Overview
Natural numbers — often called counting numbers — form the simplest infinite set of numbers used to count discrete objects and to order positions in a sequence. In common usage they are the whole numbers 1, 2, 3, ...; in many modern treatments the set also includes 0, giving 0, 1, 2, 3, .... The symbol commonly used for the collection of all natural numbers is ℕ. Natural numbers are fundamental in arithmetic, number theory, combinatorics and the foundations of mathematics.
Basic properties
Natural numbers are whole (integral) and nonnegative under the usual convention that includes zero; alternative conventions exclude zero and start at one. They satisfy simple algebraic and order properties: every natural number has a unique successor obtained by adding one, and there is no largest natural number because repeatedly adding one produces an unbounded increasing sequence. Natural numbers are closed under addition and multiplication, but not closed under subtraction unless restricted to cases where the minuend is at least as large as the subtrahend.
Definitions and formalizations
There are several equivalent ways to define the natural numbers precisely. Peano axioms present an axiomatic approach that introduces a first element (often 0 or 1) and a successor operation and captures induction. Another approach constructs natural numbers set-theoretically, for example using von Neumann ordinals where each natural number is the set of all smaller numbers. In algebra, natural numbers are the initial object in the category of commutative semirings, which reflects their role as the simplest ring-like structure with nonnegative elements.
History and development
The concept of natural numbers predates written history and appears in counting artifacts, tally marks and early numeration systems. Distinctions between counting and measuring led to separate number concepts (integers, rationals, reals). Formal study emerged in the 19th century with efforts to provide rigorous foundations for arithmetic and to clarify basic assumptions; this work culminated in axiomatic systems and set-theoretic constructions used in modern mathematics.
Uses, examples, and importance
In everyday life natural numbers count objects ("three apples") and label positions ("first, second, third"). In mathematics they index sequences, enumerate solutions in combinatorics, and serve as the domain for arithmetic operations. Computer science treats natural numbers as the basic discrete data type for loops, array indices and resource counts. Applications also include probability (counting outcomes), cryptography (number-theoretic constructions) and logic (formal induction arguments).
Variations, conventions and notable facts
- Zero: Some authors include 0 in ℕ, others start at 1. Always check the convention in a given text.
- Infinite: No natural number is infinite; however the set of all natural numbers is an infinite set with cardinality denoted by aleph-null in set theory. For contrasts with larger infinities see discussions of cardinalities and infinite sets (infinite cardinalities).
- Related sets: Natural numbers are a subset of the integers (integers), which in turn embed into rational and real numbers.
- Terminology: "Counting numbers" or "positive integers" typically refer to the natural numbers excluding zero; "whole numbers" sometimes denotes the set including zero.
For a concise introduction to counting concepts and elementary number properties see resources on basic arithmetic (counting numbers) and beginner texts that clarify whether zero is treated as a natural number in their notation.