In computer science, aninteger is adatum ofintegral data type, adata type that represents somerange of 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 ofbinary 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 processorregister or memory address as an integer.
Thevalue of 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).[1]
An integer value is typically specified in thesource code of 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.[2]
Theinternal representation of 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 memorybytes storing the bits varies; seeendianness. Thewidth,precision, orbitness[3] of an integral type is the number of bits in its representation. An integral type withn bits can encode 2n numbers; for example an unsigned type typically represents the non-negative values 0 through2n − 1. Other encodings of integer values to bit patterns are sometimes used, for examplebinary-coded decimal orGray code, or as printed character codes such asASCII.
There are four well-knownways to represent signed numbers in a binary computing system. The most common istwo's complement, which allows a signed integral type withn bits to represent numbers from−2(n−1) through2(n−1) − 1. Two's complement arithmetic is convenient because there is a perfectone-to-one correspondence between representations and values (in particular,no separate +0 and −0), and becauseaddition,subtraction andmultiplication do 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 language may be a different size in a different language, on a different processor, or in an execution context of different bitness; see§ Words.
Someolder computer architectures used 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).
| Bits | Name | Range (assumingtwo's complement forsigned) | Decimal digits | Uses | Implementations | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| C/C++ | C# | Pascal andDelphi | Java | SQL[a] | 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 | ASCII characters,code units in theUTF-8character encoding | int8_t,signed char[b] | sbyte | Shortint | byte | tinyint | INTEGER[c] | byte | i8 |
| Unsigned: From 0 to 255, which equals 28 − 1 | 2.41 | uint8_t,unsigned char[b] | 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-2 characters,code units in theUTF-16character encoding | int16_t,short,[b]int[b] | short | Smallint | short | smallint | INTEGER[c] | short | i16 |
| Unsigned: From 0 to 65,535, which equals 216 − 1 | 4.82 | uint16_t, unsigned,[b]unsigned int[b] | ushort | Word | char[d] | 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-32 characters,true color with alpha,FourCC, pointers in32-bit computing | int32_t,int,[b]long[b] | int | LongInt;Integer[e] | int | int | INTEGER[c] | int | i32 |
| Unsigned: From 0 to 4,294,967,295, which equals 232 − 1 | 9.63 | uint32_t, unsigned,[b]unsigned int,[b]unsigned long[b] | uint | LongWord;DWord;Cardinal[e] | — | unsigned int | — | uint | u32 | |||
| 64 | word, doubleword, longword, long, long long, quad, quadword, qword, int64, i64, u64 | Signed: From−(263) to263 − 1 | 18.96 | Time (e.g. milliseconds since theUnix epoch), pointers in64-bit computing | int64_t,long,[b]long long[b] | long | Int64 | long | bigint | INTEGER[c] | long | i64 |
| Unsigned: From 0 to264 − 1 | 19.27 | uint64_t,unsigned long long[b] | ulong | UInt64;QWord | — | unsigned bigint | — | ulong | u64 | |||
| 128 | octaword, double quadword, i128, u128 | Signed: From−(2127) to2127 − 1 | 38.23 | Complex scientific calculations, | Only available as non-standard or compiler-specific extensions | cent[f] | i128 | |||||
| Unsigned: From 0 to2128 − 1 | 38.53 | ucent[f] | u128 | |||||||||
| n | n-bit integer (general case) | Signed: −(2n−1) to (2n−1 − 1) | (n − 1) log10 2 | C23:_BitInt(n),signed _BitInt(n) | Ada:range-2**(n-1)..2**(n-1)-1 | |||||||
| Unsigned: 0 to (2n − 1) | n log10 2 | C23:unsigned _BitInt(n) | Ada:range0..2**n-1,mod2**n; standard libraries' or third-party arbitrary arithmetic libraries' BigDecimal or Decimal classes in many languages such as Python, C++, etc. | |||||||||
DifferentCPUs 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 havebit-field types (a specified number of bits, usually constrained to be less than the maximum hardware-supported width) andrange types (that can represent only the integers in a specified range).
Some languages, such asLisp,Smalltalk,REXX,Haskell,Python, andRaku, supportarbitrary precision integers (also known asinfinite precision integers orbignums). 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'sBigInteger class orPerl's "bigint" package.[6] 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.
ABoolean type is a type that can represent only two values: 0 and 1, usually identified withfalse andtrue 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 anibble (when eating, being smaller than abite) ornybble (being a pun on the form of the wordbyte). One nibble corresponds to one digit inhexadecimal and holds one digit or a sign code in binary-coded decimal.
The termbyte 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 termbyte was usually not used at all in connection with bit- and word-addressed machines.
The termoctet always 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 usagebyte almost invariably means eight bits, since all other sizes have fallen into disuse; thusbyte has come to be synonymous withoctet.
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 aword is 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.[7]
Practically all new desktop processors are capable of using 64-bit words, thoughembedded processors with 8- and 16-bit word size are still common. The36-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 asint a 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.h in the form ofintptr_t.
Thebitness of 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 Windows support existing 32-bit binaries, and programs compiled for Linux'sx32 ABI run in 64-bit mode yet use 32-bit memory addresses.[8]
The standard integer size is platform-dependent.
InC, it is denoted byint and required to be at least 16 bits. Windows and Unix systems have 32-bitints on both 32-bit and 64-bit architectures.
Ashort integer can 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.[9][10] A conforming program can assume that it can safely store values between −(215−1)[11] and 215−1,[12] but it may not assume that the range is not larger. InJava, ashort isalways a 16-bit integer. In theWindows API, the datatypeSHORT is defined as a 16-bit signed integer on all machines.[7]
| Programming language | Data type name | Signedness | Size inbytes | Minimum value | Maximum value |
|---|---|---|---|---|---|
| C andC++ | short | signed | 2 | −32,767[g] | +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 |
Along integer can represent a wholeinteger whoserange is 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)[11] and 231−1,[12] but it may not assume that the range is not larger.
| Programming language | Approval Type | Platforms | Data type name | Storage inbytes | Signed range | Unsigned range |
|---|---|---|---|---|---|---|
| C ISO/ANSI C99 | International Standard | Unix, 16/32-bit systems[7] Windows, 16/32/64-bit systems[7] | long | 4 (minimum requirement 4) | −2,147,483,647 to +2,147,483,647 | 0 to 4,294,967,295 (minimum requirement) |
| C ISO/ANSI C99 | International Standard | Unix, 64-bit systems[7][10] | 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[13] (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[14] (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[15] | −2,147,483,648 to +2,147,483,647 | — |
| VBA | Company Standard | Windows,Mac OS X | Long | 4[16] | −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 | long orInt64 | 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) |
In theC99 version of theC programming language and theC++11 version ofC++, along long type 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 long type did not exist in C++03. For an ANSI/ISO compliant compiler, the minimum requirements for the specified ranges, that is, −(263−1)[11] to 263−1 for signed and 0 to 264−1 for unsigned,[12] must be fulfilled; however, extending this range is permitted.[17][18] 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 library providesstdint.h; this was introduced in C99 and C++11.
Integer literals can be written as regularArabic numerals, consisting of a sequence of digits and with negation indicated by aminus sign before the value. However, most programming languages disallow use of commas or spaces fordigit grouping. Examples of integer literals are:
4210000-233000There are several alternate methods for writing integer literals in many programming languages:
0X or0x to represent ahexadecimal value, e.g.0xDEADBEEF. Other languages may use a different notation, e.g. someassembly languages append anH orh to the end of a hexadecimal value.10_000_000, and fixed-formFortran ignores embedded spaces in integer literals. C (starting fromC23) and C++ use single quotes for this purpose.0755. This was primarily intended to be used withUnix modes; however, it has been criticized because normal integers may also lead with zero.[19] As such,Python,Ruby,Haskell, andOCaml prefix octal values with0O or0o, following the layout used by hexadecimal values.0B or0b.In many programming languages, there exist predefined constants representing the least and the greatest values representable with a given integer type.
Names for these include
MAXINT[20]java.lang.Integer.MAX_VALUE,java.lang.Integer.MIN_VALUE[21]INT_MAX, etc.[citation needed]MaxInt[citation needed]sys.maxint[citation needed]maxint[23]