integers

Integers, denoted by the symbol Z, are the set consisting of zero, the positive whole numbers, and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, .... They form a countably infinite set, meaning there is a one-to-one correspondence with the natural numbers.

The integers are closed under the operations of addition, subtraction, and multiplication: the sum, difference, or product of any two integers is again an integer. Division, however, does not always produce an integer; for example, 7 ÷ 3 equals 7/3, which is not an integer. The additive identity is 0, and every integer a has an additive inverse −a, so a + (−a) = 0. The multiplicative identity is 1.

Algebraically, the integers form an abelian group under addition and a commutative ring with unity under the usual operations. They are also an ordered ring, with a total order that satisfies the usual compatibility properties (for example, adding the same number preserves order). These structural properties underpin many number-theoretic algorithms, such as those for computing greatest common divisors.

In number theory and beyond, integers support concepts such as parity (even or odd), divisibility, and modular arithmetic, where integers are studied modulo a positive integer n. The nonnegative integers are often viewed as the natural numbers, and many authors include 0 in the natural numbers.

Applications of integers span mathematics and computer science, where they model quantities, indices, and measurements, and underpin algorithms for arithmetic, cryptography, and data integrity.