# Talk:Natural number

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The term "Cauchy sequences" in the intro should be linkified. Thanks. 71.197.166.72 (talk) 19:19, 16 November 2015 (UTC)

Hmm — I'm on the fence between wikilinking it and just removing it (replacing it by something more vague). It seems more about the real numbers than about the natural numbers; probably too detailed for the lead of this article (though it could reasonably go in the body). --Trovatore (talk) 19:22, 16 November 2015 (UTC)
From my perspective of viewing at the integers as an important root of mathematical development wrt the manifold concepts of numbers, I'd like to keep these hints on means of extensions, be it by Cauchy sequences or by square roots or by inverse elements. Maybe the intricate details of the processes of extension are beyond the assumed average of math literacy of expected readers, and so linking to Cauchy sequences might overburden some, nevertheless, I plead for keeping these hints, be it as links, rather than removing them. Purgy (talk) 10:01, 17 November 2015 (UTC)
I think that would be reasonable in the body. I would frankly prefer not to have them in the lead. Partly because of the difficulty of the material, but more because it strikes me as tangential to the subject of this article, which is the natural numbers, not the integers or the rationals or the reals.
For the lead, I would look for wording that mentions that these more complicated structures can be "built up" from the naturals, but without saying much about how. --Trovatore (talk) 17:19, 17 November 2015 (UTC)
It's fine if you want to move (or remove) the mention of Cauchy sequences, but wherever it appears first, it ought to be linkified. The worst possible combination is to leave the term in the lead and NOT link it.71.197.166.72 (talk) 05:55, 19 November 2015 (UTC)
Hmm, yeah, that's true. But there's no huge hurry about it. We can decide which we want to do, and then link it wherever it appears, if it does. --Trovatore (talk) 07:39, 19 November 2015 (UTC)

I thought adding the links does not hurt anyone, fits to not only my preference and is no prejudice to the desires of Trovatore. -- Purgy (talk) 15:56, 19 November 2015 (UTC)

## Problem with notations : the notations of this page do not respect international standard iso notation

The notation of this wikipedia page does not respect official iso notation.

As is defined in international standard ISO 31-11 :

The set of natural numbers; is the set of positive integers and zero

It is denoted by ℕ = {0, 1, 2, 3, ...}

Exclusion of zero is denoted by an asterisk: ℕ* = {1, 2, 3, ...}

This iso notation should be clearly said to be the international official notation in this page.

Other notation should be considered as non classical and non official notation sometimes used by some people.

This is important that wikipedia respects official international notation and does not propagate inofficial out of fashion notation.

--31.39.233.46 (talk) 19:45, 28 September 2016 (UTC)

ISO has no authority over mathematics. The fact that they even try is ... pathetic. What we try to do is report the usages that actually exist in the mathematical community, including the situations where they are not always consistent among different workers. --Trovatore (talk) 20:14, 28 September 2016 (UTC)
I do not want to generate the faintest impression that I would not appreciate to the highest level the standardization efforts of the ISO, but any effort to force mathematical truths, handled by the leading actors in different variants, into normed terms is ... yes, pathetic, and I consider this effort not noteworthy within an encyclopedia. I also do not agree to calling either use of the term natural numbers an inofficial out of fashion notation.
Furthermore, I am convinced that a pure mathematical view, focused on Natural Numbers, would neither refer to specific numerals, nor to the subsequent notion of integers, and especially not to specific objects like 0 and and 1, which get their important meaning as units wrt to two operations in algebraic structures built upon the naturals.
The cited standardization is not apt to unbiasedly reflect the basic defining properties of Natural Numbers, as widely employed within the mathematics community, but belongs to a specific, not unanimously shared convention, prevalent in only certain math topics.
Reflecting the status quo of use in Wikipedia is nobler a task than blindly adhering to this standard, imho. Purgy (talk) 07:33, 29 September 2016 (UTC)
What ISO 31-11 says (about “ mathematical signs and symbols for use in physical sciences and technology”) is part of the status quo and worth a small mention, which I have just added. It is perhaps unfortunate that our article on the superseding standard ISO_80000-2 does not include this specification, so we cannot refer to it. Of course individual mathematicians are not bound by the ISO! PJTraill (talk) 13:40, 29 September 2016 (UTC)

## Repetition

On 26-3-2017 I did suppress the last parenthesis in “the integers, by including (if not yet in) the neutral element and an additive inverse (−n) for each natural number n (and zero, if it is not there already, as its own additive inverse)” as being a repetition, disturbing because one wonders if it really adds something…

• because after “including (if…) the neutral element” zero, there is no question whether it is there or not, it is: is has been included; thus a second “if” as “zero, if it is not there already” does not make sense any more;
• because “its own additive inverse” is evidently implied by “neutral”.

So, the whole of the parenthesis is undesirable. Surprisingly, somebody undid my edit 15 minutes later. Would somebody please explain why I am wrong… or shall we suppress the parenthesis “(and zero, if it is not there already, as its own additive inverse)”? --Dominique Meeùs (talk) 12:48, 28 March 2017 (UTC)

I agree, and I have edited again the article: Only the added numbers must be described, not their properties (here the fact that –0 = 0). D.Lazard (talk) 18:24, 28 March 2017 (UTC)
According to the history listing I was the one reverting and apologise for it. I must have been sleeping. I somehow was under the false impression that you deleted the inclusion of zero (as happened some time earlier).−Woodstone (talk) 16:19, 30 March 2017 (UTC)

## Rationals and reals

The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n; the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals;

That last part is wrong: the multiplicative inverses of nonzero integers form a very limited subset of the rationals (roughly: it omits any fraction with a numerator that isn't 1), and the only limit of (nontrivial) Cauchy sequences in that set is 0. It's been a long time since I had to know all this stuff, but my dim recollection is that the rationals are surprisingly difficult to define succinctly—something about equivalence classes of pairs of integers—so I'm not sure what I would suggest as replacement text, but as it stands it's obviously incorrect. --2001:1970:4F68:E000:C876:D11D:CB37:83F6 (talk) 10:39, 25 May 2017 (UTC)

MOS:MATH says "The lead section should include, where appropriate ... an informal introduction to the topic, without rigor, suitable for a general audience. Here, the lack of rigour consists of omitting that rational numbers include other numbers. Nevertheless I'll try to fix this. D.Lazard (talk) 10:59, 25 May 2017 (UTC)

## Natural numbers used for ordering?

Are natural numbers really used for ordering? Is third a natural number? Numerals representing specific natural numbers can be used for ordering, but can natural numbers? 83.230.4.140 (talk) 15:54, 4 October 2017 (UTC)

I strongly object to your claim that the properties of numerals, without referring to any ordering principle from outside of this notion allow for using them for ordering purposes. It is the defining structure of successor that makes the naturals suited for ordering. Maybe, one can debate if this is the unique origin of "ordering", but it is a quite general one and apt for broad application. I claim that numerals, whenever employed for ordering purposes, inherit their ordering power from the successor principle of the natural numbers, and cannot establish any order on their own. The cited ordinals (third, fourth, ...) are obviously derived from their linguistic counterparts (three, four, ...), denoting natural numbers, and "second" ("sequitur") directly hints to the "successor" of the "first". Purgy (talk) 09:12, 5 October 2017 (UTC)

I don't see any claim in the above post, only three questions. I think your answer, Purgy, is at a higher level than the question asked. To answer the users questions. 1) Yes, the natural numbers are used for ordering. The naturals numbers are one, two, three, and so on. "Third", as in "first", "second", "third", are adjectives, which describe the position of something in an ordered set, e.g. "first person in line", "second person in line", "third person in line". A numeral is just a symbol for a number, and we use symbols to communicate ideas about numbers. I hope this helps. Rick Norwood (talk) 19:04, 11 December 2017 (UTC)

Rereading my reply I perceive a not fully intended harshness —apologies. I certainly then felt a "claim" about "numerals are for ordering, naturals are not", induced by the "really" in the first question and the first part of the last sentence, undeniably being a claim. This leads me to the consequence that the level, as well as the attempted strictness in my reply may not be fully irrational. E.g., are naturals themselves "really" useful for ordering, or isn't it their axiomatic property of having exactly one successor, that is transferred to establish the ubiquitous "numerical" ordering? Numerals require still one more step of transfer, imho. In any case, the replies hopefully suffice to answer the "biased questions" on various levels. :) Purgy (talk) 07:59, 12 December 2017 (UTC)
"Really" must be used with care, because of it strong epistemologic implication. Here, the question and the answers suppose implicitly that numerals and natural numbers belong to the "reality". In fact, this is not the case, numeral first, and natural numbers much later have been invented for expressing properties of the "real world", which are cardinality and succession (one after the other). Natural numbers have been invented for allowing working with numerals. For example, in a date, the year is an ordinal numeral (2017th year after Christ), but for computing your age, you need a subtraction, which consists of manipulating ordinal numerals as natural numbers.
Thus my answer to the original question is: (ordinal) numerals, such as "third" are symbols that have been introduced for ordering (and counting, in the case of cardinal numerals). Thus numerals are not natural numbers. Natural numbers have been introduced for modeling (and later for formalizing) the experimental properties that result of ordering and counting. One of these properties is that a unique natural number may be associated to each numeral, and thus that in turn, natural numbers may be used for ordering and counting.
Thus, behind this very elementary question are hidden the very deep questions of what is mathematics, and what is its relationship with reality. My preceding answer reflects my personal views on this question, and these views are not shared by all mathematicians. D.Lazard (talk) 09:08, 12 December 2017 (UTC)
Just to confirm that not all mathematicians agree on these things, I think the natural numbers were discovered not invented ;-) Paul August 13:02, 12 December 2017 (UTC)

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