naturales

Naturales, or natural numbers, are the set of numbers used for counting and basic ordering. In mathematics they are usually denoted by N. Depending on the convention, N either includes zero (0, 1, 2, 3, …) or starts at one (1, 2, 3, …).

Historically, their formal foundation is provided by axioms such as the Peano axioms. These specify a distinguished element (often 0) and a successor function S, where every number has a unique successor, and 0 is not the successor of any number. The induction axiom allows proofs by induction over natural numbers.

Natural numbers are closed under addition and multiplication, forming a primitive algebraic structure. They are naturally ordered, with the usual less-than relation, and the order is compatible with addition and multiplication. The set of natural numbers is infinite and serves as the basis for arithmetic.

Key concepts include the successor function, addition and multiplication as recursive operations, and the principle of mathematical induction. Natural numbers underpin many branches of mathematics, including number theory, combinatorics, and set theory, and they serve as the counting basis in everyday life and computer science.

Variants: Some authors include zero as a natural number; others do not. In some contexts the term 'whole numbers' is used to denote nonnegative integers (including zero).