Integer (computer science)

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In computer science, the term integer is used to refer to a data type which represents some finite subset of the mathematical integers. These are also known as integral data types.

1 Value and representation
2 Common integral data types
3 Data type names
4 Bytes and octets
5 Words
6 See also
7 Notes

[edit] Value and representation

The value of a datum with an integral type is the mathematical integer that it corresponds to. The representation of this datum is the way the value is stored in the computer’s memory. Integral types may be unsigned (capable of representing only non-negative integers) or signed (capable of representing negative integers as well).

The most common representation of a positive integer is a string of bits, using the binary numeral system. The order of 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 three 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, and because addition, subtraction and multiplication do not need to distinguish between signed and unsigned types. The other possibilities are sign-magnitude and ones' complement. See Signed number representations for details.

Another, rather different, representation for integers is binary-coded decimal, which is still commonly used in mainframe financial applications and in databases.

[edit] Common integral data types

Bits	Name	Range	Decimal digits	Uses
8	byte, octet	Signed: −128 to +127	3	ASCII characters, C int8_t, Java byte
8	byte, octet	Unsigned: 0 to +255	3	ASCII characters, C int8_t, Java byte
16	halfword, word	Signed: −32,768 to +32,767	5	UCS-2 characters, C int16_t, Java short, Java char
16	halfword, word	Unsigned: 0 to +65,535	5	UCS-2 characters, C int16_t, Java short, Java char
32	word, doubleword, longword	Signed: −2,147,483,648 to +2,147,483,647	10	UCS-4 characters, Truecolor with alpha, C int32_t, Java int, FourCC
32	word, doubleword, longword	Unsigned: 0 to +4,294,967,295	10
64	doubleword, longword, long long, quad, quadword	Signed: −9,223,372,036,854,775,808 to +9,223,372,036,854,775,807	19	C int64_t, Java long
64	doubleword, longword, long long, quad, quadword	Unsigned: 0 to +18,446,744,073,709,551,615	20	C int64_t, Java long
128	octaword	Signed: −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	39	C only available as non-standard compiler-specific extension
128	octaword	Unsigned: 0 to +340,282,366,920,938,463,463,374,607,431,768,211,455	39
n	n-bit integer (general case)	Signed: $( - 2 n - 1)$ to $(2 n - 1 - 1)$	$\lceil (n-1) \log_{10}{2} \rceil$
n	n-bit integer (general case)	Unsigned: 0 to $(2 n - 1)$	$\lceil n \log_{10}{2} \rceil$

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, 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. 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 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.

[edit] Data type names

Bits	Signed	Java	C#	VB.Net	SQL92	vbScript	C99
8	Yes	byte	sbyte	SByte			int8_t, signed char
16	Yes	short	short, Int16	Short, Int16	smallint, int2	int	int16_t, short, int
32	Yes	int	int, Int32	Integer, Int32	integer, int, int4	long	int32_t, long
64	Yes	long	long, Int64	Long, Int64	bigint, int8		int64_t, long long
8	No		byte	Byte	tinyint, int1	byte	uint8_t, unsigned char
16	No	char^[1]	ushort, UInt16	UShort, UInt16			uint16_t, unsigned short, unsigned int
32	No		uint, UInt32	UInteger, UInt32			uint32_t, unsigned long
64	No		ulong, UInt64	ULong, UInt64			uint64_t, unsigned long long

Note: C++ has no compiler-independent integer types with fixed bit widths. C has them only since C99, in the form (u)int(n)_t. It does specify the minimum widths for char, short, int, long, and long long (as shown in the table above). It also specifies that each of those types is no larger than the following, and that char is exactly one byte (eight bits in virtually all modern computers; the exact value is defined as CHAR_BIT in <limits.h>, also for older machines with wider bytes). Both C and C++ guarantee a char to be always 1 byte, but not necessarily 8 bits. C guarantees that char has at least 8 bits.

[edit] 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.

[edit] Words

Main article: Word (computing)

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.

As of 2006, 32-bit word sizes are most common among general-purpose computers, with 64-bit machines used mostly for large installations. 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.