integer

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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^{[citation needed]}.

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 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 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.

Common integral data types

Bits	Name	Range (assuming two's complement for signed)	Decimal digits	Uses
4	nibble, semioctet
		Unsigned: 0 to +15	2	Binary-coded decimal, single decimal digit representation.
8	byte, octet	Signed: −128 to +127	3	ASCII characters, C/C++ char, C/C++ uint8_t, int8_t, Java byte, C# byte, T-SQL tinyint
		Unsigned: 0 to +255	3
16	halfword, word, short, int	Signed: −32,768 to +32,767	5	UCS-2 characters, C/C++ short, C/C++ int (minimum), C/C++ uint16_t, int16_t, Java short, C# short, Java char
		Unsigned: 0 to +65,535	5
32	word, long, doubleword, longword	Signed: −2,147,483,648 to +2,147,483,647	10	UCS-4 characters, Truecolor with alpha, C/C++ long (on Windows and 32-bit DOS and Unix), C/C++ uint32_t, int32_t, Java int, C# int, FourCC
		Unsigned: 0 to +4,294,967,295	10
64	doubleword, longword, long long, quad, quadword, int64	Signed: −9,223,372,036,854,775,808 to +9,223,372,036,854,775,807	19	C/C++ long (on 64-bit Unix), C/C++ long long, C/C++ uint64_t, int64_t, Java long, C# long, Delphi int64
		Unsigned: 0 to +18,446,744,073,709,551,615	20
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, Visual Basic Decimal[1]
		Unsigned: 0 to +340,282,366,920,938,463,463,374,607,431,768,211,455	39
n	n-bit integer (general case)	Signed: ( − 2n − 1) to (2n − 1 − 1)		Ada range -2**(n-1)..2**(n-1)-1
		Unsigned: 0 to (2n − 1)		Ada range 0..2**n-1, Ada mod 2**n

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.

Bytes and octets

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

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 2008^[update], practically all new desktop processors are of the x86-64 family and capable of using 64-bit words, they are however often used in 32-bit mode. 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.

See also

• Signed number representations
• Integer (the mathematical concept)

Notes