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 in
mathematics by
Z (or Z in
blackboard bold, ), which stands for
Zahlen (
German for "numbers"). They are also known as the
whole numbers, although that term is also used to refer only to the positive integers (with or without
zero).
Integers can be added, subtracted and multiplied, the result being an integer.
Any two integers can be 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:
- 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 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
a and
b is equal to the greatest common divisor of the two numbers, namely,
b and
r, with smaller sum. This observation is the base for the
Euclidean algorithm for computing greatest common divisors.
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. However, these "integers" can only represent a subset of all mathematical integers, since "real-world" computers are of finite capacity. 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. On the other hand, theoretical models of digital computers, e.g., Turing machines, usually do have infinite (but only
countable) capacity.