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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 one’s complement.

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 char (minimum), Java byte
16	word	Signed: −32,768 to +UCS-2 characters, C short int (minimum), C int (minimum), Java char, Java short int
32	word, doubleword, longword	Signed: −2,147,483,648 to +2,147,483,647 Unsigned: 0 to +4,294,967,295	UCS-4 characters, C int (usual), C long int (minimum), Java int
64	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 long int (on 64-bit machines), C99 long long int (minimum), Java long 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 and REXX, support arbitrary precision integers (also known as infinite precision integers or bignums). These use as much of the computer’s memory as is necessary to store the numbers; however, a computer only has a finite amount of storage, so they too can only represent a finite subset of the mathematical integers.

A Boolean or 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.

Pointers

A pointer is often, but not always, represented by an 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

The term byte initially meant ‘the least 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; octet has thus come to be synonymous with byte.

Bytes are used as the unit of computer memory of all kinds. One might speak of a 50-byte text string, a 100 kB (kilobyte) file, a 128 MB (megabyte) RAM module, or a 30 GB (gigabyte) hard disk. The prefixes used for byte measurements are similar to the SI prefixes used for other measurements, but they do not have the same meanings (see binary prefix for further discussion).

Prefix	Name	Usual (SI) meaning	Meaning when applied to bytes
k, K	kilo	103   = 1000	210 = 1024
M	mega	106   = 10002	220 = 10242
G	giga	109   = 10003	230 = 10243
T	tera	1012 = 10004	240 = 10244
P	peta	1015 = 10005	250 = 10245

Unscrupulous hard disk manufacturers describe their products using the power-of-1000 meanings, which is the subject of a current false advertising lawsuit.

See also Binary prefix, byte.

Words

The term word initially was equivalent to ‘the logical size of an address of a location in the system memory’, and was thus CPU- and OS-specific. One could say that the IBM System/360 had 32-bit words (even though its addresses were limited to 24 bits), and the 8086 had 16-bit words. Many different word sizes have been used, including 6-, 8-, 12-, 16-, 18-, 24-, 32-, 36-, 60- and 64-bit. The meanings of terms derived from word, such as longword, doubleword, quadword, and halfword, also vary with the CPU and OS.

Currently (2004) 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 aren’t a multiple of 8 have vanished along with non-8-bit bytes.