Integer (computer science)

In computer science, anintegeris adatumofintegral data type,adata typethat represents somerangeof mathematicalintegers.Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers. Computer hardware nearly always provides a way to represent a processorregisteror memory address as an integer.

Value and representation

Thevalueof an item with an integral type is the mathematical integer that it corresponds to. Integral types may beunsigned(capable of representing only non-negative integers) orsigned(capable of representing negative integers as well).

An integer value is typically specified in thesource codeof a program as a sequence of digits optionally prefixed with + or −. Some programming languages allow other notations, such as hexadecimal (base 16) or octal (base 8). Some programming languages also permitdigit group separators.

Theinternal representationof this datum is the way the value is stored in the computer's memory. Unlike mathematical integers, a typical datum in a computer has some minimal and maximum possible value.

The most common representation of a positive integer is a string ofbits,using thebinary numeral system.The order of the memorybytesstoring the bits varies; seeendianness.Thewidth,precision,orbitnessof an integral type is the number of bits in its representation. An integral type withnbits can encode 2ⁿnumbers; for example an unsigned type typically represents the non-negative values 0 through 2ⁿ−1. Other encodings of integer values to bit patterns are sometimes used, for examplebinary-coded decimalorGray code,or as printed character codes such asASCII.

There are four well-knownways to represent signed numbersin a binary computing system. The most common istwo's complement,which allows a signed integral type withnbits to represent numbers from −2⁽ⁿ⁻¹⁾through 2⁽ⁿ⁻¹⁾−1. Two's complement arithmetic is convenient because there is a perfectone-to-one correspondencebetween representations and values (in particular, no separate +0 and −0), and becauseaddition,subtractionandmultiplicationdo not need to distinguish between signed and unsigned types. Other possibilities includeoffset binary,sign-magnitude,andones' complement.

Some computer languages define integer sizes in a machine-independent way; others have varying definitions depending on the underlying processor word size. Not all language implementations define variables of all integer sizes, and defined sizes may not even be distinct in a particular implementation. An integer in oneprogramming languagemay be a different size in a different language, on a different processor, or in an execution context of different bitness; see§ Words.

Someolder computer architecturesused decimal representations of integers, stored inbinary-coded decimal (BCD)or other format. These values generally require data sizes of 4 bits per decimal digit (sometimes called anibble), usually with additional bits for a sign. Many modern CPUs provide limited support for decimal integers as an extended datatype, providing instructions for converting such values to and from binary values. Depending on the architecture, decimal integers may have fixed sizes (e.g., 7 decimal digits plus a sign fit into a 32-bit word), or may be variable-length (up to some maximum digit size), typically occupying two digits per byte (octet).

Common integral data types

Bits	Name	Range (assumingtwo's complementforsigned)	Decimal digits	Uses	Implementations
					C/C++	C#	PascalandDelphi	Java	SQL	FORTRAN	D	Rust
4	nibble,semioctet	Signed:From −8 to 7, from −(23) to 23− 1	0.9	Binary-coded decimal,single decimal digit representation	—	—	—	—	—	—	—	—
		Unsigned:From 0 to 15, which equals 24− 1	1.2
8	byte,octet,i8, u8	Signed:From −128 to 127, from −(27) to 27− 1	2.11	ASCIIcharacters,code unitsin theUTF-8character encoding	int8_t,signed char	sbyte	Shortint	byte	tinyint	integer(1)	byte	i8
		Unsigned:From 0 to 255, which equals 28− 1	2.41		uint8_t,unsigned char	byte	Byte	—	unsigned tinyint	—	ubyte	u8
16	halfword,word,short, i16, u16	Signed:From −32,768 to 32,767, from −(215) to 215− 1	4.52	UCS-2characters,code unitsin theUTF-16character encoding	int16_t,short,int	short	Smallint	short	smallint	integer(2)	short	i16
		Unsigned:From 0 to 65,535, which equals 216− 1	4.82		uint16_t, unsigned,unsigned int	ushort	Word	char	unsigned smallint	—	ushort	u16
32	word,long,doubleword, longword, int, i32, u32	Signed:From−2,147,483,648 to 2,147,483,647,from −(231) to 231− 1	9.33	UTF-32characters,true colorwith Alpha,FourCC,pointers in32-bit computing	int32_t,int,long	int	LongInt;Integer	int	int	integer(4)	int	i32
		Unsigned:From 0 to 4,294,967,295, which equals 232− 1	9.63		uint32_t, unsigned,unsigned int,unsigned long	uint	LongWord;DWord;Cardinal	—	unsigned int	—	uint	u32
64	word, doubleword, longword, long, long long, quad, quadword, qword, int64, i64, u64	Signed:From −9,223,372,036,854,775,808 to9,223,372,036,854,775,807,from −(263) to 263− 1	18.96	Time (milliseconds since theUnix epoch), pointers in64-bit computing	int64_t,long,long long	long	Int64	long	bigint	integer(8)	long	i64
		Unsigned:From 0 to 18,446,744,073,709,551,615, which equals 264− 1	19.27		uint64_t,unsigned long long	ulong	UInt64;QWord	—	unsigned bigint	—	ulong	u64
128	octaword, double quadword, i128, u128	Signed:From −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, from −(2127) to 2127− 1	38.23	Complex scientific calculations, IPv6addresses, GUIDs	C: only available as non-standard compiler-specific extension	—	—	—	—	integer(16)	cent	i128
		Unsigned:From 0 to 340,282,366,920,938,463,463,374,607,431,768,211,455, which equals 2128− 1	38.53							—	ucent	u128
n	n-bit integer (general case)	Signed:−(2n−1) to (2n−1− 1)	(n− 1) log102		C23:_BitInt(n),signed _BitInt(n)	Ada:range -2**(n-1)..2**(n-1)-1
		Unsigned:0 to (2n− 1)	nlog102		C23:unsigned _BitInt(n)	Ada:range 0..2**n-1,mod 2**n;standard libraries' or third-party arbitrary arithmetic libraries' BigDecimal or Decimal classes in many languages such as Python, C++, etc.

DifferentCPUssupport 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 havebit-fieldtypes (a specified number of bits, usually constrained to be less than the maximum hardware-supported width) andrangetypes (that can represent only the integers in a specified range).

Some languages, such asLisp,Smalltalk,REXX,Haskell,Python,andRaku,supportarbitrary precisionintegers (also known asinfinite precision integersorbignums). Other languages that 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'sBigIntegerclass orPerl'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 decimal digits long.

ABooleanorFlagtype is a type that can represent only two values: 0 and 1, usually identified withfalseandtruerespectively. 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 anibble(when eating, being smaller than abite) ornybble(being a pun on the form of the wordbyte). One nibble corresponds to one digit inhexadecimaland holds one digit or a sign code in binary-coded decimal.

Bytes and octets

Main articles:ByteandOctet (computing)

The termbyteinitially 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 termbytewas usually not used at all in connection with bit- and word-addressed machines.

The termoctetalways refers to an 8-bit quantity. It is mostly used in the field ofcomputer networking,where computers with different byte widths might have to communicate.

In modern usagebytealmost invariably means eight bits, since all other sizes have fallen into disuse; thusbytehas come to be synonymous withoctet.

Words

Main article:Word (computer architecture)

The term 'word' is used for a small group of bits that are handled simultaneously by processors of a particulararchitecture.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-, 40-, 48-, 60-, and 64-bit. Since it is architectural, the size of awordis usually set by the first CPU in a family, rather than the characteristics of a later compatible CPU. The meanings of terms derived fromword,such aslongword,doubleword,quadword,andhalfword,also vary with the CPU and OS.

Practically all new desktop processors are capable of using 64-bit words, thoughembedded processorswith 8- and 16-bit word size are still common. The36-bit word lengthwas 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 asinta variable that will be used to store values greater than 2¹⁵−1, the program will fail on computers with 16-bit integers. That variable should have been declared aslong,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. This issue is resolved by C99 instdint.hin the form ofintptr_t.

Thebitnessof a program may refer to the word size (or bitness) of the processor on which it runs, or it may refer to the width of a memory address or pointer, which can differ between execution modes or contexts. For example, 64-bit versions ofMicrosoft Windowssupport existing 32-bit binaries, and programs compiled for Linux'sx32 ABIrun in 64-bit mode yet use 32-bit memory addresses.

Standard integer

The standard integer size is platform-dependent.

InC,it is denoted byintand required to be at least 16 bits. Windows and Unix systems have 32-bitints on both 32-bit and 64-bit architectures.

Short integer

Ashort integercan represent a whole number that may take less storage, while having a smaller range, compared with a standard integer on the same machine.

InC,it is denoted byshort.It is required to be at least 16 bits, and is often smaller than a standard integer, but this is not required.A conforming program can assume that it can safely store values between −(2¹⁵−1)and 2¹⁵−1,but it may not assume that the range is not larger. InJava,ashortisalwaysa 16-bit integer. In theWindows API,the datatypeSHORTis defined as a 16-bit signed integer on all machines.

Common short integer sizes

Programming language	Data type name	Signedness	Size inbytes	Minimum value	Maximum value
CandC++	short	signed	2	−32,767	+32,767
	unsigned short	unsigned	2	0	65,535
C#	short	signed	2	−32,768	+32,767
	ushort	unsigned	2	0	65,535
Java	short	signed	2	−32,768	+32,767
SQL	smallint	signed	2	−32,768	+32,767

Long integer

Along integercan represent a wholeintegerwhoserangeis greater than or equal to that of a standard integer on the same machine.

InC,it is denoted bylong.It is required to be at least 32 bits, and may or may not be larger than a standard integer. A conforming program can assume that it can safely store values between −(2³¹−1)and 2³¹−1,but it may not assume that the range is not larger.

Common long integer sizes

Programming language	Approval Type	Platforms	Data type name	Storage inbytes	Signedrange	Unsignedrange
CISO/ANSI C99	International Standard	Unix,16/32-bit systems Windows,16/32/64-bit systems	long	4 (minimum requirement 4)	−2,147,483,647 to +2,147,483,647	0 to 4,294,967,295 (minimum requirement)
CISO/ANSI C99	International Standard	Unix, 64-bit systems	long	8 (minimum requirement 4)	−9,223,372,036,854,775,807 to +9,223,372,036,854,775,807	0 to 18,446,744,073,709,551,615
C++ISO/ANSI	International Standard	Unix,Windows, 16/32-bit system	long	4 (minimum requirement 4)	−2,147,483,648 to +2,147,483,647	0 to 4,294,967,295 (minimum requirement)
C++/CLI	International Standard ECMA-372	Unix,Windows, 16/32-bit systems	long	4 (minimum requirement 4)	−2,147,483,648 to +2,147,483,647	0 to 4,294,967,295 (minimum requirement)
VB	Company Standard	Windows	Long	4	−2,147,483,648 to +2,147,483,647	—
VBA	Company Standard	Windows,Mac OS X	Long	4	−2,147,483,648 to +2,147,483,647	—
SQL Server	Company Standard	Windows	BigInt	8	−9,223,372,036,854,775,808 to +9,223,372,036,854,775,807	0 to 18,446,744,073,709,551,615
C#/VB.NET	ECMA International Standard	Microsoft.NET	longorInt64	8	−9,223,372,036,854,775,808 to +9,223,372,036,854,775,807	0 to 18,446,744,073,709,551,615
Java	International/Company Standard	Java platform	long	8	−9,223,372,036,854,775,808 to +9,223,372,036,854,775,807	—
Pascal	?	Windows,UNIX	int64	8	−9,223,372,036,854,775,808 to +9,223,372,036,854,775,807	0 to 18,446,744,073,709,551,615 (Qword type)

Long long

"long long" redirects here. For other uses, seeLong (disambiguation).

In theC99version of theC programming languageand theC++11version ofC++,along longtype is supported that has double the minimum capacity of the standardlong.This type is not supported by compilers that require C code to be compliant with the previous C++ standard, C++03, because thelong longtype did not exist in C++03. For an ANSI/ISO compliant compiler, the minimum requirements for the specified ranges, that is, −(2⁶³−1)to 2⁶³−1 for signed and 0 to 2⁶⁴−1 for unsigned,must be fulfilled; however, extending this range is permitted.This can be an issue when exchanging code and data between platforms, or doing direct hardware access. Thus, there are several sets of headers providing platform independent exact width types. The Cstandard libraryprovidesstdint.h;this was introduced in C99 and C++11.

Syntax

Main article:Integer literal

Integer literalscan be written as regularArabic numerals,consisting of a sequence of digits and with negation indicated by aminus signbefore the value. However, most programming languages disallow use of commas or spaces fordigit grouping.Examples of integer literals are:

• 42
• 10000
• -233000

There are several alternate methods for writing integer literals in many programming languages:

• Many programming languages, especially those influenced byC,prefix an integer literal with0Xor0xto represent ahexadecimalvalue, e.g.0xDEADBEEF.Other languages may use a different notation, e.g. someassembly languagesappend anHorhto the end of a hexadecimal value.
• Perl,Ruby,Java,Julia,D,Go,RustandPython(starting from version 3.6) allow embeddedunderscoresfor clarity, e.g.10_000_000,and fixed-formFortranignores embedded spaces in integer literals. C (starting fromC23) and C++ use single quotes for this purpose.
• InCandC++,a leading zero indicates anoctalvalue, e.g.0755.This was primarily intended to be used withUnix modes;however, it has been criticized because normal integers may also lead with zero.As such,Python,Ruby,Haskell,andOCamlprefix octal values with0Oor0o,following the layout used by hexadecimal values.
• Several languages, includingJava,C#,Scala,Python,Ruby,OCaml,C (starting from C23) and C++ can represent binary values by prefi xing a number with0Bor0b.

See also

• Arbitrary-precision arithmetic
• Binary-coded decimal(BCD)
• C data types
• Integer overflow
• Signed number representations