Talk:Empty product

section break 2

We should say that 0⁰ = 1 and it is an indeterminate form. The two statements are both true. Why would anyone have a problem with that? Michael Hardy (talk) 15:44, 17 June 2008 (UTC)

Which article are you talking about, this one or exponentiation? You're right that it makes little difference whether an indeterminate form is "defined" or not. However, I just made a quick survey of three calculus texts from my shelf, and none of them defines 0^0 at all. Do you know of any calculus text that defines 0^0? — Carl (CBM · talk) 16:12, 17 June 2008 (UTC)

At least they implicitly define it when they say that for all x,

$e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!},$

since when x = 0 then the first term is 0⁰/0!. But why should it matter what calculus textbooks say about it? Calculus texts are efforts at pedagogical improvements in ways to present standard material to students who don't want to become experts on it, not authoritative statements on issues on which there's no standard pronouncement. Michael Hardy (talk) 16:02, 18 June 2008 (UTC)

The exponentiation in the first term on the right-hand side is the real-to-integer, iterated-product version of exponentiation, not the real-to-real or complex-to-complex version, which as I say is intensionally quite distinct. But the point for WP purposes is that if there "is no standard pronouncement" then it's inappropriate for us to create one. --Trovatore (talk) 17:19, 18 June 2008 (UTC)

...and saying that it's a convention is creating one. It violates NPOV. Michael Hardy (talk) 19:28, 18 June 2008 (UTC)

At this writing the article does not use the word convention, except when talking about the intersection of no subsets of a given set. --Trovatore (talk) 19:51, 18 June 2008 (UTC)

If we are talking about fixed exponent 0 in $x^{n}$ then when x tends to 0 the result is 1. So there's no problem with the summation formula of $e^{x}$.Linkato1 (talk) 02:47, 28 November 2018 (UTC)

"Convention"?

This article now says:

This justifies the convention that 0^m = 0 when m is positive.

Does anyone really regard this as a convention rather than as a fact? I was surprised to find respectable mathematicians saying that the fact that the empty product is 1 is merely a convention, rather than a fact, but even then I didn't think someone would say that the fact stated above is a convention. Michael Hardy (talk) 23:48, 12 June 2008 (UTC)

I think that's just badly phrased and should be reworded. I'll go ahead and do it. — Carl (CBM · talk) 00:12, 13 June 2008 (UTC)

It is a reminent from: "This justifies the convention that 0^m = 1 when m is zero", which by some editors is considered a convention rather than a fact. Bo Jacoby (talk) 06:44, 13 June 2008 (UTC).

That one is a convention. But 0^2 = 0 isn't. — Carl (CBM · talk) 10:45, 13 June 2008 (UTC)

I certainly don't agree that 0⁰ = 1 is only a convention. Michael Hardy (talk) 17:30, 13 June 2008 (UTC)

That's reasonable enough; but there are many authors who do consider it only a convention. I located a nice quote from one of them for the exponentiation article: "The choice whether to define 0^0 is based on convenience, not on correctness."[1]. Although I risk repeating myself too often: of course many authors do define 0^0 to be 1; especially when 0^0 represents the empty product, this definition is very common. But many other authors leave 0^0 undefined, especially in complex analysis. Regardless of our personal feelings on the matter, we can still write a description of the subtleties that arise in the literature. — Carl (CBM · talk) 22:50, 13 June 2008 (UTC)

There's no problem and there's no convention as if we are talking about $0^{x}$ then it's always 0 as long as x is non-negative and if we are talking about $x^{0}$ it's always 1 regardless of what x is. It's only when we have $x^{y}$ and both x and y tend to 0 that we must be carefull and analyse the situation.Linkato1 (talk) 02:54, 28 November 2018 (UTC)

Intuitive justification

I've altered this section to read as follows:

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 "ENTER", 3 "ENTER", 4 "ENTER", then the display reads 84, because 7 × 3 × 4 = 84. More precisely, we specify:

• A number is displayed just after "CLEAR" is pressed;
• When a number is displayed and one enters another number, the product is displayed;
• When "CLEAR" is pressed and then some numbers are entered, their product is displayed.

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 the product of zero numbers is 1.

Someone had changed it so that the third desideratum said that when clear is pressed and then a number is entered, that number is displayed. In some ways that makes this closer to a logically rigorous argument, and in this context I think that in itself may be a mistake. But also it doesn't seem so much like a self-evident desideratum since this whole thing is supposed to be about mutliplying. Michael Hardy (talk) 01:59, 12 October 2008 (UTC)

Need for rewrite

I think this article is badly in need for rewriting to make more mathematical (and common language) sense. Take the opening phrase "an empty product, or nullary product, is the result of multiplying no numbers". No it is not, it is an expression asking for the product of an sequence (or set or multiset) of numbers (could be other things as well though), which sequence (set, etc.) happens to be empty. First, there is no such thing as multiplying no numbers, or even one or three numbers; multiplication takes two arguments (the distinction that should be made is between a multiplication and a product; possibly "multiplying together" can be used but it makes no good sense for 0 arguments). Second, and empty product is not the result of anything, it is something like "0!" whose value is 1, but it is not identical to 1 (or otherwise conversely "1 is an empty product", which seems a bad formulation). In short, one should make distinction between expressions and there values, and an empty product is an expression. But there are many more points that are just not carefully stated (and a lot non encyclopedic wording as well). I agree it takes effort to be clear and precise at the same time, but it is worth the effort. To get an idea look at empty sum, which I think is better. I plan to make changes here soon (but feel free to protest already). Marc van Leeuwen (talk) 08:15, 18 March 2010 (UTC)

I cleaned it up a bit. There was too much repetition, and I deleted some things that were copied from other wiki-pages that were somewhat off topic. MvH (talk) 00:19, 28 March 2014 (UTC)MvH

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Convention

I can understand why some might not want to call the value 1 for the empty product a "convention". When you have a unique 2-sided multiplicative identity, it's the only reasonable convention.

But it's still a convention. What if you're in a structure with an associative multiplication, but no multiplicative identity? You can still define repeated products, but now there's nothing to assign to the empty product.

If you don't specify what the empty product means (that is, give a convention), then it simply has no meaning, notwithstanding there's a best choice for the meaning to give it. --Trovatore (talk) 01:41, 28 September 2019 (UTC)

That's more than a best choice, that's the unique choice. The word "convention" just means that's from an agreement among several possibilities, not by deduction. So, if the word "convention" is used, let's say at least that the choice is unique. Vincent Lefèvre (talk) 10:05, 28 September 2019 (UTC)

Strictly speaking, there is no "deduction" possible here. That's just wrong. If you don't define it, it's not defined, period; it is not possible to "deduce" what the value is, in any way whatsoever. And no, the choice is not "unique". It's the unique good choice, but the notion of goodness is based on values external to the definition. --Trovatore (talk) 19:40, 28 September 2019 (UTC)

By deduction, I meant that there is a proof that if a value exists, then the choice is unique. The value is part of the definition of the multiplication binary operation. There is nothing external. Vincent Lefèvre (talk) 20:16, 28 September 2019 (UTC)

There is no such proof. Not from the definition of the concept per se. You could start with the definition of products as iterated multiplication, and then go ahead and specify that the empty product takes the value 17.

Of course you lose some desirable properties, but you can do it. Unless those desired properties are taken to be part of the notion of products, which is an external specification, then there is no proof that the value 17 is wrong. --Trovatore (talk) 20:24, 28 September 2019 (UTC)

Here's the proof of the uniqueness: Consider the case $a_{1}=1$, where 1 denotes the multiplicative identity (which has been assumed to exist in the first paragraph). Then $P_{0}=P_{0}\cdot 1=P_{0}\cdot a_{1}=P_{1}=a_{1}=1$. Thus necessarily $P_{0}=1$. Vincent Lefèvre (talk) 20:41, 28 September 2019 (UTC)

The insufficiently justified step is $P_{0}\cdot a_{1}=P_{1}$. There is simply no way to "prove" that, without adding an external criterion. --Trovatore (talk) 20:47, 28 September 2019 (UTC)

$P_{0}\cdot a_{1}=P_{1}$ is from the recurrence relation stated in the article, which includes $m=1$. No need for another criterion. Vincent Lefèvre (talk) 20:50, 28 September 2019 (UTC)

That relation is a desirable property, but is not the only way iterated products could be defined. Therefore it is an external criterion. --Trovatore (talk) 20:58, 28 September 2019 (UTC)

This is always how extensions have been made: one first chooses the desirable property, and one deduces which value will satisfy it. And this is the criterion chosen in the article (quite natural, since the iterated product is defined with the recurrence relation). I'm not basing my proof on something that is not in the article. Vincent Lefèvre (talk) 21:06, 28 September 2019 (UTC)

I agree it's quite natural. That's not the point. If you start by defining iterated products, there is no value for the product of no values, or even of one value. You note that there is a relationship giving the product of iterated products of two or more values each, and you can see that it would make sense to extend the definition in such a way that the relationship also extends. This is all very natural, but it isn't a proof. --Trovatore (talk) 21:33, 28 September 2019 (UTC)

The recurrence relation is the hypothesis. The proof is how the value of $P_{0}$ is obtained from this hypothesis (the case of one value is completely part of the definition of an iterated binary operation and always unambiguously defined, I hope you do not suggest to write an article for this case!). This is not much different than the notion of continuous extension (extension by continuity): the property of continuity is the hypothesis, and then one seeks to extend the function at one point by taking the limit (the value comes from the property), and determining the limit constitutes a proof. Vincent Lefèvre (talk) 21:56, 28 September 2019 (UTC)

Who says it's the hypothesis? --Trovatore (talk) 22:29, 28 September 2019 (UTC)

I think that all mathematical notation is convention. This convention is one that it would be a tremendous pain to do without, but it is still a convention. Incidentally, starting with multiplication as a binary operation, the multiplication of just one thing also needs to be defined. The recurrence is an extension of multiplication from a binary operation to an m-ary operation, and it has been found that applying it in reverse order to give meaning to P₀ and P₁ is extremely useful. So we have adopted it for our own benefit, not because we were forced to. McKay (talk) 05:30, 29 September 2019 (UTC)

Yes, all mathematical notation and definitions are conventions. Then one may wonder why saying the word "convention" in some cases but not everywhere. This is misleading. And note that P₁ is part of the core definition (otherwise, in some semigroups, trying to use the recurrence relation to deduce and define its value would give several solutions). Only for P₀, one can use the recurrence relation to unambiguously define its value (when there is one). Vincent Lefèvre (talk) 09:21, 29 September 2019 (UTC)

The point is that the obvious idea of repeated products starts with a minimum of two things to multiply together. Anything else is an extension. It's a very natural extension once you think of it, but the word "convention" is not out of place — and no, you can't "prove" it's the right one. --Trovatore (talk) 17:18, 29 September 2019 (UTC)

No, the definition does not say "two". Vincent Lefèvre (talk) 19:50, 29 September 2019 (UTC)

You're missing the point. The obvious notion of an iterated product is, you start by multiplying two things, then you multiply more. You can't multiply fewer than two things. Can't be done.

There is a natural extension, yes. But it's an extension, and a convention. --Trovatore (talk) 21:14, 29 September 2019 (UTC)

For 1 element, there are 0 binary products. For 2 elements, there is 1 binary product. For 3 elements, there are 2 binary products. For 4 elements, there are 3 binary products. And so on. This is as simple as that. No need for an extension. And see the formal definition of an iterated binary operation: this starts at 1 element, not at 2. Vincent Lefèvre (talk) 22:07, 29 September 2019 (UTC)

You said it yourself: "For 1 element, there are 0 binary products." Just so. You can't multiply fewer than two things, exactly as I said. The fact that someone has made a formal definition does not change that. You're really off-base here, Vincent. This is a convention, not the obvious notion per se, notwithstanding (and I agree) that it has a certain naturality to it. --Trovatore (talk) 22:20, 29 September 2019 (UTC)

I have never said that I wanted to multiply fewer than two things. The definition of the iterated product does not imply that. If you are going this way, as McKay say, all mathematical notation is convention. So, whether one starts at 1 element, 2 elements, 3 elements or more, this is a convention. Vincent Lefèvre (talk) 22:36, 29 September 2019 (UTC)

I think your "the" in "the" definition of iterated product is a bit -- overstated. It isn't "the" definition, it's a definition. The obvious notion really does start with 2. You can't multiply fewer than 2 things, period, but you can multiply 3 things, given that multiplication is associative.

So sure, they're both conventions, but the one for empty product is more conventional than the one for 2 or more elements. --Trovatore (talk) 22:46, 29 September 2019 (UTC)

No, one cannot multiply 3 things. The multiplication is defined on 2 inputs. Then you have an extension called iterated binary operation applied here to multiplication; this extension covers 3 inputs and more, but also 1 input. I suggest you do some effort reading that article instead of wasting my time. Vincent Lefèvre (talk) 23:53, 29 September 2019 (UTC)

Sorry, that article does not help your argument. Are you really going to bikeshed over this. This is a convention; it's not a theorem of the axioms of a monoid. Trovatore isn't wasting your time.--Jasper Deng (talk) 00:39, 30 September 2019 (UTC)