Integer (computer science)

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

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 and subtraction 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.

Common integral data types

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

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 about 2560 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.

Data type names

Bits	Signed	Java	C#	SQL92	vbScript	C
8	Yes	byte	sbyte			int8_t
16	Yes	short	short, Int16	smallint, int2	int	int16_t
32	Yes	int	int, Int32	integer, int, int4	long	int32_t
64	Yes	long	long, Int64	bigint, int8		int64_t
8	No		byte	tinyint, int1	byte	uint8_t
16	No	char	ushort, UInt16			uint16_t
32	No		uint, UInt32			uint32_t
64	No		ulong, UInt64			uint64_t

Note: C and C++ have no platform-independent integer types with fixed bit widths.

Pointers

A pointer is often, but not always, represented by an unsigned integer of specified width. This is often, but not always, the widest integer that the hardware supports directly. The value of this integer is the memory address of whatever the pointer points to.

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 (computer science)

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, but word sizes that are not a multiple of 8 have vanished along with non-8-bit bytes.