Talk:Empty product/Archive 2

Archive 1

Archive 2

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)

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)

This "Intuitive Justification" needs to be completely removed. Calculators, even theoretical ones, just do not operate this way. When you CLEAR it, it does not assume there is a null product to begin with that is getting multiplied by the first number you type, it just takes the first number you type by itself as the first number to use in the series of multiplications. This is not about references. The logic itself is completely flawed. It seems there is only one person who supports the insane logic presented here, so I am removing it.

And what do you know, as soon as you try to change it, Michael stumbles back along to put his unsupported section back in. I didn't take the time to check, but let me guess.. he wrote it? —Preceding unsigned comment added by 74.197.253.232 (talk) 02:15, 12 October 2008 (UTC)

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)

Moved from WikiProject talk

Please step in at empty product; I'm afraid I lose my temper when exposed to Bo Jacoby. --Trovatore (talk) 09:49, 12 June 2008 (UTC)

I understand your frustration in dealing with Bo — he does not work well with other editors. However, on the substance of this issue, I agree with him that 0⁰=1 even for real numbers. Just accept that the exponentiation operation is discontinuous at that point. JRSpriggs (talk) 15:07, 12 June 2008 (UTC)

The literature is somewhat conflicted over this. The issue isn't whether 0^0 can be defined, in terms of real numbers, as 1. The issue is that this definition seems to be very rarely made in practice. I spent a long time looking things up when I was working on the Exponentiation article. I don't think we should have articles that state, without qualification, that 0^0 = 1 in the context of real numbers, as if this is a well established convention. — Carl (CBM · talk) 15:29, 12 June 2008 (UTC)

It's not rare in practice at all. Every time you see it asserted that

$e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}$

and that that applies in particular when z = 0, then you're using it, since the first term of the series is then 0⁰/0!. Michael Hardy (talk) 18:54, 12 June 2008 (UTC)

The exponent there is the natural number 0, not the real number 0. --Trovatore (talk) 18:57, 12 June 2008 (UTC)

Yeah, thanks, Carl. I've said way too much in this forum on the "substance of the issue", which isn't really relevant here; anyone who wants to see why I disagree with JR can look at the old talk page at empty product. For WP purposes that's all beside the point; WP is not a tool for language reform, or mathematics reform, justified or not. --Trovatore (talk) 18:45, 12 June 2008 (UTC)

This discussion does not really belong here, but now it is here.

Trovatore wrote[[1]]: 0⁰=1 (when both instances of 0 are thought of as cardinal numbers, as opposed to, say, real numbers). Usually one does not distinguish between real zero and integer zero. A certain subset of the set of reals is identified with the set of integers, such that the set of integers is considered a subset of the set of reals. When this is done it makes no sense to distinguish between real zero and integer zero. In practice it is impossible to determine whether a zero in a formula is supposed to be 'thought of' as a cardinal number or as a real number.

Obviously some editors dislike the convention that 0⁰=1. Carl just wrote that this definition seems to be very rarely made in practice. I respectfully disagree. Everywhere when a power series or a polynomial is written as a sum, $f(x)=\sum a_{n}\cdot x^{n}$, the value of $\ f(0)$ is supposed to be $\ a_{0}.$ But $f(0)=\sum a_{n}\cdot 0^{n}=a_{0}\cdot 0^{0}+a_{1}\cdot 0^{1}+\cdots$, which is undefined unless you accept the convention that 0⁰=1. So the fact is that the definition 0⁰=1 is used in practice all the time. Consequently we should state, without qualification, that 0^0 = 1 in the context of real numbers, as this is really a very well established convention, even if you can find conflicting literature. Bo Jacoby (talk) 18:59, 12 June 2008 (UTC).

The text at Exponentiation#Zero_to_the_zero_power does a good job, I think, of explaining what's going on, and I hope new participants will read through that. Executive summary:

• In the context of natural numbers and discrete mathematics, 0^0 is nearly universally treated as 1.
• In the context of complex analysis, 0^0 is nearly universally considered undefined because of the difficulty with the complex logarithm at 0.
• All the calculus textbooks I have looked at define the exponential function a^b only for a> 0. There may be a calculus text somewhere that discusses 0^0 but I haven't found one yet.
• The series notation $\sum _{i=0}^{\infty }a_{i}x^{i}/i!$ is universally understood, but this integer exponentiation is not the same exponentiation function as the continuous one (otherwise, it would be impossible to define exp(z) in terms of its series...).

I don't think many people disagree with these bullets. The issue in the articles is getting them to accurately reflect the complexity of the situation. — Carl (CBM · talk) 19:11, 12 June 2008 (UTC)

Assuming that "WP is not a tool for language reform, or mathematics reform, justified or not", note that Trovatore's distinction: "The exponent there is the natural number 0, not the real number 0", and Carl's distinction: "this integer exponentiation is not the same exponentiation function as the continuous one" are highly controversial mathematics reforms. There is only one zero in mathematics. Exponentiation functions having different domains should be compatible, (having the same values for an argument in the intersection of the domains). Bo Jacoby (talk) 19:52, 12 June 2008 (UTC).

Actually, at this level of reading tea leaves that Bo is doing, the distinction between the real 0 and the natural 0 is pretty standard--the natural number 0 "is" the empty set, whereas the real 0 "is" the set of all Cauchy sequences that converge to zero (which zero? figure it out). But no one is asking to put that distinction in articles -- I did make a nod to it in one recent edit, but only to balance the unacceptable unqualified statement that 0^0=1. Hermeneutical analysis of whether the 0^0=1 convention is implicit in some common equalities goes beyond our scope as WP editors--as Carl correctly notes, the convention is rarely given explicitly in the real-to-real context, and therefore it is inappropriate to state 0^0=1 without qualification. --Trovatore (talk) 20:02, 12 June 2008 (UTC)

I have to agree that the real number 0, and the natural number 0 are distinct. The fact that there is a natural identification between the two allowing us to generally ignore the distinction when it doesn't matter (i.e. most of the time) is not reason to ignore the distinction when it does matter, such as for 0⁰. When dealing with finite numbers we don't tend to distinguish between cardinals and ordinals; that doesn't mean they are the same thing, nor that we shouldn't be careful to qualify any statement that risks treading on transfinite territory. I agree with Carl, this is an area with some subtlety and thus more care should be taken. -- Leland McInnes (talk) 20:57, 12 June 2008 (UTC)

Re Bo, all I was saying about integer exponentiation is that if one defines, as usual,

exp(z) = $\sum _{i=0}^{\infty }z^{n}/n!$

and then computes exp(2) via the series, this is naively a different operation that computing e² by multiplying e by itself. Indeed, the fact that exp(2) = exp(1)² is a theorem that has to be proved. — Carl (CBM · talk) 21:23, 12 June 2008 (UTC)

At Exponentiation, it says "When n is a positive integer, exponentiation corresponds to repeated multiplication $a^{n}=\underbrace {a\times \cdots \times a} _{n},$.". This is the original definition of exponentiation. We then try to extend the definition to the widest possible domain. First, we allow the exponent be any natural number by letting a⁰= 1 for any a. Then we extend the domain of the exponent to negative numbers when possible, i.e. when the base is non-zero. Then we interpolate between the integers by using roots to define the power when the exponent is fractional. Then we extend the definition to irrational exponents by continuity. The definition using "exp" and "ln" allows the base to be a complex number for certain exponents.

The point is that we do not use continuity as a constraint on permissible values. Rather we use it to help us extend the definition where it is able to do so. But 0⁰=1 is established well before we even turn to continuity for such help. JRSpriggs (talk) 06:50, 13 June 2008 (UTC)

I have searched quite a bit to find what the literature says about 0^0 in the context of continuous mathematics. What do you make of the opinion of the author cited here? — Carl (CBM · talk) 10:57, 13 June 2008 (UTC)

Carl, see Exponentiation#Powers_of_e for the proof that exp(n) = exp(1)ⁿ for integer values of n. Once this is proved one need not distinguish between exp(x) and e^x ever again. Bo Jacoby (talk) 07:30, 13 June 2008 (UTC).

That's not quite true; exp(z) is an analytic function, but e^z = exp(z log e) has branches like any other complex power function. There are situations where this distinction is relevant. — Carl (CBM · talk) 10:57, 13 June 2008 (UTC)

Carl, that is an interesting point of view. The actual mathematical practice regarding exp(z) and e^z is that no distinction is made. e^z is not a (multivalued) power function but a (singlevalued) exponential function. Bo Jacoby (talk) 01:07, 15 June 2008 (UTC).

Leland McInnes, consider the cardinal number A=|{0, 0.0, 0/2, 1-1}|. The list contains 4 elements, expressed as respectivly an integer, a decimal fraction, a rational, and a difference, but the values are all equal to zero. So the set is the set containing the number zero, and the cardinal number of this set is one. If you disagree, then you are definitely making a mathematics reform. The fact that "there is a natural identification between the real number 0, and the natural number 0 allowing us to generally ignore the distinction when it doesn't matter (i.e. most of the time)", tells us not to make a crackpot convention that violates this identification. Bo Jacoby (talk) 07:30, 13 June 2008 (UTC).

Everyone agrees that 0^0 should be 1 when the 0^0 represents the cardinality of an empty set. But there are other things that 0^0 can represent, and in those cases it is much less clear that 0^0 should be (or even is) defined. For example, 0^0 can be an indeterminate form, and it can be a shorthand for exp(0 log 0). What do you make of the author's opinion in here? — Carl (CBM · talk) 10:57, 13 June 2008 (UTC)

To CBM: The note you reference quotes an article in the Mathematics Monthly as saying "... Let's start at x=0. Here x^x is undefined.". Can you give a longer and perhaps more convincing quotation than that? JRSpriggs (talk) 16:57, 13 June 2008 (UTC)

To Bo Jacoby: I do think that there is a difference among: zero the natural number, zero the integer, zero the rational number, zero the real number, zero the complex number, zero the ordinal number, zero the cardinal number, and so forth. Each has meaning only within the context of a certain number system, and each would have to be encoded as a set (or whatever) in a different manner.

To others: Nonetheless, we have certain standard injections from one number system into another, e.g. injecting the integers into the real numbers. The operations in the higher systems are supposed to be defined to agree with the images of the corresponding operations in the lower systems. So since 0⁰=1 holds in the natural numbers, it must be required to hold in any number system into which the natural numbers have a standard injection. JRSpriggs (talk) 18:00, 13 June 2008 (UTC)

The mathematics magazine article is on JStor PDF; the quote is from page 204. Have you ever encountered a complex analysis text that claimed that 0^0 was defined to be 1, in the context of complex exponentiation? My interest here is purely in describing the actual mathematical practice regarding 0^0, as I think the exponentiation article currently does. — Carl (CBM · talk) 18:14, 13 June 2008 (UTC)

To CBM: Access to JSTOR is limited. I am not one of the small class of people who is authorized to access it. A reference which cannot be referenced is not much use. JRSpriggs (talk) 18:37, 13 June 2008 (UTC)

I'd hardly describe JStor as "cannot be referenced", as I think almost all university libraries subscribe. In any case, here is the entire paragraph. If you'd like more, I can email the entire PDF to you, just send me an email to let me know. The article is devoted to the study of the function x^x where x is real but the values are permitted to be complex.

7. Gaps and particular values Notice the apparent gaps in Figures 5 and 6 at certain x-values. Let's start at x = 0. Here, x^x is undefined. But on any thread, as x gets close to 0, the values of x log x get close to 0 (the body of the spider), and so the value of x^x = e^xlogx get close to 1. So all the threads seem to emanate from the point (0, 1+0i).

The author here simply accepts that 0^0 isn't defined, and doesn't dwell on it at all. My question about complex analysis texts still stands. — Carl (CBM · talk) 22:30, 13 June 2008 (UTC)

section break 1

When some authors do not define 0⁰, and other authors do define 0⁰, then 0⁰ has been defined. Complex analysis texts need not bother to define 0⁰, because 0⁰ has already been defined in much more elementary texts. 0⁰ can be generalized to the very important function x⁰, which is =1 for all complex values of x. This generalization is crucial for the sum notation for polynomials and power series, where the constant term a₀ is written a₀·x⁰. "0⁰" can also be generalized to the unimportant function 0^x, which satisfies 0^x=0 for x>0, 0^x=1 for x=0, and 0^x undefined elsewhere. Both functions can be generalized to y^x, which is continuous for y>0, but not for y=0. "0⁰" can not be a shorthand for "exp(0 log 0)" because log 0 is undefined. Bo Jacoby (talk) 01:07, 15 June 2008 (UTC).

Nonetheless, in complex analysis, x^y is precisely a shorthand for exp(ylogx), which is why 0^0 is typically not defined there. But that has little to do with empty products. This article is already clear enough that, when 0^0 represents an empty product, it is taken to equal 1. The rest of the details about 0^0 are given at the exponentiation article. What else does this article need to say about 0^0 in the context of empty products? — Carl (CBM · talk) 03:29, 15 June 2008 (UTC)

Agreed. In the context of empty products this article needs only to say that 0^0=1. Done. Bo Jacoby (talk) 07:56, 15 June 2008 (UTC).

That is an extremely disingenuous way of putting it. I have reverted you. --Trodvatore (talk) 08:05, 15 June 2008 (UTC)

Trovatore, you are loosing your temper, but as a WP editor you are supposed to keep cool. There is for the reader no meaning in your not taking the full logical consequence of the argument you just made. The empty product argument leads invariably to 0⁰=1. This empty product section is not about complex analysis or about exp(ylogx). Your reversion made the article confusing. Think of the reader. Bo Jacoby (talk) 09:55, 15 June 2008 (UTC).

The argument that the empty product equals 1 only determines the value of the symbolic expression 0^0 when the expression is actually intended to represent an empty product. I think that the sentence

"For this reason, authors in combinatorics and set theory frequently define 00 to be 1 when it represents an empty product."

is very accurate about what's going on with 0^0 in the context of empty products. On the other hand,

"For this reason, 0⁰ = 1."

has the same difficulty that I have argued against for some time: 0^0 is not always equal to 1; there are contexts in which it is an indeterminate form, and contexts in which it is undefined. But this article isn't the place to discuss those other contexts. I dislike making the general statement that 0^0 = 1, as if this were true regardless of context, when the literature shows a much more nuanced reality. — Carl (CBM · talk) 10:33, 15 June 2008 (UTC)

I agree very much with Carl and Trovatore. Many writers explicitly say that 0^0 is indeterminate (which is stronger than not saying anything about 0^0). Thus, we should not make an unqualified statement that 0^0 = 1. It may be easier for the reader if we would be able to say so (at least as long as the reader does not come into contact with other conventions), but the priority for Wikipedia is to reflect accurately what the literature says. -- Jitse Niesen (talk) 11:56, 15 June 2008 (UTC)

I'll add my support for Carl's and Trovatore's position. When 0⁰ is used to denote an empty product then yes, it is equal to 1. It does not, however, always denote an empty product, and thus categorical statements about it should not be used. Really, is it so bad to have the small amount of qualification used here? Can't we move on to better things, like the rather poor "Intuitive justification" and the very slim section on empty categorical products (which, to me, is the best ultimate justification). -- Leland McInnes (talk) 16:05, 15 June 2008 (UTC)

The idea, that the value of a mathematical expression depend on its use, is very foreign to mathematics and must be rejected. (2+2=4 always!). So is the idea that the value of a number depend on whether the number is considered integer or real, (like 0≠0.0). (Try something like "if 0=0.0 then print '0=0.0' else print '0≠0.0' " in your favorite programming language). That "some authors use x^y merely as a shorthand for exp(ylogx), and according to that definition 0⁰ is undefined", may be reported. Done. Bo Jacoby (talk) 07:27, 16 June 2008 (UTC).

What you think "must be rejected" is irrelevant. The text as I left it accurately reflects current mathematical usage. The arguments about how things should be are for another forum. You're wrong about that too, but this isn't the place to explain why. --Trovatore (talk) 07:50, 16 June 2008 (UTC)

Trovatore. The first argument was:" If m is positive and n is zero, then there are no such functions, because there are no elements in the latter set to map those of the former set into". This argument has the conclusion: "Thus 0^m = 0 when m is positive". This makes sense. Then comes the second argument: "However, if both sets are empty (have size 0), then there is exactly one such function — the empty function". What is the logical conclusion of that argument? Use you intelligence rather than you prejustice. Otherwise the text becomes logically incomprehensible and the reader is repelled. Join the discussion if you want to and stay out if you don't want to, but stop the edit war. I was actually compromising. Personally I would prefer to simplify the text by omitting the irrelevant fact that some authors leave 0⁰ undefined, but I respect that other editors consider it important. Bo Jacoby (talk) 09:28, 16 June 2008 (UTC).

I would also have reverted your edit if Trovatore had not done so earlier. It doesn't matter what you think that 0⁰ should be, it doesn't matter what Trovatore thinks, it doesn't matter what I think. What matters is how 0⁰ is treated in the literature. -- Jitse Niesen (talk) 11:33, 16 June 2008 (UTC)

Hi Jitse. Broken logic matters to an editor. Before Trovatores revert it was said that "some authors use x^y merely as a shorthand for exp(ylogx), and according to that definition 0⁰ is undefined", which accounts for how 0⁰ is treated in the literature. Bo Jacoby (talk) 12:10, 16 June 2008 (UTC).

Since I agree with Trovatore and Jitse that ultimately current usage, rather than "what should be", is what counts, this is ultmately somewhat offtopic. Your first example asks us to consider that 2+2=4 "always", yet 2+2=1 in Z/3Z. Sure the same symbols denote different objects, but then that's all that is being argued. Secondly you suggest I try out some simple code in my favourite programming language, so here it is in Ada:

with Ada.Text_IO;

procedure Test_zero is
  begin
    if 0 = 0.0 then
      Ada.Text_IO.Put("0 = 0.0");
    else
      Ada.Text_IO.Put("0 /= 0.0");
    end if;
  end Test_zero;

with the result that it won't compile. It gives the following error

test_zero.adb:6:14: invalid operand types for operator "="
test_zero.adb:6:14: left operand has type universal integer
test_zero.adb:6:14: right operand has type universal real

So the distinction is apparently quite clear at least as far as Ada is concerned. All of that aside, however, what counts is what sources say and do on the matter. There seems to be a clear case that in analysis 0⁰ is contentious, while in discrete maths and combinatorics, where 0⁰ clearly refers to the empty product, it uncontroversially is equal to 1. Since this article is about the empty product it makes sense to point out when 0⁰ unambiguously refers to the empty product and thus agrees with it, rather than trying to make a WP:POINT about 0⁰ here. -- Leland McInnes (talk) 13:21, 16 June 2008 (UTC)

Bo, you wrote: "For this reason 0⁰ = 1. (Some authors use x^y merely as a shorthand for exp(y·log x), and according to that definition 0⁰ is undefined)." Those two sentences contradict each other. Furthermore, the reason for some authors to say that 0⁰ is indeterminate is that 0^x and x⁰ have different limits as x → 0+ ; see for instance the "Concrete Mathematics" book quoted in Exponentiation. -- Jitse Niesen (talk) 13:29, 16 June 2008 (UTC)

Thanks to Leland for the program. Obviously Ada won't compare an integer to a real, so Ada don't tell whether 0=0.0 or not. "2+2=4 always, yet 2+2=1 in Z/3Z". That is true. But as 4=1 in Z/3Z, it is not incorrect to state that 2+2=4. My point was more elementary, however, that 2 apples + 2 apples = 4 apples for integer arithmetic, and 2 inches + 2 inches = 4 inches for real arithmetic. Any calculation done in integer arithmetic is reproduced in real arithmetic, except alas 0⁰, according to Trovatore and Carl, who answer 1 in integer arithmetic, and don't answer in real arithmetic. (Well, actually they do answer that a₀·x⁰=a₀ for all x, but they do not want to say that it is because x⁰=1 for all x, because some textbook do not define 0⁰). Bo Jacoby (talk) 14:17, 16 June 2008 (UTC).

Ada isn't the only case, just the first language I tried. Any decently strict language will give you the same result. The only reason other languages let you get away with such things is that they implicitly do a type conversion in the background. Some languages, such as Eiffel, will actually supply such conversion functions explicitly in the types. The point is that in computation real (or floating) and integer types are distinct and require conversion (i.e. the application of a function mapping one type to the other), either explicit or implicit depending on the language, to do a comparison. As for Z/3Z; depending on conceptualization, one could argue that 4 doesn't exist or is meaningless there (i.e. think of (Z/3Z,+) isomorphic to C₃). The point, really, was that one needs to be careful as to what their domain of discourse is, because meanings can change. As to integer arithmetic being faithfully reproduced in the reals; I don't see that as necessarily true at all, and a few examples are hardly proof. As soon as you step into the realm of the continuous you have to be careful of expectations; things need not behave as in the discrete case. The rather more cagey approach of analysts reflects this. -- Leland McInnes (talk) 14:44, 16 June 2008 (UTC)

Leland. Surely computers represent integers and reals as different datastructures. Some computer languages (e.g. J) make automatic type conversion and evaluates the expression 0=0.0 to be 1, meaning "true". Apart from our debated 0^0.0≠0⁰, do you know examples in mathematics where integer arithmetics is not reproduced by real arithmetics? Bo Jacoby (talk) 23:48, 16 June 2008 (UTC).

Yes, computers consider 0.0 and 0 as different objects and require the use of either the injection of intergs into reals, or the partial inverse thereof, which they call type conversion, to make a comparison. Regardless of whether your preferred language decides to automatically do this conversion or not, it occurs (it basically occurs at the compiler/interpreter stage). This is all rather irrelevant, aside from the point that we do have a case where clearly 0 and 0.0 are distinct objects. As to examples for arithmetic not behaving entirely as expected; it's not what Trovatore has in mind, nor even the reals (technically), but it is dealing with continuums: try smooth ifninitesimal analysis or synthetic differential geometry for arithmetic that doesn't behave exactly as you might expect. -- Leland McInnes (talk) 13:19, 17 June 2008 (UTC)

Jitse, the authors contradict one another, but the two sentences correctly express that fact. I too am a fan of Concrete Mathematics. Surely the exponentiation function x^y is not continuous in (x,y)=(0,0) even if it is defined for { (x,y) | (x>0 AND y complex) OR (x=0 AND y≥0} OR (x≠0 AND y integer) }. Bo Jacoby (talk) 14:17, 16 June 2008 (UTC).

Surely, we don't want the article to contradict itself because the authors contradict each other. That can easily be avoided by writing something like "some say 0^0 = 1, others say 0^0 is indeterminate". If we write "it's a fact that 0^0 = 1 (but some say it is indeterminate)", we allow our own opinion to shine through, but we should try to be impartial. I agree with your last sentence; it's not possible to define x^y continuously on a domain that includes { (x,y) | x ≥ 0 AND y ≥ 0 }. -- Jitse Niesen (talk) 13:12, 17 June 2008 (UTC)

"some say 0^0 = 1, others say 0^0 is indeterminate" is exactly what we say in the appropriate place, which is the article on exponentiation. There is a prominent hatnote pointing to that text from this article. I don't understand why this article should repeat all those details about 0^0; I think it should simply give the empty product interpretation and leave it at that. — Carl (CBM · talk) 14:44, 17 June 2008 (UTC)

Yes, we agree. There is no need to use the word 'fact'. The discussion is not about facts but about definitions. The elementary argument leads to the definitions x⁰ = 1 for all x, and 0^y = 0 for positive y. The complex analysis guys use the definition x^y = exp(y·log x) for positive x. These two definitions of x^y have different domains. They agree in the intersection of these domains, so there is no contradiction involved, and a unique value of x^y is defined in the union of these domains. I still fail to understand why it is important for people who do not define something, to insist that it is undefined, when somebody else defines it. Bo Jacoby (talk) 07:09, 18 June 2008 (UTC).

Provided it is understood that (1) this is now an off-topic discussion that has nothing to do with what should go in the article and (2) that I am not interested in discussing it at length, I will respond to that. In my opinion the two functions are intensionally different even where they are extensionally the same (to be pedantic, "extensionally the same" modulo identifying integers with their images under the natural inclusion map into the reals). The real-to-real and complex-to-complex exponential functions do not mean the same thing as multiplying together a certain number of copies of the real or complex value. (This is why the equation e^iπ=−1 is not nearly as mysterious as some writers make it seem--it would be mysterious if it meant "if you could multiply together iπ copies of e you'd get −1", but it doesn't mean that at all; it refers to a different exponential function that shares the same name and some of the same values and properties.)

Again, I have no intention of putting any of this in any article, and I do not believe this is the correct forum to discuss it; we simply reflect contemporary mathematical usage and that's an end to it. If anyone wants to discuss this further perhaps an /Arguments subpage would be tolerated by the wonks, even though it's not strictly in accordance with the putative purpose of talk pages. --Trovatore (talk) 07:54, 18 June 2008 (UTC)

The discussion is not off-topic, but Trovatore's interest or disinterest in the discussion is offtopic. The article presently says: "authors in combinatorics and set theory frequently define 0⁰ to be 1 when it represents an empty product". This is unsatisfactory, because (as I have said before) the argument of the article leads to the definition 0⁰ = 1. Identifying integers with their images under the natural inclusion map into the reals is mainstream mathematics, while the distinction between 'intensional equality' and 'extensional equality' is not. A function may have several interpretations. What it means is immaterial as long as it is well-defined. Bo Jacoby (talk) 21:01, 18 June 2008 (UTC).

Math at the top

I was thinking, might the math at the top look a little neater if we wrote it to look something like this?

${\begin{aligned}&\prod {\left(\left\{2,3,5\right\}\right)}=\prod {\left(\left\{2,3\right\}\right)}\times 5=\prod {\left(\left\{2\right\}\right)\times 3\times 5}\\&=\prod {\left(\varnothing \right)}\times 2\times 3\times 5=1\times 2\times 3\times 5\end{aligned}}$

(I can't be bothered to memorize all the math commands, so I just threw that together in MathType and copied it to the clipboard in the proper format. Feel free to change alignment or whatever, I had the first equals sign on each line aligned when I made it.)  dalahäst ^{(let's talk!)} 04:20, 10 April 2012 (UTC)

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)

The programmatic statment has considerable appeal, but it is not absolutely irrefutable. For one thing it does one useless multiplication for positive values of n, which one could eliminate by initialising the product to the first value while decreasing the value of n; this would leave the empty product undefined (or defined however one likes) and the question unsettled. In the end, there is nothing new with respect to mathematical arguments for defining the empty product.

I totally agree with this treatment. I even propose to define exponentiation aⁿ as 1 × a × a × ... instead of a × a × a ...! It is totally necessary thing to do if this definition is natural (intuitive/true) as we see here. However, 135th question at mathoverflow downvotes this logic and does not explain why. --Javalenok (talk) 10:53, 11 September 2013 (UTC)

If you read the comments to the downvoted math.stackexchange (not mathoverflow!) question, it is clear that the downvotes probably have no relation to the logic, but to the presentation of the answer.