Heltallet

Heltallet, in mathematical usage in Norwegian and Swedish, refers to the integers – the set of whole numbers that have no fractional part. In standard notation this set is denoted by Z and can be written as {..., -2, -1, 0, 1, 2, ...}. The term is used to describe any member of this set, including negative numbers, zero, and positive numbers.

Relation to other number types: The natural numbers N are a subset of the integers, typically consisting of nonnegative numbers. Some conventions include zero in N, while others start at 1. The integers are a broader category that also includes negative values.

Properties: The integers are closed under addition, subtraction, and multiplication, meaning the result of these operations on integers is always an integer. They form a ring, but not a field, because division of two integers does not always yield an integer. The integers are totally ordered by the usual less-than relation, so for any a and b exactly one of a < b, a = b, or a > b holds.

Parity and sign: Every integer is even or odd. The sign distinguishes positive integers, negative integers, and zero, with |n| representing the magnitude or absolute value.

Uses: Heltallets are fundamental in number theory and discrete mathematics. They are used for counting, indexing, modular arithmetic, and as coefficients in algebraic expressions.

Examples: -7, 0, 4, 100 are all integers.

See also: Natural numbers, rational numbers, real numbers, arithmetic, modular arithmetic.