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In computer science, an integer is a datum of integral data type, a
data type In computer science and computer programming, a data type (or simply type) is a set of possible values and a set of allowed operations on it. A data type tells the compiler or interpreter how the programmer intends to use the data. Most progra ...
that represents some range of mathematical
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. 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 processor register or memory address as an integer.


Value and representation

The ''value'' of an item with an integral type is the mathematical integer that it corresponds to. Integral types may be ''unsigned'' (capable of representing only non-negative integers) or ''signed'' (capable of representing negative integers as well). An integer value is typically specified in the
source code In computing, source code, or simply code, is any collection of code, with or without comment (computer programming), comments, written using a human-readable programming language, usually as plain text. The source code of a Computer program, p ...
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 permit
digit group separator Digit may refer to: Mathematics and science * Numerical digit, as used in mathematics or computer science ** Hindu-Arabic numerals, the most common modern representation of numerical digits * Digit (anatomy), the most distal part of a limb, such ...
s. The ''internal 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 of
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...
s, using the
binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" ( zero) and "1" (one). The base-2 numeral system is a positional notatio ...
. The order of the memory
byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
s storing the bits varies; see
endianness In computing, endianness, also known as byte sex, is the order or sequence of bytes of a word of digital data in computer memory. Endianness is primarily expressed as big-endian (BE) or little-endian (LE). A big-endian system stores the most si ...
. The ''width'' or ''precision'' of an integral type is the number of bits in its representation. An integral type with ''n'' bits can encode 2''n'' numbers; for example an unsigned type typically represents the non-negative values 0 through 2''n''−1. Other encodings of integer values to bit patterns are sometimes used, for example
binary-coded decimal In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used fo ...
or Gray code, or as printed character codes such as
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
. There are four well-known ways to represent signed numbers in a binary computing system. The most common is
two's complement Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
, 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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between representations and values (in particular, no separate +0 and −0), and because
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of ...
, subtraction and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
do not need to distinguish between signed and unsigned types. Other possibilities include offset binary, sign-magnitude, and ones' 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 one
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming l ...
may be a different size in a different language or on a different processor.


Common integral data types

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 (that can represent only the integers in a specified range). Some languages, such as
Lisp A lisp is a speech impairment in which a person misarticulates sibilants (, , , , , , , ). These misarticulations often result in unclear speech. Types * A frontal lisp occurs when the tongue is placed anterior to the target. Interdental lispi ...
,
Smalltalk Smalltalk is an object-oriented, dynamically typed reflective programming language. It was designed and created in part for educational use, specifically for constructionist learning, at the Learning Research Group (LRG) of Xerox PARC by ...
, REXX, Haskell, Python, and Raku, support ''arbitrary precision'' integers (also known as ''infinite precision integers'' or ''
bignum In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are l ...
s''). 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's class or
Perl Perl is a family of two High-level programming language, high-level, General-purpose programming language, general-purpose, Interpreter (computing), interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it ...
's "" 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. A
Boolean Any kind of logic, function, expression, or theory based on the work of George Boole is considered Boolean. Related to this, "Boolean" may refer to: * Boolean data type, a form of data with only two possible values (usually "true" and "false" ...
or
Flag A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design emp ...
type is a type that 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 In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, h ...
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 network A computer network is a set of computers sharing resources located on or provided by network nodes. The computers use common communication protocols over digital interconnections to communicate with each other. These interconnections ar ...
ing, 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 that are handled simultaneously by processors of a particular
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing buildings ...
. 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 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. Practically all new desktop processors are capable of using 64-bit words, though embedded processors with 8- and 16-bit word size are still common. The
36-bit word length 36-bit computers were popular in the early mainframe computer era from the 1950s through the early 1970s. Starting in the 1960s, but especially the 1970s, the introduction of 7-bit ASCII and 8-bit EBCDIC led to the move to machines using 8-bit ...
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 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 as , 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 in
stdint.h In the C programming language, data types constitute the semantics and characteristics of storage of data elements. They are expressed in the language syntax in form of declarations for memory locations or variables. Data types also determin ...
in the form of .


Short integer

A ''short 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. In C, it is denoted by . 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. In
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
, a is ''always'' a 16-bit integer. In the Windows API, the datatype is defined as a 16-bit signed integer on all machines.


Long integer

A ''long integer'' can represent a whole
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
whose range is greater than or equal to that of a standard integer on the same machine. In C, it is denoted by . 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.


Long long

In the C99 version of the
C programming language ''The C Programming Language'' (sometimes termed ''K&R'', after its authors' initials) is a computer programming book written by Brian Kernighan and Dennis Ritchie, the latter of whom originally designed and implemented the language, as well as ...
and the
C++11 C++11 is a version of the ISO/ IEC 14882 standard for the C++ programming language. C++11 replaced the prior version of the C++ standard, called C++03, and was later replaced by C++14. The name follows the tradition of naming language versio ...
version of C++, a long long type is supported that has double the minimum capacity of the standard long. This type is not supported by compilers that require C code to be compliant with the previous C++ standard, C++03, because the type 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 C standard library provides ''
stdint.h In the C programming language, data types constitute the semantics and characteristics of storage of data elements. They are expressed in the language syntax in form of declarations for memory locations or variables. Data types also determin ...
''; this was introduced in C99 and C++11.


Syntax

Literals for integers can be written as regular
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
, consisting of a sequence of digits and with negation indicated by a
minus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
before the value. However, most programming languages disallow use of commas or spaces for digit grouping. Examples of integer literals are: * 42 * 10000 * -233000 There are several alternate methods for writing integer literals in many programming languages: * Most programming languages, especially those influenced by C, prefix an integer literal with 0X or 0x to represent a
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, h ...
value, e.g. 0xDEADBEEF. Other languages may use a different notation, e.g. some assembly languages append an H or h to the end of a hexadecimal value. *
Perl Perl is a family of two High-level programming language, high-level, General-purpose programming language, general-purpose, Interpreter (computing), interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it ...
,
Ruby A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum (aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sapp ...
,
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
, Julia, D, Go,
Rust Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxide-hydroxide (FeO(OH), ...
and Python (starting from version 3.6) allow embedded underscores for clarity, e.g. 10_000_000, and fixed-form Fortran ignores embedded spaces in integer literals. C (starting from C23) and C++ use single quotes for this purpose. * In C and C++, a leading zero indicates an
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
value, e.g. 0755. This was primarily intended to be used with Unix modes; however, it has been criticized because normal integers may also lead with zero.ECMAScript 6th Edition draft: https://people.mozilla.org/~jorendorff/es6-draft.html#sec-literals-numeric-literals As such, Python,
Ruby A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum (aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sapp ...
, Haskell, and
OCaml OCaml ( , formerly Objective Caml) is a general-purpose, multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, D ...
prefix octal values with 0O or 0o, following the layout used by hexadecimal values. * Several languages, including
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
, C#, Scala, Python,
Ruby A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum (aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sapp ...
,
OCaml OCaml ( , formerly Objective Caml) is a general-purpose, multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, D ...
, C (starting from C23) and C++ can represent binary values by prefixing a number with 0B or 0b.


See also

*
Arbitrary-precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are l ...
*
Binary-coded decimal In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used fo ...
(BCD) *
C data types In the C programming language, data types constitute the semantics and characteristics of storage of data elements. They are expressed in the language syntax in form of declarations for memory locations or variables. Data types also determin ...
*
Integer overflow In computer programming, an integer overflow occurs when an arithmetic operation attempts to create a numeric value that is outside of the range that can be represented with a given number of digits – either higher than the maximum or lower t ...
* Signed number representations


Notes


References

{{DEFAULTSORT:Integer (Computer Science) Data types Computer arithmetic Primitive types