Talk:Natural number

Cantor in lead?

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)

How about,

"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 [1] 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 (talk • contribs) 11:29, 5 February 2012 (UTC)

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

Apples

Currently the image of apples that is used to illustrate counting has two problems: The apples look too identical, so one could say it is a picture of one apple; but $1 \ne 6$. There are six apples in the picture and they could be grouped by the eye in different ways, not only the intended one. The intended way of grouping the apples as (1 single apple, a pair of 2 apples, a row of 3 apples) could be highlighted by connecting them in a colored rectangular background or other helpful way.

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)

Tests for schoolchildren

Is there agreement - at least in America - about whether the natural numbers include zero? (I don't care about university-level math in this context: I want to know what to tell my students so they'll "get the question right" on the high-stakes statewide tests.) --Uncle Ed (talk) 12:12, 25 April 2013 (UTC)