The integers or whole numbers, usually denoted by Z (or the unicode character ℤ if your browser supports unicode display) for Zahlen (German "number"), consist of the natural numbers (0, 1, 2, ...) and the negative whole numbers (-1, -2, -3, ...).
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
- a + x = b
for the unknown x; over the natural numbers, only some of those 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.
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:
- if a < b and c < d, then a + c < b + d
- if a < b and 0 < c, then ac < bc
Like the natural numbers, the integers form a countably infinite set.
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 division makes possible the Euclidean algorithm for computing greatest common divisors, which also shows that the greatest common divisor of two integers can always be written as a sum of multiples of the two numbers.
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.