Talk:Natural number

I don't think it makes sense to discuss both the non-negative integers and the positive integers in the same article, when integers gets its own article. The difference may only be a single element (namely, zero)—but this single member is of great importance, e.g. when discussing factorization, being that, while it's the additive identity, it's also the multiplicative annihilator. You do not want to define a domain of a function and get these two mixed up.

Further, this article and Whole number both suggest that "whole number" and "natural number" are synonyms, but there is plenty of literature that use both terms, in these cases, they have distinct definitions, usually "natural number" excluding zero.

Given this fact, Wikipedia is not a dictionary suggests that when multiple distinct concepts can be isolated, they should get their own article. The only exception being if the article "discusses the etymology, translations, usage, inflections, multiple distinct meanings, synonyms, antonyms, homophones, spelling, pronunciation, and so forth of a word or an idiomatic phrase" which is clearly not the case here. The fact that the layman sometimes uses terms interchangeably, instead of in the meaning isolated by the articles, is not an excuse.

This article is not a discussion of the word, or the term, it is a discussion of a specific mathematical concept. The fact that, historically, the non-negative integers (or integers, or positive integers) has at times been called the "whole numbers" and/or "natural numbers" is a fact to list in the relevant article whatever its name may end up being; this does not make an excuse to combine the two concepts into the same article.

I see two obvious corrections, either all three concepts should share Integer, which would discuss related subsets and how that affects their mathematical properties; or the relevant portions of this article moved to Whole number. Alternatively, appropriate sources could be added to show why these two sets are, in fact, the same mathematical concept deserving of a single article. Awwright (talk) 05:37, 17 June 2024 (UTC)

:

:Look, of course the two sets are not literally the same thing. That's not the point. The point is that there's very little that we want to say about the natural numbers that depends on whether or not zero is included.

:As for "whole number", that's a term that is not much in use in research mathematics. --Trovatore (talk) 05:57, 17 June 2024 (UTC)

::I mentioned it up above, there is actually quite a lot that can be said. What you call "very little" is more than most articles, e.g. I just clicked random page and got Usina do Gasômetro, it is 3 paragraphs. There are definitely 3 paragraphs of information each about the positive integers and the non-negative integers. And there are definitely a ton of reliable sources, so they are both notable. To me it is obvious they should be separate articles. But apparently 8 people disagree. 8 vs. 2 now, maybe it is time for another split proposal. 😄 Mathnerd314159 (talk) 06:30, 17 June 2024 (UTC)

::: "Can be said" is not the same as "want to say". Sure, there's a fair amount you could potentially say about off-by-one errors, but it doesn't live naturally in an article about the natural numbers.

::: Basically no one studies "the natural numbers with zero" and "the natural numbers without zero" as distinct objects of study. Sure, occasionally you will find someone who has symbols for both of them, but that is not the same thing. The natural numbers are an incredibly rich mathematical structure, the study of which has been the principal preoccupation of the entire professional lifetimes of many many brilliant people. None of those people {{efn|This sort of categorical statement is always risky; I imagine you can find someone who has both done good work and also claims to make an important distinction, but such a person would at the very least be an outlier}} divide that study into the structure with or without zero. They pick one for definiteness, but recognize that everything they say would translate with minor changes to the other convention. --Trovatore (talk) 01:33, 18 June 2024 (UTC)

::::The situation here is kind of similar to what you describe with the off-by-one, most of the article is about nonnegative integers and then there is some stuff about positive integers unnaturally mixed in. It is true no one studies "the natural numbers with zero" and "the natural numbers without zero", but that is because they are unnatural terms. There are plenty of textbooks that define positive integers and nonnegative integers as distinct objects of study and use them precisely.

::::I don't agree that the natural numbers are a mathematical structure. A mathematical structure has one definition but the natural numbers have two - no set both contains and does not contain 0. And I would argue that each paper's picking a definition does divide the literature up. As soon as you get past the basic Peano axioms, nothing translates without major changes or adding ugly conditions like "≠0" - for example, exponentiation on positive integers is well-defined, but 0^0 is not. If it really was completely equivalent there would not be a debate, there would be a theorem. Mathnerd314159 (talk) 04:40, 18 June 2024 (UTC)

:::::OK, you're again descending into quibbles that make it hard for me to believe you're taking this seriously. --Trovatore (talk) 05:18, 18 June 2024 (UTC)

::::: Mathnerd, you're wasting your time here. You've repeated the same couple of points now ad nauseam, while throwing in a mishmash of irrelevant apples-to-oranges comparisons, non sequiturs, and straw men, but it's not convincing anyone. If 8 people trying to explain why this seems like a bad idea was too few for you to get the point, you are welcome to canvass WT:WPM where you can probably get another 10 or so Wikipedians to voice their disagreement with you. Or you can take your discussion to twitter or something. It's not going to accomplish anything though. –jacobolus (t) 07:06, 18 June 2024 (UTC)

:::::The fact that a few things depend upon one's choice of convention doesn't mean that there are two separate concepts or that splitting the explanation across two pages would help anyone learn. XOR'easter (talk) 16:56, 18 June 2024 (UTC)

{{notelist}}

So regarding the edits by User:121.211.95.94... they actually seem reasonable? Specifically the edits are:

{{TextDiff|Sometimes, the whole numbers are the natural numbers as well as zero.|Sometimes, the whole numbers are the positive integers as well as zero (that is, the non-negative integers).}}

{{TextDiff|For example, the integers are made by adding 0 and negative numbers.|For example, the integers are made by adding 0 (if not already included) and negative numbers.}}

These both seem in keeping with the concept that 0 may or may not be considered a natural number. I've been reading the article several times like @Remsense suggested and they still look like good edits. Well, the whole numbers could be shortened to "Sometimes, the whole numbers are the non-negative integers." as the non-negative integers are already defined. Mathnerd314159 (talk) 05:51, 29 April 2025 (UTC)

:I suspect the point here is that sources that use the phrase "whole numbers" in this meaning generally do not include zero in the natural numbers, since then there would be no reason to use both terms. If that is the intent, I have to say I think phrasing could be found that makes the point more directly and concisely; I also struggled to figure out what people's objection was to 121's changes. Maybe something like {{tq|some texts use the term whole numbers to refer to the set with zero included, and natural numbers to refer to the set without zero}}. Just a first cut; I don't completely love it but I think it's better than what's there now.

:As an aside, I would prefer not to say "integers" too often in the lead section. I'm not worried about logical circularity, but I think it might be hard to follow for some readers — they come here to learn about natural numbers, and they get shunted off into a discussion of integers, and it might be extra stuff to keep track of. --Trovatore (talk) 06:50, 29 April 2025 (UTC)

:The problem with IP's formulation is that it uses the integers just before defining them. D.Lazard (talk) 07:35, 29 April 2025 (UTC)

::Aye, this is what I saw. It's a shame we never got to that point because someone decided it was investigation o'clock. Remsense ‥ 论 07:44, 29 April 2025 (UTC)

While elementary education often presents real numbers in terms of their decimal representations, once you try to put things on a rigorous basis they turn out to be cumbersome, and actual mathematics texts use simpler abstract definitions. The article is written from the perspective of elementary education and does not even acknowledge the existence of alternatives. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:14, 29 April 2025 (UTC)

:Here, there is no question of point of view. This is only is a question of WP:TECHNICAL: Because of its subject, this article is intended for readers with low mathematical background. Such readers may have heard of real numbers, and generally think of them as infinite decimals. But, probably, they do not care of the distinction between terminating and non-terminating decimals. This distinction and the existence of better mathematical definitions of real numbers are clearly too technical for a sentence that says just that the real numbers extend the natural numbers.

:In any case, the lead is not the place for a "rigorous basis" nor for abstract definitions of the real numbers (given in the linked article). D.Lazard (talk) 13:54, 29 April 2025 (UTC)

::Would you object to throwing in informally? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:48, 29 April 2025 (UTC)

:::Not useful, since real numbers can be formally defined by their infinite decimal representation. Moreover introducing "formally" in the lead of this article may confuse many readers who have no idea of the meaning of this jargon term. D.Lazard (talk) 15:15, 29 April 2025 (UTC)

So I want to point out [https://arxiv.org/abs/1102.0418v1 this paper by Harremoës], which as the IP requested does indeed question whether 0 should be considered an ordinal or cardinal number, in fact coming to the conclusion that 0 is not an ordinal number. Specifically the line seems to be "there is no need for the label 'zeroth' in this system because an empty set has no element that should be assigned a label like 'zero' or 'zeroth'". As far as cardinal numbers, I think 0 probably is generally included as a cardinal number (cardinal number says so), but probably if you poked around enough in old set theory textbooks you might find one that uses a convention that the empty set doesn't exist.

So anyway, in terms of the article, the "usually" phrasing definitely seems necessary. But unfortunately, regarding the paper by Harremoës, it is a self-published arXiV paper, so unless there is consensus that he is a subject matter expert, it is probably not reliable enough to cite. You can see [https://ieeexplore.ieee.org/author/37283348500 his bio], he has a PhD and edited (is still editing?) [https://web.archive.org/web/20221223043104/http://www.harremoes.dk/Peter/ 3 journals] and has given invited talks at conferences and so on, but I don't know if that's enough. Mathnerd314159 (talk) 04:28, 31 May 2025 (UTC)

:A self published paper is rarely a reliable source. Anyway, all formal definitions (Peano axioms in particular) define primarily natural numbers as ordinal numbers, and the fact that they can serve also as cardinal numbers is a theorem. So, there is not mathematical distinction between finite cardinal and ordinl numbeers.

:About "there is no need for the label 'zeroth' in this system because an empty set has no element that should be assigned a label like 'zero' or 'zeroth'": it would be problematic if mechanical counters could not be initialized to 0. So, at least in common practice, there is a need for 0 as an ordinal number. D.Lazard (talk) 11:09, 24 June 2025 (UTC)

I was getting ready to revert [https://en.wikipedia.org/w/index.php?title=Natural_number&diff=1297121231&oldid=1296025814 this change] by {{u|Physicsworld8}}, because it removes the direct link to the ISO sample doc, leaving a paywalled version. But then I provisionally changed my mind, in case this is a copyright issue. Of course I think it's terrible behavior on the part of ISO to promulgate allegedly public standards and then charge for access to them, and for that matter I don't like them (ISO) much in the first place and especially don't like them having the arrogance to stick their noses in mathematical usage. However, that's not a question for Wikipedia policy. Or even if it's not a copyright issue, I'm not sure the draft is considered a reliable source.

Frankly a better option would be to find a way to get rid of the ISO cite altogether. Surely we can source this notation to a textbook somewhere? --Trovatore (talk) 08:15, 24 June 2025 (UTC)

:My understanding is that iteh.ai is a legitimate licensee of ISO - SIST is a [https://www.iso.org/member/299538.html member body of ISO] and SIST [https://ecommerce.sist.si/ just reframes the iteh site]. Presumably they have gotten permission to show PDF previews, like other providers - they are just more generous previews. In this case it was a happy coincidence that the preview included the relevant info and I didn't have to figure out how to get the full standard. AFAICT it is not a draft, it is a sample of the official standard. It might be possible to get rid of the cite there but the ISO standard is also referenced directly in "Emergence as a term", so that cite isn't going away. Anyways there is WP:PAYWALL which says "Do not reject reliable sources just because they are difficult or costly to access."

:As far as the physicsworld8 edit I don't really understand the motivation behind it, even besides deleting the source, it also deletes the chapter/section. This user seems to be a newly-created single-purpose account which modified ISO standard references at 1 edit per 8 minutes for 2 hours. I would say not to think too hard about it and just click that revert button. Mathnerd314159 (talk) 21:24, 24 June 2025 (UTC)

::I would prefer to substitute the ISO cite in any case, paywalled or not, because I don't think ISO has any business in mathematics. We should find a real mathematical source. --Trovatore (talk) 21:38, 24 June 2025 (UTC)

:::Well the sentence is actually about several notations: superscript *, +, subscript >0, >=0, and subscript 0 superscript +. The 1978 (first) ISO standard has superscript *, I would suspect this notation might have originated with ISO. The current ISO standard has superscript * and subscript >0 / >=0, this subscript usage I am not sure where it came from. For the remaining notations I would suspect that the only available sources are notation sections in textbooks that use the notation.

:::If you don't like the standard, there is a book "Mathematical Expressions" by Jukka K. Korpela that cites the ISO standard and explains how to use the ISO notation with LaTeX and so forth, but personally I think just citing the standard is better.

:::As far as finding a "real source" that actually discusses the different notations as a subject, good luck - I had enough hardships finding sources for the definition of natural numbers, and even Enderton is a short note. Mathnerd314159 (talk) 22:01, 24 June 2025 (UTC)