View source for Integer (computer science)

In computer science, an '''integer''' is a [[data|datum]] of '''integral data type''', a [[data type]] which represents some [[Finite set|finite]] [[subset]] of the mathematical [[integer]]s. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits.  The size of the grouping varies so the set of integer sizes available varies between different types of computers. Computer hardware, including [[virtual machine]]s, nearly always provide a way to represent a processor [[word size|register]] or memory address as an integer.

== Value and representation ==
The ''value'' of an item with an integral type is the mathematical integer that it corresponds to. Integral types may be ''unsigned'' (capable of representing only non-negative integers) or ''signed'' (capable of representing negative integers as well).<ref>{{cite web |url=http://www.swarthmore.edu/NatSci/echeeve1/Ref/BinaryMath/NumSys.html |title=Representation of numbers |last=Cheever |first=Eric |publisher=Swarthmore College |accessdate=2011-09-11}}</ref>

An integer value is typically specified in the [[source code]] of a program as a sequence of digits optionally prefixed with + or −. Some programming languages allow other notations, such as hexadecimal (base 16) or octal (base 8). Some programming languages also permit [[digit group separator]]s.<ref>{{cite web|author=Madhusudhan Konda |url=http://radar.oreilly.com/2011/09/java7-features.html |title=A look at Java 7's new features - O'Reilly Radar |publisher=Radar.oreilly.com |date=2011-09-02 |accessdate=2013-10-15}}</ref>

The ''internal representation'' of this datum is the way the value is stored in the computer's memory. Unlike mathematical integers, a typical datum in a computer has some minimal and maximum possible value. 

The most common representation of a positive integer is a string of [[bit]]s, using the [[binary numeral system]].  The order of the memory [[byte]]s storing the bits varies; see [[endianness]].  The ''width'' or ''precision'' of an integral type is the number of bits in its representation.  An integral type with ''n'' bits can encode 2<sup>''n''</sup> numbers; for example an unsigned  type typically represents the non-negative values 0 through 2<sup>''n''</sup>−1. Other encodings of integer values to bit patterns are sometimes used, for example [[Binary-coded decimal]] or [[Gray code]], or as printed character codes such as [[ASCII]].

There are four well-known [[signed number representations|ways to represent signed numbers]] in a binary computing system.  The most common is [[two's complement]], which allows a signed integral type with ''n'' bits to represent numbers from −2<sup>(''n''−1)</sup> through 2<sup>(''n''−1)</sup>−1.  Two's complement arithmetic is convenient because there is a perfect [[bijection|one-to-one correspondence]] between representations and values (in particular, no separate +0 and −0), and because [[addition]], [[subtraction]] and [[multiplication]] do not need to distinguish between signed and unsigned types. Other possibilities include [[offset binary]], [[sign-magnitude]], and [[ones' complement]].

Some computer languages define integer sizes in a machine-independent way; others have varying definitions depending on the underlying processor word size. Not all language implementations define variables of all integer sizes, and defined sizes may not even be distinct in a particular implementation. An integer in one [[programming language]] may be a different size in a different language or on a different processor.

== Common integral data types ==
{| class="wikitable"
|-
! rowspan="2" | Bits 
! rowspan="2" | Name
! rowspan="2" | Range (assuming [[two's complement]] for signed)
! rowspan="2" | Decimal Digits (approx.)
! rowspan="2" | Uses
!scope=col colspan="5" | Implementations
|-
! [[C (programming language)|C]]/[[C++]] !! C# !! Pascal and Delphi !! Java !! [[SQL]]{{cref2|group=notesql|a}}
|-
|rowspan=2| 4
|rowspan=2| [[nibble]], semioctet
|''Signed:'' From −8 to 7, from −(2<sup>3</sup>) to 2<sup>3</sup>&nbsp;−&nbsp;1
|1
| rowspan="2" | [[Binary-coded decimal]], single decimal digit representation.
| rowspan="2" | n/a
| rowspan="2" | n/a
| rowspan="2" | n/a
| rowspan="2" | n/a
| rowspan="2" | n/a
|-
|''Unsigned:'' From 0 to 15 which equals 2<sup>4</sup>&nbsp;−&nbsp;1
|2
|-
|rowspan=2| 8
|rowspan=2| [[byte]], [[octet (computing)|octet]]
|''Signed:'' From [[127 (number)|−128 to 127]], from −(2<sup>7</sup>) to 2<sup>7</sup>&nbsp;−&nbsp;1
| 3
| rowspan="2" | [[ASCII]] characters
| <code>int8_t</code>, <code>char</code>{{cref2|group=notesc|b|8 char}}
| <code>sbyte</code>
| <code>Shortint</code>
| <code>byte</code>
| <code>tinyint</code>
|-
|''Unsigned:'' From 0 to 255 which equals 2<sup>8</sup>&nbsp;−&nbsp;1
| 3
| <code>uint8_t</code>, <code>char</code>{{cref2|group=notesc|b|8 char}}
| <code>byte</code>
| <code>Byte</code>
| n/a
| <code>unsigned tinyint</code>
|-
|rowspan=2| 16
|rowspan=2| halfword, [[Word (data type)|word]], short
|''Signed:'' From [[30000 (number)|−32,768 to 32,767]], from −(2<sup>15</sup>) to 2<sup>15</sup>&nbsp;−&nbsp;1
| 5
| rowspan="2" | [[Universal Character_Set|UCS-2]] characters
| <code>int16_t</code>, <code>short</code>{{cref2|group=notesc|b|16 short}}, <code>int</code>{{cref2|group=notesc|b|16 int}}
| <code>short</code>
| <code>Smallint</code>
| <code>short</code>
| <code>smallint</code>
|-
|''Unsigned:'' From 0 to 65,535 which equals 2<sup>16</sup>&nbsp;−&nbsp;1
| 5
| <code>uint16_t</code>
| <code>ushort</code>
| <code>Word</code>
| <code>char{{cref2|group=notejava|d}}</code>
| <code>unsigned smallint</code>
|-
|rowspan=2| 32
|rowspan=2| word, [[Integer (computer science)#Long integer|long]], doubleword, longword, int
|''Signed:'' From [[2147483647 (number)|−2,147,483,648 to 2,147,483,647]], from −(2<sup>31</sup>) to 2<sup>31</sup>&nbsp;−&nbsp;1
| 10
| rowspan="2"| [[UTF-32]] characters, [[True Color|True color]] with alpha, [[FourCC]], pointers in [[32-bit computing]]
| <code>int32_t</code>, <code>int</code>{{cref2|group=notesc|b|32 int}}, <code>long</code>{{cref2|group=notesc|b|32 long}}
| <code>int</code>
| <code>LongInt</code>; <code>Integer</code>{{cref2|group=notedelphi|c|signed}}
| <code>int</code>
| <code>int</code>
|-
|''Unsigned:'' From 0 to 4,294,967,295 which equals 2<sup>32</sup>&nbsp;−&nbsp;1
| 10
| <code>uint32_t</code>
| <code>uint</code>
| <code>LongWord</code>; <code>DWord</code>; <code>Cardinal</code>{{cref2|group=notedelphi|c|unsigned}}
| n/a
| <code>unsigned int</code>
|-
| rowspan="2" | 64
| rowspan="2" | word, doubleword, longword, long long, quad, quadword, qword, int64
|''Signed:'' From −9,223,372,036,854,775,808 to [[9223372036854775807|9,223,372,036,854,775,807]], from −(2<sup>63</sup>) to 2<sup>63</sup>&nbsp;−&nbsp;1
| 19
| rowspan="2" | Time (milliseconds since the [[Unix epoch]]), pointers in [[64-bit computing]]
| <code>int64_t</code>, <code>long</code>{{cref2|group=notesc|b|64 long}}, <code>long long</code>{{cref2|group=notesc|b|64 long long}}
| <code>long</code>
| <code>Int64</code>
| <code>long</code>
| <code>bigint</code>
|-
|''Unsigned:'' From 0 to 18,446,744,073,709,551,615 which equals 2<sup>64</sup>&nbsp;−&nbsp;1
|20
| <code>uint64_t</code>
| <code>ulong</code>
| <code>UInt64</code>; <code>QWord</code>
| n/a
| <code>unsigned bigint</code>
|-
|rowspan=2| 128
|rowspan=2| octaword, double quadword
|''Signed:'' From −170,141,183,460,469,231,731,687,303,715,884,105,728 to 170,141,183,460,469,231,731,687,303,715,884,105,727, from −(2<sup>127</sup>) to 2<sup>127</sup>&nbsp;−&nbsp;1
| 39
|rowspan=2| Complicated scientific calculations
[[IPv6]] addresses
|rowspan=2| C: only available as non-standard compiler-specific extension
|rowspan=2| n/a
|rowspan=2| n/a
|rowspan=2| n/a
| rowspan="2" | n/a
|-
|''Unsigned:'' From 0 to 340,282,366,920,938,463,463,374,607,431,768,211,455 which equals 2<sup>128</sup>&nbsp;−&nbsp;1
|39
|-
| rowspan="2" | ''n''
| rowspan="2" | ''n''-bit integer<br /> (general case)
| ''Signed:'' (−(2<sup>''n''−1</sup>)) to (2<sup>''n''−1</sup>&nbsp;−&nbsp;1)
| colspan="2" | ⌈(''n''&nbsp;−&nbsp;1) log<sub>10</sub>&nbsp;2⌉
| colspan="5" | [[Ada (programming language)|Ada]]: <code>range -2**(n-1)..2**(n-1)-1</code>
|-
| ''Unsigned:'' 0 to (2<sup>''n''</sup>−1)
| colspan="2" | ⌈''n'' log<sub>10</sub>&nbsp;2⌉
| colspan="5" | Ada: <code>range 0..2**n-1</code>, <code>mod 2**n</code>; standard libraries' or third-party arbitrary arithmetic libraries' BigDecimal or Decimal classes in many languages such as Python, C++, etc.
|}
{{cnote2 begin | liststyle=lower-alpha}}
{{cnote2 | group=notesql | a | Not all SQL dialects have unsigned datatypes.<ref name=SybDT/><ref name=MySQLDT />}}
{{cnote2 | group=notesc | b | n=0 | The sizes of <code>char</code>, <code>short</code>, <code>int</code>, <code>long</code> and <code>long long</code> in C/C++ are dependent upon the implementation of the language.}}
{{cnote2 | group=notedelphi | c | n=0 | The sizes of Delphi's <code>Integer</code> and <code>Cardinal</code> are not guaranteed, varying from platform to platform; usually defined as <code>LongInt</code> and <code>LongWord</code> respectively.}}
{{cnote2 | group=notejava | d | Java does not directly support arithmetic on <code>char</code> types. The results must be cast back into <code>char</code> from an <code>int</code>.}}
{{cnote2 end}}

Different [[Central processing unit|CPUs]] support different integral data types. Typically, hardware will support both signed and unsigned types but only a small, fixed set of widths.

The table above lists integral type widths that are supported in hardware by common processors.  High level programming languages provide more possibilities. It is common to have a 'double width' integral type that has twice as many bits as the biggest hardware-supported type.  Many languages also have ''bit-field'' types (a specified number of bits, usually constrained to be less than the maximum hardware-supported width) and ''range'' types (which can represent only the integers in a specified range).

Some languages, such as [[Lisp programming language|Lisp]], [[Smalltalk]], [[REXX]] and [[Haskell (programming language)|Haskell]], support ''arbitrary precision'' integers (also known as ''infinite precision integers'' or ''[[bignum]]s''). Other languages which do not support this concept as a top-level construct may have libraries available to represent very large numbers using arrays of smaller variables, such as Java's <code>BigInteger</code> class or [[Perl]]'s "<code>bigint</code>" package.<ref>{{cite web |url=http://download.oracle.com/javase/6/docs/api/java/math/BigInteger.html |title=BigInteger (Java Platform SE 6) |publisher=Oracle |accessdate=2011-09-11 }}</ref> These use as much of the computer's memory as is necessary to store the numbers; however, a computer has only a finite amount of storage, so they too can only represent a finite subset of the mathematical integers. These schemes support very large numbers, for example one kilobyte of memory could be used to store numbers up to 2466 decimal digits long.

A [[Boolean datatype|Boolean]] or [[Flag (computing)|Flag]] type is a type which can represent only two values: 0 and 1, usually identified with ''false'' and ''true'' respectively.  <!-- Pascal has them the other way around --> This type can be stored in memory using a single bit, but is often given a full byte for convenience of addressing and speed of access.

A four-bit quantity is known as a ''[[nibble]]'' (when eating, being smaller than a ''bite'') or ''nybble'' (being a pun on the form of the word ''byte'').  One nibble corresponds to one digit in [[hexadecimal]] and holds one digit or a sign code in binary-coded decimal.

=== Bytes and octets ===
{{Main|Byte}}
The term ''byte'' initially meant 'the smallest addressable unit of memory'.  In the past, 5-, 6-, 7-, 8-, and 9-bit bytes have all been used.  There have also been computers that could address individual bits ('bit-addressed machine'), or that could only address 16- or 32-bit quantities ('word-addressed machine').  The term ''byte'' was usually not used at all in connection with bit- and word-addressed machines.

The term ''octet'' always refers to an 8-bit quantity.  It is mostly used in the field of [[computer network]]ing, where computers with different byte widths might have to communicate.  

In modern usage ''byte'' almost invariably means eight bits, since all other sizes have fallen into disuse; thus ''byte'' has come to be synonymous with ''octet''.

=== Words ===
{{Main|Word (data type)}}
The term 'word' is used for a small group of bits which are handled simultaneously by processors of a particular [[computer architecture|architecture]].  The size of a word is thus CPU-specific.  Many different word sizes have been used, including 6-, 8-, 12-, 16-, 18-, 24-, 32-, 36-, 39-, 48-, 60-, and 64-bit. Since it is architectural, the size of a ''word'' is usually set by the first CPU in a family, rather than the characteristics of a later compatible CPU. The meanings of terms derived from ''word'', such as ''longword'', ''doubleword'', ''quadword'', and ''halfword'', also vary with the CPU and OS.<ref name="agnerfog" />

Practically all new desktop processors are capable of using 64-bit words, though [[Embedded system|embedded processors]] with 8- and 16-bit word size are still common.  The [[36-bit|36-bit word length]] was common in the early days of computers.

One important cause of non-portability of software is the incorrect assumption that all computers have the same word size as the computer used by the programmer. For example, if a programmer using the C language incorrectly declares as <code>int</code> a variable that will be used to store values greater than 2<sup>15</sup>−1, the program will fail on computers with 16-bit integers. That variable should have been declared as <code>long</code>, which has at least 32 bits on any computer. Programmers may also incorrectly assume that a pointer can be converted to an integer without loss of information, which may work on (some) 32-bit computers, but fail on 64-bit computers with 64-bit pointers and 32-bit integers.

===Short integer===
A ''short integer'' can represent a whole number which may take less storage, while having a smaller range, compared with a standard integer on the same machine.

In [[C (programming language)|C]], it is denoted by <code>short</code>. It is required to be at least 16 bits, and is often smaller than a standard integer, but this is not required.<ref name="c99" /><ref name="drdobbsinteger" /> A conforming program can assume that it can safely store values between −(2<sup>15</sup>−1) and 2<sup>15</sup>−1, but it may not assume that the range isn't larger. In [[Java (programming language)|Java]], a <code>short</code> is ''always'' a 16-bit integer. In the [[Windows API]], the datatype <code>SHORT</code> is defined as a 16-bit signed integer on all machines.<ref name="agnerfog" />

====Common short integer sizes====
{| class="wikitable"
|- 
! [[Programming language]]
! [[Platform (computing)|Platform]]s
! Data type name
! [[Signedness]]
! Storage in [[bytes]]
! Minimum value
! Maximum value
|-
| rowspan="2" | [[C (programming language)|C]] and [[C++]]
| rowspan="2" | common implementations
| <code>short</code>
| signed
| 2
| −32,768
| +32,767
|-
| <code>unsigned short</code>
| unsigned
| 2
| 0
| 65,535
|-
| rowspan="2" | [[C Sharp (programming language)|C#]]
| rowspan="2" | [[.NET Framework|.NET CLR/CTS]]
| <code>short</code>
| signed
| 2
| −32,768
| +32,767
|-
| <code>ushort</code>
| unsigned
| 2
| 0
| 65,535
|-
| [[Java (programming language)|Java]]
| [[Java platform]]
| <code>short</code>
| signed
| 2
| −32,768
| +32,767
|}

===Long integer===
A ''long integer'' can represent a whole [[integer]] whose [[range (computer science)|range]] is greater than or equal to that of a standard integer on the same machine.

In [[C (programming language)|C]], it is denoted by <code>long</code>. It is required to be at least 32 bits, and may or may not be larger than a standard integer. A conforming program can assume that it can safely store values between −(2<sup>31</sup>−1) and 2<sup>31</sup>−1, but it may not assume that the range isn't larger.

====Common long integer sizes====
{| class="wikitable"
|- 
! [[Programming language]]
! Approval Type
! [[Platform (computing)|Platform]]s
! Data type name
! Storage in [[bytes]]
! [[Signedness|Signed]] range
! [[Signedness|Unsigned]] range
|- 
| [[C (programming)|C]] ISO/ANSI C99
| International Standard
| [[Unix]],16/32-bit systems<ref name="agnerfog" /><br>[[Windows]],16/32/64-bit systems<ref name="agnerfog" />
| <code>long</code><sup>†</sup>
| 4<br>(minimum requirement 4)
| −2,147,483,648 to +2,147,483,647
| 0 to 4,294,967,295<br>(minimum requirement)
|- 
| [[C (programming)|C]] ISO/ANSI C99
| International Standard
| [[Unix]],<br>64-bit systems<ref name="agnerfog" /><ref name="drdobbsinteger" />
| <code>long</code><sup>†</sup>
| 8<br>(minimum requirement 4)
| −9,223,372,036,854,775,808 to +9,223,372,036,854,775,807
| 0 to 18,446,744,073,709,551,615
|- 
| [[C++]] ISO/ANSI
| International Standard
| [[Unix]], [[Windows]],<br>16/32-bit system
| <code>long</code><sup>†</sup>
| 4 <ref>{{cite web
 | title=Fundamental types in C++|url=http://cppreference.com/wiki/language/types|publisher=cppreference.com|accessdate=5 December 2010}}</ref><br>(minimum requirement 4)
| −2,147,483,648 to +2,147,483,647<br>
| 0 to 4,294,967,295<br>(minimum requirement)
|- 
| [[C++/CLI]]
| International Standard<br>[[ECMA-372]]
| [[Unix]], [[Windows]],<br>16/32-bit systems
| <code>long</code><sup>†</sup>
| 4 <ref>{{cite web| url=http://www.ecma-international.org/publications/files/ECMA-ST/ECMA-372.pdf |title=Chapter 8.6.2 on page 12|publisher=ecma-international.org}}</ref><br>(minimum requirement 4)
| −2,147,483,648 to +2,147,483,647<br>
| 0 to 4,294,967,295<br>(minimum requirement)
|- 
| [[Visual Basic|VB]]
| Company Standard
| [[Microsoft Windows|Windows]]
| <code>Long</code>
| 4 <ref>VB 6.0 help file</ref>
| −2,147,483,648 to +2,147,483,647
| N/A
|- 
| [[Visual Basic for Applications|VBA]]
| Company Standard
| [[Microsoft Windows|Windows]], [[Mac OS]]
| <code>Long</code>
| 4 <ref>{{cite web |url=http://msdn2.microsoft.com/en-us/library/aa164754(office.10).aspx |title=The Integer, Long, and Byte Data Types (VBA) |accessdate=2006-12-19 |publisher=microsoft.com}}</ref>
| −2,147,483,648 to +2,147,483,647
| N/A
|- 
| [[Microsoft SQL Server|SQL Server]]
| Company Standard
| [[Microsoft Windows|Windows]]
| <code>BigInt</code>
| 8
| −9,223,372,036,854,775,808 to +9,223,372,036,854,775,807
| 0 to 18,446,744,073,709,551,615
|- 
| [[C Sharp (programming language)|C#]]/ [[Visual Basic .NET|VB.NET]]
| ECMA International Standard
| [[Microsoft .NET]]
| <code>long</code> or <code>Int64</code>
| 8
| −9,223,372,036,854,775,808 to +9,223,372,036,854,775,807
| 0 to 18,446,744,073,709,551,615
|- 
| [[Java (programming language)|Java]]
| International/Company Standard
| [[Java platform]]
| <code>long</code>
| 8
| −9,223,372,036,854,775,808 to +9,223,372,036,854,775,807
| N/A
|- 
| [[Pascal (programming language)|Pascal]]
| ?
| [[Microsoft Windows|Windows]], [[UNIX]]
| <code>int64</code>
| 8
| −9,223,372,036,854,775,808 to +9,223,372,036,854,775,807
| 0 to 18,446,744,073,709,551,615(Qword type)
|}
<sup>†</sup> the term <code>long int</code> is equivalent<ref>{{cite web |url=http://www.open-std.org/JTC1/SC22/WG14/www/docs/n1570.pdf |title=ISO/IEC 9899:201x |accessdate=2013-03-27 |publisher=open-std.org}}</ref>

===Long long===
In the [[C99]] version of the [[C (programming language)|C programming language]] and the [[C++11]] version of [[C++]], a <CODE>long long</CODE> type is supported that has double the minimum capacity of the standard <CODE>long</CODE>, 64 bits. This type is not supported by compilers that require C code to be compliant with the previous C++ standard, C++03, because the <code>long long</code> type did not exist in C++03. For an ANSI/ISO compliant compiler the minimum requirements for the specified ranges, that is −(2<sup>31</sup>) to 2<sup>31</sup>−1 for signed and 0 to 2<sup>32</sup>−1 for unsigned, must be fulfilled; however, extending this range is permitted.<ref>{{cite web| url=http://www.ericgiguere.com/articles/ansi-c-summary.html|title=The ANSI Standard: A Summary for the C Programmer|first=Eric|last=Giguere|date=December 18, 1987|accessdate=2010-09-04}}</ref><ref>{{cite web| url=http://flash-gordon.me.uk/ansi.c.txt |title=American National Standard Programming Language C specifies the syntax and semantics of programs written in the C programming language.|accessdate=2010-09-04}}</ref> This can be an issue when exchanging code and data between platforms, or doing direct hardware access.  Thus, there are several sets of headers providing platform independent exact width types.  The C [[standard library]] provides ''[[stdint.h]]''; this was introduced in C99 and C++11.

== See also ==
*[[Signed number representations]]
*[[Binary-coded decimal]], a representation of integers commonly used in mainframe financial applications and in databases
*[[Integer overflow]]

== Notes ==
{{Reflist|30em|refs=<ref name=drdobbsinteger>{{cite web |url=http://www.drdobbs.com/184401323 |title=The New C: Integers in C99, Part 1 |date=2000-12-01 |accessdate=2010-09-04 |last=Meyers |first=Randy |publisher=drdobbs.com }}</ref><ref name=c99>{{cite web| url=http://www.ericgiguere.com/articles/ansi-c-summary.html|title=The ANSI Standard: A Summary for the C Programmer |first=Eric |last=Giguere |date=1987-12-18 |accessdate=2010-09-04}}</ref><ref name=agnerfog>{{cite web|url=http://www.agner.org/optimize/calling_conventions.pdf |title=Calling conventions for different C++ compilers and operating systems: Chapter 3, Data Representation |date=2010-02-16 |accessdate=2010-08-30 |last=Fog |first=Agner}}</ref><ref name=SybDT>{{cite web|url=http://infocenter.sybase.com/help/index.jsp?topic=/com.sybase.infocenter.dc36271.1550/html/blocks/blocks20.htm |title=Sybase Adaptive Server Enterprise 15.5 : Exact Numeric Datatypes}}</ref><ref name=MySQLDT>{{cite web|url=http://dev.mysql.com/doc/refman/5.6/en/numeric-types.html |title=MySQL 5.6 Numeric Datatypes}}</ref>}}

{{Data types}}

{{DEFAULTSORT:Integer (Computer Science)}}
[[Category:Data types]]
[[Category:Computer arithmetic]]
[[Category:Primitive types]]