Integers can be added and subtracted, multiplied, and compared. The main reason for introducing the negative numbers is that it becomes possible to solve all equations of the form
Mathematicians express the fact that all the usual laws of arithmetic are valid in the integers by saying that (Z, +, *) is a commutative ring.
The ordering on Z is given by ... < 2 < 1 < 0 < 1 < 2 < ... and it turns Z into a totally ordered set without upper or lower bound. We call an integer positive if it is greater than zero; zero itself is not considered to be positive. The order is compatible with the algebraic operations in the following way:
Like the natural numbers, the integers form a countably infinite set.
The integers do not form a field since for instance there is no integer x such that 2x = 1. The unique smallest field containing the integers is given by the rational numbers.
An important property of the integers is division with remainder: given two integers a and b with b≠0, we can always find integers q and r such that
All of this can be abbreviated by saying that Z is a Euclidean domain. It implies that Z is a principal ideal domain and that whole numbers can be written as products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.
The branch of mathematics which studies the integers is called number theory.
An integer is often one of primitive datatypes in computer languages typically with 4 bytes length. Integers are often used as an index for an array.
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