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).[1]
An integer value is typically specified in the source code of a program as a sequence of digits, without spaces or thousands separators, optionally prefixed with + or −. Some programming languages allow other notations, such as hexadecimal (base 16) or octal (base 8).
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 maximum is sometimes called MAXINT or—as in the C standard library limits.h header—INT_MAX.[2]
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. Other encodings of integer values to bit patterns are sometimes used, for example Binary-coded decimal or Gray code, or as printed character codes such as ASCII.
There are four 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 (in particular, no separate +0 and −0), and because addition, subtraction and multiplication do not need to distinguish between signed and unsigned types. The other possibilities are offset binary, sign-magnitude and ones' complement. See Signed number representations for details.
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 may be a different size in a different language or on a different processor.