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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
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630:. 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. â Carl
369:= 0 then the first term is 0/0!. 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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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
257:? 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? â Carl
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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
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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
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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.
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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.
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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
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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 7Â ĂÂ 3Â ĂÂ 4Â =Â 84. More precisely, we
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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
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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.
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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 â and no, you can't "prove" it's the right one.
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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
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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.
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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
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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.
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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. --
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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
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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.
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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
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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
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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.
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That relation is a desirable property, but is not the only way iterated products could be defined. Therefore it is an external criterion. --
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say, all mathematical notation is convention. So, whether one starts at 1 element, 2 elements, 3 elements or more, this is a convention.
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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
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So sure, they're both conventions, but the one for empty product is more conventional than the one for 2 or more elements. --
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a convention, not the obvious notion per se, notwithstanding (and I agree) that it has a certain naturality to it. --
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I think your "the" in "the" definition of iterated product is a bit -- overstated. It isn't "the" definition, it's
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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No, one cannot multiply 3 things. The multiplication is defined on 2 inputs. Then you have an extension called
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it is an indeterminate form. The two statements are both true. Why would anyone have a problem with that?
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is obtained from this hypothesis (the case of one value is completely part of the definition of an
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much like a self-evident desideratum since this whole thing is supposed to be about mutliplying.
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with the recurrence relation). I'm not basing my proof on something that is not in the article.
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before doing mass systematic removals. This message is updated dynamically through the template
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is extremely useful. So we have adopted it for our own benefit, not because we were forced to.
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the limit (the value comes from the property), and determining the limit constitutes a proof.
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then when x tends to 0 the result is 1. So there's no problem with the summation formula of
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I think that's just badly phrased and should be reworded. I'll go ahead and do it. â Carl
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When "CLEAR" is pressed and then some numbers are entered, their product is displayed.
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choice, but the notion of goodness is based on values external to the definition. --
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When a number is displayed and one enters another number, the product is displayed;
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The point is that the obvious idea of repeated products starts with a minimum of
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There is a natural extension, yes. But it's an extension, and a convention. --
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is zero", which by some editors is considered a convention rather than a fact.
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and both x and y tend to 0 that we must be carefull and analyse the situation.
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https://web.archive.org/web/20150217225003/http://planetmath.org/emptyproduct
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then it's always 0 as long as x is non-negative and if we are talking about
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Sorry, that article does not help your argument. Are you really going to
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The recurrence relation is the hypothesis. The proof is how the value of
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There's no problem and there's no convention as if we are talking about
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over this. This is a convention; it's not a theorem of the axioms of a
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is from the recurrence relation stated in the article, which includes
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It is a reminent from: "This justifies the convention that 0 = 1 when
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is an extension of multiplication from a binary operation to an
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There is no such proof. Not from the definition of the concept
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it's always 1 regardless of what x is. It's only when we have
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When you have finished reviewing my changes, please set the
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The exponentiation in the first term on the right-hand side
355:{\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!},}
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for additional information. I made the following changes:
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I certainly don't agree that 0Â =Â 1 is only a convention.
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Here's the proof of the uniqueness: Consider the case
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A number is displayed just after "CLEAR" is pressed;
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That one is a convention. But 0^2 = 0 isn't. â Carl
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324:
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313:
309:
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296:
275:At least they
273:
272:
255:exponentiation
231:
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194:
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40:WikiProjects
1759:Jasper Deng
900:Sourcecheck
365:since when
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1816:Categories
1577:semigroups
986:reasonable
980:Convention
970:Report bug
760:calculator
758:Imagine a
425:convention
406:convention
277:implicitly
1719:Trovatore
1680:Trovatore
1642:Trovatore
1607:Trovatore
1523:Trovatore
1463:Trovatore
1425:Trovatore
1319:Trovatore
1068:Trovatore
1032:Trovatore
997:Trovatore
953:this tool
946:this tool
816:empty sum
573:Bo Jacoby
429:Trovatore
391:Trovatore
187:Archive 2
182:Archive 1
1751:bikeshed
959:Cheers.â
763:specify:
735:Linkato1
498:Linkato1
170:Archives
1440:defined
886:checked
863:my edit
215:90 days
139:on the
1762:(talk)
1755:monoid
1060:per se
894:failed
36:scale.
1694:McKay
1635:fewer
1556:McKay
1459:proof
811:value
1741:talk
1723:talk
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1684:talk
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1646:talk
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1592:talk
1560:talk
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1467:talk
1461:. --
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1036:talk
1028:good
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890:true
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892:or
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634:CBM
591:CBM
553:CBM
261:CBM
236:and
131:Low
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