# Talk:Natural number

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Field: Number theory
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## 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)

## 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.

I thought the same thing; their (apparently exact) similarity hides the issue of "differences among identical objects", which gets into some issues about the Peano axioms and the reflexive definition of equality: "For every natural number x, x = x. That is, equality is reflexive." http://en.wikipedia.org/wiki/Peano_axioms Does 1 apple = 1 apple? What if the apples are of different size? Or of different type? Then 1 apple might not equal 1 apple ... Bruce Schuman (talk) 14:28, 3 August 2013 (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)

## 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)

I don't know about America but in Australia the students get taught that natural numbers don't include zero, because whole numbers are natural numbers INCLUDING the number zero. — Preceding unsigned comment added by 121.45.228.1 (talk) 11:23, 5 February 2014 (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)