In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
, the hexadecimal (also base-16 or simply hex) numeral system is a
positional numeral system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
that represents numbers using a
radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
(base) of 16. Unlike the
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
system representing numbers using 10 symbols, hexadecimal uses 16 distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from 10 to 15.
Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of
binary-coded values. Each hexadecimal digit represents four
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 represente ...
s (binary digits), also known as a
nibble
In computing, a nibble (occasionally nybble, nyble, or nybl to match the spelling of byte) is a four-bit aggregation, or half an octet. It is also known as half-byte or tetrade. In a networking or telecommunication context, the nibble is oft ...
(or nybble). For example, an 8-bit
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 ...
can have values ranging from 00000000 to 11111111 in binary form, which can be conveniently represented as 00 to FF in hexadecimal.
In mathematics, a subscript is typically used to specify the base. For example, the decimal value would be expressed in hexadecimal as . In programming, a number of notations are used to denote hexadecimal numbers, usually involving a prefix. The prefix
0x
is used in
C, which would denote this value as
0x
.
Hexadecimal is used in the transfer encoding Base16, in which each byte of the
plaintext
In cryptography, plaintext usually means unencrypted information pending input into cryptographic algorithms, usually encryption algorithms. This usually refers to data that is transmitted or stored unencrypted.
Overview
With the advent of comp ...
is broken into two 4-bit values and represented by two hexadecimal digits.
Representation
Written representation
In most current use cases, the letters A–F or a–f represent the values 10–15, while the
numerals
A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to:
* Numeral system used in mathematics
* Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' and ''first'' in English)
* Numerical d ...
0–9 are used to represent their decimal values.
There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used.
Seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
s use mixed-case AbCdEF to make digits that can be distinguished from each other.
There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following
hex dump
In computing, a hex dump is a hexadecimal view (on screen or paper) of computer data, from memory or from a computer file or storage device. Looking at a hex dump of data is usually done in the context of either debugging, reverse engineering or ...
, each 8-bit
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 ...
is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.
00000000 57 69 6b 69 70 65 64 69 61 2c 20 74 68 65 20 66
00000010 72 65 65 20 65 6e 63 79 63 6c 6f 70 65 64 69 61
00000020 20 74 68 61 74 20 61 6e 79 6f 6e 65 20 63 61 6e
00000030 20 65 64 69 74 0a
Distinguishing from decimal
In contexts where the
base is not clear, hexadecimal numbers can 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: 159
10 is decimal 159; 159
16 is hexadecimal 159, which equals 345
10. Some authors prefer a text subscript, such as 159
decimal and 159
hex, or 159
d and 159
h.
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
introduced the use of a particular typeface to represent a particular radix in his book ''The TeXbook''. Hexadecimal representations are written there in a
typewriter typeface:
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
*
Unix
Unix (; trademarked as UNIX) is a family of multitasking, multiuser computer operating systems that derive from the original AT&T Unix, whose development started in 1969 at the Bell Labs research center by Ken Thompson, Dennis Ritchie, and ot ...
(and related) shells,
AT&T
AT&T Inc. is an American multinational telecommunications holding company headquartered at Whitacre Tower in Downtown Dallas, Texas. It is the world's largest telecommunications company by revenue and the third largest provider of mobile tel ...
assembly language and likewise 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 its syntactic descendants such as
C++
C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
,
C#,
Go,
D,
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 List ...
,
JavaScript
JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of Website, websites use JavaScript on the Client (computing), client side ...
,
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
and
Windows PowerShell
PowerShell is a task automation and configuration management program from Microsoft, consisting of a command-line shell and the associated scripting language. Initially a Windows component only, known as Windows PowerShell, it was made open-sourc ...
) 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 _is_a_string_containing_11_characters_with_two_embedded_Esc_characters._To_output_an_integer_as_hexadecimal_with_the_printf.html" ;"title="m\x1B[25;1H"
is a string containing 11 characters with two embedded Esc characters. To output an integer as hexadecimal with the printf">m\x1B[25;1H" is a string containing 11 characters with two embedded Esc characters. To output an integer as hexadecimal with the printf function family, the format conversion code
%X
or
%x
is used.
* In URIs (including URLs), character encoding, character codes are written as hexadecimal pairs prefixed with
%
:
http://www.example.com/name%20with%20spaces
where
%20
is the code for the
space (blank) character,
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 of ...
code point 20 in hex, 32 in decimal.
* In
XML
Extensible Markup Language (XML) is a markup language and file format for storing, transmitting, and reconstructing arbitrary data. It defines a set of rules for encoding documents in a format that is both human-readable and machine-readable. T ...
and
XHTML
Extensible HyperText Markup Language (XHTML) is part of the family of XML markup languages. It mirrors or extends versions of the widely used HyperText Markup Language (HTML), the language in which Web pages are formulated.
While HTML, prior ...
, characters can be expressed as hexadecimal
numeric character reference
A numeric character reference (NCR) is a common markup construct used in SGML and SGML-derived markup languages such as HTML and XML. It consists of a short sequence of characters that, in turn, represents a single character. Since WebSgml, XML ...
s using the notation
&#x''code'';
, for instance
’
represents the character U+2019 (the right single quotation mark). If there is no the number is decimal (thus
’
is the same character).
* In the
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
standard, a character value is represented with
U+
followed by the hex value, e.g.
U+20AC
is the
Euro sign
The euro sign () is the currency sign used for the euro, the official currency of the eurozone and unilaterally adopted by Kosovo and Montenegro. The design was presented to the public by the European Commission on 12 December 1996. It consists ...
(€).
*
Color references in HTML,
CSS
Cascading Style Sheets (CSS) is a style sheet language used for describing the presentation of a document written in a markup language such as HTML or XML (including XML dialects such as SVG, MathML or XHTML). CSS is a cornerstone techno ...
and
X Window
The X Window System (X11, or simply X) is a windowing system for bitmap displays, common on Unix-like operating systems.
X provides the basic framework for a GUI environment: drawing and moving windows on the display device and interacting wit ...
can be expressed with six hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with
#
: white, for example, is represented as
#FFFFFF
. CSS also allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange: ).
* In
MIME
Multipurpose Internet Mail Extensions (MIME) is an Internet standard that extends the format of email messages to support text in character sets other than ASCII, as well as attachments of audio, video, images, and application programs. Message ...
(e-mail extensions)
quoted-printable
Quoted-Printable, or QP encoding, is a binary-to-text encoding system using printable ASCII characters (alphanumeric and the equals sign =) to transmit 8-bit data over a 7-bit data path or, generally, over a medium which is not 8-bit clean. Hist ...
encoding, character codes are written as hexadecimal pairs prefixed with
=
:
Espa=F1a
is "España" (F1 is the code for ñ in the ISO/IEC 8859-1 character set).)
* In Intel-derived
assembly language
In computer programming, assembly language (or assembler language, or symbolic machine code), often referred to simply as Assembly and commonly abbreviated as ASM or asm, is any low-level programming language with a very strong correspondence be ...
s and Modula-2, hexadecimal is denoted with a suffixed or :
FFh
or
05A3H
. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write
0FFh
instead of
FFh
. Some other implementations (such as NASM) allow C-style numbers (
0x42
).
* Other assembly languages (
6502
The MOS Technology 6502 (typically pronounced "sixty-five-oh-two" or "six-five-oh-two") William Mensch and the moderator both pronounce the 6502 microprocessor as ''"sixty-five-oh-two"''. is an 8-bit microprocessor that was designed by a small te ...
,
Motorola
Motorola, Inc. () was an American Multinational corporation, multinational telecommunications company based in Schaumburg, Illinois, United States. After having lost $4.3 billion from 2007 to 2009, the company split into two independent p ...
),
Pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Fren ...
,
Delphi
Delphi (; ), in legend previously called Pytho (Πυθώ), in ancient times was a sacred precinct that served as the seat of Pythia, the major oracle who was consulted about important decisions throughout the ancient classical world. The oracle ...
, some versions of
BASIC
BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming languages designed for ease of use. The original version was created by John G. Kemeny and Thomas E. Kurtz at Dartmouth College ...
(
Commodore
Commodore may refer to:
Ranks
* Commodore (rank), a naval rank
** Commodore (Royal Navy), in the United Kingdom
** Commodore (United States)
** Commodore (Canada)
** Commodore (Finland)
** Commodore (Germany) or ''Kommodore''
* Air commodore, a ...
),
GameMaker Language,
Godot and
Forth
Forth or FORTH may refer to:
Arts and entertainment
* ''forth'' magazine, an Internet magazine
* ''Forth'' (album), by The Verve, 2008
* ''Forth'', a 2011 album by Proto-Kaw
* Radio Forth, a group of independent local radio stations in Scotla ...
use
$
as a prefix:
$5A3
.
* Some assembly languages (Microchip) use the notation
H'ABCD'
(for ABCD
16). Similarly,
Fortran 95 uses Z'ABCD'.
*
Ada
Ada may refer to:
Places
Africa
* Ada Foah, a town in Ghana
* Ada (Ghana parliament constituency)
* Ada, Osun, a town in Nigeria
Asia
* Ada, Urmia, a village in West Azerbaijan Province, Iran
* Ada, Karaman, a village in Karaman Province, ...
and
VHDL
The VHSIC Hardware Description Language (VHDL) is a hardware description language (HDL) that can model the behavior and structure of digital systems at multiple levels of abstraction, ranging from the system level down to that of logic gates ...
enclose hexadecimal numerals in based "numeric quotes":
16#5A3#
. For bit vector constants
VHDL
The VHSIC Hardware Description Language (VHDL) is a hardware description language (HDL) that can model the behavior and structure of digital systems at multiple levels of abstraction, ranging from the system level down to that of logic gates ...
uses the notation
x"5A3"
.
*
Verilog
Verilog, standardized as IEEE 1364, is a hardware description language (HDL) used to model electronic systems. It is most commonly used in the design and verification of digital circuits at the register-transfer level of abstraction. It is also ...
represents hexadecimal constants in the form
8'hFF
, where 8 is the number of bits in the value and FF is the hexadecimal constant.
* The
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 Alan Ka ...
language uses the prefix
16r
:
16r5A3
*
PostScript
PostScript (PS) is a page description language in the electronic publishing and desktop publishing realm. It is a dynamically typed, concatenative programming language. It was created at Adobe Systems by John Warnock, Charles Geschke, Doug Br ...
and the
Bourne shell
The Bourne shell (sh) is a Shell (computing), shell Command-line interface#Command-line interpreter, command-line interpreter for computer operating systems.
The Bourne shell was the default Unix shell, shell for Version 7 Unix. Unix-like syste ...
and its derivatives denote hex with prefix
16#
:
16#5A3
. For PostScript, binary data (such as image
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device.
In most digital display devices, pixels are the smal ...
s) can be expressed as unprefixed consecutive hexadecimal pairs:
AA213FD51B3801043FBC
...
*
Common Lisp
Common Lisp (CL) is a dialect of the Lisp programming language, published in ANSI standard document ''ANSI INCITS 226-1994 (S20018)'' (formerly ''X3.226-1994 (R1999)''). The Common Lisp HyperSpec, a hyperlinked HTML version, has been derived fro ...
uses the prefixes
#x
and
#16r
. Setting the variables *read-base* and *print-base* to 16 can also be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.
*
MSX BASIC
MSX BASIC is a dialect of the BASIC programming language. It is an extended version of Microsoft's MBASIC Version 4.5, adding support for graphic, music, and various peripherals attached to MSX microcomputers. Generally, MSX BASIC is designed to ...
,
QuickBASIC
Microsoft QuickBASIC (also QB) is an Integrated Development Environment (or IDE) and compiler for the BASIC programming language that was developed by Microsoft. QuickBASIC runs mainly on DOS, though there was also a short-lived version for the c ...
,
FreeBASIC
FreeBASIC is a free and open source multiplatform compiler and programming language based on BASIC licensed under the GNU GPL for Microsoft Windows, protected-mode MS-DOS (DOS extender), Linux, FreeBSD and Xbox. The Xbox version is no longer m ...
and
Visual Basic Visual Basic is a name for a family of programming languages from Microsoft. It may refer to:
* Visual Basic .NET (now simply referred to as "Visual Basic"), the current version of Visual Basic launched in 2002 which runs on .NET
* Visual Basic (cl ...
prefix hexadecimal numbers with
&H
:
&H5A3
*
BBC BASIC
BBC BASIC is a version of the BASIC programming language released in 1981 as the native programming language for the BBC Micro home/personal computer, providing a standardized language for a UK computer literacy project of the BBC. It was wri ...
and
Locomotive BASIC
Locomotive Basic is a proprietary dialect of the BASIC programming language written by Locomotive Software on the Amstrad CPC (where it was built-in on ROM) and the later Locomotive BASIC-2 as a GEM application on the Amstrad PC1512 and 1640. ...
use
&
for hex.
*
TI-89
The TI-89 and the TI-89 Titanium are graphing calculators developed by Texas Instruments (TI). They are differentiated from most other TI graphing calculators by their computer algebra system, which allows symbolic manipulation of algeb ...
and 92 series uses a
0h
prefix:
0h5A3
*
ALGOL 68
ALGOL 68 (short for ''Algorithmic Language 1968'') is an imperative programming language that was conceived as a successor to the ALGOL 60 programming language, designed with the goal of a much wider scope of application and more rigorously de ...
uses the prefix
16r
to denote hexadecimal numbers:
16r5a3
. Binary, quaternary (base-4) and octal numbers can be specified similarly.
* The most common format for hexadecimal on IBM mainframes (
zSeries
IBM Z is a family name used by IBM for all of its z/Architecture mainframe computers.
In July 2017, with another generation of products, the official family was changed to IBM Z from IBM z Systems; the IBM Z family now includes the newest mo ...
) and midrange computers (
IBM i
IBM i (the ''i'' standing for ''integrated'') is an operating system developed by IBM for IBM Power Systems. It was originally released in 1988 as OS/400, as the sole operating system of the IBM AS/400 line of systems. It was renamed to i5/OS in ...
) running the traditional OS's (
zOS,
zVSE,
zVM,
TPF,
IBM i
IBM i (the ''i'' standing for ''integrated'') is an operating system developed by IBM for IBM Power Systems. It was originally released in 1988 as OS/400, as the sole operating system of the IBM AS/400 line of systems. It was renamed to i5/OS in ...
) is
X'5A3'
, and is used in Assembler,
PL/I
PL/I (Programming Language One, pronounced and sometimes written PL/1) is a procedural, imperative computer programming language developed and published by IBM. It is designed for scientific, engineering, business and system programming. I ...
,
COBOL
COBOL (; an acronym for "common business-oriented language") is a compiled English-like computer programming language designed for business use. It is an imperative, procedural and, since 2002, object-oriented language. COBOL is primarily us ...
,
JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.
* Any
IPv6 address
An Internet Protocol Version 6 address (IPv6 address) is a numeric label that is used to identify and locate a network interface of a computer or a network node participating in a computer network using IPv6. IP addresses are included in the pac ...
can be written as eight groups of four hexadecimal digits (sometimes called
hextet
In computing, a hextet, or a ''chomp'', is a sixteen-bit aggregation, or four nibbles. As a nibble typically is notated in hexadecimal format, a hextet consists of 4 hexadecimal digits. A hextet is the unofficial name for each of the 8 blocks in a ...
s), where each group is separated by a colon (
:
). This, for example, is a valid IPv6 address: or abbreviated by removing zeros as (
IPv4 address
Internet Protocol version 4 (IPv4) is the fourth version of the Internet Protocol (IP). It is one of the core protocols of standards-based internetworking methods in the Internet and other packet-switched networks. IPv4 was the first version d ...
es are usually written in decimal).
*
Globally unique identifier
A universally unique identifier (UUID) is a 128-bit label used for information in computer systems. The term globally unique identifier (GUID) is also used.
When generated according to the standard methods, UUIDs are, for practical purposes, un ...
s are written as thirty-two hexadecimal digits, often in unequal hyphen-separated groupings, for example .
Other symbols for 10–15 and mostly different symbol sets
The use of the letters ''A'' through ''F'' to represent the digits above 9 was not universal in the early history of computers.
* During the 1950s, some installations, such as Bendix-14, favored using the digits 0 through 5 with an
overline
An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a '' vinculum'', a notation for grouping symbols which is expressed in m ...
to denote the values 10–15 as , , , , and .
* The
SWAC (1950)
and
Bendix G-15
The Bendix G-15 is a computer introduced in 1956 by the Bendix Corporation, Computer Division, Los Angeles, California. It is about and weighs about . The G-15 has a drum memory of 2,160 29-bit words, along with 20 words used for special purpos ...
(1956)
computers used the lowercase letters ''u'', ''v'', ''w'', ''x'', ''y'' and ''z'' for the values 10 to 15.
* The
ORDVAC
The ORDVAC (''Ordnance Discrete Variable Automatic Computer)'', is an early computer built by the University of Illinois for the Ballistic Research Laboratory at Aberdeen Proving Ground. A successor to the ENIAC (along with EDVAC built earlier). ...
and
ILLIAC I
The ILLIAC I (Illinois Automatic Computer), a pioneering computer in the ILLIAC series of computers built in 1952 by the University of Illinois, was the first computer built and owned entirely by a United States educational institution.
Computer ...
(1952) computers (and some derived designs, e.g.
BRLESC
The BRLESC I (Ballistic Research Laboratories Electronic Scientific Computer) was one of the last of the first-generation electronic computers. It was built by the United States Army's Ballistic Research Laboratory (BRL) at Aberdeen Proving Gro ...
) used the uppercase letters ''K'', ''S'', ''N'', ''J'', ''F'' and ''L'' for the values 10 to 15.
* The Librascope
LGP-30
The LGP-30, standing for Librascope General Purpose and then Librascope General Precision, was an early off-the-shelf computer. It was manufactured by the Librascope company of Glendale, California (a division of General Precision Inc.), and so ...
(1956) used the letters ''F'', ''G'', ''J'', ''K'', ''Q'' and ''W'' for the values 10 to 15.
* On the
PERM
Perm or PERM may refer to:
Places
*Perm, Russia, a city in Russia
** Permsky District, the district
**Perm Krai, a federal subject of Russia since 2005
**Perm Oblast, a former federal subject of Russia 1938–2005
**Perm Governorate, an administra ...
(1956) computer, hexadecimal numbers were written as letters ''O'' for zero, ''A'' to ''N'' and ''P'' for 1 to 15. Many machine instructions had mnemonic hex-codes (''A''=add, ''M''=multiply, ''L''=load, ''F''=fixed-point etc.); programs were written without instruction names.
* The
Honeywell
Honeywell International Inc. is an American publicly traded, multinational conglomerate corporation headquartered in Charlotte, North Carolina. It primarily operates in four areas of business: aerospace, building technologies, performance ma ...
Datamatic D-1000
The DATAmatic 1000 is an obsolete computer system from Honeywell introduced in 1957. It uses vacuum tubes and crystal diodes for logic, and featured a unique magnetic tape format for storage.
The CPU uses a 48-bit word (plus four check bits). A ...
(1957) used the lowercase letters ''b'', ''c'', ''d'', ''e'', ''f'', and ''g'' whereas the
Elbit 100 (1967) used the uppercase letters ''B'', ''C'', ''D'', ''E'', ''F'' and ''G'' for the values 10 to 15.
* The
Monrobot XI
The Monroe Calculating Machine Mark XI (or Monrobot XI) was a general-purpose stored-program electronic digital computer introduced in 1960 by the Monroe Calculating Machine Division of Litton Industries. The system was marketed for "primarily ...
(1960) used the letters ''S'', ''T'', ''U'', ''V'', ''W'' and ''X'' for the values 10 to 15.
* The
NEC
is a Japanese multinational corporation, multinational information technology and electronics corporation, headquartered in Minato, Tokyo. The company was known as the Nippon Electric Company, Limited, before rebranding in 1983 as NEC. It prov ...
parametron
Parametron is a logic circuit element invented by Eiichi Goto in 1954.Information Processing Society of Japan Parametron/ref> The parametron is essentially a resonant circuit with a nonlinear reactive element which oscillates at half the driving f ...
computer NEAC 1103 (1960) used the letters ''D'', ''G'', ''H'', ''J'', ''K'' (and possibly ''V'') for values 10–15.
* The Pacific Data Systems 1020 (1964) used the letters ''L'', ''C'', ''A'', ''S'', ''M'' and ''D'' for the values 10 to 15.
* Bruce Alan Martin of
Brookhaven National Laboratory
Brookhaven National Laboratory (BNL) is a United States Department of Energy national laboratory located in Upton, Long Island, and was formally established in 1947 at the site of Camp Upton, a former U.S. Army base and Japanese internment c ...
considered the choice of A–F "ridiculous". In a 1968 letter to the editor of the
CACM, he proposed an entirely new set of symbols based on the bit locations.
![Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations](https://upload.wikimedia.org/wikipedia/commons/a/a0/Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg)
* New numeric symbols and names were introduced in the
Bibi-binary
The Bibi-binary system for numeric notation (in French système Bibi-binaire, or abbreviated "système Bibi") is a hexadecimal numeral system first described in 1968 by singer/mathematician Robert "Boby" Lapointe (1922–1972). At the time, it ...
notation by
Boby Lapointe
Robert Jean-François Joseph Pascal Lapointe (; 16 April 1922 – 29 June 1972), better known by his stage name Boby Lapointe (), was a French actor and singer, noted for his humorous texts, alliterationsSee the titles: ''Aubade à Lydie en do'' ...
in 1968.
![Base-16 digits](https://upload.wikimedia.org/wikipedia/commons/3/30/Base-16_digits.png)
* Ronald O. Whitaker of Rowco Engineering Co., in 1972, proposed a triangular font that allows "direct binary reading" in order to "permit both input and output from computers without respect to encoding matrices."
[ (1 page)][ (7 pages)]
* Some
seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.
Verbal and digital representations
Since there were no traditional numerals to represent the quantities from ten to fifteen, alphabetic letters were re-employed as a substitute. Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. Some people read hexadecimal numbers digit by digit, like a phone number, or using the
NATO phonetic alphabet
The (International) Radiotelephony Spelling Alphabet, commonly known as the NATO phonetic alphabet, is the most widely used set of clear code words for communicating the letters of the Roman alphabet, technically a ''radiotelephonic spellin ...
, the
Joint Army/Navy Phonetic Alphabet
The Allied military phonetic spelling alphabets prescribed the words that are used to represent each letter of the alphabet, when spelling other words out loud, letter-by-letter, and how the spelling words should be pronounced for use by the Alli ...
, or a similar ''ad-hoc'' system. In the wake of the adoption of hexadecimal among
IBM System/360
The IBM System/360 (S/360) is a family of mainframe computer systems that was announced by IBM on April 7, 1964, and delivered between 1965 and 1978. It was the first family of computers designed to cover both commercial and scientific applica ...
programmers, Magnuson (1968)
[ suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A" was pronounced "ann", B "bet", C "chris", etc.] Another naming system was elaborated by Babb (2015), based on a joke in Silicon Valley
Silicon Valley is a region in Northern California that serves as a global center for high technology and innovation. Located in the southern part of the San Francisco Bay Area, it corresponds roughly to the geographical areas San Mateo County ...
. Yet another naming-system was published online by Rogers (2007) that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below.
Others have proposed using the verbal Morse Code conventions to express four-bit hexadecimal digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hexadecimal digit for decimal 15.
Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 102310 on ten fingers. Another system for counting up to FF16 (25510) is illustrated on the right.
Signs
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210 and so on.
Hexadecimal can also be used to express the exact bit patterns used in the processor
Processor may refer to:
Computing Hardware
* Processor (computing)
**Central processing unit (CPU), the hardware within a computer that executes a program
*** Microprocessor, a central processing unit contained on a single integrated circuit (I ...
, so a sequence of hexadecimal digits may represent a signed or even a floating-point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
value. This way, the negative number −4210 can be written as FFFF FFD6 in a 32-bit CPU register
A processor register is a quickly accessible location available to a computer's processor. Registers usually consist of a small amount of fast storage, although some registers have specific hardware functions, and may be read-only or write-only. ...
(in 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 ...
), as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found i ...
).
Hexadecimal exponential notation
Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. P notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
uses the letter ''P'' (or ''p'', for "power"), whereas ''E'' (or ''e'') serves a similar purpose in decimal E notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
. The number after the ''P'' is ''decimal'' and represents the ''binary'' exponent. Increasing the exponent by 1 multiplies by 2, not 16: . Usually, the number is normalized so that the hexadecimal digits start with (zero is usually with no ''P'').
Example: represents .
P notation is required by the IEEE 754-2008
The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operation ...
binary floating-point standard, and can be used for floating-point literals in the C99
C99 (previously known as C9X) is an informal name for ISO/IEC 9899:1999, a past version of the C programming language standard. It extends the previous version ( C90) with new features for the language and the standard library, and helps impl ...
edition 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 ...
.
Using the ''%a'' or ''%A'' conversion specifiers, this notation can be produced by implementations of the ''printf
The printf format string is a control parameter used by a class of functions in the input/output libraries of C and many other programming languages. The string is written in a simple template language: characters are usually copied literal ...
'' family of functions following the C99 specification and
Single Unix Specification (IEEE Std 1003.1) POSIX
The Portable Operating System Interface (POSIX) is a family of standards specified by the IEEE Computer Society for maintaining compatibility between operating systems. POSIX defines both the system- and user-level application programming interf ...
standard.
Conversion
Binary conversion
Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (410).
This example converts 11112 to base ten. Since each position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:
* 00012 = 110
* 00102 = 210
* 01002 = 410
* 10002 = 810
Therefore:
With little practice, mapping 11112 to F16 in one step becomes easy: see table in written representation. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.[
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
The conversion from hexadecimal to binary is equally direct.]
Other simple conversions
Although quaternary
The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS). It follows the Neogene Period and spans from 2.58 million years ...
(base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits and each quaternary digit corresponds to a pair of binary digits. In the above example 5 E B 5 216 = 11 32 23 11 024.
The 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, ...
(base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.
Division-remainder in source base
As with all bases there is a simple algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans only decimal and for most computers only binary (which can be converted by far more efficient methods) can be easily handled with this method.
Let d be the number to represent in hexadecimal, and the series hihi−1...h2h1 be the hexadecimal digits representing the number.
# i ← 1
# hi ← d mod 16
# d ← (d − hi) / 16
# If d = 0 (return series hi) else increment i and go to step 2
"16" may be replaced with any other base that may be desired.
The following is a JavaScript
JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of Website, websites use JavaScript on the Client (computing), client side ...
implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work with bitwise operators
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
.
function toHex(d)
function toChar(n)
Conversion through addition and multiplication
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value — before carrying out multiplication and addition to get the final representation.
For example, to convert the number B3AD to decimal, one can split the hexadecimal number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16''p'' (''p'' being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:
which is 45997 in base 10.
Tools for conversion
Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hexadecimal.
In Microsoft Windows
Windows is a group of several proprietary graphical operating system families developed and marketed by Microsoft. Each family caters to a certain sector of the computing industry. For example, Windows NT for consumers, Windows Server for serv ...
, the Calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
utility can be set to Programmer mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (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, ...
) and 2 (binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
), the bases most commonly used by programmers. In Programmer Mode, the on-screen numeric keypad
A numeric keypad, number pad, numpad, or ten key,
is the palm-sized, usually-17-key section of a standard computer keyboard, usually on the far right. It provides calculator-style efficiency for entering numbers. The idea of a 10-key nu ...
includes the hexadecimal digits A through F, which are active when "Hex" is selected. In hex mode, however, the Windows Calculator supports only integers.
Elementary arithmetic
Elementary operations such as addition, subtraction, multiplication and division can be carried out indirectly through conversion to an alternate numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
, such as the commonly-used decimal system or the binary system where each hex digit corresponds to four binary digits.
Alternatively, one can also perform elementary operations directly within the hex system itself — by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division
In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps ...
and the traditional subtraction algorithm.
Real numbers
Rational numbers
As with other numeral systems, the hexadecimal system can be used to represent rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, although repeating expansions are common since sixteen (1016) has only a single prime factor; two.
For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1
. Because the radix 16 is a perfect square (42), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic number
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
:142857 × 1 = 142857
:14 ...
s (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wri ...
and sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.1 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.062510 (one-sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560.
Irrational numbers
The table below gives the expansions of some common irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s in decimal and hexadecimal.
Powers
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
Cultural history
The traditional Chinese units of measurement
Chinese units of measurement, known in Chinese as the ''shìzhì'' ("market system"), are the traditional units of measurement of the Han Chinese. Although Chinese numerals have been decimal (base-10) since the Shang dynasty, Shang, several Chine ...
were base-16. For example, one jīn (斤) in the old system equals sixteen tael
Tael (),["Tael" entry]
at the suanpan
The suanpan (), also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely ''Supplementary Notes on the Art of Figures'' written by Xu Yue. However, the exact design ...
(Chinese abacus
The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
) can be used to perform hexadecimal calculations such as additions and subtractions.
As with the duodecimal
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wri ...
system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals. Some proposals unify standard measures so that they are multiples of 16.
An early such proposal was put forward by John W. Nystrom in ''Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base'', published in 1862.
Nystrom among other things suggested hexadecimal time
Hexadecimal time is the representation of the time of day as a hexadecimal number in the interval [0, 1).
The day is divided into 1016 (1610) hexadecimal hours, each hour into 10016 (25610) hexadecimal minutes, and each minute into 1016 ...
, which subdivides a day by 16,
so that there are 16 "hours" (or "10 ''tims''", pronounced ''tontim'') in a day.
The word ''hexadecimal'' is first recorded in 1952. It is macaronic in the sense that it combines Greek language, Greek ἕξ (hex) "six" with Latinate ''-decimal''.
The all-Latin alternative '':wikt:sexadecimal, sexadecimal'' (compare the word ''sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
'' for base 60) is older, and sees at least occasional use from the late 19th century.
It is still in use in the 1950s in Bendix documentation.
Schwartzman (1994) argues that use of ''sexadecimal'' may have been avoided because of its suggestive abbreviation to ''sex''.
Many western languages since the 1960s have adopted terms equivalent in formation to ''hexadecimal'' (e.g. French ''hexadécimal'', Italian ''esadecimale'', Romanian ''hexazecimal'', Serbian ''хексадецимални'', etc.)
but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic ''sextándakerfi'', Russian ''шестнадцатеричной'' etc.)
Terminology and notation did not become settled until the end of the 1960s.
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
in 1969 argued that the etymologically correct term would be ''senidenary'', or possibly ''sedenary'', a Latinate term intended to convey "grouped by 16" modelled on ''binary'', ''ternary'' and ''quaternary'' etc.
According to Knuth's argument, the correct terms for ''decimal'' and ''octal'' arithmetic would be ''denary'' and ''octonary'', respectively.
Alfred B. Taylor used ''senidenary'' in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".
The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of the
publication of the Fortran IV manual for IBM System/360
The IBM System/360 (S/360) is a family of mainframe computer systems that was announced by IBM on April 7, 1964, and delivered between 1965 and 1978. It was the first family of computers designed to cover both commercial and scientific applica ...
, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants.IBM System/360 FORTRAN IV Language
(1966), p. 13.
As noted above, alternative notations were used by NEC
is a Japanese multinational corporation, multinational information technology and electronics corporation, headquartered in Minato, Tokyo. The company was known as the Nippon Electric Company, Limited, before rebranding in 1983 as NEC. It prov ...
(1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martin
in his letter to the editor of the CACM complains that
Martin's argument was that use of numerals 0 to 9 in nondecimal numbers "imply to us a base-ten place-value scheme":
"Why not use entirely new symbols (and names) for the seven or fifteen nonzero digits needed in octal or hex. Even use of the letters A through P would be an improvement, but entirely new symbols could reflect the binary nature of the system".
He also argued that "re-using alphabetic letters for numerical digits represents a gigantic backward step from the invention of distinct, non-alphabetic glyphs for numerals sixteen centuries ago" (as Brahmi numerals
The Brahmi numerals are a numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). They are a non positional decimal system. They are the direct graphic ancestors of the modern Hindu–Arabic numeral s ...
, and later in a Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
),
and that the recent 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 of ...
standards (ASA X3.4-1963 and USAS X3.4-1968)
"should have preserved six code table positions following the ten decimal digits
-- rather than needlessly filling these with punctuation characters"
(":;<=>?") that might have been placed elsewhere among the 128 available positions.
Base16 (transfer encoding)
Base16 (as a proper name without a space) can also refer to a binary to text encoding
A binary-to-text encoding is code, encoding of data (computing), data in plain text. More precisely, it is an encoding of binary data in a sequence of character (computing), printable characters. These encodings are necessary for transmission of ...
belonging to the same family as Base32
Base32 is the base-32 numeral system. It uses a set of 32 digits, each of which can be represented by 5 bits (25). One way to represent Base32 numbers in a human-readable way is by using a standard 32-character set, such as the twenty-two upper- ...
, Base58
A binary-to-text encoding is encoding of data in plain text. More precisely, it is an encoding of binary data in a sequence of printable characters. These encodings are necessary for transmission of data when the channel does not allow binary da ...
, and Base64
In computer programming, Base64 is a group of binary-to-text encoding schemes that represent binary data (more specifically, a sequence of 8-bit bytes) in sequences of 24 bits that can be represented by four 6-bit Base64 digits.
Common to all bina ...
.
In this case, data is broken into 4-bit sequences, and each value (between 0 and 15 inclusively) is encoded using one of 16 symbols from the 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 of ...
character set. Although any 16 symbols from the ASCII character set can be used, in practice the ASCII digits '0'–'9' and the letters 'A'–'F' (or the lowercase 'a'–'f') are always chosen in order to align with standard written notation for hexadecimal numbers.
There are several advantages of Base16 encoding:
* Most programming languages already have facilities to parse ASCII-encoded hexadecimal
* Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64 respectively
* The symbols 0–9 and A–F are universal in hexadecimal notation, so it is easily understood at a glance without needing to rely on a symbol lookup table
* Many CPU architectures have dedicated instructions that allow access to a half-byte (otherwise known as a "nibble
In computing, a nibble (occasionally nybble, nyble, or nybl to match the spelling of byte) is a four-bit aggregation, or half an octet. It is also known as half-byte or tetrade. In a networking or telecommunication context, the nibble is oft ...
"), making it more efficient in hardware than Base32 and Base64
The main disadvantages of Base16 encoding are:
* Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte. In contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively.
* Possible added complexity of having to accept both uppercase and lowercase letters
Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the W3C
The World Wide Web Consortium (W3C) is the main international standards organization for the World Wide Web. Founded in 1994 and led by Tim Berners-Lee, the consortium is made up of member organizations that maintain full-time staff working to ...
standard for URL percent encoding, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers.
See also
* Base32
Base32 is the base-32 numeral system. It uses a set of 32 digits, each of which can be represented by 5 bits (25). One way to represent Base32 numbers in a human-readable way is by using a standard 32-character set, such as the twenty-two upper- ...
, Base64
In computer programming, Base64 is a group of binary-to-text encoding schemes that represent binary data (more specifically, a sequence of 8-bit bytes) in sequences of 24 bits that can be represented by four 6-bit Base64 digits.
Common to all bina ...
(content encoding schemes)
* Hexadecimal time
Hexadecimal time is the representation of the time of day as a hexadecimal number in the interval
* IBM hexadecimal floating-point">, 1).
The day is divided into 1016 (1610) hexadecimal hours, each hour into 10016 (25610) hexadecimal minutes, and each minute into 1016 ...
* IBM hexadecimal floating-point
* Hex editor
* Hex dump
* Bailey–Borwein–Plouffe formula (BBP)
* Hexspeak
* P notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
References
{{reflist, refs=
[{{cite web , title=Computer Arithmetic , at=The Early Days of Hexadecimal , author-first=John J. G. , author-last=Savard , date=2018 , orig-year=2005 , work=quadibloc , url=http://www.quadibloc.com/comp/cp02.htm , access-date=2018-07-16 , url-status=live , archive-url=https://web.archive.org/web/20180716102439/http://www.quadibloc.com/comp/cp02.htm , archive-date=2018-07-16]
[{{cite book , title=G15D Programmer's Reference Manual , chapter=2.1.3 Sexadecimal notation , publisher=]Bendix Computer
Bendix Corporation is an American manufacturing and engineering company which, during various times in its existence, made Automotive industry, automotive brake shoes and systems, vacuum tubes, aircraft brakes, aeronautical hydraulics and electr ...
, Division of Bendix Aviation Corporation
Bendix Corporation is an American manufacturing and engineering company which, during various times in its existence, made automotive brake shoes and systems, vacuum tubes, aircraft brakes, aeronautical hydraulics and electric power systems, av ...
, location=Los Angeles, CA, USA , page=4 , url=http://bitsavers.trailing-edge.com/pdf/bendix/g-15/G15D_Programmers_Ref_Man.pdf , access-date=2017-06-01 , url-status=live , archive-url=https://web.archive.org/web/20170601222212/http://bitsavers.trailing-edge.com/pdf/bendix/g-15/G15D_Programmers_Ref_Man.pdf , archive-date=2017-06-01 , quote=This base is used because a group of four bits can represent any one of sixteen different numbers (zero to fifteen). By assigning a symbol to each of these combinations we arrive at a notation called sexadecimal (usually hex in conversation because nobody wants to abbreviate sex). The symbols in the sexadecimal language are the ten decimal digits and, on the G-15 typewriter, the letters u, v, w, x, y and z. These are arbitrary markings; other computers may use different alphabet characters for these last six digits.
[{{cite web , title=ILLIAC Programming – A Guide to the Preparation of Problems For Solution by the University of Illinois Digital Computer , author-first1=S. , author-last1=Gill , author-first2=R. E. , author-last2=Neagher , author-first3=D. E. , author-last3=Muller , author-first4=J. P. , author-last4=Nash , author-first5=J. E. , author-last5=Robertson , author-first6=T. , author-last6=Shapin , author-first7=D. J. , author-last7=Whesler , editor-first=J. P. , editor-last=Nash , edition=Fourth printing. Revised and corrected , date=1956-09-01 , publisher=Digital Computer Laboratory, Graduate College, ]University of Illinois
The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the University ...
, location=Urbana, Illinois, USA , pages=3–2 , url=http://www.textfiles.com/bitsavers/pdf/illiac/ILLIAC/ILLIAC_programming_Sep56.pdf , website=bitsavers.org , access-date=2014-12-18 , url-status=live , archive-url=https://web.archive.org/web/20170531153804/http://www.textfiles.com/bitsavers/pdf/illiac/ILLIAC/ILLIAC_programming_Sep56.pdf , archive-date=2017-05-31
[{{cite book , title=ROYAL PRECISION Electronic Computer LGP – 30 PROGRAMMING MANUAL , publisher= Royal McBee Corporation , location=Port Chester, New York , date=April 1957 , url=http://ed-thelen.org/comp-hist/lgp-30-man.html#R4.13 , access-date=2017-05-31 , url-status=live , archive-url=https://web.archive.org/web/20170531153004/http://ed-thelen.org/comp-hist/lgp-30-man.html , archive-date=2017-05-31 (NB. This somewhat odd sequence was from the next six sequential numeric keyboard codes in the ]LGP-30
The LGP-30, standing for Librascope General Purpose and then Librascope General Precision, was an early off-the-shelf computer. It was manufactured by the Librascope company of Glendale, California (a division of General Precision Inc.), and so ...
's 6-bit character code.)
[{{cite web , title=Die PERM und ALGOL , url=http://www.manthey.cc/sites/seminars/src/History.pdf , author-first1=Steffen , author-last1=Manthey , author-first2=Klaus , author-last2=Leibrandt , date=2002-07-02 , access-date=2018-05-19 , language=de]
Binary arithmetic
Hexadecimal numeral system
Power-of-two numeral systems
Positional numeral systems