heltal

Heltal is the set of integers, a term used in Swedish (and in other Scandinavian languages) to describe the numbers ...,-2,-1,0,1,2,3,... The standard mathematical symbol for this set is Z (often written in bold as ℤ). Heltal include both positive and negative whole numbers as well as zero.

Heltal are closed under addition, subtraction and multiplication: adding or multiplying any two integers yields another integer, and subtracting one integer from another also results in an integer. Division, however, is not closed in the set, since a divided by b is not always an integer (except in special cases). Every integer a has an additive inverse −a, and 0 acts as the additive identity while 1 acts as the multiplicative identity. The usual order relation defines a total order on the integers.

Subsets and related sets: the even integers form a subgroup 2Z, and more generally nZ denotes multiples of n. The nonnegative integers {0,1,2,...} are a commonly used subset in many contexts. Prime numbers refer to positive integers greater than 1 with exactly two positive divisors.

Algebraic structure: Z forms a ring with unity and is an integral domain. It is also a principal ideal domain and a Euclidean domain with respect to the absolute value, which underpins the fundamental theorem of arithmetic: every integer greater than 1 factors uniquely into primes up to units ±1.

Applications: integers are fundamental in number theory and algebra, and are used widely in counting, arithmetic, modular arithmetic and computer science. Different conventions may treat natural numbers differently, with some including zero and others not.