# Talk:Natural number/Archive 2

 ← Archive 1 Archive 2

## A successor

Maybe the Peano axiom "Every natural number a has a natural number successor, denoted by S(a)." should rather read "Every natural number a has exactly one natural number successor, denoted by S(a)." I know, that the last (mathematical induction) axiom implies this, but the proposed form (in contrast to the current form) is in accord with the use of the term successor here ("the successor").

Or change the use of the term successor so that it is not silently assumed that the successor is only one, could be even better. (something like a+1 instead of S(a) and "if a property is possessed by 'some' successor" instead of "... possessed by the successor" in the last axiom) --trosos 213.220.249.112 15:55, 13 January 2007 (UTC)

By the definition above, zero seems to be a natural number as it has a successor. Yet stretching this a bit further makes -1 a natural number as it too has a successor. I recall Peano got round the problem by first having an axiom defining zero as a natural number. Then the successor function is used to build the set and -1 never gets a look in. John H, Morgan (talk) 17:26, 2 December 2008 (UTC)

## |N

I've seen the set |N discussed in the sci.math newsgroup occasionally; is this another notation for N, or does it have some other special meaning? Or is this an attempt to represent a special character within the limitations of ASCII? Should it be mentioned in the article? I found this informal definition in a newsgroup post:

The informal definition |N = {1,2,3,...} is usually taken to mean that |N is a set S such that
(1) 1 is a member of S
(2) for each member n of S, n+1 is also a member of S, and
(3) |N is a subset of every set, S, with properties (1) and (2).

I'm not clear on (3), except that I think it means that S can be any set containing consecutive naturals (and possibly other members as well), and therefore |N is a subset of any such set. — Loadmaster 23:13, 9 April 2007 (UTC)

It's an attempt to mimic blackboard bold in ASCII. It's almost always more confusion than it's worth. No, I don't think it deserves mention in the article. --Trovatore 23:16, 9 April 2007 (UTC)

## Definition

Couldn't the first sentence just be "A natural number is a number greater than zero with no decimal separator"? For people who just want to know the basic definition without reading the entire article? —Preceding unsigned comment added by 86.76.137.45 (talk)

No. That is a definition by non-essentials. You should not confuse a number with a particular representation of that number (in this case the decimal representation). JRSpriggs 10:38, 22 April 2007 (UTC)

## Peano axioms and isomorphism

The section on the Peano axioms claims that "All systems that satisfy these axioms are isomorphic". This would seem to contradict both the incompleteness theorem and the Löwenheim–Skolem theorem. 72.75.107.59 (talk) 01:29, 19 January 2008 (UTC)

Those refer to first-order logic. What the claim means is that all structures that satisfy the full Peano axioms, in the sense of second-order logic, are isomorphic. --Trovatore (talk) 01:33, 19 January 2008 (UTC)
Aren't the axioms listed there ("these axioms") all first-order? 72.75.107.59 (talk) 01:40, 19 January 2008 (UTC)
No, the full axiom of induction is not first-order. It becomes a first-order axiom schema if you limit the properties being considered to ones that can be defined by a first-order formula. --Trovatore (talk) 01:56, 19 January 2008 (UTC)

## Number 4

this is topical: Mathematics to Retire Number 4--Billymac00 (talk) 14:58, 4 April 2008 (UTC)

This is funny, but I don't think it should be in the article. Oleg Alexandrov (talk) 15:24, 4 April 2008 (UTC)

## THE Natural number

I am new to wiki and not sure if this is the proper way of posting a question, but I thought THE natural number was e (2.71828 18284 59045 23536...) since it appears in nature so often. (i.g. birthrates) Why is it not even noted on this page?

Just thought It should be noted since a single letter is hard to search for. see [1]

Fozforic (talk) 14:00, 23 September 2008 (UTC)

The term natural number, in mathematics, universally refers to the concept treated here (either the nonnegative integers or the positive integers, depending on your taste), and certainly does not include e, which however is the base of the natural logarithm function. The word natural appears in the names of both concepts, but that shouldn't be taken to indicate any close connection between them. This is in general the way mathematical nomenclature goes — multi-word terms mean what they're defined to mean, and their names should usually be taken as historical artifacts rather than as something you expect to be able to figure out the definition from. --Trovatore (talk) 17:02, 23 September 2008 (UTC)
This raises the question whether for the benefit of really ignorant people we should add a hat-note to disambiguate this, saying for example "For the base of the natural logarithms, see e (mathematical constant).". JRSpriggs (talk) 05:05, 24 September 2008 (UTC)
No, a search for e will get the reader to the desired page. --Salix alba 06:04, 24 September 2008 (UTC)

## citation needed

i removed the citation needed on "Some authors who exclude zero from the naturals use the term whole numbers, denoted \mathbb{W}, for the set of nonnegative integers. Others use the notation \mathbb{P} for the positive integers." if a citation is needed for that than there is quite a bit else that needs citation in this article. these symbols are very often found in math textbooks.

I thought that $\mathbb{P}$ referred to the set of prime numbers….
Kinkydarkbird (Talk Page) 09:23, 9 January 2009 (UTC)
i've seen it for both, and i have at least one text with me that uses $\mathbb{P}$ for the positive integers. —Preceding unsigned comment added by 71.192.103.225 (talk) 06:05, 26 January 2009 (UTC)

## Natural numbers as sets

The article mentions the alternative encoding of natural numbers as sets by 0 = {} and n+1 = {n}. I have heard that this encoding was used by Peano. Does someone know the reference? --Jan 91.180.52.246 (talk) 22:25, 5 January 2009 (UTC)

That would have been an odd thing for Peano to do, since his axiomata begin with $\mathbb{N}_1$. I believe that the construction that you here mention was proposed by Russell. —SlamDiego←T 06:04, 20 January 2009 (UTC)

## Non-word?

Wow, somebody really needs to learn more English morphology. Regardless of whether “definitionally” was superfluous (I was trying to capture that for which some other editor had been reaching with the mistaken claim that the inclusion of zero were “explicit”), it's a perfectly proper English word. —SlamDiego←T 06:00, 20 January 2009 (UTC)

## 0 is not the empty set

In the chapter "History of natural numbers and the status of zero" it says that 0 is the empty set. Sorry, but that is wrong. —Preceding unsigned comment added by 80.165.82.22 (talk) 19:49, 18 February 2009 (UTC)

What do you mean by "wrong"? There are multiple ways to define natural numbers in terms of set theory. Not all of them define 0 as the empty set, but the one that's generally considered "standard" for use in mathematics (whether you're doing mathematical logic or teaching undergraduates) very definitely does. And all of this is explained pretty clearly in the article. --75.36.134.30 (talk) 15:05, 26 February 2009 (UTC)

## Redundancy?

A question for you all: should we delete the text "This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null" in the Notation section? I tried editing it out for the following reasons:

• The content is repeated in the Generalizations section, so it is redundant.
• It goes against a suggestion in the Wikipedia Manual of Style for mathematics articles: "A general approach is to start simple, then move toward more abstract and technical statements as the article proceeds."

My deletion was reverted by another Wikipedian (see the history), who later gave the following explanation for the revert:

• Since "countably infinite" and "aleph-null" are defined in terms of the set of natural numbers, it seems appropriate to me to mention that fact in the section dealing with notations for the natural numbers and related things.
• A little redundancy helps communication.
• The section on generalization seems too far down in the article to introduce these ideas.

Any opinions on this? FactSpewer (talk) 03:27, 15 April 2009 (UTC)

## Quaeler would you please choose your formulation and reedit the page?

You deleted

Which one we should use depends on which is better suited for our purpose. E.g. in number theory, only the positive integers are denoted by natural numbers whereas in algebra it is convenient to denote the non-negative integers as the natural numbers.

and below

Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense.

and commented in the history

18:54, 24 June 2009 Quaeler (talk | contribs) m (21,121 bytes) (Reverted good faith edits by Achim1999; Personalization of prose unencyclopedic; heavy mathematical drill down in opening paragraph makes the article less accessible to the average reader.. using [[WP:TW) (undo)

Well, I "personalization" with the words "we" and "our" because I thought this is good style! If you prefer you can use the passive form like

Which one should be used depends on which is better ....

And to your point "heavy mathematical drill down": You should immediateley delete the term "Ramsey theory" Argh! This is much heavier, why not using "combinatorics" ?

Moreover you (ther reader) could ask why this set is called "Natural" numbers? This never is done in this article! I open the eyes of the reader hopefully by stating ".... whether 0 is natural in the intuitive sense."

I hope in good faith you will take up these ideas and improve the article concerning these points and not only delete important(?) information. Regards, Achim1999 (talk) 19:33, 24 June 2009 (UTC)

I see by your talk page that you're no stranger to introducing verbiage that people have issue with; similarly, i found that your edits did nothing to improve the readability of the page and introduced no 'improvement'. Additionally, articles are not a place to wax philosophic as your text was doing. To be more to the point, i don't see that the article needs improvement in the areas which you attempted to introduce it and, as such, see no place to "take up these ideas and improve the article". Quaeler (talk) 20:52, 24 June 2009 (UTC)
The sentence "Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense." has no meaningful content and is stylistically inappropriate for an encyclopedia. JRSpriggs (talk) 20:56, 24 June 2009 (UTC)

Are you pulling my leg? 2) The sentence "Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense." should be taken, to tell the reader, somewhere in the main article if you like, WHY is this set of numbers called NATURAL numbers! 1) The sentence "Which one we should use depends on which is better suited for our purpose. E.g. in number theory, only the positive integers are denoted by natural numbers whereas in algebra it is convenient to denote the non-negative integers as the natural numbers." (you may reformulate it in passive form, if you dislike the we/our style) gives the reader an explanation why there are two different sets used. And therefore exactly this is the position to place it, IMHO. And this should be undoubtfully an improvement. Please judge because on the matters concerning the facts and not because your personal feelings are influenced by reading e.g. my talk-page or because you know I favour to behind-ask critically given states, if you feel to act like as a reviewer. :-( Regards, Achim1999 (talk) 21:17, 24 June 2009 (UTC)

To be pedantic, the order of my analysis was:
1. Notice new edits on this article because it's on my watchlist.
2. Reviewed the edits.
3. Thought to myself, 'wow - this shouldn't be in the article'
4. Rolled back the edits; due to my tools, this also opens your talk page.
6. Thought to myself, 'oh - that makes a little more sense'
and not:
1. Came across your talk page.
3. Thought to myself, 'i should go scrutinize his edits'.
So i have originally judged the matter based on solely on your work, and not the impression your talk page later propped up. Quaeler (talk) 00:22, 25 June 2009 (UTC)

It seems to me you favor meta-discussion to avoid discussion on contens like just happens. Sorry, therefore I have read your talk page and get the impression that you already are in the set of guys you are hunting for in the eyes of some other faithful WP-users. Therefore my advice: think twice about your article-editing-habit here on WP! BTW: I still hope that "Quaeler" is not the name/word in german for your intention here on WP. :-/ Regards, Achim1999 (talk) 10:04, 25 June 2009 (UTC)

## About zero as natural number

In the article I read "To be unambiguous about whether zero is included or not, sometimes an index "0" is added ..." The funny thing is this isn't unambiguous as in my country (Belgium), zero is a natural number, and this symbol $\mathbb{N}_0$ means without zero, because $\mathbb{N}$ represents the natural numbers, which is with zero. Also for other number sets a sub-index 0 means "without zero" (in Belgium). Something extra I forgot first to mention: some people may wonder why things like $\mathbb{Z}^+$ includes zero, but zero is in Belgium a positive AND a negative number so it is logic that in that case things like $\mathbb{Z}^+$ means the positive numbers, which is with zero. In other countries in Europe is this not always the case, it depends.193.190.253.144 (talk) 21:47, 7 June 2008 (UTC)

Interesting -- it strikes me as really counterintuitive to write $\mathbb{N}_0$ for the specific purpose of excluding zero. However I also have not come across this notation as a way of allowing zero, which is what the article currently claims -- can anyone find this attested somewhere? --Trovatore (talk) 22:49, 7 June 2008 (UTC)
193's statement is at least partially confirmed by the Dutch Wikipedia. In the article Natuurlijk getal (Natural number) the notations $\Z^+$ and $\Z^-$ are given as including 0, while $\Z_0^+$ and $\Z_0^-$ are given as excluding 0. Also Positief getal (Positive number), Negatief getal (Negative number), and 0 (getal) (0 (number)) state that in Belgium the number 0 is considered both positive and negative. Dutch is one of the official languages of Belgium. The French Wikipedia is equivocal; for example, it defines Nombre positif (Positive number) as: un nombre qui est supérieur (supérieur ou égal) à zéro ("a number that is greater than (greater than or equal to) zero"), without explaining when the parenthesis is supposed to be in effect. Next thing, the French article states: Zéro est un nombre réel positif, et est un entier naturel. Lorsqu'un nombre est positif et non nul, il est dit strictement positif. ("Zero is a real positive number, and is a natural number. When a number is positive and non-zero, it is called strictly positive.") The Spanish Wikipedia, at Número positivo, has a similar text but is slightly clearer in expressing that the meaning is ambiguous. In Germany and Italy the number 0 is unequivocally neither positive nor negative.  --Lambiam 07:56, 8 June 2008 (UTC)
In French-speaking European countries, the word "positive" used to be used much as it is in English. "x is positive" meant x > 0. Now there are competing conventions. What used to be called positif ou nul (nonnegative) is now often positif (positive), and what used to be positif is now often strictement positif (strictly positive). (The word nul is an adjective that means "being zero".) I suspect that the change came about mostly because of the choices made by Bourbaki. The newer terminology is not universal, but it does predominate in France. Claims that the older usage has disappeared are probably exaggerations. Also, Canada has not followed French-speaking European countries with regard to the meaning of "positive". 128.32.238.145 (talk) 22:49, 16 November 2008 (UTC)

Since there is a lot of confusion in this area, I suggest we follow ISO so long as there is no reason to deviate from it. In particular, ISO 31-11 says that $\mathbb N$ includes 0, and also defines $\mathbb N^*=\mathbb N\setminus\{0\}$. Normally I would have suggested to use $\mathbb N_0$ instead of $\mathbb N$, for clarity. I believe this is relatively standard practice in Germany. But as it seems this would only increase the confusion for Belgian and Dutch readers, it's probably best not to do that. --Hans Adler (talk) 11:10, 28 April 2009 (UTC)

There's an ISO for math??? That's terrible; that should not exist.
Of course I personally include zero in the naturals, but it's not because of any silly standards org that no one's ever heard of (in mathematics). I think the correct solution for the article is just to report that some include 0 and some don't, which is the truth. ISO should just be ignored, as is the practice of mathematicians generally. --Trovatore (talk) 18:07, 28 April 2009 (UTC)
I think such standards actually have the potential to be quite influential. What I learned at school seemed to be directly inspired by DIN 5473. I agree that such standards don't have much influence on university maths, and note my disclaimer about reasons to deviate from them. But keep in mind that most people who deal with concepts such as the natural numbers after school are engineers, for whom industry standards generally are relevant. Another point is that the committees behind such standards often do research that is similar to what we as Wikipedia editors are doing when we try to standardise our notation across articles: They try to find out which usage is more common, and they try to identify historical trends. Taken all this together, I think all else being equal we should prefer the usage prescribed by an ISO standard.
The DIN norms for maths, which no doubt influenced the ISO norms, are due to the Ausschuss für Einheiten und Formelgrößen, which was founded in 1907 by 10 scientific organisations including the Deutsche Physikalische Gesellschaft and the Verein Deutscher Ingenieure. [2] The mathematical norms function like a manual of style: If they manage to convince people or organisations, they will follow them. After reading our article ISO 31-11 I must say it seems to be perfectly sane. It would seem strange for me to not follow them simply to prove one's independence. Of course one problem with these standards is that the practice of only making them available for a lot of money is incompatible with scientific culture. --Hans Adler (talk) 20:17, 28 April 2009 (UTC)
Our standards should be the same as they always are — usage in the literature and the mathematical community, period. What some standards body claims is irrelevant. I have no specific gripe about any of the choices I saw in a brief scan; that isn't the point. Our job is to reflect the prevailing usage, and at the moment I believe it is still correct to say that both usages of natural number are current. Therefore we should say so. --Trovatore (talk) 20:44, 28 April 2009 (UTC)
I didn't want to say otherwise, but I see now how what I wrote can be misunderstood that way. I came here because anonymous editors at countable set have been removing the 0 from the natural numbers. (An ISO standard is probably a much more convincing argument for such users than a literature survey, but that's of course not relevant here.) However, I do think that we shouldn't have to define what $\mathbb N$ means in every single article that uses this symbol; we should have a general convention for this, and $\mathbb N / \mathbb N^*$ is probably the best choice.
This discussion seems to have established (to my surprise) that $\mathbb N_0$ is ambiguous. Therefore it seems that the current text introducing the symbols needs changing. I am proposing to say that the natural numbers with/without 0 are denoted by $\mathbb N / \mathbb N^*$, and that it is also common to denote them by $\mathbb N_0 / \mathbb N$. If it can be sourced, I would also say that $\mathbb N_0$ is ambiguous because it can denote $\mathbb N^*$. --Hans Adler (talk) 21:17, 28 April 2009 (UTC)
Since we seem to agree that in current use the term "natural number" has become ambiguous, so is necessarily the notation $\mathbb N.$ We should not try to hide that unfortunate fact. −Woodstone (talk) 22:39, 28 April 2009 (UTC)
I don't want to hide the ambiguity. But like every good textbook we can say, the symbol is ambiguous and this is how I use it. Only that we need to be a bit more subtle and can't say explicitly how we use the symbol, both for stylistic reasons and because we will never get all articles consistent with whatever convention we choose. When we use the symbol $\mathbb N$ in Wikipedia, then in many cases (probably the majority) it makes a difference whether 0 is included or not. See WP:WikiProject Mathematics/Conventions for how we have dealt with some similar problems. I believe that use of the symbol (but not of the term "natural number") needs to be standardised across Wikipedia. This talk page seems to be the best place to talk about this, although depending on the outcome of the discussion here it should of course be proposed on the Conventions page as well. Perhaps something like the following works:
A recent trend is to denote the natural numbers including 0 by $\mathbb N$ and the natural numbers without 0 by $\mathbb N^*$. [3] This trend is reflected in recommendations for scientific writings such as ISO 31-11. However, the traditional practice of writing $\mathbb N$ for the natural numbers without 0 is still widespread. --Hans Adler (talk) 23:44, 28 April 2009 (UTC)

Yesterday evening, I stumpled about a paragraph in "Numbers" from Ebbinghaus et a., Springer, 1991, which states that 1stly R. Dedekind liked to start the Natural Numbers with one and 2ndly that G. Peano acknowledged that he was influenced by Dedekind's postulates when he defined his axioms for the Natural Numbers. Thus it appears to me that already in the beginning of the foundation of formal logic / axiomatization (around 1880) this "0 is / is not in N"-war started. ;-) Regards,Achim1999 (talk) 15:12, 8 July 2009 (UTC)

## Formal Language

I know this isn't the Simple wiki, but even so, is it really necessary to say "Either a member of the set of the positive integers, or a member of the set of the non-negative integers", instead of leaving "sets" out of it and just saying "Either a positive integer or a non-negative integer"? 174.46.172.13 (talk) 10:35, 30 June 2009 (UTC)

I understand your point, and I tried to address it. Hans Adler 10:55, 30 June 2009 (UTC)

## Do number theorists start the natural numbers with 1?

Although some number theory texts do start with 1, there are also many distinguished number theory texts that start with 0, and some of them are not so recent. For instance, see

• Serre, A course in arithmetic, Springer-Verlag, 1973, p. 115.
• Weil, Basic number theory, second edition, Springer, 1973, p. XIII.
• Ramakrishnan and Valenza, Fourier analysis on number fields, Springer-Verlag, 1999, p. 138.
• Rotman, Advanced modern algebra, Pearson, 2002, p.1. (This isn't really exclusively a number theory text, but the definition occurs in a section called "Some number theory".)

Therefore I suggest removing the bit in the history section about number theorists keeping the older tradition of starting with 1. --FactSpewer (talk) 20:44, 8 November 2008 (UTC)

I don't understand in what way Woodstone's rewrite of the opening paragraph is "more balanced". I feel that the original wording matches actual usage better, because it is true that currently "natural number" can mean either an element of {1,2,...} or an element of {0,1,2,...}. The first sentence of the revision suggests that the former is a thing of the past, which is not the case. The second sentence of the revision suggests that all authors in mathematical logic, set theory, and computer science start with 0 (I would hesitate to make such a claim), and is ambiguous on whether starting at 1 is done by everyone else, or by no one else, or ... . --FactSpewer (talk) 04:34, 28 April 2009 (UTC)

After a neutral "0 is in or not", the preceding version had a phrase: "the latter is especially preferred in (some fields)". So only a positive statement for including 0. I wanted to add something positive for not including 0. It cannot be doubted that 0 is a relatively new concept compared to the positive natural numbers. So "originally start from 1" is true. Some specific fields now often include 0, is true as well. I agree this is not universal in those fields, so we could add a remark about that, and make explicit that other fields stick to the original definition. −Woodstone (talk) 07:44, 28 April 2009 (UTC)
OK, thank you; I understand your thinking now. Probably whoever wrote especially preferred meant not that it was better, but that it was more common in those fields; but I agree with you that that wording could be construed as making a value judgment, so let's avoid it. I'll just word the opening sentence to make it clear that both conventions are used now; and I'll leave the history to the history section, which agrees with what you say, and expands upon it in detail. --FactSpewer (talk) 04:15, 12 May 2009 (UTC)

Divisibility and 0 bites each other very much in theory. :) Thus 0 was never paid attention in classical number theory and it was not missed! Achim1999 (talk) 18:55, 24 June 2009 (UTC)

Division works somewhat differently in number theory and analysis. In analysis one is simply not permitted to divide by zero. In number theory division is replaced by the notions of divisibility and congruence, where divisibility and zero get along just fine. The positive integers under the divides relation m|n form a distributive lattice that is not complete, whereas the nonnegative integers form a complete distributive lattice with 1 at the bottom and 0 at the top (the opposite of the usual convention for naming top and bottom in a lattice). In particular, without zero one can take the GCD of any infinite set of positive integers but not the LCM, whereas with zero one can take both the GCD and LCM of any infinite set of nonnegative integers. Hence number theory is better off when it includes zero because doing so expands the available operations.
In computer science the natural numbers always start from 0, as a consequence of the convention of breaking the cycle ... < 110 < 111 < 000 < 001 < 010 < ... between 111 and 000 when interpreting bit strings as unsigned integers, more precisely the integers mod 2n for Ironically computer keyboards break it between 000 and 001 (so to speak) as a result of typewriters having always done so, putting 0 at the right. Rotary telephones also did so, but that was because dialling n produced n pulses and if 0 had produced no pulses instead of ten pulses the switchboard would never hear 0. --Vaughan Pratt (talk) 05:40, 17 August 2009 (UTC)

## Only mathematical well-defined statements wanted in the definition!

Sorry, do be pressed to open this section: But we dislike to see subjective, social unnecessary statments in the beginning (perhaps also in the wohle article?) of this scientific-supposed-to-be article! What N is, is stated precisely. No need for your personal oppinion to be added!

Regards, Achim1999 (talk) 11:49, 5 July 2009 (UTC)

I support the clarification there that the distinction is a matter of each author's convention. Without that clarification the "or" is more difficult to understand. — Carl (CBM · talk) 12:33, 5 July 2009 (UTC)
Achim, your argument makes no sense, and your attempt to enforce your removal of a stylistically necessary clarification by means of wiki-lawyering [4] was completely out of order. Given your obvious problems with the English language you should definitely not be edit-warring over style. You are probably not aware of it, but that's exactly what you are doing. Hans Adler 13:36, 5 July 2009 (UTC)
People who can not or want not to argue by contens but prefer to argue by pointing out wording, style and typos should be better ignored with their talk. :-(

Regards, Achim1999 (talk) 21:30, 5 July 2009 (UTC)

"..depending on context." is an empty phrase which is always correct. Thus adding this gives no further information but only blows up the writing. Escpecially we are here in a mathematical definition, hence one expects short, simple and clear wording, and no story-writers using fill-words. Argh!

Regards, Achim1999 (talk) 21:30, 5 July 2009 (UTC)

In dingo culture, a human being is either a man from asia or a woman from africa. What is unclear? And what becomes clearer by appending "depending on the context." to this sentence? *shaking my head* Regards, Achim1999 (talk) 21:35, 5 July 2009 (UTC)

Better wording-suggestion:

In mathematics, a natural number is either from the set {1, 2, 3, ...}, hence a positive integer, or from the set {0, 1, 2, ...}, then called a non-negative integer.

Regards, Achim1999 (talk) 21:58, 5 July 2009 (UTC)

That's not a good solution to the problem, and since your made-up example seems to confirm my suspicion that you are completely missing the issue, here it is: The purpose of "depending on context" or "depending on the author's convention" is to make it immediately clear that "either ... or" separates two different conventions rather than being part of a single, universally accepted definition of the natural numbers. This is an unusual situation, especially for such an elementary and well-known notion. We can expect that most readers 1) are not mathematicians, and 2) are familiar with only one convention and will be surprised to learn there are two. Without very clear hints they may well not understand this, and instead think that for some reason which they don't understand we are describing the convention they know in impenetrable mathematical jargon.
Note that your dingo example has exactly the same problem. Without "depending on context" it would be possible that a dingo can say "two human beings" to refer to a man from Asia and a woman from Africa. Adding "depending on context" or "depending on the dingo's convention" would clarify that there are two dingo dialects: one in which "human being" can only refer to a man from Asia, and one in which "human being" can only refer to woman from Africa.
Oh, and "depending on context" is of course not always correct. E.g. "an irreducible natural number is either the number 1 or a prime number" is correct, while "an irreducible natural number is either the number 1 or a prime number, depending on context" is plain wrong.
Your proposed reformulation is only marginally better in this respect than the version you have twice reverted to. It is also written in a very clumsy style, and any attempts to fix it would probably lead to what we have now (with or without the clarification).
By the way, I have two questions for you regarding "we dislike to see subjective, social unnecessary statments in the beginning":
1. Are you a single person or is "Achim1999" a group account? If you are a single person, who else do you think you are speaking for?
2. After my explanation, do you still believe that the explanation is unnecessary? If so, could you please elaborate what it is you don't like about it; I don't think there are many editors here who share your opinion.
Hans Adler 23:32, 5 July 2009 (UTC)
There are two conventions on what a natural number is:...? Septentrionalis PMAnderson 00:40, 6 July 2009 (UTC)

Thanks for finally give information of the contens and context which you think is ambigious. This appending "depending on the context." makes nothing clearer to me. Because this interpretation has nothing to do with english language, it occurs in german too and probably in many other language this is possible. But now I know what the misinterpretation could be. I only wonder if you think "Without very clear hints" is necessary why you are reluctant to give these very clear hints and make a better formulation which avoids exatly this supposed-to-be misunderstanding. E.g.:

In mathematics, there are two convention for a natural number: either it is a positive integer from the set {1, 2, 3, ...} or a non-negative integer from the set {0, 1, 2, ...}.

or perhaps

In mathematics, there are two convention for a natural number: either it is from the set {1, 2, 3, ...}, hence a positive integer, or from the set {0, 1, 2, ...}, then called a non-negative integer.

It should be easy, even for you ;-), to quickly generate contexts, such that "an irreducible natural number is either the number 1 or a prime number, depending on context." is a correct and useful statement. :)

1. Achim1999 is no group-account. I even did not know that this is possible here.
2. I don't know. I have not made up my mind. But I was already pointed to this guidline(?) by other people / editors(?) here.
3. I never thought of your misinterpretation-possibility. I still have problems to realize how to interpretate this sentence as a single unique definition. Perhaps this is due to the fact, that one choice is a subset of the other.
4. I still don't know whether there is a clarification necessary, sorry. (see my answer 3).
5. I think appending "depending on context" makes nothing clearer -- I think you believe it should cause a grouping of some words in this sentence in your understanding. But this I can't realize, honestly.

Regards, Achim1999 (talk) 11:17, 6 July 2009 (UTC)

Group accounts are physically possible, in the sense that we cannot prevent someone else from sitting down with your password; they are forbidden, and grounds for blocking. Septentrionalis PMAnderson 16:32, 6 July 2009 (UTC)

BTW: I still wonder why you have pressed others (at least me) to make this discussion, and refuse to suggest a clear formulation which hits your point (two conventions are here in use) and therefore avoid such a waste of time. Regards, Achim1999 (talk) 11:41, 6 July 2009 (UTC)

I have pressed you? You reverted me twice for a reason that I suspect can be understood by nobody other than yourself ("we [sic!] dislike to see subjective, social unnecessary statments"), then you started the discussion here, and you unnecessarily notified me of it on my talk page. So far you have been reverted once by me, once (essentially) by JRSpriggs, and you have been told you are wrong by CBM. That's a score of 3 professional mathematicians and experienced Wikipedia editors disagreeing with you against a total of 1 editors (including you) agreeing with you. I think it's pretty clear at this stage that if you still don't understand things after I have taken the time to explain them to you, then it's entirely your problem. Wikipedia isn't school, and other editors here are not your teachers. We are not payed for explaining things to you that you don't want to understand. Hans Adler 15:05, 6 July 2009 (UTC)

Sadly, you seems really unwilling to stay factually. On such a base it makes no sense for me to discuss further! Sadly also, you like to judge wrong versus right by 3:1 opinnions getting in a few days. I can only hope that you act as a professional mathematician much more factually. Noone wanted you to act as a teacher -- at least I did not, but this seems also an off-topic attitude by you. Well, meanwhile other people took hand on this sentence and this very special interpretation-issue is gone. Unbelievable that you refused to make constructive changes which hit your point of (necessary?) clarification of those sentence. :-( Regards, Achim1999 (talk) 11:40, 7 July 2009 (UTC)

You were unable to express your concerns comprehensibly, which made it impossible for me to take them into account. Now the article has been changed to give slightly more detail. I think we had something very similar recently and somebody objected, but I don't have the time to look up the history now. In any case a first sentence with two parenthetical phrases is rubbish. And I have no idea why your strange objection doesn't apply to the present formulation.
If you have trouble communicating in English don't blame others for the fact. Thanks for the promise to end this silly discussion, by the way. Hans Adler 15:20, 7 July 2009 (UTC)

Yesterday a user added "... according to a more recent definition now also in common use." (I removed the parentheses). I'm really unhappy with the elastic wording "more recent". This should be made preciser. I believe it was first invented/defined by a formal approach in the 19th century to lay the foundation of numbers-notion (G. Frege / R. Dedekind). On the other hand the sentence should not become too long. Regards, Achim1999 (talk) 13:54, 8 July 2009 (UTC)

## Keep wording short, clear and precise -- a main feature of good encyclopidae!

Well, just again this non-factual (I heard, there is a WP-guidline to generally not name people by calling their account names, especially if their action can be considered bad) liking author, disqualifing himself with this kind of attitude in the long run, sprang in action and triggers me to write this general comment regarding all WP-articles -- but I like to give two examples first to make my point hopefully clear:

Facts to this special example:

1. Since many months this section "algebraic properties" were here located without significant changes.
2. Some days ago, I noticed, that a,b, and c are in the mathematical interpretation so-called free variables. This I dislike, because in the langugual interpretation given by the context they should be considered to hold only natural numbers.
3. Therefore I added a very precise, short, mathematical statement as lead-in to all the other mathematical expressions in the following list which is in table-form and which is everything in this section.
4. Suddenly people spring in action, to avoid this mathematical clarification, which is also easily understandable by typcial to-be-expected non-mathematical readers, only to rephrase it with superfluous words like story-writes, using the interpretable word "properties" without any need! I did not expect others to act like very good mathematical writes, like say, D. E. Knuth, but thoughtless (in the most positive assumption) changes should be avoided which only blows up articles and even decrease readability.
5. Argh! :-( Sorry.

And a further example in this article which happend recently: Another guy/account disliked the word "computist". Even me, who probably had never heard this word, knew immeditely what the original text/author wants to express with this wording. Then this guy sprang into action who dislike this wording and must replace it by computer. But in common use this has today almost always a different meaning, thus this guy had to add an addtion to clarify this. Ironically he added in his comment to his change, that in pre-computer-age this word "computer" was used with a unique meaning of computist. But it now seems that he prefers fighting for the ambigusity of "computer" and supports it here by reusing it where totally unnecessary. My well reasoned revert, in fact I used his(!) reasons of usage in past to revert it, he reverted aagin but without any comments. Sorry, this I can not call faithful editing by heart. And to be frank, more the opposite creaps in my mind from learning other changes in other articles here in WP.

These kinds of attitude/behaviour of editing will surely not encourage more experts from special fields to sacrify their time to write WP-articles, a wish WP wants strongly.

BTW: Sorry, to sound in this writing to act like a well-sounded teacher. :-/

Regards, Achim1999 (talk) 10:48, 15 July 2009 (UTC)

No such guideline exists. Having to hunt through page histories to find out who you're talking about is very annoying, so giving names is almost always appropriate. Algebraist 10:52, 15 July 2009 (UTC)
I was told by a deserved editor on a user-talk page, that WP prefer to avoid explicit naming if criticizing editors at least. And it seems in retrospect that I criticized another deserved editor in that case. Maybe he meant only critizing of deserved editors and his wording looked to me more than a proposed policy. Sadly I have deleted all user-talk-pages (except my own) from my watch-list.

Regards, Achim1999 (talk) —Preceding undated comment added 11:19, 15 July 2009 (UTC).

You were told wrongly. Algebraist 11:22, 15 July 2009 (UTC)
Why do you think inline symbols are more readable than clear English text? Algebraist 11:49, 15 July 2009 (UTC)
Surely not generally, but you should think why they exist (resp. were invented). And in this case, section "Algebraic properties", they cause much better readability than prose. Anyway, I think it is not worth to discuss with people like CBM · talk who acts before thinking.
Have fun to change this article in the future without my support. Regards, Achim1999 (talk) 17:26, 18 July 2009 (UTC)

Sometimes making a small edit draws attention of other people to lingering problems in a page. I'm not thrilled at all with the table in the "algebraic properties" section. I don't see any reason why we cannot use prose, perhaps a bulleted list. I'll see what I can do. — Carl (CBM · talk) 11:46, 15 July 2009 (UTC)

Yes, prose would be an improvement. Algebraist
I made some changes. I'm still not thrilled with the "algebraic properties" section being followed by the "properties" section; they should probably be merged. Also, there should probably be a section on the key property of natural numbers, which is that they support proofs by induction. There is currently just a link to mathematical induction buried in the article. — Carl (CBM · talk) 12:04, 15 July 2009 (UTC)

## Conventions

Saying that starting at 1 is the traditional convention, and that starting at 0 is the convention following a formal definition in the 19th century, is not entirely accurate. The issue is more complicated than this, and is best left to the discussions later in the article, where it is amply explained. For example, Peano's original definition started at 1, not 0 (it is only that some logicians modified his definition later on to start at 0, out of convenience). And both definitions occur in the modern literature. --WardenWalk (talk) 19:20, 1 August 2009 (UTC)

Could someone with more experience of these things than I (or access to a good academic library) please add a reference to a work which treats 0 as a natural number? The article makes it sound as though this is the more common modern day convention, but my 2nd edition copy of the Penguin Dictionary of Mathematics (for example) only lists the ℤ>0 version (which it also identifies as ℤ+, to further muddy the waters). Thanks. Aoeuidhtns (talk) —Preceding undated comment added 12:46, 5 February 2010 (UTC).

Literally any book on set theory or mathematical logic. I checked Levy, Jech, and Kunen's set theory books just now, and also Shoenfield; Hinman; and Boolos, Boolos and Jeffrey on the logic side. All of these begin the natural numbers at 0. I don't think I have seen any text in logic that begins the natural numbers with 1, although I can't say I've read every single book in the area. — Carl (CBM · talk) 13:26, 5 February 2010 (UTC)

I think some mention of convention by country should be mentioned. For instance here in the UK it is traditional to start the natural numbers at 1 whereas in France it is traditional to include 0. (I do not know of any references for this but it seems to be common knowledge amongst my professors and doctors (I'm doing a post grad in maths)). Porkbroth (talk) 20:08, 2 March 2010 (UTC)

I move to change the article to be in agreement with MathWorld, the standard online reference for mathematics. This would involve using "+" to indicate positive (instead of non-negative) and "*" to indicate non-negative. We shouldn't have the top 2 references that come up when you Google "natural numbers" disagree! Also, see: The MathWorld page on Positive Integers. watson (talk) 02:32, 1 October 2010 (UTC)

MathWorld is very very bad at terminology. We should not rely on it whatsoever. --Trovatore (talk) 02:41, 1 October 2010 (UTC)
Agreed. I don't see how mathworld is a standard reference for mathematics any more than wiki. If anything the opposite is true, mathworld being by and large the work of a single individual, whereas wiki is a cooperative effort involving a number of individuals. Tkuvho (talk) 04:36, 1 October 2010 (UTC)

## Nominal numbers

I agree with Hans Adler that nominal numbers should be dealt with separately. Note to Fullmetalactor: I would say that the 23 in "I live on 23rd street" is functioning as an ordinal number (by the way, be careful when correcting spelling!) Ebony Jackson (talk) 07:29, 17 August 2009 (UTC)

## Symbol unicode number

Would someone please mentions the symbol "ℕ" Unicode number next to it ? --DynV (talk) 07:16, 25 October 2009 (UTC)

You've made this request in at least two articles, maybe more, but I'm afraid I don't understand it. Could you please explain more clearly what you mean? --Trovatore (talk) 10:39, 25 October 2009 (UTC)

Certain symbols appear for some readers as a "ℕ" and for other readers as the intended symbol. When [itex] is used, this problem does not occur. So in case of doubt, let us please use [itex]. Bob.v.R (talk) 14:05, 25 October 2009 (UTC)

To Trovatore: Perhaps DynV means that he cannot read that character with his browser. It might look like an empty box or a question mark to him. (I presume he just cut and pasted the source to get it here.) He wants to know what numerical value is the code (the Unicode) for the symbol. JRSpriggs (talk) 19:43, 25 October 2009 (UTC)
That would be 2115 (hex), or entity "&#x2115;", showing as ℕ. −Woodstone (talk) 20:31, 25 October 2009 (UTC)

## merger

I think that Addition of natural numbers could be merged here; it has very little content and I don't forsee it growing much. — Carl (CBM · talk) 16:46, 21 January 2010 (UTC)

The "definition" of addition in Peano arithmetic (although called "in the natural numbers") should be moved there or to a related article, rather than here. I think perhaps a disambiguation might be left between addition in the natural numbers, pointing here, and addition in Peano arithmetic, pointing there. — Arthur Rubin (talk) 17:33, 21 January 2010 (UTC)
The recursive definitions of addition and multiplication certainly deserve to be in Peano arithmetic, but they are already there, in the section "Arithmetic". — Carl (CBM · talk) 17:37, 21 January 2010 (UTC)
Noted, sorry. I still think that addition of natural numbers should be left as a disambiguation page after the merger. — Arthur Rubin (talk) 17:52, 21 January 2010 (UTC)
Apparently I was grumpy this morning, I apologize for that. I do think a disambiguation page would be OK, and it will be good to save the edit history in any case. — Carl (CBM · talk) 19:11, 21 January 2010 (UTC)

## Smallest

When someone says "smallest group containing" or "smallest ring containing" they always mean in the sense of group or ring inclusion. For example, $\mathbb{Q}[\sqrt{2}]$ is the smallest ring containing $\mathbb{Q}$ and $\sqrt{2}$, although $\mathbb{Q}[\sqrt{2}]$ is still a countable ring. The integers are that smallest group containing the natural numbers in the sense that for any injective homorphism φ from the natural numbers to a group, there is an extension of φ which is an injective homorphism from the integers to that group. Indeed, that integers are precisely the group obtained from the natural numbers via the standard construction of extending a commutative cancellative semigroup into a group. — Carl (CBM · talk) 13:31, 5 February 2010 (UTC)

I'm slightly amazed by the mention of Cantor and set theory in the first paragraph. What does this have to do particularly with natural numbers? The article is called "natural number", not "the set of natural numbers". And even articles that are expressly about sets don't need to mention set theory explicitly. I am tempted to throw out the phase entirely (for now I just repaired it; it was calling the numbers themselves a set, which cannot be meant here), and to reformulate the following sentence so that it avoids the term set and the curly braces. Marc van Leeuwen (talk) 12:28, 22 April 2011 (UTC)

The reason for this is mainly historical: the page used to start by talking about the set of natural numbers, without any further explanation of "set". I am not sure if it would go over well to delete this altogether, though. Tkuvho (talk) 12:36, 22 April 2011 (UTC)
(edit conflict) I agree that the name-dropping of Cantor might be distracting. Since the next sentence in the lede already linked to set, I removed the sentence about Cantor entirely. Is that OK with everyone? — Carl (CBM · talk) 12:38, 22 April 2011 (UTC)
Well, perhaps not. Do we really need a further explanation of set there in the lede? — Carl (CBM · talk) 12:38, 22 April 2011 (UTC)

## Zero

The lead says "natural numbers are the ordinary counting numbers 1, 2, 3, ... (sometimes zero is also included)," but the bulk of the article uses the usual mathematical definition including zero (e.g. Natural number#Algebraic properties, Natural number#Properties, Natural numbers#Formal definitions, etc.). I suggest rewording to indicate zero is almost always included. -- 202.124.74.200 (talk) 07:29, 28 August 2011 (UTC)

The inclusion of 0 in the term natural numbers is a relatively recent development. Traditionally it was not included. I propose to leave the lead neutral about this issue. −Woodstone (talk) 15:56, 28 August 2011 (UTC)
Relatively recent, but quite common for about two centuries now, surely? And it would be nice if the lead was rewritten to be neutral, rather than taking an exclude-zero stand which contradicted the include-zero body of the article. -- 202.124.72.202 (talk) 23:32, 28 August 2011 (UTC)
"The natural numbers are inconsistently defined. This inconsistency requires those using the term to be specific about which set of numbers they mean. Some define the natural numbers as the set of counting numbers, excluding zero {1, 2, 3, 4, ...}. Others define the set as including zero {0, 1, 2, 3, 4, ...}."
This describes the inconsistency and the requirement for clarity when dealing with this set, and takes no bias in definition. Cliff (talk) 18:43, 30 August 2011 (UTC)
I am not in favour of starting out with focus on a controversy in the lead. The current phrasing is clear and neutral enough. You may want to replace "sometimes", by something stronger, like "regularly". −Woodstone (talk) 10:10, 31 August 2011 (UTC)

I made a bold change to address the problem. The article now starts: "[...] are the ordinary counting numbers 0, 1, 2, 3, ... (traditionally zero is omitted)." Hans Adler 12:52, 31 August 2011 (UTC)

I tried another bold change. I rearranged the lede to put the stuff I find more important first, and put the long paragraph about zero last. Really the issue of zero is not the first thing we want people to worry about when they read the article. I also tried tweaking the wording some. I am afraid that a naive reader may not realize that "traditionally" means "in older texts, but not necessarily in newer ones" - they may think it means "since long ago, and continuing to today", i.e. "by tradition". — Carl (CBM · talk) 13:04, 31 August 2011 (UTC)
Brilliant! It's much better this way. Hans Adler 13:30, 31 August 2011 (UTC)
The article states " Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number.", with a footnote stating "This is common in texts about Real analysis. See, for example, Carothers (2000) p.3 [5] or Thomson, Bruckner and Bruckner (2000), p.2". Thanks to the miracle of Google books, I can see Carothers, page 3, which states "n is the set of natural numbers (positive integers)". There's nothing here about zero as a natural number. - Crosbie 07:43, 1 September 2012 (UTC)
Basically, "in numbers" is used for meaning "which come in more than one". Therefore the greek numbers started with two, but this does not mean that the Greeks did not know how to handle quantities one, or zero. If we are using the word "number" as meaning "a quantity", zero has to be included. If we are dealing with real numbers by contrast, zero is just another version of the infinite. Askedonty (talk) 20:49, 5 November 2012 (UTC)

## Notation

In the article it is written that $\,\mathbb N_0=\aleph_0=\omega$, but should it not be $|\,\mathbb N_0|=\aleph_0=\omega$? If it is correct, then it should be more clarified that $\aleph_0$ can denote also a set of that size and not just the cardinality. — Preceding unsigned comment added by Tagib (talkcontribs) 11:29, 5 February 2012 (UTC)

Please read the sentence in which that formula occurs. — Carl (CBM · talk) 20:43, 21 July 2012 (UTC)

## Whole = integer

In the article:

others use whole number in a way that includes both 0 and the negative integers, i.e., as an equivalent of the integer term.[citation needed]

The Hungarian term for numbers in {..., -2, -1, 0, 1, 2, ...} is egész, which means — see a Hungarian–English dictionary — whole. So at least Hungarians tend to interpret/use whole number as integer. Consider this a citation.— Preceding unsigned comment added by 46.107.101.192 (talk) 23:22, 26 March 2013‎ (UTC)

Well, the Czech word for integer (celé číslo) also literally translates as whole number, but it’s just that: a literal translation. This does not count as a use of the actual English expression, as literal translations of mathematical terms often give nonsensical results: for example, you cannot cite German Körper as evidence that body is a valid English synonym for field.—Emil J. 12:35, 25 April 2013 (UTC)
Speaking of German, ganze Zahl also literally means whole number, of course. Integers are in fact called “whole numbers” in quite a few (most?) languages. That’s how the English term came about in the first place, as the Latin adjective integer means whole.—Emil J. 12:47, 25 April 2013 (UTC)

## Counting number and whole number

This article misleadingly gave the impression that counting number is always defined to include zero, that whole number is always defined to include zero (sometimes with the addition of negative integers), and that integer is sometimes defined in a different way to the usual definition. MathWorld says that there are also authors that define counting number and whole number to exclude zero, the Wikipedia article for Whole number agrees that there are three possibilities for that term, and I think there is general agreement about integer. So I have edited this article.

Note also that:

• Whole number has a link to Natural number#History of natural numbers and the status of zero, so it is good to put all relevant information in that section.
• There are additional references in a wikitext comment within the source of Whole number.
• If there are reliable sources, this article could explain what combinations of definitions are used (i.e. to explain how authors use the terms to distinguish the different sets of numbers). This is partly done in the Notation section, but without a citation.
• Although this article mentions the usage of the definitions in set theory, logic and computer science, it says nothing about the usage in number theory.

JonH (talk) 22:08, 6 August 2013 (UTC)

I like your changes, mostly because they're simpler than the previous text. I think, after a minimal mention to clear up confusions, the less time spent on the "status of zero" and on the locutions "whole number" and "counting number", the better. So that's basically to say I'm not enthusiastic about your last two bullets above — I think we should focus on the math, not the terminology. --Trovatore (talk) 22:12, 6 August 2013 (UTC)
I see what you mean. I think it is important that readers are told about the possibility of being confused; but perhaps the details of the confusion are not so important. JonH (talk) 22:26, 6 August 2013 (UTC)
Yes, that sounds about right. --Trovatore (talk) 07:32, 7 August 2013 (UTC)

## Minus one twelfth

FWIW - I don't know if my edit (based on a recent WP:Reliable Source) (recently reverted by User:Marc van Leeuwen) is entirely ok or not - but seems worth a discussion:

Copied from the Natural number lead:

Interestingly, the summation of all natural numbers to infinity is "minus one-twelfth".< ref name="NYT-20140203">Overbye, Dennis (February 3, 2014). "In the End, It All Adds Up to –1/12". New York Times. Retrieved February 3, 2014.</ref>

$\sum_{n=1}^{\infty} n = - \frac {1}{12}$

ALSO - A relevant video (07:49) by Numberphile "proving" the notion is at the following => http://www.youtube.com/watch?v=w-I6XTVZXww - A related discussion is ongoing at Talk:Infinity#Minus one twelfth - Comments Welcome of course - in any case - Enjoy! :) Drbogdan (talk) 17:10, 4 February 2014 (UTC)

This is an obvious fallacy and hence does not belong in the article. — Anita5192 (talk) 17:16, 4 February 2014 (UTC)
Agree it does not belong here. But if you read 1 + 2 + 3 + 4 + ⋯ you see that the methods of zeta function regularization and Ramanujan summation assign the series a value of −1/12.--Salix alba (talk): 16:16, 6 February 2014 (UTC)

Not this again, please. Sławomir Biały (talk) 01:13, 10 February 2014 (UTC)