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Integer
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). Like the natural numbers, the integers form a countably infinite set. The branch of mathematics which includes the study of the integers is called number theory.
Algebraic properties
Like the natural numbers, Z is 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, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.
The following table lists some of the basic properties of addition and multiplication for any integers a, b and c.
addition | multiplication | |
closure: | a + b is an integer | a × b is an integer |
associativity: | a + (b + c) = (a + b) + c | a × (b × c) = (a × b) × c |
commutativity: | a + b = b + a | a × b = b × a |
existence of an identity element: | a + 0 = a | a × 1 = a |
existence of inverse elements: | a + (−a) = 0 | |
distributivity: | a × (b + c) = (a × b) + (a × c) |
In the language of abstract algebra, the first five properties listed above for addition, say that Z together with addition is an abelian group. The lack of inverse elements for multiplication (a × b = 1 implies that a = b = 1 or a = b = −1), means that Z together with multiplication is not a group. All the properties from the above table taken together say that Z together with addition and multiplication is a ring. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the rational numbers. This follows from the fact the that the set of rational numbers (which are essentially just ratios, or "divisions", of integers) is the closure of Z under division.
Although ordinary division is not defined for Z, it does possess an important property called the division algorithm: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The integer q is called the quotient and r is called the remainder, resulting from division of a by b. This is the basis for the Euclidean algorithm for computing greatest common divisors.
Again in the language of abstract algebra, the above says 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.
Order theoretic properties
Z is a totally ordered set without upper or lower bound. The ordering of Z is given by
- ... < −2 < −1 < 0 < 1 < 2 < ...
An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive.
The ordering of integers 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
Integers in computing
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).
Topics in mathematics related to quantity |