Integer (computer science)

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.

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 +32,767 Unsigned: 0 to +65,535	UCS-2 characters, C short int (minimum), C int (minimum), 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, True color with alpha, 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
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 int __attribute__ ((mode(TI))) (on 64-bit machines using gcc)
n	n-bit integer	Signed: to Unsigned: 0 to

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 avaliable to represent very large numbers using arrays of smaller variables, such as Java's BigInteger class. 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 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.

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; byte has thus come to be synonymous
with octet.

Bytes are used as the unit of computer storage 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 written the same as SI prefixes used for other measurements, but they often have somewhat different values (see binary prefix for further discussion).

In particular, hard disk manufacturers describe their products using the SI units, making their disks sound larger than one might expect. As drives become larger, the difference is growing (see table below).

Prefix	Name	SI Meaning	Binary meaning	Size difference
k or K	kilo	103   = 1000	210 = 1024	2.40%
M	mega	106   = 10002	220 = 10242	4.86%
G	giga	109   = 10003	230 = 10243	7.37%
T	tera	1012 = 10004	240 = 10244	9.95%
P	peta	1015 = 10005	250 = 10245	12.59%

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-, 39-, 48-, 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.

As of 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.