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Natural number

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Number used for counting

Natural numbers can be used for counting: one apple; two apples are one apple added to another apple, three apples are one apple added to two apples, ...

Inmathematics, thenatural numbers are the numbers0,1,2,3, and so on, possibly excluding 0.[a][1] The termspositive integers,non-negative integers,whole numbers, andcounting numbers are also used.[2][3] Theset of the natural numbers is commonly denoted with a boldN or ablackboard bold⁠N{\displaystyle \mathbb {N} }⁠.

The natural numbers are used for counting, and for labeling the result of a count, like "there areseven days in a week", in which case they are calledcardinal numbers. They are also used to label places in an ordered series, like "thethird day of the month", in which case they are calledordinal numbers. Natural numbers may also be used to label, like thejersey numbers of a sports team; in this case, they have no specific mathematical properties and are callednominal numbers.[4]

Two naturaloperations are defined on natural numbers,addition andmultiplication.Arithmetic is the study of the ways to perform these operations.Number theory is the study of the properties of these operations and their generalizations. Much ofcombinatorics involves counting mathematical objects, patterns and structures that are defined using natural numbers.

Manynumber systems are built from the natural numbers and contain them. For example, theintegers are made by including 0 and negative numbers. Therational numbers add fractions, and thereal numbers add all infinite decimals.Complex numbers add thesquare root of−1.[5] This makes up natural numbers as foundational for all mathematics.[6]

Terminology and notation

The termnatural numbers has two common definitions: either0, 1, 2, ... or1, 2, 3, .... Because there is no universal convention, the definition can be chosen to suit the context of use.[1][7] To eliminate ambiguity, the sequences1, 2, 3, ... and0, 1, 2, ... are often called thepositive integers and thenon-negative integers, respectively.

The phrasewhole numbers is frequently used for the natural numbers that include 0, although it may also mean all integers, positive and negative.[8][2] In primary education,counting numbers usually refer to the natural numbers starting at 1,[3] though this definition can vary.[9][10]

Theset of all natural numbers is typically denotedN or inblackboard bold asN.{\displaystyle \mathbb {N} .}[7][11][b] Whether 0 is included is often determined by the context but may also be specified by usingN{\displaystyle \mathbb {N} } orZ{\displaystyle \mathbb {Z} } (the set of all integers) with a subscript or superscript. Examples includeN1{\displaystyle \mathbb {N} _{1}},[13] orZ+{\displaystyle \mathbb {Z} ^{+}}[14] (for the set starting at 1) andN0{\displaystyle \mathbb {N} _{0}}[15] orZ0+{\displaystyle \mathbb {Z} ^{0+}}[16] (for the set including 0).

Intuitive concept

An intuitive and implicit understanding of natural numbers is developednaturally through using numbers for counting, ordering and basic arithmetic. Within this are two closely related aspects of what a natural number is: thesize of a collection; andaposition in a sequence.

Size of a collection

Natural numbers can be used to answer questions like: "how many apples are on the table?".[17] A natural number used in this way describes a characteristic of acollection of objects. This characteristic, thesize of a collection is calledcardinality and a natural number used to describe or measure it is called a cardinal number.

A group of apples and group of oranges with the same cardinality.

Two collections have the same size or cardinality if there is aone-to-one correspondence between the objects in each collection to the objects in the other. For example, in the image to the right every apple can be paired off with one orange and every orange can be paired off with one apple. From this, even without counting or using numbers it can be seen that the group of apples has thesame cardinality as the group of oranges, meaning they are both assigned the same cardinal number.

The natural number 3 is the thing used for the particular cardinal number described above and for the cardinal number of any other collection of objects that could be paired off in the same way to one of these groups.

Position in a sequence

The natural numbers have a fixed progression, which is the familiar sequence beginning with 1, 2, 3, and so on. A natural number can be used to denote a specific position in any other sequence, in which case it is called anordinal number. To have a specific position in a sequence means to come either before or after every other position in the sequence in a defined way, which is the concept oforder.

The natural number 3 then is the thing thatcomes after 2 and 1, andbefore 4, 5 and so on. The number 2 is the thing thatcomes after 1, and 1 is the first element in the sequence. Each number represents the relation that position bears to the rest of the infinite sequence.[18]

Counting

The process of counting involves both the cardinal and ordinal use of the natural numbers and illustrates the way the two fit together. To count the number of objects in a collection, each object is paired off with a natural number, usually by mentally or verbally saying the name of the number and assigning it to a particular object. The numbers must be assigned in order starting with 1 (they are ordinal) but the order of the objects chosen is arbitrary as long as each object is assigned one and only one number. When all of the objects have been assigned a number, the ordinal number assigned to the final object gives the result of the count, which is the cardinal number of the whole collection.

History

Ancient roots

Further information:Prehistoric counting
TheIshango bone (on exhibition at theRoyal Belgian Institute of Natural Sciences)[19][20][21] is believed to have been used 20,000 years ago for natural number arithmetic.

The most primitive method of representing a natural number is to use one's fingers, as infinger counting. Putting down atally mark for each object is another primitive method. 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 ofnumerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancientEgyptians developed a powerful system of numerals with distincthieroglyphs for 1, 10, and all powers of 10 up to over 1 million. A stone carving fromKarnak, dating back from around 1500 BCE and now at theLouvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. TheBabylonians had aplace-value 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.[22]

A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0digit in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.[c] TheOlmec andMaya civilizations used 0 as a separate number as early as the1st century BCE, but this usage did not spread beyondMesoamerica.[24][25] The use of a numeral 0 in modern times originated with the Indian mathematicianBrahmagupta in 628 CE. However, 0 had been used as a number in the medievalcomputus (the calculation of the date of Easter), beginning withDionysius Exiguus in 525 CE, without being denoted by a numeral. StandardRoman numerals do not have a symbol for 0; instead,nulla (or the genitive formnullae) fromnullus, the Latin word for "none", was employed to denote a 0 value.[26]

The first systematic study of numbers asabstractions is usually credited to theGreek philosophersPythagoras andArchimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[d]Euclid, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).[28] However, in the definition ofperfect number which comes shortly afterward, Euclid treats 1 as a number like any other.[29]

Independent studies on numbers also occurred at around the same time inIndia, China, andMesoamerica.[30]

Emergence as a term

Nicolas Chuquet used the termprogression naturelle (natural progression) in 1484.[31] The earliest known use of "natural number" as a complete English phrase is in 1763.[32][33] The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.[33]

Starting at 0 or 1 has long been a matter of definition. In 1727,Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.[34] In 1889,Giuseppe Peano used N for the positive integers and started at 1,[35] but he later changed to using N0 and N1.[36] Historically, most definitions have excluded 0,[33][37][38] but many mathematicians such asGeorge A. Wentworth,Bertrand Russell,Nicolas Bourbaki,Paul Halmos,Stephen Cole Kleene, andJohn Horton Conway have preferred to include 0.[39][33] This approach gained wider adoption in the 1960s[33] and was formalized inISO 31-11 (1978), which defines natural numbers to include zero, a convention retained in the currentISO 80000-2 standard.[40]

Formal construction

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers.Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.[41]Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".[e]

Theconstructivists saw a need to improve upon the logical rigor in thefoundations of mathematics.[f] In the 1860s,Hermann Grassmann suggested arecursive definition for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications.

Set-theoretical definitions of natural numbers were initiated byFrege. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, includingRussell's paradox. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.[44]

In 1881,Charles Sanders Peirce provided the firstaxiomatization of natural-number arithmetic.[45][46] In 1888,Richard Dedekind proposed another axiomatization of natural-number arithmetic,[47] and in 1889, Peano published a simplified version of Dedekind's axioms in his bookThe principles of arithmetic presented by a new method (Latin:Arithmetices principia, nova methodo exposita). This approach is now calledPeano arithmetic. It is based on anaxiomatization of the properties ofordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic isequiconsistent with several weak systems ofset theory. One such system isZFC with theaxiom of infinity replaced by its negation.[48] Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms includeGoodstein's theorem.[49]

Formal definitions

Formal definitions of the natural numbers take the existing, intuitive notion of natural numbers and the rules of arithmetic and define them both in the more fundamental terms of mathematical logic. The two standard methods for doing this are: thePeano axioms; andset theory.

The Peano axioms (named forGiuseppe Peano) do not explicitly define what the natural numbersare, but instead comprise a list of statements oraxioms that must be true of natural numbers, however they are defined. In contrast, set theory defines each natural number as a particularset, in which a set can be generally understood as a collection of distinct objects orelements. While the two approaches are different, they are consistent in that the natural number sets collectivelysatisfy the Peano axioms.

Peano axioms

Main article:Peano axioms

The five Peano axioms are the following:[50][g]

  1. 0 is a natural number.
  2. Every natural number has a successor which is also a natural number.
  3. 0 is not the successor of any natural number.
  4. If the successor ofx{\displaystyle x} equals the successor ofy{\displaystyle y}, thenx{\displaystyle x} equalsy{\displaystyle y}.
  5. Theaxiom 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 ofx{\displaystyle x} isx+1{\displaystyle x+1}.

Set-theoretic definition

Main articles:Set-theoretic definition of natural numbers andvon Neumann ordinal

In set theory each natural numbern is defined as an explicitly defined set, whose elements allow counting the elements of other sets. A variety of different constructions have been proposed, however the standard solution (due toJohn von Neumann)[51] is to define each natural numbern as a set containingn elements in the following way:

  • Call0 = { }, theempty set.
  • Define thesuccessorS(a) of any seta byS(a) =a ∪ {a}.
  • By theaxiom of infinity, there exist sets which contain 0 and areclosed under the successor function. Such sets are said to beinductive. The intersection of all inductive sets is still an inductive set.
  • This intersection is the set of thenatural numbers.

This produces an iterative definition of the natural numbers satisfying the Peano axioms, sometimes calledvon Neumann ordinals:

  • 0 = { }
  • 1 = 0 ∪ {0} = {0} = {{ }}
  • 2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}}
  • 3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}}
  • n =n−1 ∪ {n−1} = {0, 1, ...,n−1} = {{ }, {{ }}, ..., {{ }, {{ }}, ...}}

In this definition each natural number is equal to the set of all natural numbers less than it. Given a natural numbern, the sentence "a setS hasn elements" can be formally defined as "there exists abijection fromn toS." This formalizes the operation ofcounting the elements ofS. Also,n ≤m if and only ifn is asubset ofm. In other words, theset inclusion defines the usualtotal order on the natural numbers. This order is awell-order.

Another construction sometimes calledZermelo ordinals[52] defines0 = { } andS(a) = {a} and is now largely only of historical interest.

Properties

This section uses the conventionN=N0=N∗∪{0}{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}.

Addition

Given the setN{\displaystyle \mathbb {N} } of natural numbers and thesuccessor functionS:N→N{\displaystyle S\colon \mathbb {N} \to \mathbb {N} } sending each natural number to the next one, one can defineaddition of natural numbers recursively by settinga + 0 =a anda +S(b) =S(a +b) for alla,b. Thus,a + 1 =a + S(0) = S(a+0) = S(a),a + 2 =a + S(1) = S(a+1) = S(S(a)), and so on. Thealgebraic structure(N,+){\displaystyle (\mathbb {N} ,+)} is acommutativemonoid withidentity element 0. It is afree monoid on one generator. This commutative monoid satisfies thecancellation property, so it can be embedded in agroup. The smallest group containing the natural numbers is theintegers.

If 1 is defined asS(0), thenb + 1 =b +S(0) =S(b + 0) =S(b). That is,b + 1 is simply the successor ofb.

Multiplication

Analogously, given that addition has been defined, amultiplication operator×{\displaystyle \times } can be defined viaa × 0 = 0 anda × S(b) = (a ×b) +a. This turns(N∗,×){\displaystyle (\mathbb {N} ^{*},\times )} into afree commutative monoid with identity element 1; a generator set for this monoid is the set ofprime numbers.

Relationship between addition and multiplication

Addition and multiplication are compatible, which is expressed in thedistribution law:a × (b +c) = (a ×b) + (a ×c). These properties of addition and multiplication make the natural numbers an instance of acommutativesemiring. 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 thatN{\displaystyle \mathbb {N} } is notclosed under subtraction (that is, subtracting one natural from another does not always result in another natural), means thatN{\displaystyle \mathbb {N} } isnot aring; instead it is asemiring (also known as arig).

If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin witha + 1 =S(a) anda × 1 =a. Furthermore,(N∗,+){\displaystyle (\mathbb {N^{*}} ,+)} has no identity element.

Order

In this section, juxtaposed variables such asab indicate the producta ×b,[53] and the standardorder of operations is assumed.

Atotal order on the natural numbers is defined by lettinga ≤b if and only if there exists another natural numberc wherea +c =b. This order is compatible with thearithmetical operations in the following sense: ifa,b andc are natural numbers anda ≤b, thena +c ≤b +c andac ≤bc.

An important property of the natural numbers is that they arewell-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by anordinal number; for the natural numbers, this is denoted asω (omega).

Division

In this section, juxtaposed variables such asab indicate the producta ×b, and the standardorder 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 ofdivision with remainder orEuclidean division is available as a substitute: for any two natural numbersa andb withb ≠ 0 there are natural numbersq andr such that

a=bq+r and r<b.{\displaystyle a=bq+r{\text{ and }}r<b.}

The numberq is called thequotient andr is called theremainder of the division ofa by b. The numbersq andr are uniquely determined bya and b. This Euclidean division is key to the several other properties (divisibility), algorithms (such as theEuclidean algorithm), and ideas in number theory.

Algebraic properties satisfied by the natural numbers

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:

  • Closure under addition and multiplication: for all natural numbersa andb, botha +b anda ×b are natural numbers.[54]
  • Associativity: for all natural numbersa,b, andc,a + (b +c) = (a +b) +c anda × (b ×c) = (a ×b) ×c.[55]
  • Commutativity: for all natural numbersa andb,a +b =b +a anda ×b =b ×a.[56]
  • Existence ofidentity elements: for every natural numbera,a + 0 =a anda × 1 =a.
    • If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural numbera,a × 1 =a. However, the "existence of additive identity element" property is not satisfied
  • Distributivity of multiplication over addition for all natural numbersa,b, andc,a × (b +c) = (a ×b) + (a ×c).
  • No nonzerozero divisors: ifa andb are natural numbers such thata ×b = 0, thena = 0 orb = 0 (or both).

Generalizations

Natural numbers are broadly used in two ways: to quantify and to order. A number used to represent the quantity of objects in a collection ("there are 6 coins on the table") is called acardinal numeral, while a number used to order individual objects within a collection ("she finished 6th in the race") is anordinal numeral.

These two uses of natural numbers apply only tofinite sets.Georg Cantor discovered at the end of the 19th century that both uses of natural numbers can be generalized toinfinite sets, but that they lead to two different concepts of "infinite" numbers, thecardinal numbers and theordinal numbers.

Other generalizations of natural numbers are discussed inNumber § Extensions of the concept.

See also

Set inclusions between thenatural numbers (ℕ), theintegers (ℤ), therational numbers (ℚ), thereal numbers (ℝ), and thecomplex numbers (ℂ)


Notes

  1. ^It depends on authors and context whether 0 is considered a natural number.
  2. ^Older texts have occasionally employedJ as the symbol for this set.[12]
  3. ^ A tablet found at Kish ... 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.[23]
  4. ^This convention is used, for example, inEuclid's Elements, see D. Joyce's web edition of Book VII.[27]
  5. ^The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."[42][43]
  6. ^"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606)
  7. ^Hamilton (1988, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.  0 is a natural number."
    Halmos (1974, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I)  0 ∈ ω (where, of course,0 = ∅" (ω is the set of all natural numbers).
    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)

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  54. ^Fletcher, Harold; Howell, Arnold A. (9 May 2014).Mathematics with Understanding. Elsevier. p. 116.ISBN 978-1-4832-8079-0....the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
  55. ^Davisson, Schuyler Colfax (1910).College Algebra. Macmillian Company. p. 2.Addition of natural numbers is associative.
  56. ^Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962).Laidlaw mathematics series. Vol. 8. Laidlaw Bros. p. 25.

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