Natural Number · aiwiki.page

A natural number is a number used to count objects or indicate positions in an ordered list. The natural numbers are commonly written $1,2,3,\ldots$ , although a widely used alternative includes zero, giving $0,1,2,3,\ldots$ . They form a basic number system in mathematics, underlying elementary arithmetic and the construction of larger number systems. Neither convention includes negative integers or noninteger fractions. (openstax.org)

Definition and notation

The set of natural numbers is usually denoted $\mathbb N$ . Because its starting point varies between texts, an explicit definition is important. Unambiguous descriptions include

\mathbb Z_{\geq0}=\{0,1,2,\ldots\}, \qquad \mathbb Z_{>0}=\{1,2,3,\ldots\},

where $\mathbb Z$ denotes the integers. This article uses the zero-inclusive convention unless otherwise stated. Some elementary textbooks reserve “natural numbers” for the positive integers and call the zero-inclusive set “whole numbers.” (openstax.org)

A number is distinct from its written representation. For example, the decimal numeral $12$ uses positional notation: its digits represent one ten and two units. Changing the notation does not change the quantity represented. Thus the natural numbers are mathematical objects rather than a particular collection of written symbols. (openstax.org)

Arithmetic and order

Addition and multiplication of natural numbers always produce natural numbers: the set is closed under these operations. Both operations are associative and commutative, and multiplication distributes over addition. Zero is the additive identity, while one is the multiplicative identity. These laws provide the arithmetic structure subsequently extended to integers and rational numbers. (ams.org)

Subtraction and division are not closed operations on the natural numbers. For example, $3-5=-2$ requires a negative integer, while $3/2$ requires a noninteger rational number. These limitations motivate extensions of the number system. Under the usual inclusions,

\mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R,

with $\mathbb R$ the real numbers. (openstax.org)

The usual order is discrete: there is no natural number strictly between $n$ and $n+1$ . Every natural number has a successor, so there is no greatest natural number. A deeper property is the well-ordering principle: every nonempty subset of $\mathbb N$ has a least element. This supports arguments that choose the smallest possible counterexample to a statement. (math.purdue.edu)

Axioms and induction

The Peano axioms describe the natural numbers through an initial element, a successor operation, and an induction principle. In a zero-based formulation, zero is a natural number; each natural number has a natural-number successor; zero is not a successor; and distinct numbers have distinct successors. Finally, any subset containing zero and containing the successor of each of its members contains every natural number. These axioms characterize the progression underlying counting. (smith-at-sfsu.net)

The last condition yields mathematical induction, a method of mathematical proof. To establish a statement $P(n)$ for all natural numbers, one proves $P(0)$ , then proves that $P(n)$ implies $P(n+1)$ . Strong induction instead allows the step to use all preceding cases. Unlike checking numerous examples, induction establishes a statement throughout the infinite domain. (webapps.math.uci.edu)

Arithmetic can also be defined using recursion. Writing $S(n)$ for the successor, addition and multiplication satisfy

a+0=a,\qquad a+S(b)=S(a+b),

a\cdot0=0,\qquad a\cdot S(b)=a\cdot b+a.

These definitions build operations from initial values and successive steps, rather than assuming their familiar rules in advance. Induction can then establish those rules. (ams.org)

Set-theoretic construction

In set theory, natural numbers can be represented by sets. The von Neumann construction begins with the empty set and defines

0=\varnothing,\qquad S(n)=n\cup\{n\}.

Consequently,

1=\{0\},\quad 2=\{0,1\},\quad 3=\{0,1,2\}.

Each number is the set of all preceding numbers, simultaneously representing a finite size and a position in the sequence. The construction is associated with John von Neumann. (smith-at-sfsu.net)

This representation is not the only possible one. Another construction uses successively nested singleton sets. Different representations can realize the same successor structure and arithmetic; their purpose is to place natural numbers within a foundational framework, not to identify ordinary counting with particular physical objects. (smith-at-sfsu.net)

Infinity and divisibility

The natural numbers provide the standard model of a countably infinite set. An infinite set is countably infinite when its elements can be paired one-to-one with the natural numbers. For example, $n\mapsto2n$ pairs $\mathbb N$ with the nonnegative even integers, despite the latter being a proper subset. The integers and rational numbers are also countable, whereas the real numbers are uncountable. (dpmms.cam.ac.uk)

In number theory, divisibility organizes the multiplicative structure of the positive natural numbers. A prime number is an integer greater than one whose only positive divisors are one and itself. The fundamental theorem of arithmetic states that every integer greater than one has a factorization into primes unique apart from their order. For example, $60=2^2\cdot3\cdot5$ . One is excluded from the primes, and zero is outside this theorem’s domain. The primes themselves form an infinite set, a result traditionally associated with Euclid. (nrich.maths.org)