Integer (computer science)

In computer science, an integer is a datum of integral data type, a data type which represents some finite subset of the mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. ^[1]

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).^[2]

An integer value is typically specified in the (source code of the) program as a sequence of digits, without spaces or thousands separators, optionally prefixed with + or -. Some programming languages allow other notations, such as hexadecimal (base 16) or octal (base 8).

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. Typically all integers from the minimum through the maximum can be represented.

The maximum is sometimes called MAXINT or—as in the C standard library limits.h header—INT_MAX.^[3]

The most common representation of a positive integer is a string of bits, using the binary numeral system. The order of the memory bytes 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 2ⁿ numbers; for example an unsigned type typically represents the non-negative values 0 through 2ⁿ−1.

There are four different ways to represent negative numbers in a binary numeral system. The most common is two’s complement, which allows a signed integral type with n bits to represent numbers from −2⁽ⁿ⁻¹⁾ through 2⁽ⁿ⁻¹⁾−1. Two’s complement arithmetic is convenient because there is a perfect 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. The other possibilities are offset binary, sign-magnitude and ones' complement. See Signed number representations for details.

Common integral data types

Bits	Name	Range (assuming two's complement for signed)	Decimal Digits (approx.)	Uses	Implementations
					C/C++	C#	Delphi	Java	SQL[a]
4	nibble, semioctet	Signed: From − 8 to 7, from − (23) to 23 − 1	1	Binary-coded decimal, single decimal digit representation.
		Unsigned: From 0 to 15 which equals 24 − 1	2
8	byte, octet	Signed: From − 128 to 127, from − (27) to 27 − 1	3	ASCII characters	int8_t, char	sbyte	Shortint	byte	tinyint
		Unsigned: From 0 to 255 which equals 28 − 1	3		uint8_t, char	byte	Byte	n/a	unsigned tinyint
16	halfword, word, short	Signed: From − 32,768 to 32,767, from − (215) to 215 − 1	5	UCS-2 characters	int16_t, short[b], int[b]	short	Smallint	short	smallint
		Unsigned: From 0 to 65,535 which equals 216 − 1	5		uint16_t	ushort	Word	char[d]	unsigned smallint
32	word, long, doubleword, longword, int	Signed: From − 2,147,483,648 to 2,147,483,647, from − (231) to 231 − 1	10	UCS-4 characters, Truecolor with alpha, FourCC, ActionScript int	int32_t, int[b], long[b]	int	LongInt; Integer[c]	int	int
		Unsigned: From 0 to 4,294,967,295 which equals 232 − 1	10		uint32_t	uint	LongWord; Cardinal[c]	n/a	unsigned int
64	word, doubleword, longword, long long, quad, quadword, int64	Signed: From − 9,223,372,036,854,775,808 to 9,223,372,036,854,775,807, from − (263) to 263 − 1	19	Very large numbers	int64_t, long[b], long long[b]	long	Int64	long	bigint
		Unsigned: From 0 to 18,446,744,073,709,551,615 which equals 264 − 1	20		uint64_t	ulong	n/a	n/a	unsigned bigint
128	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 − (2127) to 2127 − 1	39	C: only available as non-standard compiler-specific extension
		Unsigned: From 0 to 340,282,366,920,938,463,463,374,607,431,768,211,455 which equals 2128 − 1	39
n	n-bit integer (general case)	Signed: ( − 2n − 1) to (2n − 1 − 1)	$\lceil (n-1) \log_{10}{2} \rceil$		Ada range -2**(n-1)..2**(n-1)-1
		Unsigned: 0 to (2n − 1)	$\lceil n \log_{10}{2} \rceil$		Ada range 0..2**n-1, Ada mod 2**n

1. ^ Not all SQL dialects have unsigned datatypes.^[4][5]
2. The sizes of short, int, and long in C/C++ are dependent upon the implementation of the language:
   • On older, 16-bit operating systems, int was 16-bit and long was 32-bit.
   • On 32-bit Unix, DOS, and Windows, int and long are 32-bits, while long long is 64-bits. This is also true for 64-bit processors running 32-bit programs.^[6]
   • On 64-bit Unix, int is 32-bits, while long and long long are 64-bits.^[7][8][9]
3. The sizes of Delphi's Integer and Cardinal are not guaranteed, varying from platform to platform; usually defined as LongInt and LongWord respectively.
4. ^ Java does not directly support arithmetic on char types. The results must be cast back into char from an int.

Different 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, Smalltalk, REXX and Haskell, support arbitrary precision integers (also known as infinite precision integers or bignums). 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 BigInteger class or Perl's "bigint" package.^[10] 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 or Flag type is a type which can represent only two values: 0 and 1, usually identified with false and true respectively. 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 article: 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 networking, 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 article: Word (data type)

The term 'word' is used for a small group of bits which are handled simultaneously by processors of a particular 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.^[7]

Practically all new desktop processors are capable of using 64-bit words, though embedded processors with 8- and 16-bit word size are still common. The 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 int a variable that will be used to store values greater than 2¹⁶-1, the program will fail on computers with 16-bit integers. That variable should have been declared as long, 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.

A short integer in one programming language may be a different size in a different language or on a different processor. In some languages this size is fixed across platforms, while in others it is machine-dependent. In some languages this datatype does not exist at all.

In C, it is denoted by short. It is required to be at least 16 bits, and is often smaller than a standard integer, but this is not required.^[9][8] A conforming program can assume that it can safely store values between −(2¹⁵−1) and 2¹⁵−1, but it may not assume that the range isn't larger. In Java, a short is always a 16-bit integer. In the Windows API, the datatype SHORT is defined as a 16-bit signed integer on all machines.^[7]

Common short integer sizes

Programming language	Platforms	Data type name	Signedness	Storage in bytes	Minimum value	Maximum value
C and C++	common implementations	short	signed	2	−32,768	32,767
		unsigned short	unsigned	2	0	65,535
C#	.NET CLR/CTS	short	signed	2	−32,768	32,767
		ushort	unsigned	2	0	65,535
Java	Java platform	short	signed	2	−32,768	32,767

Long integer

A long integer can represent a whole integer number whose range is greater than or equal to that of a standard integer on the same machine.

A long integer in one programming language may be different in size from a long integer in a different language or processor. In some languages this size is fixed across platforms, while in others it is machine dependent. In some languages this data type does not exist at all.

A long integer commonly requires double the storage capacity of a standard integer, although this is not always the case.

C and C++

In the C99 version of the C programming language and the C++11 version of C++, a long long type is supported that doubles the minimum capacity of the standard long to 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 long long type did not exist in C++03. For an ANSI/ISO compliant compiler the minimum requirements for the specified ranges, that is −(2³¹) to 2³¹−1 for signed and 0 to 2³²−1 for unsigned, must be fulfilled; however, extending this range is permitted.^{[11] [12]} 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.

Common long integer sizes

Programming language	Approval Type	Platforms	Data type name	Storage in bytes	Signed range	Unsigned range
C ISO/ANSI C99	International Standard	Unix,16/32-bit systems[7] Windows,16/32/64-bit systems[7]	long†	4 (minimum requirement 4)	−2,147,483,648 to 2,147,483,647	0 to 4,294,967,295 (minimum requirement)
C ISO/ANSI C99	International Standard	Unix, 64-bit systems[7][8]	long†	8 (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, 16/32-bit system	long†	4 [13] (minimum requirement 4)	−2,147,483,648 to 2,147,483,647	0 to 4,294,967,295 (minimum requirement)
C++/CLI	International Standard ECMA-372	Unix, Windows, 16/32-bit systems	long†	4 [14] (minimum requirement 4)	−2,147,483,648 to 2,147,483,647	0 to 4,294,967,295 (minimum requirement)
VB	Company Standard	Windows	Long	4 [15]	−2,147,483,648 to 2,147,483,647	N/A
VBA	Company Standard	Windows, Mac OS	Long	4 [16]	−2,147,483,648 to 2,147,483,647	N/A
SQL Server	Company Standard	Windows	BigInt	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#/ VB.NET	ECMA International Standard	Microsoft .NET	long or Int64	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	International/Company Standard	Java platform	long	8	−9,223,372,036,854,775,808 to +9,223,372,036,854,775,807	N/A
Pascal	?	Windows	int64	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)

^† the term long int is equivalent but it is used rarely

See also

• Signed number representations
• Binary-coded decimal, another, rather different, representation for integers which is still commonly used in mainframe financial applications and in databases.

Notes

1. ^ "C provides six operators for bit manipulation; these may only be applied to integral operands, that is, char, short, int and long, whether signed or unsigned." section 2.9. Kernighan & Ritchie, The C Programming Language - Second Edition, 43rd printing (c) 2008
2. ^ Cheever, Eric. "Representation of numbers". Swarthmore College. http://www.swarthmore.edu/NatSci/echeeve1/Ref/BinaryMath/NumSys.html. Retrieved 2011-09-11. 
3. ^ Juric, Zeljko. "values.h". TiCalc.org. http://tigcc.ticalc.org/doc/values.html. Retrieved 2011-09-11. 
4. ^ "Sybase Adaptive Server Enterprise 15.5 : Exact Numeric Datatypes". http://infocenter.sybase.com/help/index.jsp?topic=/com.sybase.infocenter.dc36271.1550/html/blocks/blocks20.htm. 
5. ^ "MySQL 5.6 Numeric Datatypes". http://dev.mysql.com/doc/refman/5.6/en/numeric-types.html. 
6. ^ http://jk-technology.com/c/inttypes.html (c) 2008
7. ^ ^{a b c d e f} Fog, Agner (2010-02-16). "Calling conventions for different C++ compilers and operating systems: Chapter 3, Data Representation". http://www.agner.org/optimize/calling_conventions.pdf. Retrieved 2010-08-30. 
8. ^ ^{a b c} Meyers, Randy (2000-12-01). "The New C: Integers in C99, Part 1". drdobbs.com. http://www.drdobbs.com/184401323. Retrieved 2010-09-04. 
9. ^ ^{a b} Giguere, Eric (1987-12-18). "The ANSI Standard: A Summary for the C Programmer". http://www.ericgiguere.com/articles/ansi-c-summary.html. Retrieved 2010-09-04. 
10. ^ "BigInteger (Java Platform SE 6)". Oracle. http://download.oracle.com/javase/6/docs/api/java/math/BigInteger.html. Retrieved 2011-09-11. 
11. ^ Giguere, Eric (December 18, 1987). "The ANSI Standard: A Summary for the C Programmer". http://www.ericgiguere.com/articles/ansi-c-summary.html. Retrieved 2010-09-04. 
12. ^ "American National Standard Programming Language C specifies the syntax and semantics of programs written in the C programming language.". http://flash-gordon.me.uk/ansi.c.txt. Retrieved 2010-09-04. 
13. ^ "Fundamental types in C++". cppreference.com. http://cppreference.com/wiki/language/types. Retrieved 5 December 2010. 
14. ^ "Chapter 8.6.2 on page 12". ecma-international.org. http://www.ecma-international.org/publications/files/ECMA-ST/ECMA-372.pdf. 
15. ^ VB 6.0 help file
16. ^ "The Integer, Long, and Byte Data Types (VBA)". microsoft.com. http://msdn2.microsoft.com/en-us/library/aa164754(office.10).aspx. Retrieved 2006-12-19.