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Natural number

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This article is about “positive integers” and “non-negative integers”. For all the numbers …, −2, −1, 0, 1, 2, …, see Integer.
Natural numbers can be used for counting (one apple, two apples, three apples, …)


Some authors 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, ….[1][2][3][4] 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.[5][6] These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.


In common language, for example in primary school, natural numbers may be called counting numbers[7] 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.


Ancient roots

The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)[8][9][10] 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 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.




Independent studies also occurred at around the same time in India, China, and Mesoamerica.[17]

Modern definitions

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man".

In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics.[18] 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.

Set-theoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in one-to-one 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 one-to-one correspondence with that set is said to have that number of elements.[19]


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 theorists[21] and logicians.[22] Other mathematicians also include 0[4] although many have kept the older tradition and take 1 to be the first natural number.[23] Computer scientists often start from zero when enumerating items like loop counters and string- or array- elements.[24][25]


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 (an N in blackboard bold) to refer to the set of all natural numbers. 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 0.[26]


0 = ℕ0 = {0, 1, 2, …}
* = ℕ+ = ℕ1 = ℕ>0 = {1, 2, …}.






Analogously, given that addition has been defined, a multiplication × 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




In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.





a = bq + r and r < b.


Algebraic properties satisfied by the natural numbers




Two generalizations of natural numbers arise from the two uses:


Many well-ordered sets with cardinal number 0 have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality 0 (i.e., the initial ordinal) is ω.


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 naïve integers don't fill up . Other generalizations are discussed in the article on numbers.

Formal definitions

Peano axioms

Main article: Peano axioms

Many properties of the natural numbers can be derived from the Peano axioms.[27][28]

  • Axiom One: 0 is a natural number.
  • Axiom Two: Every natural number has a successor.
  • Axiom Three: 0 is not the successor of any natural number.
  • Axiom Four: If the successor of x equals the successor of y, then x equals y.
  • Axiom Five (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 Five by an axiom schema one obtains a (weaker) first-order theory called Peano Arithmetic.

Constructions based on set theory

von Neumann construction

In the area of mathematics called set theory, a special case of the von Neumann ordinal construction [29] defines the natural numbers as follows:

  • Set 0 = { }, the empty set,
  • 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.
  • 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 nm if and only if n is a subset of m.

Also, with this definition, different possible interpretations of notations like n (n-tuples versus mappings of n into ) coincide.


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} = {{ }},
  • n = {n−1} = {{{…}}}, etc.

See also


  1. ^ Weisstein, Eric W., "Natural Number", MathWorld.
  3. ^ Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)
  4. ^ 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: contains an "initial" number 0; …'. They follow that with their version of the Peano Postulates. (p. 15)
  5. ^ 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)
  6. ^ Bluman (2010): "Numbers make up the foundation of mathematics." (p. 1)
  7. ^ Weisstein, Eric W., "Counting Number", MathWorld.
  8. ^ Introduction, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
  9. ^ Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
  10. ^ The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. UNESCO's Portal to the Heritage of Astronomy
  11. ^ Georges Ifrah, The Universal History of Numbers, Wiley, 2000, ISBN 0-471-37568-3
  12. ^ "A history of Zero". MacTutor History of Mathematics. Retrieved 2013-01-23. … 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 
  13. ^ Mann, Charles C. (2005), 1491: New Revelations Of The Americas Before Columbus, Knopf, p. 19, ISBN 9781400040063 .
  14. ^ Evans, Brian (2014), "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations", The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context, John Wiley & Sons, ISBN 9781118853979 .
  17. ^ Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 [1972], ISBN 0-19-506135-7
  18. ^ "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)
  19. ^ Eves 1990, Chapter 15
  20. ^ 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.
  21. ^ Bagaria, Joan. "Set Theory". The Stanford Encyclopedia of Philosophy (Winter 2014 Edition). 
  22. ^ Goldrei, Derek (1998). "3". Classic set theory : a guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 0-412-60610-0. 
  23. ^ This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2000, p. 2).
  24. ^ Brown, Jim (1978). "In Defense of Index Origin 0". ACM SIGAPL APL Quote Quad 9 (2): 7 – 7. doi:10.1145/586050.586053. Retrieved 19 January 2015. 
  25. ^ Hui, Roger. "Is Index Origin 0 a Hindrance?". JSoftware / Roger Hui. Retrieved 19 January 2015.  External link in |website= (help)
  26. ^ Weisstein, Eric W., "Cardinal Number", MathWorld.
  27. ^ G.E. Mints (originator), "Peano axioms", Encyclopedia of Mathematics (Springer, in cooperation with the European Mathematical Society), retrieved 8 October 2014 
  28. ^ Hamilton (1988) calls them "Peano's Postulates" and begins with "1.  0 is a natural number." (p. 117f)
    Halmos (1960) 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). (p. 46)
    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)
  29. ^ Von Neumann 1923


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