ints

Integers, denoted Z, are the set of whole numbers together with their negatives, including zero. They exclude fractions and decimals. Integers are closed under addition and subtraction. For any integers a and nonzero b, there exist unique integers q and r such that a = bq + r and 0 ≤ r < |b|; this is the division algorithm and defines the remainder.

Mathematically, the integers form an infinite, countable, totally ordered ring with unity. They are closed under addition and multiplication, and subtraction provides additive inverses. They are not a field because most nonzero integers do not have multiplicative inverses within the set. The absolute value maps an integer to its distance from zero, and parity distinguishes even from odd numbers.

Subsets include even and odd integers. Primes are integers greater than 1 with exactly two positive divisors; gcd and lcm describe common factors. Bezout's identity states that for any integers a and b, there exist integers x and y with ax + by = gcd(a,b).

In computing, integers are represented as fixed-width data types (for example 8, 16, 32, or 64 bits). They can be signed or unsigned. Two's complement is a common representation for signed integers. Ranges are limited, and arithmetic may overflow, wrapping around modulo 2^n in unsigned arithmetic or causing language-specific behavior in signed arithmetic.

Applications of integers include counting, indexing, and numerous algorithms in mathematics and computer science, such as modular arithmetic, primality testing, and cryptography. See also number theory and integer arithmetic for related topics.