Integer: Meaning (information, definition, explanation, facts)

The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. 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).

Properties

Like the natural numbers, the integers are "closed" under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, the integers gain an important "new" feature. That is, unlike the natural numbers, the integers are also closed under subtraction.

Note however that the integers are not closed under the operation of division, since the quotient of two integers (e.g. 1 divided by 2), need not be an integer.

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.

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 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. On the other hand, theoretical models of digital computers, e.g., Turing machines, usually do have infinite (but only countable) capacity.

For more information, see Integer (computer science).