Natural number - Revision history

Karl Jones at 18:38, 23 October 2016

2016-10-23T18:38:38Z

Karl Jones at 20:26, 30 August 2016

2016-08-30T20:26:49Z

Karl Jones: /* External links */

2016-04-20T22:44:39Z

‎External links

Karl Jones: /* See also */

2015-09-02T14:15:19Z

‎See also

Karl Jones: Undecidable problem

2015-08-30T19:28:42Z

Undecidable problem

Karl Jones: First

2015-08-30T18:40:59Z

First

New page

In [[mathematics]], the '''natural numbers''' (sometimes called the '''whole numbers''') are those used for [[counting]] (as in "there are six coins on the table") and [[ordering]] (as in "this is the third largest city in the country").

== Counting and ordering ==

In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".

== Nominal numbers ==

Another use of natural numbers is for what linguists call nominal numbers, such as the model number of a product, where the "natural number" is used only for naming (as distinct from a serial number where the order properties of the natural numbers distinguish later uses from earlier uses) and generally lacks any meaning of number as used in mathematics.

== Basis for other numbers sets ==

The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including an unresolved negation operation; the rational numbers, by including with the integers an unresolved division operation; the real numbers by including with the rationals the termination of Cauchy sequences; the complex numbers, by including with the real numbers the unresolved square root of minus one; the hyperreal numbers, by including with real numbers the infinitesimal value epsilon; vectors, by including a vector structure with reals; matrices, by having vectors of vectors; the nonstandard integers; and so on.[5][6] Therefore, the natural numbers are canonically embedded (identified) in the other number systems.

== Number theory ==

Properties of the natural numbers, such as divisibility and the distribution of [[Prime number|prime numbers]], are studied in [[number theory]].

== Combinatorics ==

Problems concerning counting and ordering, such as partitioning and enumerations, are studied in [[combinatorics]].

== Zero ==

There is no universal agreement about whether to include [[zero]] in the set of natural numbers.

Some authors begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2, 3, ....

This distinction is of no fundamental concern for the natural numbers as such, since their core construction is the unary operation successor.

Including the number 0 just supplies an identity element for the (binary) operation of addition, which makes up together with the multiplication the usual arithmetic in the natural numbers, to be completed within the integers and the rational numbers, only.

== Counting numbers versus real numbers ==

In common language, for example in primary school, natural numbers may be called counting numbers to distinguish them from the real numbers which are used for measurement.

== See also ==

* [[Counting]]
* [[Mathematics]]
* [[Number theory]]
* [[Ordering]]

== External links ==

* [https://en.wikipedia.org/wiki/Natural_number Natural number] @ Wikipedia

← Older revision		Revision as of 18:38, 23 October 2016
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−	In [[mathematics]], the '''natural numbers''' (sometimes called the '''whole numbers''') are those used for [[counting]] (as in "there are six coins on the table") and [[Total order\|ordering]] (as in "this is the third largest city in the country").	+	[[File:Three_apples.png\|300px\|thumb\|Natural numbers can be used for counting (one apple, two apples, three apples, ...).]]In [[mathematics]], the '''natural numbers''' (sometimes called the '''whole numbers''') are those used for [[counting]] (as in "there are six coins on the table") and [[Total order\|ordering]] (as in "this is the third largest city in the country").

	== Counting and ordering ==		== Counting and ordering ==

← Older revision		Revision as of 22:44, 20 April 2016
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	* [https://en.wikipedia.org/wiki/Natural_number Natural number] @ Wikipedia		* [https://en.wikipedia.org/wiki/Natural_number Natural number] @ Wikipedia
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		+	[[Category:Mathematics]]
		+	[[Category:Numbers]]

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-In [[mathematics]], the '''natural numbers''' (sometimes called the '''whole numbers''') are those used for [[counting]] (as in "there are six coins on the table") and [[ordering]] (as in "this is the third largest city in the country").
+In [[mathematics]], the '''natural numbers''' (sometimes called the '''whole numbers''') are those used for [[counting]] (as in "there are six coins on the table") and [[Total order|ordering]] (as in "this is the third largest city in the country").
 == Counting and ordering ==
@@ Line 37: / Line 37: @@
 == See also ==
 * [[Counting]]
 * [[Mathematics]]
 * [[Number theory]]
-* [[Ordering]]
+* [[Peano axioms]]
 * [[Prime number]]
 * [[Undecidable problem]]

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 * [[Number theory]]
 * [[Ordering]]
 == External links ==
 * [https://en.wikipedia.org/wiki/Natural_number Natural number] @ Wikipedia