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