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.

This article is about "positive integers" and "non-negative integers". For all the numbers ..., −2, −1, 0, 1, 2, ..., see Integer.

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 number sets are built by successively extending the set of natural numbers: the integers, by including an additive identity 0 (if not yet in) 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 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 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.

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

Naturals with zero: $\;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}$

The set of all natural numbers is standardly denoted $N$ or $\mathbb {N} .$ ^[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]

Alternatively, since the natural numbers naturally form a subset of the integers (often denoted $\mathbb {Z}$ ), 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 " $*$ " or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:^[1]

under addition and multiplication: for all natural numbers $a$ and $b$ , both $a + b$ and $a \times b$ are natural numbers.^[35]

Closure

: for all natural numbers $a$ , $b$ , and $c$ , $a + (b + c) = (a + b) + c$ and $a \times (b \times c) = (a \times b) \times c$ .^[36]

Associativity

: for all natural numbers $a$ and $b$ , $a + b = b + a$ and $a \times b = b \times a$ .^[37]

Commutativity

of multiplication over addition for all natural numbers $a$ , $b$ , and $c$ , $a \times (b + c) = (a \times b) + (a \times c)$ .

Distributivity

No nonzero : if $a$ and $b$ are natural numbers such that $a \times b = 0$ , then $a = 0$ or $b = 0$ (or both).

zero divisors

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 $\emptyset$ . 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 ( $ℵ 0$ ).

empty set

Natural numbers are also used as : "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the empty set $\emptyset$ . 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 without 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.

linguistic ordinal numbers

Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.

The least ordinal of cardinality $ℵ 0$ (that is, the initial ordinal of $ℵ 0$ ) is $ω$ but many well-ordered sets with cardinal number $ℵ 0$ 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 non-standard model of arithmetic satisfying the Peano Arithmetic (that is, 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. Other generalizations are discussed in Number § Extensions of the concept.

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

Call $0 = { }$ , the .

empty set

Define the successor $S (a)$ of any set $a$ by $S (a) = a \cup {a}$ .

By the , 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.

axiom of infinity

This intersection is the set of the natural numbers.

– Representation of a number as a product of primes

Canonical representation of a positive integer

– Mathematical set that can be enumerated

Countable set

– Function of the natural numbers in another set

Sequence

– Generalization of "n-th" to infinite cases

Ordinal number

– Size of a possibly infinite set

Cardinal number

– Axiom(s) of Set Theory

Set-theoretic definition of natural numbers

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Natural number"

. apronus.com.

Natural number

Closure

Associativity

Commutativity

Distributivity

zero divisors

empty set

linguistic ordinal numbers

empty set

axiom of infinity

Canonical representation of a positive integer

Countable set

Sequence

Ordinal number

Cardinal number

Set-theoretic definition of natural numbers

"Natural number"

"Axioms and construction of natural numbers"