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natural number
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{{short descriptionkind of number used for counting}}{{aboutâ€œpositive integersâ€ and â€œnonnegative integersâ€all the numbers â€¦, âˆ’2, âˆ’1, 0, 1, 2, â€¦Integer}}File:Three apples(1).svgrightthumbNatural numbers can be used for counting (one appleappleFile:Numbersystems.svgthumbReal numbers (â„) include the rational (â„š), which include the integers (â„¤), which include the natural numbers (â„•)]]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 definitions, including the standard ISO 800002, begin the natural numbers with {{num0}}, corresponding to the nonnegative integers {{math1=0, 1, 2, 3, â€¦}}, whereas others start with 1, corresponding to the positive integers {{math1={{num1}}, {{num2}}, {{num3}}, â€¦}}.{{MathWorldtitle=Natural Numberid=NaturalNumber}}{{Citation url =weblink title = natural number work = MerriamWebster.com publisher = MerriamWebster accessdate = 4 October 2014}}{{harvtxtCarothers2000}} says: "â„• is the set of natural numbers (positive integers)" (p. 3){{harvtxtMac LaneBirkhoff1999}} include zero in the natural numbers: 'Intuitively, the set {{mathâ„• {{=}} {{mset0, 1, 2, ...}}}} of all natural numbers may be described as follows: {{mathâ„•}} 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).BOOK, Jack G. Ganssle & Michael Barr, Embedded Systems Dictionary, 1578201209, 2003,weblink 138 (integer), 247 (signed integer), & 276 (unsigned integer), integer 1. n. Any whole number., The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (âˆ’n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on.{{harvtxtMendelson2008}} says: "The whole fantastic hierarchy of number systems is built up by purely settheoretic means from a few simple assumptions about natural numbers." (Preface, p. x){{harvtxtBluman2010}}: "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{{MathWorldtitle=Counting Numberid=CountingNumber}} both to intuitively exclude the negative integers and zero, and also 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. the content below is remote from Wikipedia
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History
Ancient roots
File:Os d'Ishango IRSNB.JPGthumbuprightThe Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)WEB,weblink Introduction,weblink March 4, 2016, Royal Belgian Institute of Natural Sciences, Brussels, Belgium, Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. UNESCOUNESCOThe most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a placevalue system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context.Georges Ifrah, The Universal History of Numbers, Wiley, 2000, {{isbn0471375683}}A much later advance was the development of the idea that {{num0}} can be considered as a number, with its own numeral. The use of a 0 digit in placevalue notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number.WEB,weblink A history of Zero, â€¦ a tablet found at Kish â€¦ thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place, MacTutor History of Mathematics, 20130123, The Olmec and Maya civilizations used 0 as a separate number as early as the {{nowrap1st century BC}}, but this usage did not spread beyond Mesoamerica.{{citationtitle=1491: New Revelations Of The Americas Before Columbusfirst=Charles C.last=Mannpublisher=Knopfyear=2005isbn=9781400040063page=19url=https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19}}.{{citationtitle=The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Contextfirst=Brianlast=Evanspublisher=John Wiley & Sonsyear=2014isbn=9781118853979contribution=Chapter 10. PreColumbian Mathematics: The Olmec, Maya, and Inca Civilizationscontributionurl=https://books.google.com/books?id=3CPwAgAAQBAJ&pg=PT73}}. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); instead nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.WEB, Michael Deckers,weblink Cyclus Decemnovennalis Dionysii â€“ Nineteen year cycle of Dionysius, Hbar.phys.msu.ru, 20030825, 20120213, The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.Independent studies also occurred at around the same time in India, China, and Mesoamerica.Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 [1972], {{isbn0195061357}}Modern definitions
{{refimprove sectiondate=October 2014}}In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school{{whichdate=March 2017}} of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri PoincarÃ© was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man".The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886."{{Citation last=Gray first=Jeremy authorlink=Jeremy Gray year=2008 title=Plato's Ghost: The Modernist Transformation of Mathematics publisher=Princeton University Press publicationplace= page=153 url=https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22#v=snippet&q=%22God%20made%20the%20integers%2C%20all%20else%20is%20the%20work%20of%20man.%22&f=false accessdate= }}Weber, Heinrich L. 18912. Kronecker. Jahresbericht der Deutschen MathematikerVereinigung 2:523. (The quote is on p. 19.)In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics."Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." {{harvEves1990p=606}} In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural but a consequence of definitions. Later, two classes of such formal definitions were constructed; later, they were shown to be equivalent in most practical applications.Settheoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in onetoone correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox. Therefore, this formalism was modified so that a natural number is defined as a particular set, and any set that can be put into onetoone correspondence with that set is said to have that number of elements.{{harvnbEves1990loc=Chapter 15}}The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano; this approach is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every nonzero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982.With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theoristsWEB, Bagaria, Joan, Set Theory,weblink The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), and logicians.BOOK, Goldrei, Derek, Classic set theory : a guided independent study, 1998, Chapman & Hall/CRC, Boca Raton, Fla. [u.a.], 0412606100, 33, 1. ed., 1. print, 3, Other mathematicians also include 0 although many have kept the older tradition and take 1 to be the first natural number.This is common in texts about Real analysis. See, for example, {{harvtxtCarothers2000p=3}} or {{harvtxtThomsonBrucknerBruckner2000p=2}}. Computer scientists often start from zero when enumerating items like loop counters and string or array elements.JOURNAL, Brown, Jim, In Defense of Index Origin 0, ACM SIGAPL APL Quote Quad, 1978, 9, 2, 7, 10.1145/586050.586053,weblink 19 January 2015, WEB, Hui, Roger, Is Index Origin 0 a Hindrance?,weblink jsoftware.com, 19 January 2015,Notation
File:U+2115.svgrightthumbuprightThe doublestruck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbolsList of mathematical symbolsMathematicians use N or {{mathâ„•}} (an N in blackboard bold) to refer to the set of all natural numbers. Older texts have also occasionally employed J as the symbol for this set.BOOK,weblink Principles of Mathematical Analysis, Rudin, W., McGrawHill, 1976, 9780070542358, New York, 25, This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is alephnaught ({{mathâ„µ0}}).{{MathWorld urlname=CardinalNumber title=Cardinal Number}}To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "{{math*}}" or subscript "{{math>0}}" is added in the latter case:BOOK, ISO 800002:2009,weblink International Organization for Standardization, Standard number sets and intervals, 6,
{{math1=â„•0 = â„•0 = {{mset0, 1, 2, â€¦}}}}
{{math1=â„•* = â„•+ = â„•1 = â„•>0 = {{mset1, 2, â€¦}}}}.
Alternatively, natural numbers may be distinguished from positive integers with the index notation, but it must be understood by context that since both symbols are used, the natural numbers contain zero.BOOK, Grimaldi, Ralph P., A review of discrete and combinatorial mathematics, 2003, AddisonWesley, Boston, MA, 9780201726343, 133, 5th,
{{math1=â„• = {{mset0, 1, 2, â€¦}} }}.
{{math1=â„¤+= {{mset1, 2, â€¦}} }}.
Properties
Addition
One can recursively define an addition operator on the natural numbers by setting {{matha + 0 {{=}} a}} and {{matha + S(b) {{=}} S(a + b)}} for all {{matha}}, {{mathb}}. Here {{mathS}} should be read as "successor". This turns the natural numbers {{math(â„•, +)}} into a commutative monoid with identity element 0, the socalled free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word group). The smallest group containing the natural numbers is the integers.If 1 is defined as {{mathS(0)}}, then {{mathb + 1 {{=}} b + S(0) {{=}} S(b + 0) {{=}} S(b)}}. That is, {{mathb + 1}} is simply the successor of {{mathb}}.Multiplication
Analogously, given that addition has been defined, a multiplication operator Ã— can be defined via {{matha Ã— 0 {{=}} 0}} and {{matha Ã— S(b) {{=}} (a Ã— b) + a}}. This turns {{math(â„•*, Ã—)}} into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.Relationship between addition and multiplication
Addition and multiplication are compatible, which is expressed in the distribution law: {{matha Ã— (b + c) {{=}} (a Ã— b) + (a Ã— c)}}. These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that {{mathâ„•}} is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that {{mathâ„•}} is not a ring; instead it is a semiring (also known as a rig).If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and Ã— are as above, except that they begin with {{matha + 1 {{=}} S(a)}} and {{matha Ã— 1 {{=}} a}}.Order
In this section, juxtaposed variables such as {{mathab}} indicate the product {{matha Ã— b}}, and the standard order of operations is assumed.A total order on the natural numbers is defined by letting {{matha â‰¤ b}} if and only if there exists another natural number {{mathc}} where {{matha + c {{=}} b}}. This order is compatible with the arithmetical operations in the following sense: if {{matha}}, {{mathb}} and {{mathc}} are natural numbers and {{matha â‰¤ b}}, then {{matha + c â‰¤ b + c}} and {{mathac â‰¤ bc}}.An important property of the natural numbers is that they are wellordered: every nonempty set of natural numbers has a least element. The rank among wellordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as {{mathÏ‰}} (omega).Division
In this section, juxtaposed variables such as {{mathab}} indicate the product {{matha Ã— b}}, and the standard order of operations is assumed.While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers {{matha}} and {{mathb}} with {{mathb â‰ 0}} there are natural numbers {{mathq}} and {{mathr}} such that
{{matha {{=}} bq + r}} and {{mathr < b}}.
The number {{mathq}} is called the quotient and {{mathr}} is called the remainder of the division of {{matha}} by {{mathb}}. The numbers {{mathq}} and {{mathr}} are uniquely determined by {{matha}} and {{mathb}}. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.Algebraic properties satisfied by the natural numbers
The addition (+) and multiplication (Ã—) operations on natural numbers as defined above have several algebraic properties: Closure under addition and multiplication: for all natural numbers {{matha}} and {{mathb}}, both {{matha + b}} and {{matha Ã— b}} are natural numbers.
 Associativity: for all natural numbers {{matha}}, {{mathb}}, and {{mathc}}, {{matha + (b + c) {{=}} (a + b) + c}} and {{matha Ã— (b Ã— c) {{=}} (a Ã— b) Ã— c}}.
 Commutativity: for all natural numbers {{matha}} and {{mathb}}, {{matha + b {{=}} b + a}} and {{matha Ã— b {{=}} b Ã— a}}.
 Existence of identity elements: for every natural number a, {{matha + 0 {{=}} a}} and {{matha Ã— 1 {{=}} a}}.
 Distributivity of multiplication over addition for all natural numbers {{matha}}, {{mathb}}, and {{mathc}}, {{matha Ã— (b + c) {{=}} (a Ã— b) + (a Ã— c)}}.
 No nonzero zero divisors: if {{matha}} and {{mathb}} are natural numbers such that {{matha Ã— b {{=}} 0}}, then {{matha {{=}} 0}} or {{mathb {{=}} 0}} (or both).
Generalizations
Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers. A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be countably infinite and to have cardinality alephnull ({{mathâ„µ0}}).
 Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any wellordered countably infinite set. This assignment can be generalized to general wellorderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a wellordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two wellordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as {{mathÏ‰}}; this is also the ordinal number of the set of natural numbers itself.
Formal definitions
Peano axioms
Many properties of the natural numbers can be derived from the five Peano axioms:{{Citation url =weblink author = G.E. Mints (originator) title = Peano axioms work = Encyclopedia of MathematicsSpringer Science+Business Media>Springer, in cooperation with the European Mathematical Society accessdate = 8 October 2014}}{{harvtxtHamilton1988}} calls them "Peano's Postulates" and begins with "1.{{spaces2}}0 is a natural number." (p. 117f){{harvtxtHalmos1960}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces2}}{{math0 âˆˆ Ï‰}} (where, of course, {{math0 {{=}} âˆ…}}" ({{mathÏ‰}} is the set of all natural numbers). (p. 46){{harvtxtMorash1991}} gives "a twopart axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers)
Constructions based on set theoryVon Neumann constructionIn the area of mathematics called set theory, a special case of the von Neumann ordinal construction {{HarvnbVon Neumann1923}} defines the natural numbers as follows:
*{{math0 {{=}} {{mset }}}},
*{{math1 {{=}} 0 âˆª {{mset0}} {{=}} {{mset0}} {{=}} {{mset{{mset }}}}}},
*{{math2 {{=}} 1 âˆª {{mset1}} {{=}} {{mset0, 1}} {{=}} {{mset{{mset }}, {{mset{{mset }}}}}}}},
*{{math3 {{=}} 2 âˆª {{mset2}} {{=}} {{mset0, 1, 2}} {{=}} {{mset{{mset }}, {{mset{{mset }}}}, {{mset{{mset }}, {{mset{{mset }}}}}}}}}},
*{{mathn {{=}} nâˆ’1 âˆª {{msetnâˆ’1}} {{=}} {{mset0, 1, â€¦, nâˆ’1}} {{=}} {{mset{{mset }}, {{mset{{mset }}}}, â€¦, {{mset{{mset }}, {{mset{{mset }}}}, â€¦}}}}}}, etc.
With this definition, a natural number {{mathn}} is a particular set with {{mathn}} elements, and {{mathn â‰¤ m}} if and only if {{mathn}} is a subset of {{mathm}}.Also, with this definition, different possible interpretations of notations like {{mathâ„n}} ({{mathn}}tuples versus mappings of {{mathn}} into {{mathâ„}}) coincide.Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.Other constructionsAlthough the standard construction is useful, it is not the only possible construction. Zermelo's construction goes as follows:
*{{math0 {{=}} {{mset }}}},
*{{math1 {{=}} {{mset0}} {{=}} {{mset{{mset }}}}}},
*{{math2 {{=}} {{mset1}} {{=}} {{mset{{mset{{mset }}}}}}}},
*{{mathn {{=}} {{msetnâˆ’1}} {{=}} {{mset{{mset{{msetâ€¦}}}}}}}}, etc.
Each natural number is then equal to the set containing just the natural number preceding it.
See also
Notes{{Reflist}}References
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