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 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. 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 (if not yet in) the neutral element 0 and an additive inverse (−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. 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 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.
1.1 Ancient roots 1.2 Modern definitions
2 Notation 3 Properties
3.1 Addition 3.2 Multiplication 3.3 Relationship between addition and multiplication 3.4 Order 3.5 Division 3.6 Algebraic properties satisfied by the natural numbers
4 Generalizations 5 Formal definitions
5.1 Peano axioms 5.2 Constructions based on set theory
5.2.1 Von Neumann construction 5.2.2 Other constructions
6 See also 7 Notes 8 References 9 External links
History Ancient roots
The 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
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The double-struck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbols).
Mathematicians use N or ℕ (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. 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 aleph-naught (ℵ0). 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 "*" or subscript ">0" is added in the latter case:
ℕ0 = ℕ0 = 0, 1, 2, … ℕ* = ℕ+ = ℕ1 = ℕ>0 = 1, 2, … .
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.
ℕ = 0, 1, 2, … . ℤ+= 1, 2, … .
Properties Addition One can recursively define an addition operator on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Here S should be read as "successor". This turns the natural numbers (ℕ, +) 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 S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b. Multiplication Analogously, given that addition has been defined, a multiplication operator × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (ℕ*, ×) 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: a × (b + c) = (a × b) + (a × 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 ℕ is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that ℕ 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 × are as above, except that they begin with a + 1 = S(a) and a × 1 = a. Order In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed. A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ 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 ω (omega). Division In this section, juxtaposed variables such as ab indicate the product a × 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 a and b with b ≠ 0 there are natural numbers q and r such that
a = bq + r and r < b.
The number q is called the quotient and r is called the remainder of
the division of a by b. The numbers q and r are uniquely
determined by a and b. This
Closure under addition and multiplication: for all natural numbers a
and b, both a + b and a × b are natural numbers.
Associativity: for all natural numbers a, b, and c, a + (b + c) = (a +
b) + c and a × (b × c) = (a × b) × c.
Commutativity: for all natural numbers a and b, a + b = b + a and a ×
b = b × a.
Existence of identity elements: for every natural number a, a + 0 = a
and a × 1 = a.
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 (ℵ0). 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 ω; this is also the ordinal number of the set of natural numbers itself.
Many well-ordered sets with cardinal number ℵ0 have an ordinal
number greater than ω (the latter is the lowest possible). The least
ordinal of cardinality ℵ0 (i.e., the initial ordinal) is ω.
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
0 is a natural number. Every natural number has a successor. 0 is not the successor of any natural number. If the successor of
equals the successor of
. 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 + 1
. Replacing Axiom Five by an axiom schema one obtains a (weaker) first-order theory called Peano Arithmetic. Constructions based on set theory Main article: Set-theoretic definition of natural numbers Von Neumann construction In the area of mathematics called set theory, a special case of the von Neumann ordinal construction  defines the natural numbers as follows:
Set 0 = , the empty set, Define S(a) = a ∪ a for every set a. S(a) is the successor of a, and 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:
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 = , , …, , , … , etc.
With this definition, a natural number n is a particular set with n elements, and n ≤ m if and only if n is a subset of m. Also, with this definition, different possible interpretations of notations like ℝn (n-tuples versus mappings of n into ℝ) 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 constructions Although the standard construction is useful, it is not the only possible construction. Zermelo's construction goes as follows:
Set 0 = Define S(a) = a , It then follows that
0 = , 1 = 0 = , 2 = 1 = , n = n−1 = … , etc.
Each natural number is then equal to the set containing just the natural number preceding it.
Set-theoretic definition of natural numbers
Canonical representation of a positive integer
^ a b "Standard number sets and intervals". ISO 80000-2:2009.
International Organization for Standardization. p. 6.
^ Weisstein, Eric W. "Natural Number". MathWorld.
^ "natural number", Merriam-Webster.com, Merriam-Webster, retrieved 4
^ Carothers (2000) says: "ℕ is the set of natural numbers (positive
integers)" (p. 3)
^ a b Mac Lane & Birkhoff (1999) include zero in the natural
numbers: 'Intuitively, the set ℕ = 0, 1, 2, ... of all natural
numbers may be described as follows: ℕ contains an "initial" number
0; ...'. They follow that with their version of the Peano Postulates.
^ Jack G. Ganssle & Michael Barr (2003). Embedded Systems
Dictionary. p. 138 (integer), 247 (signed integer), & 276
(unsigned integer). ISBN 1578201209. integer 1. n. Any whole
^ 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)
^ Bluman (2010): "Numbers make up the foundation of mathematics." (p.
^ Weisstein, Eric W. "
Bluman, Allan (2010), Pre-Algebra DeMYSTiFieD (Second ed.),
Carothers, N.L. (2000), Real analysis, Cambridge University Press,
Clapham, Christopher; Nicholson, James (2014), The Concise Oxford
Dedekind, Richard (2007), Essays on the Theory of Numbers, Kessinger Publishing, LLC, ISBN 0-548-08985-X
Eves, Howard (1990), An Introduction to the History of Mathematics
(6th ed.), Thomson, ISBN 978-0-03-029558-4
Halmos, Paul (1960), Naive Set Theory, Springer Science & Business
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Von Neumann, John (January 2002) , "On the introduction of
transfinite numbers", in Jean van Heijenoort, From
Hazewinkel, Michiel, ed. (2001) , "Natural number", Encyclopedia
Springer Science+Business Media
v t e
Natural numbers (
displaystyle mathbb N
) Integers (
displaystyle mathbb Z
) Rational numbers (
displaystyle mathbb Q
) Constructible numbers Algebraic numbers (
displaystyle mathbb A
𝔸) Periods Computable numbers Definable real numbers Arithmetical numbers Gaussian integers
Real numbers (
displaystyle mathbb R
) Complex numbers (
displaystyle mathbb C
) Quaternions (
displaystyle mathbb H
) Octonions (
displaystyle mathbb O
Split Composition algebras
displaystyle mathbb R
: Split-complex numbers Split-quaternions Split-octonions over
displaystyle mathbb C
: Bicomplex numbers Biquaternions Bioctonions
Dual numbers Dual quaternions Hyperbolic quaternions Sedenions (
displaystyle mathbb S
𝕊) Split-biquaternions Multicomplex numbers
Cardinal numbers Irrational numbers Fuzzy numbers Hyperreal numbers Levi-Civita field Surreal numbers Transcendental numbers Ordinal numbers p-adic numbers Supernatural numbers Superreal numbers
v t e
Classes of natural numbers
Powers and related numbers
Achilles Power of 2 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Perfect power Powerful Prime power
Of the form a × 2b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall
Other polynomial numbers
Carol Hilbert Idoneal Kynea Leyland Lucky numbers of Euler Repunit
Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Padovan Pell Perrin
Possessing a specific set of other numbers
Knödel Riesel Sierpinski
Expressible via specific sums
Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme
Generated via a sieve
Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star
Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal
Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral
Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula
Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal
Centered pentachoric Squared triangular
Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Somer–Lucas pseudoprime Strong pseudoprime
Bell Cake Catalan Dedekind Delannoy Euler Fuss–Catalan Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus
By properties of σ(n)
Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect
By properties of Ω(n)
Almost prime Semiprime
By properties of φ(n)
Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
By properties of s(n)
Amicable Betrothed Deficient Semiperfect
Dividing a quotient
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson
Other prime factor or divisor related numbers
Blum Erdős–Nicolas Erdős–Woods Friendly Frugal Giuga Harmonic divisor Lucas–Carmichael Pronic Regular Rough Smooth Sociable Sphenic Størmer Super-Poulet Zeisel
Automorphic Cyclic Digit-reassembly Dudeney Equidigital Extravagant Factorion Friedman Happy Harshad Kaprekar Keith Lychrel Missing-digit sum Narcissistic Palindromic Pandigital Parasitic Pernicious Polydivisible Primeval Repdigit Repunit Self Self-descriptive Smarandache–Wellin Strictly non-palindromic Strobogrammatic Sum-product Transposable Trimorphic Undulating Vampire