Täisarvu

Täisarv (Estonian for integer) refers to any whole number that can be positive, negative, or zero. The set of all täisarvud is denoted by Z and includes numbers such as ..., −3, −2, −1, 0, 1, 2, 3, .... Each täisarv is determined by its sign and its magnitude, the latter given by the absolute value.

Täisarvud are closed under the usual arithmetic operations of addition and multiplication: the sum or product of any two täisarvud is a täisarv. Every täisarv a has an additive inverse −a, and subtraction is defined as addition with this inverse. The order relation on täisarvud is the standard total order: for any a and b, exactly one of a < b, a = b, or a > b holds.

Algebraically, the set Z with the usual addition and multiplication forms a commutative ring with identity. It is also a Euclidean domain, a principal ideal domain, and an integral domain. These properties underpin results such as the division algorithm, the existence of greatest common divisors, and the fundamental theorem of arithmetic (every integer greater than 1 factors uniquely into primes up to sign).

Täisarvud are used in counting discrete quantities, indexing, and mathematical theory, including number theory and computer science. They support concepts such as parity (even or odd), divisibility, and modular arithmetic, where calculations are performed modulo a fixed integer. In notation, a täisarv is frequently written as a ∈ Z, and is commonly represented in decimal form with a possible leading minus sign for negative values.

Natural numbers are sometimes defined as a subset of täisarvud, depending on convention, typically the non-negative integers.