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In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively. A later notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.

There is no universal agreement about whether to include zero in the set of natural numbers. 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, ...}.

History

The most primitive method of representing a natural number is to put down a dot for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a dot for each object in 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 place-value system based essentially on the numerals for 1 and 10.

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 0 digit in place-value 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. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. 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.

The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes. Many Greek mathematicians did not consider 1 to be "a number", so to them 2 was the smallest number.

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

Several set-theoretical definitions of natural numbers were developed in the 19th century. With these definitions it was convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists, logicians, and computer scientists. Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number. The term counting number is also used to refer to the natural numbers, with or without zero, though in modern usage it is convenient to use this term to refer to the case where zero is excluded. Some authors use the term whole number to mean a natural number while others use whole number to mean counting number; while still others use whole number to refer to any integer, whether positive, zero, or negative.

Notation

Mathematicians use N or ${\displaystyle \mathbb {N} }$ (an N in blackboard bold, displayed as ℕ in Unicode) 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 ${\displaystyle (\aleph _{0})}$.

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 "${\displaystyle *}$" or subscript "${\displaystyle 1}$" is added in the latter case:

${\displaystyle \mathbb {N} ^{0}=\mathbb {N} _{0}=\{0,1,2,\ldots \}}$

${\displaystyle \mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{1}=\mathbb {N} _{>0}=\{1,2,\ldots \}.}$

Some authors who exclude 0 from the naturals may distinguish the set of nonnegative integers by referring to the latter as the natural numbers with zero, whole numbers, or counting numbers, denoted W. Others use the notation P for the positive integers if there is no danger of confusing this with the prime numbers. In that case, a popular notation is to use a script P for positive integers (which extends to using script N for negative integers, and script Z for 0).

Set theorists often denote the set of all natural numbers including 0 by a lower-case Greek letter omega: ω. This stems from the identification of an ordinal number with the set of ordinals that are smaller. Moreover, adopting the von Neumann definition of ordinals and defining cardinal numbers as minimal ordinals among those with same cardinality leads to the identity ${\displaystyle \,\mathbb {N} _{0}=\aleph _{0}=\omega }$.

Algebraic properties

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

• 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.
• Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).
• No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0.

Properties

One can recursively define an addition 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.

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. 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, 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.

For the remainder of the article, 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 with 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 expressed as ω.

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 division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

Generalizations

Two generalizations of natural numbers arise from the two uses:

• A natural number can be used to express the size of a finite set; more generally a cardinal number is a measure for the size of a set also suitable for infinite sets; this refers to a concept of "size" such that if there is a bijection between two sets they have the same size. The set of natural numbers itself and any other countably infinite set has cardinality aleph-null (${\displaystyle \aleph _{0}}$).
• Linguistic ordinal numbers "first", "second", "third" can be assigned to the elements of a totally ordered finite set, and also to the elements of well-ordered countably infinite sets like the set of natural numbers itself. This can be generalized to ordinal numbers which describe the position of an element in a well-ordered set in general. An ordinal number is also used to describe the "size" of a well-ordered set, in a sense different from cardinality: if there is an order isomorphism between two well-ordered sets they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ${\displaystyle \omega }$; this is also the ordinal number of the set of natural numbers itself.

Many well-ordered sets with cardinal number ${\displaystyle \aleph _{0}}$ have an ordinal number greater than ${\displaystyle \omega }$ (the latter is the lowest possible). The least ordinal of cardinality ${\displaystyle \aleph _{0}}$ (i.e., the initial ordinal) is ${\displaystyle \omega }$.

For finite well-ordered sets, there is 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.

Hypernatural numbers are part of a non-standard model of arithmetic due to Skolem.

Other generalizations are discussed in the article on numbers.

Formal definitions

Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano axioms state conditions that any successful definition must satisfy. Certain constructions show that, given set theory, models of the Peano postulates must exist.

Peano axioms

The Peano axioms give a formal theory of which the natural numbers may serve as a model. The axioms are:

• There is a specific number, which here will be called the "first number".
• Every number a has a number as its successor, denoted by S(a). Intuitively, S(a) is a + 1.
• There is no number whose successor is the first number.
• S is injective, i.e. distinct numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
• If a property is possessed by the first number and also by the successor of every number that possesses this property, then it is possessed by all numbers. (This axiomatizes the proof technique of mathematical induction within the system.)

The "first number" in the above definition need not correspond to the number zero. All systems that satisfy these axioms are elementarily equivalent in first-order logic; however, there exist models for the Peano axioms that are uncountable; these are called non-standard models for arithmetic and are guaranteed by the upward Löwenheim–Skolem theorem. The name "0" or "1" is typically used for the first element, but the set of integers greater than any chosen integer satisfy the axioms. In Peano's original formulation, the first number was denoted 1.

Constructions based on set theory

A standard construction in set theory, a special case of the von Neumann ordinal construction, is to define the natural numbers as follows:

Set 0 := { }, the empty set,

and 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'.) Then the intersection of all 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.

Each natural number is then equal to the set of all natural numbers less than it, so that

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

and so on.

When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n, and n ≤ m (in the naïve sense) if and only if n is a subset of m.

Also, with this definition, different possible interpretations of notations like R (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 what it means to be one of these sets. For a set n to be a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements.

Other constructions

Although the standard construction is useful, it is not the only possible construction. For example:

one could define 0 = { }

and S(a) = {a},

producing

• 0 = { }
• 1 = {0} ={{ }}
• 2 = {1} = {{{ }}}, etc.

Each natural number is then equal to the set of the natural number preceding it.

It is also possible to define 0 = {{ }}

and S(a) = a ∪ {a}

producing

• 0 = {{ }}
• 1 = {{ }, 0} = {{ }, {{ }}}
• 2 = {{ }, 0, 1}, etc.

The oldest and most "classical" set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and Russell, under which each concrete natural number n is defined as the set of all sets with n elements. This may appear circular, but can be made rigorous with care. Define 0 as {{ }} (clearly the set of all sets with zero elements) and define S(A) (for any set A) as {x ∪ {y} | x ∈ A ∧ y ∉ x} (see set-builder notation). Then 0 will be the set of all sets with zero elements, 1 = S(0) will be the set of all sets with one element, 2 = S(1) will be the set of all sets with two elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under S (that is, if the set contains an element n, it also contains S(n)). One could also define "finite" independently of the notion of "natural number", and then define natural numbers as equivalence classes of finite sets under the equivalence relation of equipollence. This definition does not work in the usual systems of axiomatic set theory because the collections (classes) involved are too large (it will not work in any set theory with the axiom of separation); but it does work in New Foundations (and in related systems known to be relatively consistent) and in some systems of type theory.

See also

• Canonical representation of a positive integer
• Countable set
• Integer
• Number#Classification of numbers for other number systems (rational, real, complex etc.)

Notes

1. "natural number", Merriam-Webster.com, Merriam-Webster, retrieved 4 October 2014
2. "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
3. Michael L. Gorodetsky (2003-08-25). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Retrieved 2012-02-13.
4. This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.
5. This is common in texts about Real analysis. See, for example, Carothers (2000) p.3 or Thomson, Bruckner and Bruckner (2000), p.2.
6. Weisstein, Eric W. "Counting Number". MathWorld. {{cite web}}: Unknown parameter |id2= ignored (help); Unknown parameter |title2= ignored (help)
7. Weisstein, Eric W. "Cardinal Number". MathWorld.
8. Von Neumann 1923
9. Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl (1884). Breslau.
10. Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.

References

• Landau, Edmund (1966). Foundations of Analysis (3 ed.). Chelsea Pub Co. ISBN 0-8218-2693-X.
• Dedekind, Richard (1963). Essays on the Theory of Numbers. Dover. ISBN 0-486-21010-3.
  • Dedekind, Richard (2007). Essays on the Theory of Numbers. Kessinger Publishing, LLC. ISBN 0-548-08985-X.
• N.L. Carothers, Real analysis, Cambridge University Press, 2000, ISBN 0-521-49756-6.
• Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner. Elementary Real Analysis, ClassicalRealAnalysis.com, 2000, ISBN 0-13-019075-6.
• Von Neumann, Johann (1923), "Zur Einführung der trasfiniten Zahlen", Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, 1: 199–208
  • Von Neumann, John (January 2002) , "On the introduction of transfinite numbers", in Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923 harvnb error: no target: CITEREFvon_Neumann1923 (help).

External links

• "Natural number", Encyclopedia of Mathematics, EMS Press, 2001
• Axioms and Construction of Natural Numbers
• Essays on the Theory of Numbers by Richard Dedekind at Project Gutenberg
• "Natural Number". MathWorld.

• Cardinal numbers
• Elementary mathematics
• Integers
• Number theory
• Numbers