# Difference between revisions of "Natural number"

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− | In [[mathematics]], a '''natural number''' is a a number from the set {0,1,2,...} or | + | In [[mathematics]], a '''natural number''' is a a number from the set {0,1,2,...} or {1,2,3...}, depending on the context and the reference.<ref>0 is usually included in the list of natural numbers in modern textbooks (Bourbaki 1968, Halmos 1974). + Older books sometimes exclude [[zero]], as there is a long history of people thinking that zero is unnatural or not really a number. Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P." [http://mathworld.wolfram.com/NaturalNumber.html (Wolfram)] </ref> Natural numbers were used initially for counting ("there are three cows in this field"), but they took on the purpose of ordering as well ("She is the 2nd fastest person alive). These are specific instances of the more general notions of [[cardinality]] and [[ordinality]] which slowly become more complicated as one treats [[infinite]] numbers as well. The set of natural numbers is [[countable]]- via [[bijection]], this property can be used to prove the countability of the [[integer]]s and [[rational number]]s. |

==Axiomatization== | ==Axiomatization== |

## Revision as of 10:11, 31 August 2008

In mathematics, a **natural number** is a a number from the set {0,1,2,...} or {1,2,3...}, depending on the context and the reference.^{[1]} Natural numbers were used initially for counting ("there are three cows in this field"), but they took on the purpose of ordering as well ("She is the 2nd fastest person alive). These are specific instances of the more general notions of cardinality and ordinality which slowly become more complicated as one treats infinite numbers as well. The set of natural numbers is countable- via bijection, this property can be used to prove the countability of the integers and rational numbers.

## Axiomatization

In the late 19th century, Giuseppe Peano (August 27, 1858 – April 20, 1932) elaborated *the* axiomatic system for the Natural Numbers, later named Peano Axioms in his honor.

## Reference

- ↑ 0 is usually included in the list of natural numbers in modern textbooks (Bourbaki 1968, Halmos 1974). + Older books sometimes exclude zero, as there is a long history of people thinking that zero is unnatural or not really a number. Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P." (Wolfram)