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{{about|"positive integers" and "non-negative integers"|all the numbers ..., âˆ’2, âˆ’1, 0, 1, 2, ...|Integer}}{{redirect|â„•|the cryptocurrency|Namecoin}}{{short description|A kind of number, used for counting}}File:Three apples(1).svg|right|thumb|Natural numbers can be used for counting (one appleappleIn 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 mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes (labels or "names"); that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.Some definitions, including the standard ISO 80000-2, begin the natural numbers with {{num|0}}, corresponding to the non-negative integers {{math|1=0, 1, 2, 3, â€¦}}, whereas others start with 1, corresponding to the positive integers {{math|1={{num|1}}, {{num|2}}, {{num|3}}, â€¦}}.{{MathWorld|title=Natural Number|id=NaturalNumber}}{{Citation| url =weblink| title = natural number| work = Merriam-Webster.com| publisher = Merriam-Webster| accessdate = 4 October 2014}}{{harvtxt|Carothers|2000}} says: "â„• is the set of natural numbers (positive integers)" (p. 3){{harvtxt|Mac Lane|Birkhoff|1999}} include zero in the natural numbers: 'Intuitively, the set {{math|â„• {{=}} {{mset|0, 1, 2, ...}}}} of all natural numbers may be described as follows: {{math|â„•}} 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, 978-1-57820-120-4, 2003,weblink 138 (integer), 247 (signed integer), & 276 (unsigned integer), integer 1. n. Any whole number., The natural numbers are a basis from which many other number sets may be built by extension: the integers (Grothendieck group), 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.{{harvtxt|Mendelson|2008}} 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){{harvtxt|Bluman|2010}}: "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{{MathWorld|title=Counting Number|id=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.

History

Modern definitions

• Naturals with zero: ;{0,1,2,...}=mathbb{N}_0={mathbb{N}}cup{0}
• Naturals without zero: {1,2,...}=mathbb{N}^=mathbb{N}smallsetminus{0}.

Notation

File:U+2115.svg|right|thumb|upright|The double-struck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbolsList of mathematical symbolsMathematicians use N or {{math|â„•}} (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., McGraw-Hill, 1976, 978-0-07-054235-8, New York, 25, To be unambiguous about whether 0 is included or not, sometimes a subscript (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 80000-2:2009,weblink International Organization for Standardization, Standard number sets and intervals, 6,
{{math|1=â„•0 = â„•0 = â„• âˆª {{mset|0}} = {{mset|0, 1, 2, â€¦}}}} {{math|1=â„•* = â„•+ = â„•1 = â„•>0 = {{mset|1, 2, 3, â€¦}}}}.
Alternatively, since natural numbers naturally embed in the integers, they may be referred to as the positive, or the non-negative integers, respectively.BOOK, Grimaldi, Ralph P., A review of discrete and combinatorial mathematics, 2003, Addison-Wesley, Boston, 978-0-201-72634-3, 133, 5th,
{1, 2, 3,dots} = mathbb Z^+ {0, 1, 2,dots} = mathbb Z^{ge 0}

Properties

Infinity

The set of natural numbers is an infinite set. This kind of infinity is, by definition, called countable infinity. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. This is also expressed by saying that the cardinal number of the set is aleph-naught ({{math|â„µ0}}).{{MathWorld |urlname=CardinalNumber |title=Cardinal Number}}

One can recursively define an addition operator on the natural numbers by setting {{math|a + 0 {{=}} a}} and {{math|a + S(b) {{=}} S(a + b)}} for all {{math|a}}, {{math|b}}. Here {{math|S}} should be read as "successor". This turns the natural numbers {{math|(â„•, +)}} into a commutative monoid with identity element 0, the so-called 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 {{math|S(0)}}, then {{math|b + 1 {{=}} b + S(0) {{=}} S(b + 0) {{=}} S(b)}}. That is, {{math|b + 1}} is simply the successor of {{math|b}}.

Multiplication

Analogously, given that addition has been defined, a multiplication operator Ã— can be defined via {{math|a Ã— 0 {{=}} 0}} and {{math|a 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: {{math|a 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 {{math|â„•}} is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that {{math|â„•}} 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 {{math|a + 1 {{=}} S(a)}} and {{math|a Ã— 1 {{=}} a}}.

Order

In this section, juxtaposed variables such as {{math|ab}} indicate the product {{math|a Ã— b}}, and the standard order of operations is assumed.A total order on the natural numbers is defined by letting {{math|a â‰¤ b}} if and only if there exists another natural number {{math|c}} where {{math|a + c {{=}} b}}. This order is compatible with the arithmetical operations in the following sense: if {{math|a}}, {{math|b}} and {{math|c}} are natural numbers and {{math|a â‰¤ b}}, then {{math|a + c â‰¤ b + c}} and {{math|ac â‰¤ bc}}.An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as {{math|Ï‰}} (omega).

Division

In this section, juxtaposed variables such as {{math|ab}} indicate the product {{math|a 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 {{math|a}} and {{math|b}} with {{math|b â‰  0}} there are natural numbers {{math|q}} and {{math|r}} such that
{{math|a {{=}} bq + r}}      and      {{math|r < b}}.
The number {{math|q}} is called the quotient and {{math|r}} is called the remainder of the division of {{math|a}} by {{math|b}}. The numbers {{math|q}} and {{math|r}} are uniquely determined by {{math|a}} and {{math|b}}. 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 {{math|a}} and {{math|b}}, both {{math|a + b}} and {{math|a Ã— b}} are natural numbers.
• Associativity: for all natural numbers {{math|a}}, {{math|b}}, and {{math|c}}, {{math|a + (b + c) {{=}} (a + b) + c}} and {{math|a Ã— (b Ã— c) {{=}} (a Ã— b) Ã— c}}.
• Commutativity: for all natural numbers {{math|a}} and {{math|b}}, {{math|a + b {{=}} b + a}} and {{math|a Ã— b {{=}} b Ã— a}}.
• Existence of identity elements: for every natural number a, {{math|a + 0 {{=}} a}} and {{math|a Ã— 1 {{=}} a}}.
• Distributivity of multiplication over addition for all natural numbers {{math|a}}, {{math|b}}, and {{math|c}}, {{math|a Ã— (b + c) {{=}} (a Ã— b) + (a Ã— c)}}.
• No nonzero zero divisors: if {{math|a}} and {{math|b}} are natural numbers such that {{math|a Ã— b {{=}} 0}}, then {{math|a {{=}} 0}} or {{math|b {{=}} 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 aleph-null ({{math|â„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 well-ordered countably infinite set. This assignment can be generalized to general well-orderings 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 well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as {{math|Ï‰}}; this is also the ordinal number of the set of natural numbers itself.
Many well-ordered sets with cardinal number {{math|â„µ0}} have an ordinal number greater than {{math|Ï‰}} (the latter is the lowest possible). The least ordinal of cardinality {{math|â„µ0}} (i.e., the initial ordinal) is {{math|Ï‰}}.For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.A countable non-standard model of arithmetic satisfying the Peano Arithmetic (i.e., the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.Georges Reeb used to claim provocatively that The naÃ¯ve integers don't fill up {{math|â„•}}. Other generalizations are discussed in the article on numbers.

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 Mathematics
Springer Science+Business Media>Springer, in cooperation with the European Mathematical Society| accessdate = 8 October 2014}}{{harvtxt|Hamilton|1988}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number." (p. 117f){{harvtxt|Halmos|1960}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces|2}}{{math|0 âˆˆ Ï‰}} (where, of course, {{math|0 {{=}} âˆ…}}" ({{math|Ï‰}} is the set of all natural numbers). (p. 46){{harvtxt|Morash|1991}} gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers)
1. 0 is a natural number.
2. Every natural number has a successor.
3. 0 is not the successor of any natural number.
4. If the successor of x equals the successor of y , then x equals y.
5. The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of x is x + 1. Replacing axiom 5 by an axiom schema, one obtains a (weaker) first-order theory called Peano arithmetic.

Constructions based on set theory

Von Neumann ordinals

In the area of mathematics called set theory, a specific construction due to John von Neumann{{Harvp|Von Neumann|1923}}{{harvp|Levy|1979|page=52}} attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s. defines the natural numbers as follows:
• Set {{math|0 {{=}} {{mset| }}}}, the empty set,
• Define {{math|S(a) {{=}} a âˆª {{mset|a}}}} for every set {{math|a}}. {{math|S(a)}} is the successor of {{math|a}}, and {{math|S}} is called the successor function.
• By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be 'inductive'. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.
• It follows that each natural number is equal to the set of all natural numbers less than it:

*{{math|0 {{=}} {{mset| }}}}, *{{math|1 {{=}} 0 âˆª {{mset|0}} {{=}} {{mset|0}} {{=}} {{mset|{{mset| }}}}}}, *{{math|2 {{=}} 1 âˆª {{mset|1}} {{=}} {{mset|0, 1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}, *{{math|3 {{=}} 2 âˆª {{mset|2}} {{=}} {{mset|0, 1, 2}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}}}, *{{math|n {{=}} nâˆ’1 âˆª {{mset|nâˆ’1}} {{=}} {{mset|0, 1, â€¦, nâˆ’1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, â€¦, {{mset|{{mset| }}, {{mset|{{mset| }}}}, â€¦}}}}}}, etc.
With this definition, a natural number {{math|n}} is a particular set with {{math|n}} elements, and {{math|n â‰¤ m}} if and only if {{math|n}} is a subset of {{math|m}}. The standard definition, now called definition of von Neumann ordinals, is: "each ordinal is the well-ordered set of all smaller ordinals."Also, with this definition, different possible interpretations of notations like {{math|â„n}} ({{math|n}}-tuples versus mappings of {{math|n}} into {{math|â„}}) 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.

Zermelo ordinals

Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo's construction goes as follows:
• Set {{math|0 {{=}} {{mset| }}}}
• Define {{math|S(a) {{=}} {{mset|a}}}},
• It then follows that

*{{math|0 {{=}} {{mset| }}}}, *{{math|1 {{=}} {{mset|0}} {{=}} {{mset|{{mset| }}}}}}, *{{math|2 {{=}} {{mset|1}} {{=}} {{mset|{{mset|{{mset| }}}}}}}}, *{{math|n {{=}} {{mset|nâˆ’1}} {{=}} {{mset|{{mset|{{mset|â€¦}}}}}}}}, etc. Each natural number is then equal to the set containing just the natural number preceding it. (This is the definition of Zermelo ordinals.)

{{Reflist}}

References

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{{commons category|Natural numbers}} {{Number systems}}{{Classes of natural numbers}}

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