Hexadecimal

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In mathematics and computer science, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a through f) to represent values ten to fifteen. For example, the hexadecimal number 2AF3 is equal, in decimal, to (2 × 16³) + (10 × 16²) + (15 × 16) + 3, or 10,995.

Each hexadecimal digit represents four binary digits (bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a human-friendly representation of binary coded values in computing and digital electronics. For example, byte values can range from 0 to 255 (decimal) but may be more conveniently represented as two hexadecimal digits in the range 00 through FF. Hexadecimal is also commonly used to represent computer memory addresses.

Representing hexadecimal

0_hex = 0_dec = 0_oct 0 0 0 0
1_hex = 1_dec = 1_oct 0 0 0 1
2_hex = 2_dec = 2_oct 0 0 1 0
3_hex = 3_dec = 3_oct 0 0 1 1

4_hex = 4_dec = 4_oct 0 1 0 0
5_hex = 5_dec = 5_oct 0 1 0 1
6_hex = 6_dec = 6_oct 0 1 1 0
7_hex = 7_dec = 7_oct 0 1 1 1

8_hex = 8_dec = 10_oct 1 0 0 0
9_hex = 9_dec = 11_oct 1 0 0 1
A_hex = 10_dec = 12_oct 1 0 1 0
B_hex = 11_dec = 13_oct 1 0 1 1

C_hex = 12_dec = 14_oct 1 1 0 0
D_hex = 13_dec = 15_oct 1 1 0 1
E_hex = 14_dec = 16_oct 1 1 1 0
F_hex = 15_dec = 17_oct 1 1 1 1

In situations where there is no context, a hexadecimal number might be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 159₁₀ is decimal 159; 159₁₆ is hexadecimal 159, which is equal to 345₁₀. Other authors prefer a text subscript, such as 159_decimal and 159_hex, or 159_d and 159_h.

In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:

• In URLs, character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space (blank) character (code value 20 in hex, 32 in decimal).
• In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation &#xcode;, where code is the 1- to 6-digit hex number assigned to the character in the Unicode standard. Thus ’ represents the curled right single quote (Unicode value 2019 in hex, 8217 in decimal).
• Color references in HTML and CSS can be expressed with six hexdigits (two each for the red, green, and blue components, in that order) prefixed with #: white, for example, is represented #FFFFFF .[1] CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange).
• *nix (UNIX and related) shells, and likewise the C programming language, which was designed for UNIX (and the syntactical descendants of C[2]) use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits: '\x1B' represents the Esc control character; "\x1B[0m\x1B[25;1H" is a string containing 11 characters (plus a trailing NUL to mark the end of the string) with two embedded Esc characters.[3] To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used.
• In the Unicode standard, a character value is represented with U+ followed by the hex value: U+20AC is the Euro sign (€).
• In MIME (e-mail extensions) quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits (in ASCII) prefixed by an equal to sign =, as in Espa=F1a to send "España" (Spain). (Hexadecimal F1, equal to decimal 241, is the code number for the lower case n with tilda in the ISO/IEC 8859-1 character set.)
• In Intel-derived assembly languages, hexadecimal is indicated with a suffixed H or h: FFh or 05A3H. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit: 0FFh
• Other assembly languages (6502, AT&T, Motorola), Pascal, Delphi, some versions of BASIC (Commodore) and Forth use $ as a prefix: $5A3.
• Some assembly languages (Microchip) use the notation H'ABCD' (for ABCD₁₆).
• Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. VHDL also supports x"5A3".
• Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant.
• Modula 2 and some other languages use # as a prefix: #05A3
• The Smalltalk programming language uses the prefix 16r: 16r5A3
• Postscript indicates hex with prefix 16#: 16#5A3. Binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC...
• Common Lisp use the prefixes #x and #16r.
• QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3
• BBC BASIC and Locomotive BASIC use & for hex.[4]
• TI-89 and 92 series uses a 0h prefix: 0h5A3
• Notations such as X'5A3' are sometimes seen, such as in PL/I. This is the most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (iSeries) running traditional OS's (zOS, zVSE, zVM, TPF, OS/400), and is used in Assembler, PL/1, Cobol, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well.
• Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.[5] Hexadecimal representations are written there in a typewriter typeface: 5A3

There is no universal convention to use lowercase or uppercase for the letter digits, and each is prevalent or preferred in particular environments by community standards or convention.