Integer
An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component.
For example, 21, 4, 0, and −2048 are integers, while 9.75, 5½, and Template:Sqrt are not.
(TO DO: expand, organize, cross-reference, illustrate.)
Contents
Description
The set of integers consists of zero (Template:Num), the natural numbers (Template:Num, Template:Num, Template:Num, …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e. −1, −2, −3, ...).
This is often denoted by a boldface Z ("Z") or blackboard bold <math>\mathbb{Z}</math> (Unicode U+2124 Template:Unicode) standing for the German word Zahlen (Template:IPA-de, "numbers").
Template:Unicode is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers.
In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers.
In fact, the (rational) integers are the algebraic integers that are also rational numbers.
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, Template:Num, Z (unlike the natural numbers) is also closed under subtraction.
The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring.
This universal property, namely to be an initial object in the category of rings, characterizes the ring Z.
Order-theoretic properties
Z is a totally ordered set without upper or lower bound. The ordering of Z is given by:
- … −3 < −2 < −1 < 0 < 1 < 2 < 3 < …
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.
Computer science
See Integer (computer science).
An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity.
Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.)
Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).
Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).
Cardinality
The cardinality of the set of integers is equal to <math>\aleph_0</math> (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N.
See also
- 0.999...
- Canonical representation of a positive integer
- Hyperinteger
- Integer-valued function
- Integer lattice
- Integer part
- Integer sequence
- List of mathematical symbols
- Number
- Profinite integer
- Rational number
- Real number
