, or whole numbers
, consist of the natural numbers
(0, 1, 2, ...) and the negative whole numbers (-1, -2, -3, ...).
of all integers is usually denoted by Z
(more properly, a Z in blackboard bold
), which stands for Zahlen
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.
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
- a = b q + r
and such that 0 <= r
| (see absolute value
is called the quotient
is called the remainder
resulting from division of a
. The numbers q
are uniquely determined by a
. 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.
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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