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In mathematics, the natural 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 common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers". Some authors begin the natural numbers with , corresponding to the non-negative integers , whereas others start with 1, corresponding to the positive integers .〔 〕〔 〕〔 says: "ℕ is the set of natural numbers (positive integers)" (p. 3)〕〔 include zero in the natural numbers: 'Intuitively, the set of all ''natural numbers'' may be described as follows: contains an "initial" number 0; …'. They follow that with their version of the Peano Postulates. (p. 15)〕 Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers). The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including an additive inverse (-n) for each natural number n (and zero, if it is not there already, as its own additive inverse); the rational numbers, by including a multiplicative inverse (1/n) for each integer number n; the real numbers by including with the rationals the (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one; and so on.〔 says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." (Preface, p. x)〕〔: "Numbers make up the foundation of mathematics." (p. 1)〕 These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers to contrast the discreteness of counting to the continuity of measurement, established by the real numbers. The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense. ==History== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「natural number」の詳細全文を読む スポンサード リンク
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