Integer

The integers consist of the natural numbers (0, 1, 2, ...) and their negatives (-1, -2, -3, ...; -0 is equal to 0 and therefore not included as a separate integer). The set of all integers is usually denoted by Z (or Z in blackboard bold, ), which stands for Zahlen (German for "numbers").

Integers can be added and subtracted, multiplied, and compared. Introducing the negative integers makes it possible to solve all equations of the form

a + x = b

(where a and b are constant natural numbers) for the unknown x; if x is constrained to the natural numbers, only some of these equations are solvable.

Mathematicians express the fact that all the usual laws of arithmetic are valid in the integers by saying that (Z, +, *) is a commutative ring.

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by

... < -2 < -1 < 0 < 1 < 2 < ...

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:

1. if a < b and c < d, then a + c < b + d
2. if a < b and 0 < c, then ac < bc

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 smallest field containing the integers is 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

a = b q + r

and such that 0 <= r < |b| (see absolute value). q is called the quotient and r is called the remainder resulting from division of a by b. The numbers q and r are uniquely determined by a and b. This shows that the greatest common divisor of two integers can always be written as a sum of multiples of the two numbers, and makes the Euclidean algorithm for computing greatest common divisors possible.

All of this can be abbreviated by saying that Z is a Euclidean domain. This 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 the primitive datatypes in computer languages. Note, however, that a computer can only represent a subset of all mathematical integers, given that computers are finite machines. Integer datatypes are typically implemented using a fixed number of bits, and even variable-length representations eventually run out of storage space when trying to represent especially large numbers. See integer (computer science) for more detailed discussion.