Natural number

In mathematics, the natural numbers are the numbers 1, 2, 3, etc., possibly including 0 as well.^{[under discussion]} Some definitions, including the standard ISO 80000-2,^[1] 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, ...^[2][a] Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).^[4] In common language, particularly in primary school education, natural numbers may be called counting numbers^[5] to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement—a hallmark characteristic of real numbers.

The natural numbers can be used for counting (as in "there are six coins on the table"), in which case they serve as cardinal numbers. They may also be used for ordering (as in "this is the third largest city in the country"), in which case they serve as ordinal numbers. Natural numbers are sometimes used as labels—also known as nominal numbers, (e.g. jersey numbers in sports)—which do not have the properties of numbers in a mathematical sense.^[3][6]

The natural numbers form a set, often symbolized as ${\textstyle \mathbb {N} }$. Many other

${\displaystyle 1/n}$ for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including the limits of Cauchy sequences^[b] of rationals; the complex numbers, by adjoining to the real numbers a square root of −1 (and also the sums and products thereof); and so on.^[c][d] This chain of extensions canonically embeds the natural numbers in the other number systems.

Properties of the natural numbers, such as

.

History

Ancient roots

Royal Belgian Institute of Natural Sciences)^[7][8][9]

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 in

tally 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 of

A much later advance was the development of the idea that 

The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.^[f] 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).^[16] However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other.^[17]

Independent studies on numbers also occurred at around the same time in India, China, and Mesoamerica.^[18]

Modern definitions

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.^[19] Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".^[g]

The

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 still, they were shown to be equivalent in most practical applications.

The second class of definitions was introduced by

With all these definitions, it is convenient to include 0 (corresponding to the

Notation

The set of all natural numbers is standardly denoted N or ${\displaystyle \mathbb {N} .}$^[3][30] Older texts have occasionally employed J as the symbol for this set.^[31]

Since natural numbers may contain 0 or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as:^[1][32]

• Naturals without zero: ${\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}$
• Naturals with zero: ${\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}$

Alternatively, since the natural numbers naturally form a subset of the integers (often denoted ${\displaystyle \mathbb {Z} }$), they may be referred to as the positive, or the non-negative integers, respectively.^[33] To be unambiguous about whether 0 is included or not, sometimes a superscript "${\displaystyle *}$" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:^[1]

${\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}$

${\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}$

Properties

This section uses the convention ${\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}$.

Addition

Given the set ${\displaystyle \mathbb {N} }$ of natural numbers and the successor function ${\displaystyle S\colon \mathbb {N} \to \mathbb {N} }$ sending each natural number to the next one, one can define

${\displaystyle (\mathbb {N} ,+)}$ is a .

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 ${\displaystyle \times }$ can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns ${\displaystyle (\mathbb {N} ^{*},\times )}$ into a

.

Relationship between addition and multiplication

Addition and multiplication are compatible, which is expressed in the

. 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 ${\displaystyle \mathbb {N} }$ is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that ${\displaystyle \mathbb {N} }$ 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. Furthermore, ${\displaystyle (\mathbb {N^{*}} ,+)}$ has no identity element.

Order

In this section, juxtaposed variables such as ab indicate the product a × b,^[34] and the standard order of operations is assumed.

A

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

(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 or Euclidean division 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

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

The number q is called the

), 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 numbers a and b, both a + b and a × b are natural numbers.^[35]
• Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.^[36]
• Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.^[37]
• Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
  • If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number a, a × 1 = a. However, the "existence of additive identity element" property is not satisfied
• 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 (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. The numbering of cardinals usually begins at zero, to accommodate the empty set ${\displaystyle \emptyset }$. 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. The numbering of ordinals usually begins at zero, to accommodate the order type of the empty set
  ${\displaystyle \emptyset }$. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any
  limit points. 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.

The least ordinal of cardinality ℵ₀ (that is, the initial ordinal of ℵ₀) is ω but many well-ordered sets with cardinal number ℵ₀ have an ordinal number greater than 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

ultrapower construction. Other generalizations are discussed in Number § Extensions of the concept

.

Georges Reeb used to claim provocatively that "The naïve integers don't fill up ${\displaystyle \mathbb {N} }$".^[38]

Formal definitions

There are two standard methods for formally defining natural numbers. The first one, named for

Peano arithmetic, based on few axioms called Peano axioms

.

The second definition is based on

one to one correspondence

between the two sets n and S.

The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not provable inside Peano arithmetic. A probable example is Fermat's Last Theorem.

The definition of the integers as sets satisfying Peano axioms provide a

consistent

(as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong.

Peano axioms

Main article: Peano axioms

The five Peano axioms are the following:^[39][j]

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 of ${\displaystyle x}$ equals the successor of ${\displaystyle y}$, then ${\displaystyle x}$ equals ${\displaystyle y}$.
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 ${\displaystyle x}$ is ${\displaystyle x+1}$.

Set-theoretic definition

Main article: Set-theoretic definition of natural numbers

Intuitively, the natural number n is the common property of all

one to one correspondence". This does not work in set theory, as such an equivalence class would not be a set (because of Russell's paradox

). The standard solution is to define a particular set with n elements that will be called the natural number n.

The following definition was first published by

von Neumann ordinals

.

The definition proceeds as follows:

• Call 0 = { }, the empty set.
• Define the successor S(a) of any set a by S(a) = a ∪ {a}.
• By the axiom of infinity, there exist sets which contain 0 and are closed under the successor function. Such sets are said to be inductive. The intersection of all inductive sets is still an inductive set.
• This intersection is the set of the natural numbers.

It follows that the natural numbers are defined iteratively as follows:

• 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.

It can be checked that the natural numbers satisfy the Peano axioms.

With this definition, given a natural number n, the sentence "a set S has n elements" can be formally defined as "there exists a

set inclusion defines the usual total order on the natural numbers. This order is a well-order

.

It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the

von Neumann definition of ordinals for defining all ordinal numbers

, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."

If one does not accept the axiom of infinity, the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.

There are other set theoretical constructions. In particular, Ernst Zermelo provided a construction that is nowadays only of historical interest, and is sometimes referred to as Zermelo ordinals.^[41] It consists in defining 0 as the empty set, and S(a) = {a}.

With this definition each natural number is a

cardinalities

is not directly accessible; only the ordinal property (being the nth element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.

See also

• Canonical representation of a positive integer
   – Representation of a number as a product of primes
• Countable set – Mathematical set that can be enumerated
• Sequence – Function of the natural numbers in another set
• Ordinal number – Generalization of "n-th" to infinite cases
• Cardinal number – Size of a possibly infinite set
• Set-theoretic definition of natural numbers – Axiom(s) of Set Theory

Number systems Complex ${\displaystyle :\;\mathbb {C} }$ Real ${\displaystyle :\;\mathbb {R} }$ Rational ${\displaystyle :\;\mathbb {Q} }$ Integer ${\displaystyle :\;\mathbb {Z} }$ Natural ${\displaystyle :\;\mathbb {N} }$ Zero : 0 One : 1 Prime numbers Composite numbers Negative integers Fraction Finite decimal Dyadic (finite binary) Repeating decimal Irrational Algebraic irrational Transcendental Imaginary

Notes

1. ^ Carothers (2000, p. 3) says, "${\displaystyle \mathbb {N} }$ is the set of natural numbers (positive integers)." Both definitions are acknowledged whenever convenient, and there is no general consensus on whether zero should be included in the natural numbers.^[3]
2. ^ Any Cauchy sequence in the Reals converges,
3. ^ Mendelson (2008, p. x) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."
4. ^ Bluman (2010, p. 1): "Numbers make up the foundation of mathematics."
5. ^ 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.^[11]
6. ^ This convention is used, for example, in Euclid's Elements, see D. Joyce's web edition of Book VII.^[15]
7. ^ 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."^[20][21]
8. ^ "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)
9. ^ Mac Lane & Birkhoff (1999, p. 15) include zero in the natural numbers: 'Intuitively, the set ${\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}$ of all natural numbers may be described as follows: ${\displaystyle \mathbb {N} }$ contains an "initial" number 0; ...'. They follow that with their version of the
   Peano's axioms
   .
10. ^ Hamilton (1988, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.  0 is a natural number."
    Halmos (1960, 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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18. ISBN 0-19-506135-7
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24. ISSN 0024-6093
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27. S2CID 40187000
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32. ISBN 978-0-201-72634-3
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33. ISBN 978-0-201-72634-3
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34. ^ Weisstein, Eric W. "Multiplication". mathworld.wolfram.com. Retrieved 27 July 2020.
35. ISBN 978-1-4832-8079-0
    . ...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
36. ^ Davisson, Schuyler Colfax (1910). College Algebra. Macmillian Company. p. 2. Addition of natural numbers is associative.
37. ^ Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962). Laidlaw mathematics series. Vol. 8. Laidlaw Bros. p. 25.
38. doi:10.14321/realanalexch.42.2.0193
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39. ^ Mints, G.E. (ed.). "Peano axioms". Encyclopedia of Mathematics. Springer, in cooperation with the European Mathematical Society. Archived from the original on 13 October 2014. Retrieved 8 October 2014.
40. ^ von Neumann (1923)
41. ^ ^{a b} Levy (1979), p. 52

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External links

Wikimedia Commons has media related to Natural numbers.

• "Natural number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Axioms and construction of natural numbers". apronus.com.