naturalnumber

Natural numbers, usually denoted N, are the numbers used for counting and ordering discrete objects. In mathematics they form a subset of the integers and are typically defined as starting at 0 or at 1 and increasing by 1. The most common convention today is to include 0, so N = {0,1,2,...}; in some contexts N = {1,2,3,...}.

Natural numbers are closed under addition and multiplication, and these operations satisfy the standard algebraic laws: associativity, commutativity, and distributivity. They are totally ordered by the usual ≤ relation, and the natural numbers are well-ordered: every nonempty subset has a least element. This underpins the principle of mathematical induction.

Foundationally, the natural numbers are often described by the Peano axioms, which introduce a first element (often 0), a successor function S, and axioms ensuring uniqueness of successors, that 0 is not a successor, and the induction principle.

In set theory, natural numbers can be constructed as von Neumann ordinals: 0 = ∅, 1 = {0}, 2 = {0,1}, and so on; each n is the set of its predecessors. They also appear in type theories and other foundations of mathematics.

Natural numbers serve as the basis for arithmetic and number theory and provide the starting point for defining integers, rationals, reals, and complex numbers. They are essential in computing as well, where they commonly represent unsigned integers. The set of natural numbers is infinite and contains infinitely many primes and other arithmetic structures.

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