integer

An integer is a whole number, which may be positive, negative, or zero. The set of all integers is denoted by Z and can be written as {..., -2, -1, 0, 1, 2, ...}. Integers extend the natural numbers by including additive inverses: for every n > 0 there is −n, and 0 is its own inverse.

Arithmetic on integers is closed under addition and multiplication, meaning the sum or product of integers is an integer. Subtraction is defined as adding the additive inverse, and division is not closed, since the quotient of two integers need not be an integer. The integers form a commutative ring with unity: they have additive identity 0, multiplicative identity 1, and satisfy associativity, commutativity, and distributivity of multiplication over addition. They are also totally ordered in a way that is compatible with these operations.

Key properties include parity (even and odd) and divisibility. Every nonzero integer factors uniquely into primes up to sign, a statement known as the Fundamental Theorem of Arithmetic for positive integers, with negative integers carrying a sign factor. Absolute value |n| gives the distance from zero, and the distance between m and n is |m − n|.

Integers underpin much of arithmetic and number theory, and they appear in computer science, cryptography, and modular arithmetic. They provide the basic counting system for signed quantities and serve as a foundational example of a commutative ring.