmetic is restored. The fact that some mathematical texts don't define 0⁰ is already included in the exponentiation article, and I have now added the Knuth reference. Please refer to your 'many authors' one by one and let the readers count and evaluate them. No WP article on Swokowsky was found. Bo Jacoby 11:28, 15 December 2006 (UTC)

Huh? The only thing I withdrew was the claim that Knuth was not a mathematician. The rest of it stands. --Trovatore 17:17, 15 December 2006 (UTC)

Let me have a say here as well, since Trovatore seems to be struggling alone against two editors (or so) who are dogmatically defending an indefensible position. Trovatore has made the convincing argument that, at best, 0⁰ is an issue not agreed upon by mathematicians. This fact alone should cause us to default to the less dogmatic position of calling it undefined except in certain special cases. I will not get roped in by specious and tangential arguments to the contrary and I refuse to respond to such arguments. Unless it can be proven that a vast majority of mathematicians consider 0⁰ to be well-defined as 1 no matter what set of numbers we are using, we have no choice but to go with the least bold and contentious statement. When consensus cannot be found, we are forced to go with the statement that offends the least and still manages to be true, like, "Some mathematicians think this...Others think this..." How hard is that? (For the record, it is silliness to take the homomorphic image of any object and try to imbue it with any special properties of the source object that are not respected by the homomorphism. It is a categorically unwise thing to do--in both senses of the word! Since when should exponentiation among the reals be defined by a homomorphism from the integers? Trovatore has made this point repeatedly, but to no avail apparently.) VectorPosse 18:06, 15 December 2006 (UTC)

When did preserving images of functions become "special properties of the source object"? Of all the properties of a structure that should be preserved by a natural homomorphism, I'd put that pretty near the top. EdC 21:36, 16 December 2006 (UTC)

[edit] differing conventions in continuous and discrete mathematics

Normally, 0^0 is taken by convention to be 1 in the context of discrete mathematics and taken by convention to be undefined in continuous mathematics such as calculus and complex analysis. These conventions, especially the one in discrete math, are chosen entirely due to their utility in shortening the statements of certain identities, not because 0^0 "is" 1. Outside of discrete math, these identities are less useful (the binomial theorem, for example, has no "counting" interpretation for noninteger exponents) and so their conventions are not adopted. In particular, when exponentiation x^y is defined as e^{y*ln(x)}, there is no reason at all to put 0^0 = 1.

The exponentiation article should describe both conventions, and this article should point out that the "empty product" interpretation of 0^0 only makes sense in a discrete context. This last part is, I think, essentially what Trovatore has suggested. CMummert 17:59, 15 December 2006 (UTC)

While it is good to see the section here on 0^0 being expanded, that section belongs in the exponentiation article. This article is about the empty product, and the empty product is just one way of interpreting the string of symbols 0^0. CMummert 18:15, 15 December 2006 (UTC)

I'll probably regret this, but I'm going to weigh in here on Trovatore's (and CMummert's) side. In complex analysis, the natural function that springs to mind is z^z = e^zlogz. Now if you've studied complex analysis at all you realize that the logarithm function has a really nasty pole at 0, and that the residue theorem is true. Claiming that z^z approaches 1 as z approaches 0 makes absolutely no sense at all in this context, because we can make it approach a whole lot of different values ... the logarithm cannot be made into a single-valued function in any neighborhood that includes the origin. So for any particular z ≠ 0 we can add any integral power of e^z2πi to the purported "value" of this function at the origin by making the path along which the origin is approached into an infinitely curly spiral, for instance.

You empty product guys are trying to map the Riemann surface for the logarithm function onto a flat piece of paper, and that simply cannot be done. DavidCBryant 19:31, 15 December 2006 (UTC)

Not a pole. Charles Matthews 13:51, 16 December 2006 (UTC)

I'm not making any claims about the limit of z^z. I'm making a claim about the value of 0⁰. Surely you must know of plenty of functions where the limit approaching a point is not the value attained at that point. EdC 21:36, 16 December 2006 (UTC)

Also, when did complex analysis become relevant to an elementary function in real arithmetic? EdC 21:36, 16 December 2006 (UTC)

Where else do we find calculations in discrete mathematics that does not give the same result as the corresponding calculation in continuous mathematics ? Nowhere ! In no case do we need to distinguish between integer 0 and real 0, ("0.0"). Let us keep it that way and reject Trovatore's proposal. The value x^y is defined as e^y·log(x) for nonzero values of x, but not for x=0, because log(0) is not defined and cannot be defined. This does not imply, however, that 0^y cannot be defined either, but nonzero and zero values of x must be treated separately, as it is done in exponentiation. There you also find a list of mathematicians supporting either point of view: "0⁰=1" (Libri, Möbius, Knuth), or "0⁰ undefined", (Cauchy). Add to that list the names of mathematicians that support your point of view, and let readers decide for themselves who they want to follow. I follow Knuth, in case anybody doubted. His analysis is better that the arguments seen on this talk page. Even we 'empty product guys' know that x^y is singular in (0,0), no matter whether it is defined in (0,0) or not. We are not trying to map the Riemann surface of the logarithm onto a flat piece of paper. Bo Jacoby 23:33, 15 December 2006 (UTC)

Really now, Bo Jacoby, this is getting strange. Your last edit summary claims that "no modern author disagrees with Knuth." The precise quotation is: "We have no modern authors opposing Knuth". Bo Jacoby 13:07, 16 December 2006 (UTC). Do you really want to make such an audacious claim with absolutely no proof? I pulled one modern calculus textbook off my shelf, the one by James Stewart, at random and it claims that 0⁰ is indeterminate. (This book is used at an enormous number of universities around the world!) Then I checked Larson, Hostetler, Edwards, and it says the same thing. It's only due to lack of time that I didn't check more. But it is fact that every modern calculus text I've seen calls this indeterminate. These are calculus books, dealing with continuous phenomena, whereas Knuth's book is about discrete mathematics. In which area do you think a result about the real numbers falls? Please stop being antagonistic. VectorPosse 01:08, 16 December 2006 (UTC)

Of course I agree that modern calculus textbooks leave 0^0 undefined. Knuth himself says so: "Some textbooks leave the quantity 0^0 undefined". That is not the point. Now he continues: "But this is a mistake" ! So the point is to find authors who refute Knuth. When I claim that there is no such author, it is based on the fact that I have requested references from Trovatore and CMummert without getting answer. When looking up the names James Stewart, Larson, Hostetler, Edwards, I found no mathematician in WP. Please provide quotes to show that you authors are aware of Knuths point of view, rather than just repeating an old mistake. And I do protest against Trovatore's edit war on exponentiation. Bo Jacoby 08:48, 16 December 2006 (UTC). PS. Answering your last question. Knuth's fine book (ISBN 0-201-55802-5) is called 'Concrete mathematics'. The preface explains: "But what exactly is Concrete Mathematics? It is a blend of CONtinuous and disCRETE mathematics". So it covers both areas. I have spent quite some time studying the book, and I recommend you guys to do the same. Bo Jacoby 10:06, 16 December 2006 (UTC).

That's misdirecting the nature of the debate. There does not need to be a single book saying that Knuth is wrong. All we need is the large body of literature that says sometimes it is convenient to define 0^0 as one thing or another, or that sometimes it should be left undefined. Knuth's book is one of the well-known sources where a strongly opinionated statement such as 0^0 must be such-and-such is made. There are other cases when an idiosyncratic author makes a bold statement like that, and in all those cases it should be carefully considered whether this reflects merely the author's own opinion or a consensus of a larger mathematical community. --C S (Talk) 11:30, 16 December 2006 (UTC)

WP should not disseminate a mistake, even if it is common in a 'large body of literature'. Knuth is respected in the mathematical community and not just 'an idiosyncratic author'. Some books define 0⁰ and other books leave it undefined, but few books, if any? say that sometimes it should be defined and sometimes it should be left undefined. There is no benefit of undefining what has once been well defined, (unless the definition leads to contradictions, like any definition of 1/0 does, but that is not the case here). We agree that (for x,y>0) lim_x,y→0y^x is undefined, and that lim_y→0lim_x→0y^x = 1 and that lim_x→0lim_y→0y^x = 0 . We need to define 0⁰=1 in order to save the binomial formula from being 'arbitrarily restricted'. We ought to spare the readers of the encyclopedia from unnecessary complications. I am sad that Trovatore has vandalized exponentiation. Bo Jacoby 12:54, 16 December 2006 (UTC)

Well, that seems to be just wrong. NPOV says that Knuth cannot be the ace of trumps. Charles Matthews 13:51, 16 December 2006 (UTC)

What is wrong, Charles? I don't say that Knuth is the ace of trumps. I merely say that if there are authors of mathematics supporting Trovatore's point of view, we need to know their names. NPOV does not mean that only one of the two conflicting views has to provide references. Libri, Möbius and Knuth supports the view that 0^0=1. Cauchy disagrees. Knuth knew Cauchy's works, but Cauchy did not know Knuth's works. Many textbooks follow Cauchy and leave 0^0 undefined. Do they discuss Knuth's arguments at all? Do they follow anybody else but Cauchy and each others? Do they say who they are following?

Drexel Math Forum [2] quotes named mathematicians in favour of 0^0=1, (Rotando & Korn, Kahan, Euler), but only anonymously, 'some Calculus textbooks', in favour of 0^0 undefined. "The controversy raged throughout the nineteenth century, but was mainly conducted in the pages of the lesser journals: Grunert's Archiv and Schlomilch's Zeitschrift. Consensus has recently been built around setting the value of 0^0 = 1" .

Is there any contemporary author supporting Trovatore's point of view?

Bo Jacoby 23:52, 16 December 2006 (UTC)

Your theory boils down to: If Knuth says it, and no one bothers specifically to say he was wrong, then we have to say he was right. That's utter nonsense. Calculus textbooks say 0⁰ is undefined; we report that usage, whether or not they bothered to say "by the way, Knuth says otherwise, but he's wrong".

I want to say I have the greatest respect and admiration for Knuth, even though, as a Platonist-leaning set theorist myself, I obviously don't agree with his foundational views on mathematics. But "idiosyncratic" is an absolutely accurate word to apply to him -- and his greatest admirers will not trouble to contest the point. It's not a bad thing to be idiosyncratic. But we don't have to copy his idiosyncracies here (though we can certainly report his views). --Trovatore 00:03, 17 December 2006 (UTC)

Snd that Euler was engaging in eighteenth-century vagueness was made clear as long ago as 1921. Cauchy denied the equality, saying (correctly) that 0⁰ can be evaluated as zero, finite or infinite depending on how you do it. Moritz, R. E.. "Discussions: On the Geometrical Representation of Indeterminate Forms". JSTOR. Septentrionalis PMAnderson 01:37, 17 December 2006 (UTC)

As far as I can tell, the author you cited is confusing indeterminate forms with undefined values. (Learn to be very careful when someone cites teaching experience in a learned journal; if students make errors, it's highly unlikely that their teacher understands the topic more than superficially.) I'd be very interested to see how Cauchy believes it is possible to evaluate 0⁰ as anything other than 1. --EdC 03:20, 17 December 2006 (UTC)

I scanned through PManderson's cited article, but not anything written by Cauchy. It seems likely that Cauchy claimed that it isn't possible to evaluate 0^0 at all, not that he claimed it can be evaluated to something other than one. This is the predominant viewpoint except in discrete mathematics, where the desire to simplify exposition has lead to the convention that 0^0 should be interpreted as one. I don't give this convention any more ontological status than the convention in measure theory that $0 \cdot \infty = 0$ . CMummert 03:31, 17 December 2006 (UTC)

[edit] Why the empty product equals 1

Here is my opinion on why it is so. The value of the empty product is a convention adopted because it has lovely consequences, not because it has to be that way. We mathematicians could define the empty product as 3.713243 tomorrow if wanted, but that would only make life and mathematical writing harder for us so we won't. We agree to define it as 1 because it makes the identity

$\left(\prod_{i\in A} i\right) \times \left(\prod_{i\in B} i\right) = \prod_{i\in A\cup B} i$

true for all multisets A and B, not just for non-empty multisets. This identity is implicitly present in a vast number of places, so making it more generally true makes a lot of other things more generally true and consistent. Technical point: the identity above makes sense for commutative monoids, which include the real numbers etc. For non-commutative monoids, use sequences instead of multisets and concatenation instead of union; the idea is the same. --McKay 06:23, 14 December 2006 (UTC)

[edit] Reference for empty product (and empty sum)

Does anyone know a print reference where the convention that an empty product is one is mentioned? All the references in the article right now are related to computing 0^0. Once that material is moved to exponentiation there will be no references left here. I don't have the right level of undergraduate textbook to find such a convention discussed. Also, the empty sum article is an unreferenced stub. CMummert 15:49, 16 December 2006 (UTC)

OK, here are some references: Michael Hardy 00:23, 17 December 2006 (UTC)

Serge Lang, Algebra, Springer-Verlag, 2002, ISBN 038795385X, page 9.

"The empty product is the unit element."

---

Christine Lescop, Global Surgery Formula for the Casson-Walker Invariant, Annals of Mathematics Studies, 140, Princeton University Press, 1996, ISBN 0691021325, page 29

"An empty product equals one (an empty sum equals zero),"

---

Jaroslav Nešetřil, Jiří Matoušek, Invitation to Discrete Mathematics, Oxford University Press, 1998, ISBN 0198502079 page 12

"An empty product ... is always defined as 1 (not 0 as for an empty sum)."

---

A. Shen, N.K. Vereshchagin, Computable Functions, American Mathematical Society, 2003, page 120

"The empty product is also allowed; it is assumed to be equal to 1."

---

Nicolas Bourbaki, Elements of Mathematics, Springer, 1989, ISBN 3540642439, page 83

"the empty product ... is the identity element"

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David Lawrence Johnson, Elements of Logic Via Numbers and Sets, Springer, 1989, ISBN 3540761233, page 14.

"the empty product is taken to be 1."

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Daniel Zwillinger, CRC Standard Mathematical Tables and Formulae, CRC Press, 2002, page 17.

"Note that, since the empty product is 1, it follows that 0! = 1."

---

A.E. Ingham, contributor Contributor R C Vaughan, The Distribution of Prime Numbers, Cambridge University Press, 1990, ISBN 0521397898, page 1

an 'empty' sum (i.e. a sum containing no terms) it to have the value 0, and an 'empty' product the value 1."

[edit] Remarks on "binomial theorem" argument

This is a little OT, but so is most of this talk page at the moment (should probably be moved to an Arguments subpage). Anyway I want to comment on the claim that we need to define 0⁰=1 "to make the binomial theorem work".

First we need to consider which binomial theorem we are talking about. The original binomial theorem uses natural-number exponents and is completely unproblematic -- all you need to make it work is to agree that the real number zero, raised to the power of the natural number zero, is the real number 1: 0.0⁰=1.0. As I've said, this seems completely natural to me and I have no problem with it.

Supposedly there is a version with complex exponents, but I don't know what sort of restrictions or caveats you need to make it true. Presumably you have to make careful choices of branch cuts in the various terms. Anyway we're discussing reals here, so I won't consider complex numbers in this discussion.

So the relevant binomial theorem is the one with real exponents, giving an infinite series. This is most useful, ordinarily, in the following form:

$(1+y)^r=\sum_{k=0}^\infty {r \choose k} y^{k}$

Here the k is a natural number, so again we have no problem with y=0.

But, you might say, what if I don't want to divide through by x^r? What if I want to use the following form:

$(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^{r-k} y^{k}$

and I want to let x=0?

Well, this version is false! Try computing (0+1)^0.5 and you'll see what I mean. --Trovatore 22:21, 16 December 2006 (UTC)

Right, that version is wrong and that's all there is to it. If you state theorems incorrectly, anything can happen. --McKay 03:52, 18 December 2006 (UTC)

In the binomial formula

$(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^{r-k} y^{k}$

set x = +1.0, y = −1.0, r = 0.0

$(+1.0-1.0)^{0.0}=\sum_{k=0}^\infty {0.0 \choose k} (+1.0)^{0.0-k}(-1.0)^{k}$

The only nonzero term on the right hand side is for k = 0

$0.0^{0.0}={0.0 \choose 0} (+1.0)^{0.0-0}(-1.0)^{0}=(1)(1)(1)=1$

$\ 0.0^{0.0}=1$

Q.E.D.

Bo Jacoby 22:58, 16 December 2006 (UTC)

Except that at that level of generality, the binomial theorem is not in fact true. As my example above illustrates.

(Oh, of course with whatever hypotheses are necessary to call it a theorem, it's true. What I mean is that there are more hypotheses needed to make it true, than might be obvious. In particular it is not true for all real x, y, r.) --Trovatore 23:34, 16 December 2006 (UTC)

We are discussing mathematical design rather that philosophical truth. We may define $00$ freely, but there is a price to pay for bad design. Knuth writes: "Anybody who wants the binomial theorem $(x + y)^n = \sum_{k = 0}^n {n\choose k} x^k y^{n - k}$ to hold for at least one nonnegative integer $n$ must believe that $00 = 1$ ". Bo Jacoby 00:29, 17 December 2006 (UTC).

If n is a nonnegative integer, there is no problem. As I said, I find 0.0⁰=1.0 to be completely convincing and unproblematic.

There is indeed a price to pay for bad design. Ignoring types (natural v. real) is bad design. You said at some point that the programmer's distinction between reals and integers was "a source of error", which it can be if you do something aggressively dumb (say, passing values by way of untyped pointers), but is far more often a protection against conceptual error (when the compiler warns you that you're assigning a double value to an int variable, it's likely to mean you were thinking of the problem wrong, or perhaps storing a quantity in the wrong variable). --Trovatore 00:37, 17 December 2006 (UTC)

Being allowed to ignore types in expressions is good design. So letting the boolean expression (0=0.0) evaluate to TRUE is good design. Being allowed to substitute equal expressions is good design. So letting the boolean expression (0^0=0^0.0) evaluate to TRUE is good design. We agree that 0^0 must evaluate to 1. How can you avoid letting 0^0.0 evaluate to 1 ? Your position is unbelievably unscientific. Some of the united states of America are troubled by the so called intelligent design, which, by argument of neutral point of view (NPOV) is taught in biology classes on equal footing with scientific theories of evolution. In the same way the mathematicians here on WP are troubled by this unintelligent design, having no purpose other than to disseminate an old mistake forever, and having no argument except NPOV. There is no advantage in not defining 0^0.0=1, and the price for not doing so is high: useful formulas and rules of calculation are no longer valid and the articles are polluted by useless exceptions from general rules, making the articles dull and uninteresting and incomprehensible. No named authors argue for the unintelligent design, only the believing readers of a bunch of text books. It is fascinating and sad. Bo Jacoby 15:39, 17 December 2006 (UTC).

[edit] History and References for 0^0=1

Please have a look on the interesting FAQ of sci.math sci.math FAQ: What is 0^0?. Hope this will help to solve the conflict.

Also notice the following quotation: "toute quantité élevée à la puissance zéro remplace l'unité." (every quantity raised to the power zero stands for the unity) - Laplace (Œuvres complètes de Laplace, 1912. p. 39) pom 23:28, 16 December 2006 (UTC)

The sci.math FAQ might be a reference worthy of being cited, but I'm afraid I can't accept it as a definitive reference for closing any question once and for all. --Trovatore 23:37, 16 December 2006 (UTC)

The interest was in the historical background (specialy Cauchy vs Libri and Euler) and references given there:

Knuth. Two notes on notation. (AMM 99 no. 5 (May 1992), 403-422),
H. E. Vaughan. The expression ' 0^0 '. Mathematics Teacher 63 (1970), pp.111-112.
Louis M. Rotando and Henry Korn. The Indeterminate Form 0^0 . Mathematics Magazine, Vol. 50, No. 1 (January 1977), pp. 41-42.
L. J. Paige,. A note on indeterminate forms. American Mathematical Monthly, 61 (1954), 189-190; reprinted in the Mathematical Association of America's 1969 volume, Selected Papers on Calculus, pp. 210-211.
Baxley &Hayashi. A note on indeterminate forms. American Mathematical Monthly, 85 (1978), pp. 484-486.

I surely would not consider a FAQ as a reference. pom 23:43, 16 December 2006 (UTC)

I looked at several of those references this morning, and although their titles are interesting the articles in the American Mathematical Monthly don't make any claims about whether 0^0 is 1 or not. Instead, they show that the indeterminate form 0^0 equals 1 in some special cases, while acknowledging that it is indeterminate in general. Knuth does make an argument for why the convention 0^0 = 1 is useful in discrete math, but says that the indeterminate form convention is "right" (his word) as well.

The simple fact of the matter is that there are two conventions for 0^0, as I explained above, and in order to be neutral this article should describe both of them rather than claiming that one is "correct" while the other is not. CMummert 00:01, 17 December 2006 (UTC)

Yes, it should describe both; but it should also explain why those opinions are held. (For example, my opinion is that clearly 0⁰=1 on the naturals and rationals, and that the reals – being ontologically posterior to the rationals – necessarily inherit the values of operations. As I understand it, Trovatore's position is that the reals inherit values of operations from the rationals through the medium of continuity, and thus values in discontinuous neighbourhoods become undefined (I won't ask what happens to the floor function in the transition from the rationals to the reals... oh, bother.)) --EdC 00:37, 17 December 2006 (UTC)

The floor function is specifically designed to be discontinuous. There is no question of finding a "natural" floor function; it just is what it has been defined to be, and there are no disagreements on what the definition is. (By the way, rarely does anyone bother to define it first on the rationals; usually it's defined directly on the reals without going through the rationals.) It does make sense to discuss what is the natural notion of exponentiation in a given context (though it's not guaranteed that there will be a unique correct answer). --Trovatore 00:49, 17 December 2006 (UTC)

"There is no question of finding a "natural" floor function; it just is what it has been defined to be" – see, that's how I feel about the exponential. Still, at least the floor function shows that indeterminate $\not\to$ undefined. EdC 03:31, 17 December 2006 (UTC)

The exponential function is defined differently by different mathematicians, so that's no help; we're discussing how it should be defined, which is not an issue with floor. Your last sentence is especially confusing -- there's nothing indeterminate about the floor of any value. What do you mean? --Trovatore 03:38, 17 December 2006 (UTC)

Sorry, I meant discontinuous, I guess. EdC 10:24, 17 December 2006 (UTC)

Yes, of course the reasons should be discussed. When I moved the material into exponentiation just now, I made an effort to list the justifications for each convention in a neutral manner. CMummert 00:50, 17 December 2006 (UTC)

However 0⁰ is only related to the subject of this page (empty product) when the exponent is regarded as the natural number 0. Otherwise it belongs somewhere else. --McKay 03:55, 18 December 2006 (UTC)

[edit] conceptual justification

I removed the conceptual justification because it was complicated and didn't justify the need for the definition. Ossido 09:58, 20 December 2006 (UTC)

I've put it back. It certainly does justify the definition and it's very simple. If it doesn't accomodate the learning style of user:Ossido, it does some others'. Michael Hardy 18:49, 20 December 2006 (UTC)

Removed it since there are no references. Nishantman 15:56, 12 October 2007 (UTC)

Is this the kind of thing references are needed for? What about a particular example of a quadratic equation in the article titled quadratic equation? Anyone can check the arithmetic. Michael Hardy 21:49, 16 October 2007 (UTC)

[edit] nonsense

Nullary arithmetic product-motivation: example:

2*3/2*3*5 = 1/5

This deletion of all factors is equivalent to dividing by all factors. The numerator becomes here a product of no numbers, i.e. equal to 1.

comment:

Deletion is not equivalent to division, and is not even a defined mathematical operation. If it was, why would the 1 appear in the numerator?

In terms of fundamental/basic sets, and simplifying by substitution of the product of 2*3, the numerator n = {6}, the denominator d = {6,6,6,6,6}. Dividing all the elements by 6, n={1}, d ={1,1,1,1,1} = {5}. Division scales the measure of the sets, but it does not eliminate (delete) the set! The statement including "product of no numbers", makes no sense logically or mathematically. phyti64.24.149.78 02:02, 22 March 2007 (UTC)

[edit] Programmatic justification

A programmatic justification for the empty product can be given, as in this C code example:

float product(float vals[], int n)
 {
     float   prod;
 
     prod = 1.0;
     for (i = 0;  i < n;  i++)
         prod *= vals[i];
     return prod;
 }

Obviously, when n is zero, the product of zero values is 1. — Loadmaster 21:23, 2 August 2007 (UTC)

This seems essentially the same as what this article says in the "intuitive justification" section. Michael Hardy 21:54, 2 August 2007 (UTC)

Categories: Wikipedia articles incorporating text from PlanetMath

Talk:Empty product

From Wikipedia, the free encyclopedia

Contents

[edit] Eequor's allegations of fallaciousness

[edit] back to the conceptual rationale

[edit] Conceptual issues

[edit] technical justification

[edit] 0⁰ is indeterminate?

[edit] Thank you

[edit] Cancelling numbers

[edit] empty product of sets

[edit] associativity of Cartesian product

[edit] Why I reverted the deletion of material by User:WAS 4.250

[edit] content in question

[edit] Comments

[edit] Convention or fact

[edit] conceptual rationale etc.

[edit] zero to the zero

[edit] differing conventions in continuous and discrete mathematics

[edit] Why the empty product equals 1

[edit] Reference for empty product (and empty sum)

[edit] Remarks on "binomial theorem" argument

[edit] History and References for 0^0=1

[edit] conceptual justification

[edit] nonsense

[edit] Programmatic justification

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Talk:Empty product

From Wikipedia, the free encyclopedia

Contents

[edit] Eequor's allegations of fallaciousness

[edit] back to the conceptual rationale

[edit] Conceptual issues

[edit] technical justification

[edit] 00 is indeterminate?

[edit] Thank you

[edit] Cancelling numbers

[edit] empty product of sets

[edit] associativity of Cartesian product

[edit] Why I reverted the deletion of material by User:WAS 4.250

[edit] content in question

[edit] Comments

[edit] Convention or fact

[edit] conceptual rationale etc.

[edit] zero to the zero

[edit] differing conventions in continuous and discrete mathematics

[edit] Why the empty product equals 1

[edit] Reference for empty product (and empty sum)

[edit] Remarks on "binomial theorem" argument

[edit] History and References for 0^0=1

[edit] conceptual justification

[edit] nonsense

[edit] Programmatic justification

Views

Navigation

Interaction

Search

[edit] 0⁰ is indeterminate?