Talk:Empty product

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Contents 1 Eequor's allegations of fallaciousness 1.1 back to the conceptual rationale 1.2 Conceptual issues 2 technical justification 3 00 is indeterminate? 4 Thank you 5 Cancelling numbers 6 empty product of sets 7 associativity of Cartesian product 8 Why I reverted the deletion of material by User:WAS 4.250 8.1 content in question 8.2 Comments 9 Convention or fact 10 conceptual rationale etc.

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)

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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)

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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)

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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)

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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)

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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|ηυωρ]] 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|ηυωρ]] 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|ηυωρ]] 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)

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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 suggests to put and not ). 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.)

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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)

[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 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

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)