Heltallene

Heltallene, or the integers, are the set of whole numbers with their opposites, together with zero: {..., -2, -1, 0, 1, 2, ...}. They are usually denoted by Z, from the German Zahl. As a subset of the real numbers, the integers extend the natural numbers by adjoining negative counterparts.

Key properties: the integers are closed under addition and multiplication, contain the additive identity 0, and every element has an additive inverse, making (Z, +) an infinite cyclic group. Under the standard order, Z is a totally ordered, discrete, and Archimedean set. They are countably infinite and form a fundamental building block in number theory.

Algebraic structure: Z is a commutative ring with unity. It is a Euclidean domain (using the absolute value as the Euclidean function) and thus a principal ideal domain and a unique factorization domain. The only units (multiplicative inverses) in Z are 1 and −1, so Z is not a field. Every nonzero integer can be factored uniquely into primes up to order and sign, as stated by the Fundamental Theorem of Arithmetic.

Relation to other number systems: the natural numbers N are contained in Z (depending on convention, 0 may be included in N). The integers sit inside the rational numbers Q and reals R, providing a discrete foundation for arithmetic operations. Division of integers generally yields a quotient with remainder, since not every nonzero integer has a multiplicative inverse in Z.

Applications: Heltallene are essential in number theory, algebra, and computer science, underpinning concepts such as parity, divisibility, prime numbers, and modular arithmetic.