Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0.^[a]^[1] The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used.^[2]^[3] The set of the natural numbers is commonly denoted by a bold $N$ or a blackboard bold ⁠ $\mathbb {N}$ ⁠.

The natural numbers are used for counting, and for labeling the result of a count, such as: "there are seven days in a week", in which case they are called cardinal numbers. They are also used to label places in an ordered series, such as: "the third day of the month", in which case they are called ordinal numbers.^[4]

Natural numbers are commonly expressed in writing using ten symbols called numerals ("0 1 2 3 4 5 6 7 8 9"). These numerals can also be used as unique identifiers or labels (like the jersey numbers of a sports team) that are referred to as nominal numbers,^[5] which resemble natural numbers but have no specific mathematical properties.

Natural numbers can be compared by magnitude, with larger numbers coming after smaller ones in the list 1, 2, 3, .... Two basic arithmetical operations are defined on natural numbers: addition and multiplication. However, the inverse operations, subtraction and division, only sometimes give natural-number results: subtracting a larger natural number from a smaller one results in a negative number and dividing one natural number by another commonly leaves a remainder.

The most common number systems used throughout mathematics – the integers, rational numbers, real numbers, and complex numbers – contain the natural numbers, and can be formally defined in terms of natural numbers.^[6]

Arithmetic is the study of the ways to perform basic operations on these number systems. Number theory is the study of the properties of these operations and their generalizations. Much of combinatorics involves counting mathematical objects, patterns and structures that are defined using natural numbers.

Intuitive concept

An intuitive and implicit understanding of natural numbers is developed naturally through using numbers for counting, ordering and basic arithmetic.^[7] Within this are two closely related aspects of what a natural number is: the size of a collection;^[8] and a position in a sequence.

Size of a collection

Natural numbers can be used to answer questions like "how many apples are on the table?"^[9] A natural number used in this way describes a characteristic of a finite collection of objects. This characteristic, the size of a collection, is called cardinality and a natural number used to describe or measure it is a cardinal number.

Two finite collections have the same size or cardinality if they have a one-to-one correspondence, meaning the objects can be arranged in pairs (one from each collection), with every object in exactly one pair. In the adjacent image every apple is paired with exactly one orange and every orange is paired with exactly one apple. As such, the group of apples has the same cardinality as the group of oranges, or put more simply the number of apples is the same as the number of oranges.

Because this equality can be established without counting or using any prior notion of number,^[10]^[11] it can form the definition of a cardinal number.^[12] In this case, the number of apples, oranges – and of any other collection that could be paired off to either group – is 3.

If two collections do not have the same cardinality, pairing will leave one of the collections with objects that are unpaired and this can be used to define a size relationship between them. The collection in which all objects are paired is said to be "smaller" and the one left with unpaired objects "larger", than the other.

Position in a sequence

A sequence is a list of objects in a specific order. More precisely, a sequence is a function that assigns an object to each position in that list. The positions themselves are labeled using a well-ordered set; every element always has a clear next element.^[13] Every well-ordered set has an order type, which is the ordinal number that describes its shape of ordering.^[14] The position labels here are not counts or size like with the cardinal numbers, just ordered elements.^[15]

The natural numbers are the most common choice for labeling infinite sequences because they form the simplest infinite well-ordered set, with order type $ω$ . They start at either $0$ or $1$ and continue in their familiar fixed order – $1$ , $2$ , $3$ , and so on – with no end point. Each natural number labels a specific position in the sequence based on where it falls relative to all other positions. For example, $1$ is the first position, $2$ is the position right after $1$ , and $3$ is the position after both $1$ and $2$ and before $4$ , $5$ , and so on. This ordering matches the usual ordering, smaller numbers before larger ones. But the natural numbers are simply the most familiar example; any well-ordered set would work equally well for indexing a sequence, for example the set of letters a, b, c, and so on.^[16]

Terminology and notation

The term natural numbers has two common definitions: either $0, 1, 2, ...$ or $1, 2, 3, ...$ . Because there is no universal convention, the definition can be chosen to suit the context of use.^[1]^[17] To eliminate ambiguity, the sequences $1, 2, 3, ...$ and $0, 1, 2, ...$ are often called the positive integers and the non-negative integers, respectively.

The phrase whole numbers is frequently used for the natural numbers that include $0$ , although it may also mean all integers, positive and negative.^[18]^[2] In primary education, counting numbers usually refer to the natural numbers starting at $1$ ,^[3] though this definition can vary.^[19]^[20]

The set of all natural numbers is typically denoted $N$ or in blackboard bold as ⁠ $\mathbb {N}$ ⁠.^[17]^[21]^[b] Whether $0$ is included is often determined by the context but may also be specified by using $\mathbb {N}$ or $\mathbb {Z}$ (the set of all integers) with a subscript or superscript. Examples include ⁠ $\mathbb {N} _{1}$ ⁠,^[23] or $\mathbb {Z} ^{+}$ ^[24] (for the set starting at $1$ ) and $\mathbb {N} _{0}$ ^[25] or $\mathbb {Z} ^{0+}$ ^[26] (for the set including $0$ ).

Numeral

A numeral is a symbol or grouping of symbols used to express a natural number in writing, and a particular set of symbols with specific rules for using them is a numeral system. Each symbol in a numeral system represents a unique natural number - said to be its value - and can be used alone as a numeral, or in a string with other symbols which together form a numeral.

The decimal system which uses Arabic numerals and positional notation rules is the universal standard for representing natural numbers in mathematics and in common use. In part because of this universal standard, the distinction between an abstract number (a value) and its symbol (a numeral) is generally unimportant so numerals are frequently referred to simply as "numbers". This is sometimes done even where the distinction is relevant, as with binary numerals which are often called "binary numbers".

Use of natural numbers

Natural numbers are used for counting and the four basic operations of arithmetic: addition, subtraction, multiplication, and division.

Counting

Counting is the process of iterating through the natural numbers in sequential order starting at $1$ . It can be done using numbers alone (as in "counting to 10"), or by applying the count to objects (as in "counting the students in the class").

When applied to a collection of objects, counting determines the cardinality of the collection by establishing a one-to-one correspondence between the objects and a sequence of natural numbers.^[27] This involves consecutively "tagging" each object with a number while maintaining a running partition of the tagged objects from those not yet tagged.^[28] The numbers must be assigned in order starting at $1$ – so they are ordinal numbers – but the order of the objects chosen is arbitrary, as long as each object receives one and only one number. The cardinality principle is the theorem that the ordinal number assigned to the final object does not depends on the order chosen for the objects, and gives the result of the count: the cardinal number of the collection.^[29]

Formal definitions

Formal definitions take the existing, intuitive notion of natural numbers together with the rules of arithmetic and define them both in the more fundamental terms of mathematical logic. Formal systems typically assume that the defining characteristic of natural numbers is their fixed order^[30]^[31]^{[page needed]} and establish this order using the primitive notion of a successor. Every natural number has a successor, which is another unique natural number that it is followed by.

Two standard formal definitions are based on the Peano axioms and set theory. The Peano axioms (named for Giuseppe Peano) do not explicitly define what the natural numbers are, but instead comprise a list of statements or axioms that must be true of natural numbers, however they are defined. In contrast, set theory defines each natural number as a particular set, in which a set can be generally understood as a collection of distinct objects or elements. While the two methods are different, they are consistent in that the natural number sets collectively satisfy the Peano axioms.

Peano axioms

The five Peano axioms are:^[32]^[c]

$0$ is a natural number.
Every natural number has a successor which is also a natural number.
$0$ is not the successor of any natural number.
If the successor of $x$ equals the successor of $y$ , then $x$ equals $y$ .
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 $x + 1$ .

Set-theoretic definition

In set theory each natural number $n$ is defined as a specific set. A variety of constructions have been proposed, however the standard solution (due to John von Neumann)^[33] is:

Call $0 = { }$ , the empty set.
Define the successor $S (a)$ of any set $a$ by $S (a) = a \cup {a}$ .
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.

This produces an iterative definition of the natural numbers called the von Neumann ordinals: ${\begin{alignedat}{2}0&&&{}=\{\}&&{}=\varnothing \\1&=0\cup \{0\}&&{}=\{0\}&&=\{\varnothing \}\\2&=1\cup \{1\}&&{}=\{0,1\}&&{}=\{\varnothing ,\{\varnothing \}\}\\3&=2\cup \{2\}&&{}=\{0,1,2\}&&{}=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}\\n&=n-1\cup \{n-1\}&&{}=\{0,1,...,n-1\}&&{}=\{\varnothing ,\{\varnothing \},...,\{\varnothing ,\{\varnothing \},...\}\}\\\end{alignedat}}$

In this construction every natural number $n$ is a set containing $n$ elements, where each element is a natural number less than $n$ . From this, the intuitive concepts of cardinality and order can be formally defined as:

Cardinality: a set $S$ has $n$ elements if there is a one-to-one correspondence or bijection from $n$ to $S$ .
Order: $n \leq m$ if and only if $n$ is a subset of $m$ .

Another construction sometimes called Zermelo ordinals^[34] defines $0 = { }$ and $S (a) = {a}$ and is now largely only of historical interest.

Properties

This section uses the convention that $0$ is a natural number: ⁠ $\mathbb {N} =\mathbb {N} _{0}$ ⁠.

Addition

Given the set $\mathbb {N}$ of natural numbers and the successor function $S\colon \mathbb {N} \to \mathbb {N}$ sending each natural number to the next one, addition (⁠ $+$ ⁠) is defined by: ${\begin{aligned}a+0&=a&{\textrm {(1)}}\\a+S(b)&=S(a+b)&{\textrm {(2)}}\\\end{aligned}}$ In the statements above, (1) explicitly defines addition for the first natural number and (2) gives a recursive definition for each subsequent number in terms of previous definitions, as illustrated below. ${\begin{alignedat}{2}&a+1=a+S(0)=S(a+0)=S(a)\\&a+2=a+S(1)=S(a+1)=S(S(a))\\&a+3=a+S(2)=S(a+2)=S(S(S(a)))\end{alignedat}}$ In this way, addition can be seen as repeated application of the successor function. Intuitively, ⁠ $a+b$ ⁠ is evaluated by applying the successor function to ⁠ $a$ ⁠ as many times as it must be applied to ⁠ $0$ ⁠ to produce ⁠ $b$ ⁠.

The algebraic structure $(\mathbb {N} ,+)$ is a commutative monoid with identity element ⁠ $0$ ⁠. It is a

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