{{Short description|Base-16 numeric representation}} {{Redirect-multi|3|Sexadecimal|Hex digit|Hex format|base 60|Sexagesimal|binary-coded hexadecimals|Nibble|hexadecimal file formats|Hex file (disambiguation)}} {{Use dmy dates|date=March 2020|cs1-dates=y}} {{Numeral systems}} '''Hexadecimal''' ('''hex''' for short) is a positional numeral system for representing a numeric value as base 16. For the most common convention, a digit is represented as "0" to "9" like for decimal and as a letter of the alphabet from "A" to "F" (either upper or lower case) for the digits with decimal value 10 to 15.
As typical computer hardware is binary in nature and that hex is power of 2, the hex representation is often used in computing as a dense representation of binary information. A hex digit represents 4 contiguous bits {{Ndash}}known as a nibble.<ref>{{Cite news |title=The hexadecimal system |url=https://www.ionos.co.uk/digitalguide/server/know-how/hexadecimal-system/ |access-date=2022-08-26 |newspaper=Ionos Digital Guide |language=en |archive-date=2022-08-26 |archive-url=https://web.archive.org/web/20220826084201/https://www.ionos.co.uk/digitalguide/server/know-how/hexadecimal-system/ |url-status=live }}</ref> An 8-bit byte is two hex digits, such as {{code|2C}}.
Special notation is often used to indicate that a number is hex. In mathematics, a subscript is typically used to specify the base. For example, the decimal value {{val|491|fmt=commas}} would be expressed in hex as {{hexadecimal|491}}. In computer programming, various notations are used. In C and many related languages, the prefix <code>0x</code> is used. For example, <code>0x{{hexadecimal|491|no}}</code>.
==Written representation== ===Common convention=== Typically, a hex representation convention allows either lower or upper case letters and treats the letter the same regardless of its case.
Often when rendering non-textual data, a value stored in memory is displayed as a sequence of hex digits with spaces between values. For instance, in the following hex dump, each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number. <div class="nowrap"><syntaxhighlight lang="hexdump"> 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 2C 20 69 6E 63 6C 75 64 69 6E 67 00000040 20 79 6F 75 20 28 61 6E 64 20 6D 65 29 21 </syntaxhighlight></div>
===Identification=== There are several conventions for expressing that a number is represented as hex. <!-- NOTICE: Order from most to least likely to be encountered by lay people -->
* A decimal subscript can give the base explicitly. For example 159<sub>10</sub> indicates decimal 159, 159<sub>16</sub> indicates hex 159. Some prefer a text subscript, such as 159<sub>decimal</sub> and 159<sub>hex</sub>, or 159<sub>d</sub> and 159<sub>h</sub>
* {{anchor|_nix}}In C and many languages influenced by it, the prefix <code>0x</code> indicates that the numeric literal after it is in hex, a character of a string or character literal can be expressed as hex with the prefix <code>\x</code> (for example <code>'\x1B'</code> represents the Esc control character) and to output an integer as hex via printf-like function, the format conversion code <code>%X</code> or <code>%x</code> is used
* In URIs (including URLs), character codes are written as hex pairs prefixed with {{code|%}}: {{code|<nowiki>http://www.example.com/name%20with%20spaces</nowiki>}} where {{code|%20}} is the code for the space (blank) character, ASCII code point 20 in hex, 32 in decimal.
* In XML and XHTML, a character can be expressed as a hex numeric character reference using the notation <code>&#x''code'';</code>, for instance <code>&#x0054;</code> represents the character U+0054 (the uppercase letter "T"). If there is no {{code|x}} the number is decimal (thus <code>&#0084;</code> is the same character).<ref>{{cite web|url=https://www.unicode.org/charts/PDF/U2000.pdf|title=The Unicode Standard, Version 7|website=Unicode|access-date=28 October 2018|archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303175510/http://www.unicode.org/charts/PDF/U2000.pdf|url-status=live}}</ref>
* {{anchor|Intel}}In Intel-derived assembly languages and Modula-2,<ref>{{cite web |title=Modula-2 – Vocabulary and representation |url=http://modula2.org/reference/vocabulary.php |website=Modula −2 |access-date=1 November 2015 |archive-date=2015-12-13 |archive-url=https://web.archive.org/web/20151213053318/http://www.modula2.org/reference/vocabulary.php |url-status=live }}</ref> hex is denoted with a suffixed {{mono|H}} or {{mono|h}}: <code>FFh</code> or <code>05A3H</code>. Some implementations require a leading zero when the first hex digit character is not a decimal digit, so one would write <code>0FFh</code> instead of <code>FFh</code>. Some other implementations (such as NASM) allow C-style numbers (<code>0x42</code>)
* {{anchor|Motorola}}Other assembly languages (6502, Motorola), Pascal, Delphi, some versions of BASIC (Commodore), GameMaker Language, Godot and Forth use <code>$</code> as a prefix: <code>$5A3</code>, <code>$C1F27ED</code>
* Some assembly languages (Microchip) use the notation <code>H'ABCD'</code> (for ABCD<sub>16</sub>); similarly, Fortran 95 uses Z'ABCD'
* Ada and VHDL enclose hex numerals in based "numeric quotes": <code>16#5A3#</code>, <code>16#C1F27ED#</code>. For bit vector constants VHDL uses the notation <code>x"5A3"</code>, <code>x"C1F27ED"</code>.<ref>{{cite web |url=https://www.fpgatutorial.com/vhdl-types-and-conversions#vhdl-assign-data |title=An Introduction to VHDL Data Types |website=FPGA Tutorial |date=10 May 2020 |access-date=2020-08-21 |archive-date=2020-08-23 |archive-url=https://web.archive.org/web/20200823094252/https://www.fpgatutorial.com/vhdl-types-and-conversions/#vhdl-assign-data |url-status=live }}</ref>
* Verilog represents hex constants in the form <code>8'hFF</code>, where 8 is the number of bits in the value and FF is the hex constant
* The Icon and Smalltalk languages use the prefix <code>16r</code>: <code>16r5A3</code>
* PostScript and the Bourne shell and its derivatives denote hex with prefix <code>16#</code>: <code>16#5A3</code>, <code>16#C1F27ED</code>
* Common Lisp uses the prefixes <code>#x</code> and <code>#16r</code>. Setting the variables *read-base*<ref>{{cite web |title=*read-base* variable in Common Lisp |url=http://www.lispworks.com/documentation/HyperSpec/Body/v_rd_bas.htm |website=CLHS |access-date=2015-01-10 |archive-date=2016-02-03 |archive-url=https://web.archive.org/web/20160203221612/http://www.lispworks.com/documentation/HyperSpec/Body/v_rd_bas.htm |url-status=live }}</ref> and *print-base*<ref>{{cite web |title=*print-base* variable in Common Lisp |url=http://www.lispworks.com/documentation/HyperSpec/Body/v_pr_bas.htm#STprint-baseST |website=CLHS |access-date=2015-01-10 |archive-date=2014-12-26 |archive-url=https://web.archive.org/web/20141226172420/http://www.lispworks.com/documentation/HyperSpec/Body/v_pr_bas.htm#STprint-baseST |url-status=live }}</ref> to 16 can also be used to switch the reader and printer of a Common Lisp system to hex representation for reading and printing numbers. Thus hex numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.
* MSX BASIC,<ref>[http://www.atarimagazines.com/compute/issue56/107_1_MSX_IS_COMING.php MSX is Coming — Part 2: Inside MSX] {{Webarchive|url=https://web.archive.org/web/20101124111223/http://www.atarimagazines.com/compute/issue56/107_1_MSX_IS_COMING.php |date=2010-11-24 }} Compute!, issue 56, January 1985, p. 52</ref> QuickBASIC, FreeBASIC and Visual Basic prefix hex numbers with <code>&H</code>: <code>&H5A3</code>
* BBC BASIC and Locomotive BASIC use <code>&</code> for hex<ref>BBC BASIC programs are not fully portable to Microsoft BASIC (without modification) since the latter takes <code>&</code> to prefix octal values. (Microsoft BASIC primarily uses <code>&O</code> to prefix octal, and it uses <code>&H</code> to prefix hex, but the ampersand alone yields a default interpretation as an octal prefix.</ref>
* TI-89 and 92 series uses a <code>0h</code> prefix: <code>0h5A3</code>, <code>0hC1F27ED</code>
* ALGOL 68 uses the prefix <code>16r</code> to denote hex numbers: <code>16r5a3</code>, <code>16rC1F27ED</code>. Binary, quaternary (base-4), and octal numbers can be specified similarly.
* The most common format for hex on IBM mainframes (zSeries) and midrange computers (IBM i) running the traditional OS's (zOS, zVSE, zVM, TPF, IBM i) is <code>X'5A3'</code> or <code>X'C1F27ED'</code>, and is used in Assembler, PL/I, COBOL, 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.
* Donald Knuth used a typewriter typeface to represent hex in his book ''The TeXbook'',<ref>{{Cite book |last=Knuth |first=Donald Ervin |title=The TeXbook |date=1986 |others=Duane Bibby |isbn=0-201-13448-9 |publisher=American Mathematical Society and Addison-Wesley Publishing Company |url=https://web.math.ucsb.edu/~bigelow/books/texbook.pdf |archive-date=19 November 2025 |archive-url=https://web.archive.org/web/20251119183650/https://web.math.ucsb.edu/~bigelow/books/texbook.pdf |location=Reading, Mass. |oclc=12973034 }}</ref> like: {{mono|5A3}}, {{mono|C1F27ED}}
===Implicit=== In some contexts, a number is always written as hex, and therefore, needs no identification notation.
* In the Unicode standard, a character value is represented with {{code|U+}} followed by the hex value, e.g. {{code|U+00A1}} is the inverted exclamation point (¡).
* Color references in HTML, CSS and X Window can be expressed with six hex digits (two each for the red, green and blue components, in that order) prefixed with {{code|#}}: <span style="color:#FF00FF">magenta</span>, for example, is represented as {{code|#FF00FF}}.<ref>{{cite web |url=http://www.web-colors-explained.com/hex.php |title=Hexadecimal web colors explained |access-date=2006-01-11 |archive-url=https://web.archive.org/web/20060422004336/http://www.web-colors-explained.com/hex.php |archive-date=2006-04-22 |url-status=dead }}</ref> CSS also allows 3-hexdigit abbreviations with one hexdigit per component: {{code|#FA3}} abbreviates {{code|#FFAA33}}, <span style="color:#FFAA33">a golden orange</span>.
* In MIME (e-mail extensions) quoted-printable encoding, character codes are written as hex pairs prefixed with {{code|1==}}: {{code|1=Espa=F1a}} is "España" (F1{{sub|hex}} is the code for ''ñ'' in the ISO/IEC 8859-1 character set).<ref>{{Cite web |url=https://www.ic.unicamp.br/~stolfi/EXPORT/www/ISO-8859-1-Encoding.html|title=ISO-8859-1 (ISO Latin 1) Character Encoding |website=www.ic.unicamp.br|access-date=2019-06-26|archive-date=2019-06-29 |archive-url=https://web.archive.org/web/20190629203430/http://www.ic.unicamp.br/~stolfi/EXPORT/www/ISO-8859-1-Encoding.html|url-status=live}}</ref>)
* PostScript binary data (such as image pixels) can be expressed as unprefixed consecutive hex pairs: {{code|AA213FD51B3801043FBC}} ... * Any IPv6 address can be written as eight groups of four hex digits (sometimes called hextets), where each group is separated by a colon ({{code|:}}). This, for example, is a valid IPv6 address: {{code|2001:0db8:85a3:0000:0000:8a2e:0370:7334}} or abbreviated by removing leading zeros as {{code|2001:db8:85a3::8a2e:370:7334}} (IPv4 addresses are usually written in decimal).
* Globally unique identifiers are written as thirty-two hex digits, often in unequal hyphen-separated groupings, for example {{code|3F2504E0-4F89-41D3-9A0C-0305E82C3301}}.
===Alternative symbols===
thumb|Bruce Alan Martin's hex notation proposal<ref name="Martin_1968" />thumb|Ronald O. Whitaker's hex notation proposal.<ref name="Whitaker_1972" /><ref name="Whitaker_1975" />
Notable other hexadecimal representations that use symbols other than letters "A" through "F" to represent the digits above 9 include:
* During the 1950s, some installations, such as Bendix-14, favored using the digits 0 through 5 with an overline to denote the values {{nowrap|10–15}} as {{overline|0}}, {{overline|1}}, {{overline|2}}, {{overline|3}}, {{overline|4}} and {{overline|5}}.
* The SWAC (1950)<ref name="Savard_2018_CA"/> and Bendix G-15 (1956)<ref name="Bendix"/><ref name="Savard_2018_CA"/> computers used the lowercase letters ''u'', ''v'', ''w'', ''x'', ''y'' and ''z'' for the values 10 to 15.
* The ORDVAC and ILLIAC I (1952) computers (and some derived designs, e.g. BRLESC) used the uppercase letters ''K'', ''S'', ''N'', ''J'', ''F'' and ''L'' for the values 10 to 15.<ref name="Illiac-I"/><ref name="Savard_2018_CA"/>
* The Librascope LGP-30 (1956) used the letters ''F'', ''G'', ''J'', ''K'', ''Q'' and ''W'' for the values 10 to 15.<ref name="RP_1957_LGP-30"/><ref name="Savard_2018_CA"/>
* On the PERM (1956) computer, hex 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.<ref name="PERM"/>
* The Honeywell Datamatic D-1000 (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.<ref name="Savard_2018_CA"/>
* The Monrobot XI (1960) used the letters ''S'', ''T'', ''U'', ''V'', ''W'' and ''X'' for the values 10 to 15.<ref name="Savard_2018_CA"/>
* The NEC parametron computer NEAC 1103 (1960) used the letters ''D'', ''G'', ''H'', ''J'', ''K'' (and possibly ''V'') for values 10–15.<ref name="NEC_1960_NEAC-1103">{{cite book |title=NEC Parametron Digital Computer Type NEAC-1103 |publisher=Nippon Electric Company Ltd. |location=Tokyo, Japan |id=Cat. No. 3405-C |date=1960 |url=http://archive.computerhistory.org/resources/text/NEC/NEC.1103.1958102646285.pdf |access-date=2017-05-31 |url-status=live |archive-url=https://web.archive.org/web/20170531112850/http://archive.computerhistory.org/resources/text/NEC/NEC.1103.1958102646285.pdf |archive-date=2017-05-31}}</ref>
* The Pacific Data Systems 1020 (1964) used the letters ''L'', ''C'', ''A'', ''S'', ''M'' and ''D'' for the values 10 to 15.<ref name="Savard_2018_CA"/>
* New numeric symbols and names were introduced in the Bibi-binary notation by Boby Lapointe in 1968.
* Bruce Alan Martin of Brookhaven National Laboratory 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.<ref name="Martin_1968">{{cite journal | title=Letters to the editor: On binary notation | first=Bruce Alan | last=Martin | publisher=Associated Universities Inc. | journal=Communications of the ACM | volume=11 | issue=10 | date=October 1968 | page=658 | doi=10.1145/364096.364107| s2cid=28248410 | doi-access=free }}</ref>
* In 1972, Ronald O. Whitaker of Rowco Engineering Co. proposed a triangular font that allows "direct binary reading" to "permit both input and output from computers without respect to encoding matrices".<ref name="Whitaker_1972">{{cite news |title=More on man/machine |department=Letters |author-first=Ronald O. |author-last=Whitaker |journal=Datamation |publisher=Technical Publishing Company |location=Indianapolis, Indiana, US |publication-place=Barrington, Illinois, US |date=January 1972 |volume=18 |number=1 |page=103 |url=http://www.bitsavers.org/magazines/Datamation/197201.pdf |access-date=2022-12-24 |url-status=live |archive-url=https://web.archive.org/web/20221205110246/http://www.bitsavers.org/magazines/Datamation/197201.pdf |archive-date=2022-12-05}} (1 page)</ref><ref name="Whitaker_1975">{{cite web |title=Combined display and range selector for use with digital instruments employing the binary numbering system |author-first=Ronald O. |author-last=Whitaker |id=US Patent 3974444A |location=Indianapolis, Indiana, US |date=1976-08-10 |orig-date=1975-02-24 |url=https://patentimages.storage.googleapis.com/88/54/da/d88ca78fe93623/US3974444.pdf |access-date=2022-12-24 |url-status=live |archive-url=https://web.archive.org/web/20221224135846/https://patentimages.storage.googleapis.com/88/54/da/d88ca78fe93623/US3974444.pdf |archive-date=2022-12-24}} (7 pages)</ref> * Some seven-segment display decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.<ref>{{Cite web |title=SN5446A, '47A, '48, SN54LS47, 'LS48, 'LS49, SN7446A, '47A, '48, SN74LS47, 'LS48, 'LS49 BCD-to-Seven-Segment Decoders/Drivers |publisher=Texas Instruments Incorporated |date=March 1988 |orig-date=1974 |id=SDLS111 |publication-place=Dallas, Texas, US |url-status=live |url=https://www.ti.com/lit/gpn/sn74ls47 |access-date=2021-09-15 |archive-url=https://web.archive.org/web/20211020192609/https://www.ti.com/lit/ds/symlink/sn74ls47.pdf?ts=1634757966777 |archive-date=2021-10-20}} (29 pages)</ref> {{Wide image|Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg|950|Bibi-binary}}
===Sign=== The hex system can express negative numbers the same way as in decimal, by putting a minus sign (−) before the number to indicate that it is negative.
===Bit pattern=== Hex can express the bit pattern in a processor, so a sequence of hex digits may represent a signed or even a floating-point value. This way, the negative number −42<sub>10</sub> can be written as FFFF FFD6 in a 32-bit CPU register (in two's complement), 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).
===Exponential notation=== Just as decimal numbers can be represented in exponential notation, so too can hex numbers. P notation uses the letter ''P'' (or ''p'', for "power"), whereas ''E'' (or ''e'') serves a similar purpose in decimal E notation. The number after the ''P'' is ''decimal'' and represents the ''binary'' exponent. Increasing the exponent by 1 multiplies by 2, not 16: {{mono|1=20p0 = 10p1 = 8p2 = 4p3 = 2p4 = 1p5}}. Usually, the number is normalized so that the hex digits start with {{mono|1.}} (zero is usually {{mono|0}} with no ''P'').
Example: {{mono|1.3DEp42}} represents {{math|1.3DE<sub>16</sub> × 2<sup>42<sub>10</sub></sup>}}.
P notation is required by the IEEE 754-2008 binary floating-point standard and can be used for floating-point literals in the C99 edition of the C programming language.<ref>{{cite web |url=http://www.iso.org/iso/iso_catalogue/catalogue_ics/catalogue_detail_ics.htm?csnumber=29237 |title=ISO/IEC 9899:1999 – Programming languages – C |publisher=Iso.org |website=ISO |date=2011-12-08 |access-date=2014-04-08 |archive-date=2016-10-10 |archive-url=https://web.archive.org/web/20161010112929/http://www.iso.org/iso/iso_catalogue/catalogue_ics/catalogue_detail_ics.htm?csnumber=29237 |url-status=live }}</ref> Using the ''%a'' or ''%A'' conversion specifiers, this notation can be produced by implementations of the ''printf'' family of functions following the C99 specification<ref name="Rationale_2003_C">{{cite web |title=Rationale for International Standard – Programming Languages – C |version=5.10 |date=April 2003 |pages=52, 153–154, 159 |url=http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf |website=Open Standards |access-date=2010-10-17 |url-status=live |archive-url=https://web.archive.org/web/20160606072228/http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf |archive-date=2016-06-06}}</ref> and Single Unix Specification (IEEE Std 1003.1) POSIX standard.<ref name="printf_2013">{{cite web |title=dprintf, fprintf, printf, snprintf, sprintf – print formatted output |work=The Open Group Base Specifications |edition=Issue 7, IEEE Std 1003.1, 2013 |date=2013 |orig-year=2001 |author=The IEEE and The Open Group |url=http://pubs.opengroup.org/onlinepubs/9699919799/functions/printf.html |access-date=2016-06-21 |url-status=live |archive-url=https://web.archive.org/web/20160621211105/http://pubs.opengroup.org/onlinepubs/9699919799/functions/printf.html |archive-date=2016-06-21}}</ref>
==Verbal representation== right|thumb|Hex finger-counting scheme Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. Some people read hex numbers digit by digit, like a phone number, or using the NATO phonetic alphabet, the Joint Army/Navy Phonetic Alphabet, or a similar ad hoc system. In the wake of the adoption of hex among IBM System/360 programmers, Magnuson (1968)<ref name=Magnuson-1968-01/> suggested a pronunciation guide that gave short names to the letters of hex – for instance, "A" was pronounced "ann", B "bet", C "chris", etc.<ref name=Magnuson-1968-01>{{cite magazine |last1=Magnuson |first1=Robert A. |title=A hexadecimal pronunciation guide |magazine=Datamation |date=January 1968 |volume=14 |issue=1 |page=45}}</ref> Another naming-system was published online by Rogers (2007)<ref name=Rogers-2007>{{cite web |first=S.R. |last=Rogers |year=2007 |title=Hexadecimal number words |website=Intuitor |language=en-US |url=http://www.intuitor.com/hex/words.html |access-date=2019-08-26 |archive-date=2019-09-17 |archive-url=https://web.archive.org/web/20190917015855/http://www.intuitor.com/hex/words.html |url-status=live }}</ref> 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. Yet another naming system was elaborated by Babb (2015), based on a joke in ''Silicon Valley''.<ref name=Babb-2015>{{cite web |first=Tim |last=Babb |year=2015 |url=https://www.bzarg.com/p/how-to-pronounce-hexadecimal/ |title=How to pronounce hexadecimal |website=Bzarg |language=en-US |access-date=2021-01-01 |archive-date=2020-11-11 |archive-url=https://web.archive.org/web/20201111174319/https://www.bzarg.com/p/how-to-pronounce-hexadecimal/ |url-status=live }}</ref>
Others have proposed using the verbal Morse code conventions to express four-bit hex 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 hex digit for decimal 15.
Systems of counting on digits have been devised for both binary and hex. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023<sub>10</sub> on ten fingers.<ref>{{cite book |last1=Clarke |first1=Arthur |last2=Pohl |first2=Frederik |title=The Last Theorem |url=https://archive.org/details/lasttheorem00clar |url-access=registration |date=2008 |publisher=Ballantine |isbn=978-0007289981 |page=[https://archive.org/details/lasttheorem00clar/page/91 91]}}</ref> Another system for counting up to FF<sub>16</sub> (255<sub>10</sub>) is illustrated on the right.
{| class="wikitable" style="display: inline-table; margin-right: 50px; text-align:right;" |+ Magnuson (1968)<ref name=Magnuson-1968-01/><br />naming method ! Hex !! Name !! Decimal |- | A || ann || 10 |- | B || bet || 11 |- | C || chris || 12 |- | D || dot || 13 |- | E || ernest || 14 |- | F || frost || 15 |- | 1A || annteen || 26 |- | A0 || annty || 160 |- | 5B || fifty bet || 91 |- | A,01C || annty<br>christeen || 40,988 |- | 1,AD0 || annteen<br>dotty || 6,864 |- | 3,A7D || thirty ann<br>seventy dot || 14,973 |}
{| class="wikitable" style="display: inline-table; margin-right: 50px; text-align:right;" |+ Rogers (2007)<ref name=Rogers-2007/><br /> naming method ! Hex !! Name !! Decimal |- | A || ten || 10 |- | B || eleven || 11 |- | C || twelve || 12 |- | D || draze || 13 |- | E || eptwin || 14 |- | F || fim || 15 |- | 10 || tex || 16 |- | 11 || oneteek || 17 |- | 1F || fimteek || 31 |- | 50 || fiftek || 80 |- | C0 || twelftek || 192 |- | 100 || hundrek || 256 |- | 1,000 || thousek || 4,096 |- | 3E || thirtek eptwin || 62 |- | E1 || eptek one || 225 |- | C4A || twelve hundrek <br>fourtek ten || 3,146 |- | 1,743 || one thousek <br>seven hundrek<br>fourtek three || 5,955 |}
==Conversion==
===Binary conversion=== [[File:Hewlett-Packard Model HP-16C Programmable RPN Calculator, HP's First and Only Calculator esp. for Programmers, built 1982-1989 (edited to rectangular, V2).jpg|thumb|The programmable RPN-calculator HP-16C Computer Scientist from 1982 was designed for programmers. One of its key features was the conversion between different numeral systems (note hex number in display).]] 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 hex than to decimal because each hex digit maps to a whole number of bits (4<sub>10</sub>). This example converts 1111<sub>2</sub> to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right: * 0001<sub>2</sub> = 1<sub>10</sub> * 0010<sub>2</sub> = 2<sub>10</sub> * 0100<sub>2</sub> = 4<sub>10</sub> * 1000<sub>2</sub> = 8<sub>10</sub>
Therefore:
{| |- | 1111<sub>2</sub>|| = 8<sub>10</sub> + 4<sub>10</sub> + 2<sub>10</sub> + 1<sub>10</sub> |- | || = 15<sub>10</sub> |}
With little practice, mapping 1111<sub>2</sub> to F<sub>16</sub> in one step becomes easy. The advantage of using hex 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 hex, it is trivial to regard the binary string as 4-digit groups and map each to a single hex digit.<ref name=Mano-Ciletti/>
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
{| | (1001011100)<sub>2</sub>|| = 512<sub>10</sub> + 64<sub>10</sub> + 16<sub>10</sub> + 8<sub>10</sub> + 4<sub>10</sub> |- | || = 604<sub>10</sub> |}
Compare this to the conversion to hex, where each group of four digits can be considered independently and converted directly:
{| |- | (1001011100)<sub>2</sub>|| = ||0010<sub> </sub>||0101<sub> </sub>||1100<sub>2</sub> |- | || = || align="center" |2|| align="center" |5||align="center" |C<sub>16</sub> |- | || = || colspan="5" |25C<sub>16</sub> |}
The conversion from hex to binary is equally direct.<ref name=Mano-Ciletti>{{cite book|title=Digital Design – With an Introduction to the Verilog HDL|edition=Fifth|last1=Mano|first1=M. Morris|last2=Ciletti|first2=Michael D.|publisher=Pearson Education|date=2013|pages=6, 8–10|isbn=978-0-13-277420-8}}</ref>
===Other simple conversions===
Although quaternary (base 4) is little used, it can easily be converted to and from hex or binary. Each hex digit corresponds to a pair of quaternary digits, and each quaternary digit corresponds to a pair of binary digits. In the above example 2 5 C<sub>16</sub> = 02 11 30<sub>4</sub>.
The octal (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 hex 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 for converting a representation of a number to hex 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 hex, and the series h<sub>i</sub>h<sub>i−1</sub>...h<sub>2</sub>h<sub>1</sub> be the hex digits representing the number.
# i ← 1 # h<sub>i</sub> ← d mod 16 # d ← (d − h<sub>i</sub>) / 16 # If d = 0 (return series h<sub>i</sub>) 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 implementation of the above algorithm for converting any number to a hex 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.
<syntaxhighlight lang="javascript"> function toHex(d) { var r = d % 16; if (d - r == 0) { return toChar(r); } return toHex((d - r) / 16) + toChar(r); }
function toChar(n) { const alpha = "0123456789ABCDEF"; return alpha.charAt(n); } </syntaxhighlight>
===Conversion through addition and multiplication=== [[Image:Hexadecimal multiplication table.svg|right|thumb|A hex multiplication table]] It is also possible to make the conversion by assigning each place in the source base the hex representation of its place value {{Ndash}}before carrying out multiplication and addition to get the final representation. For example, to convert the number B3AD to decimal, one can split the hex number into its digits: B (11<sub>10</sub>), 3 (3<sub>10</sub>), A (10<sub>10</sub>) and D (13<sub>10</sub>), and then get the final result by multiplying each decimal representation by 16<sup>''p''</sup> (''p'' being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:
{{math|B3AD {{=}} (11 × 16<sup>3</sup>) + (3 × 16<sup>2</sup>) + (10 × 16<sup>1</sup>) + (13 × 16<sup>0</sup>)}}
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 hex.
In Microsoft Windows, the Calculator, on its Programmer mode, allows conversions between hex and other common programming bases.
== Elementary arithmetic == Elementary operations such as division can be carried out indirectly through conversion to an alternate numeral system, 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 {{Ndash}}by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm.
==Real numbers==
=== Rational numbers === As with other numeral systems, the hex system can be used to represent rational numbers, although repeating expansions are common since sixteen (10<sub>16</sub>) 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 or dividing one by sixteen for hex, both of these fractions are written as <code>0.1</code>. Because the radix 16 is a perfect square (4<sup>2</sup>), fractions expressed in hex have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hex notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hex (and binary) less convenient than decimal for representing rational numbers since a larger proportion lies outside its range of finite representation.
All rational numbers finitely representable in hex are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hex 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 hex. For example, decimal 0.1 corresponds to the infinite recurring representation 0.1{{overline|9}} in hex. However, hex is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.0625<sub>10</sub> (one-sixteenth) is equivalent to 0.1<sub>16</sub>, 0.09<sub>12</sub>, and 0;3,45<sub>60</sub>.
{|class="wikitable" ! rowspan=2 style="vertical-align:bottom;" | n ! colspan="3" | Decimal<br />Prime factors of: base, b = 10: {{color|#920000|2}}, {{color|#920000|5}};<br />b − 1 = 9: {{color|#000092|3}};<br />b + 1 = 11: {{color|#004900|11}} ! colspan="3" | Hexadecimal<br />Prime factors of: base, b = 16{{sub|10}} = 10: {{color|#920000|2}}; b − 1 = 15{{sub|10}} = F: {{color|#000092|3, 5}}; b + 1 = 17{{sub|10}} = 11: {{color|#004900|11}} |- ! Reciprocal ! Prime factors ! Positional representation<br />(decimal) ! Positional representation<br />(hex) ! Prime factors ! Reciprocal |- | 2 | align="center" | 1/2 | align="center" | {{color|#920000|'''2'''}} | '''0.5''' | '''0.8''' | align="center" | {{color|#920000|'''2'''}} | align="center" | 1/2 |- | 3 | align="center" | 1/3 | align="center" | {{color|#000092|'''3'''}} | bgcolor=#e4e4e4 | '''0.'''3333... = '''0.'''{{overline|3}} | bgcolor="#e4e4e4" | '''0.'''5555... = '''0.'''{{overline|5}} | align="center" | {{color|#000092|'''3'''}} | align="center" | 1/3 |- | 4 | align="center" | 1/4 | align="center" | {{color|#920000|'''2'''}} | '''0.25''' | '''0.4''' | align="center" | {{color|#920000|'''2'''}} | align="center" | 1/4 |- | 5 | align="center" | 1/5 | align="center" | {{color|#920000|'''5'''}} | '''0.2''' | bgcolor="#e4e4e4" | '''0.'''{{overline|3}} | align="center" | {{color|#000092|'''5'''}} | align="center" | 1/5 |- | 6 | align="center" | 1/6 | align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}} | bgcolor=#e4e4e4 | '''0.1'''{{overline|6}} | bgcolor="#e4e4e4" | '''0.2'''{{overline|A}} | align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}} | align="center" | 1/6 |- | 7 | align="center" | 1/7 | align="center" | '''7''' | bgcolor=#e4e4e4 | '''0.'''{{overline|142857}} | bgcolor="#e4e4e4" | '''0.'''{{overline|249}} | align="center" | '''7''' | align="center" | 1/7 |- | 8 | align="center" | 1/8 | align="center" | {{color|#920000|'''2'''}} | '''0.125''' | '''0.2''' | align="center" | {{color|#920000|'''2'''}} | align="center" | 1/8 |- | 9 | align="center" | 1/9 | align="center" | {{color|#000092|'''3'''}} | bgcolor=#e4e4e4 | '''0.'''{{overline|1}} | bgcolor="#e4e4e4" | '''0.'''{{overline|1C7}} | align="center" | {{color|#000092|'''3'''}} | align="center" | 1/9 |- | 10 | align="center" | 1/10 | align="center" | {{color|#920000|'''2'''}}, {{color|#920000|'''5'''}} | '''0.1''' | bgcolor="#e4e4e4" | '''0.1'''{{overline|9}} | align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''5'''}} | align="center" | 1/A |- | 11 | align="center" | 1/11 | align="center" | {{color|#004900|'''11'''}} | bgcolor=#e4e4e4 | '''0.'''{{overline|09}} | bgcolor="#e4e4e4" | '''0.'''{{overline|1745D}} | align="center" | '''B''' | align="center" | 1/B |- | 12 | align="center" | 1/12 | align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}} | bgcolor=#e4e4e4 | '''0.08'''{{overline|3}} | bgcolor="#e4e4e4" | '''0.1'''{{overline|5}} | align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}} | align="center" | 1/C |- | 13 | align="center" | 1/13 | align="center" | '''13''' | bgcolor=#e4e4e4 | '''0.'''{{overline|076923}} | bgcolor="#e4e4e4" | '''0.'''{{overline|13B}} | align="center" | '''D''' | align="center" | 1/D |- | 14 | align="center" | 1/14 | align="center" | {{color|#920000|'''2'''}}, '''7''' | bgcolor=#e4e4e4 | '''0.0'''{{overline|714285}} | bgcolor="#e4e4e4" | '''0.1'''{{overline|249}} | align="center" | {{color|#920000|'''2'''}}, '''7''' | align="center" | 1/E |- | 15 | align="center" | 1/15 | align="center" | {{color|#000092|'''3'''}}, {{color|#920000|'''5'''}} | bgcolor=#e4e4e4 | '''0.0'''{{overline|6}} | bgcolor="#e4e4e4" | '''0.'''{{overline|1}} | align="center" | {{color|#000092|'''3'''}}, {{color|#000092|'''5'''}} | align="center" | 1/F |- | 16 | align="center" | 1/16 | align="center" | {{color|#920000|'''2'''}} | '''0.0625''' | '''0.1''' | align="center" | {{color|#920000|'''2'''}} | align="center" | 1/10 |- | 17 | align="center" | 1/17 | align="center" | '''17''' | bgcolor=#e4e4e4 | '''0.'''{{overline|0588235294117647}} | bgcolor="#e4e4e4" | '''0.'''{{overline|0F}} | align="center" | {{color|#004900|'''11'''}} | align="center" | 1/11 |- | 18 | align="center" | 1/18 | align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}} | bgcolor=#e4e4e4 | '''0.0'''{{overline|5}} | bgcolor="#e4e4e4" | '''0.0'''{{overline|E38}} | align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}} | align="center" | 1/12 |- | 19 | align="center" | 1/19 | 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===Irrational numbers=== The table below gives the expansions of some common irrational numbers in decimal and hex. {| class="wikitable" ! rowspan=2 | Number ! colspan=2 | Positional representation |- ! Decimal ! Hex |- | √2 (the length of the diagonal of a unit square) | {{val|1.414213562373095048}}... | 1.6A09E667F3BCD... |- | √3 (the length of the diagonal of a unit cube) | {{val|1.732050807568877293}}... | 1.BB67AE8584CAA... |- | √5 (the length of the diagonal of a 1×2 rectangle) | {{val|2.236067977499789696}}... | 2.3C6EF372FE95... |- | {{mvar|φ}} (phi, the golden ratio = {{math|(1+{{radical|5}})/2}}) | {{val|1.618033988749894848}}... | 1.9E3779B97F4A... |- | {{mvar|π}} (pi, the ratio of circumference to diameter of a circle) | {{val|3.141592653589793238462643}}<br />{{val|383279502884197169399375105}}... | 3.243F6A8885A308D313198A2E0<br />3707344A4093822299F31D008... |- | {{mvar|e}} (the base of the natural logarithm) | {{val|2.718281828459045235}}... | 2.B7E151628AED2A6B... |- | {{mvar|τ}} (the Thue–Morse constant) | {{val|0.412454033640107597}}... | 0.6996 9669 9669 6996... |- | {{mvar|γ}} (the limiting difference between the harmonic series and the natural logarithm) | {{val|0.577215664901532860}}... | 0.93C467E37DB0C7A4D1B... |}
===Powers=== The first 16 powers of 2 are below as hex to show relative simplicity compared to decimal representation.
{| class="wikitable" ! 2<sup>''x''</sup> !! Hex !! Decimal |- | 2<sup>0</sup> || style="text-align:right;" | 1 || style="text-align:right;" | 1 |- | 2<sup>1</sup> || style="text-align:right;" | 2 || style="text-align:right;" | 2 |- | 2<sup>2</sup> || style="text-align:right;" | 4 || style="text-align:right;" | 4 |- | 2<sup>3</sup> || style="text-align:right;" | 8 || style="text-align:right;" | 8 |- | 2<sup>4</sup> || style="text-align:right;" | 10 || style="text-align:right;" | 16 |- | 2<sup>5</sup> || style="text-align:right;" | 20 || style="text-align:right;" | 32 |- | 2<sup>6</sup> || style="text-align:right;" | 40 || style="text-align:right;" | 64 |- | 2<sup>7</sup> || style="text-align:right;" | 80 || style="text-align:right;" | 128 |- | 2<sup>8</sup> || style="text-align:right;" | 100 || style="text-align:right;" | 256 |- | 2<sup>9</sup> || style="text-align:right;" | 200 || style="text-align:right;" | 512 |- | 2{{sup|10}} || style="text-align:right;" | 400 || style="text-align:right;" | 1,024 |- | 2{{sup|11}} || style="text-align:right;" | 800 || style="text-align:right;" | 2,048 |- | 2{{sup|12}} || style="text-align:right;" | 1,000 || style="text-align:right;" | 4,096 |- | 2{{sup|13}} || style="text-align:right;" | 2,000 || style="text-align:right;" | 8,192 |- | 2{{sup|14}} || style="text-align:right;" | 4,000 || style="text-align:right;" | 16,384 |- | 2{{sup|15}} || style="text-align:right;" | 8,000 || style="text-align:right;" | 32,768 |- | 2{{sup|16}} || style="text-align:right;" | 10,000 || style="text-align:right;" | 65,536 |}
==Cultural history== The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hex calculations such as additions and subtractions.<ref>{{Cite web|url=http://totton.idirect.com/soroban/Hex_as/|title=算盤 Hexadecimal Addition & Subtraction on a Chinese Abacus|website=totton.idirect.com|access-date=2019-06-26|archive-date=2019-07-06 |archive-url=https://web.archive.org/web/20190706221609/http://totton.idirect.com/soroban/Hex_as/|url-status=live}}</ref>
As with the duodecimal system, there have been occasional attempts to promote hex as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals.<ref>{{cite web | url = http://www.hauptmech.com/base42 | title = Base 4^2 Hexadecimal Symbol Proposal | website = Hauptmech | access-date = 2008-09-04 | archive-date = 2021-10-20 | archive-url = https://web.archive.org/web/20211020192525/http://www.hauptmech.com/base42/wiki/index.php?title=Main_Page | url-status = live }}</ref> Some proposals unify standard measures so that they are multiples of 16.<ref>{{cite web|url=http://www.intuitor.com/hex/|title=Intuitor Hex Headquarters|website=Intuitor|access-date=28 October 2018|archive-date=2010-09-04 |archive-url=https://web.archive.org/web/20100904144850/http://www.intuitor.com/hex/|url-status=live}}</ref><ref>{{cite web|url=https://www.unicode.org/wg2/docs/n2677.pdf |title=A proposal for addition of the six Hexadecimal digits (A-F) to Unicode |last=Niemietz|first=Ricardo Cancho|date=21 October 2003 |publisher=ISO/IEC JTC1/SC2/WG2 |access-date=2024-06-25}}</ref> 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.<ref name="nystrom">{{cite book|url=https://archive.org/details/bub_gb_aNYGAAAAYAAJ|title=Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base|last=Nystrom|first=John William|publisher=Lippincott|year=1862|location=Philadelphia}}</ref> Nystrom among other things suggested hexadecimal time, which subdivides a day by 16, so that there are 16 "hours" (or "10 ''tims''", pronounced ''tontim'') in a day.<ref>Nystrom (1862), p. 33: "In expressing time, angle of a circle, or points on the compass, the unit ''tim'' should be noted as integer, and parts thereof as ''tonal fractions'', as 5·86 ''tims'' is five times and ''metonby'' [*"sutim and metonby" John Nystrom accidentally gives part of the number in decimal names; in Nystrom's pronunciation scheme, 5=su, 8=me, 6=by, c.f. [http://www.unifoundry.com/tonal/index.html unifoundry.com] {{Webarchive|url=https://web.archive.org/web/20210519080658/http://www.unifoundry.com/tonal/index.html |date=2021-05-19 }} ]."</ref>
{{anchor|Etymology}}{{wiktionary|hexadecimal}} The word ''hexadecimal'' is first recorded in 1952.<ref>C. E. Fröberg, ''Hexadecimal Conversion Tables'', Lund (1952).</ref> It is macaronic in the sense that it combines Greek ἕξ (hex) "six" with Latinate ''-decimal''. The all-Latin alternative ''sexadecimal'' (compare the word ''sexagesimal'' for base 60) is older, and sees at least occasional use from the late 19th century.<ref> ''The Century Dictionary'' of 1895 has ''sexadecimal'' in the more general sense of "relating to sixteen". An early explicit use of ''sexadecimal'' in the sense of "using base 16" is found also in 1895, in the ''Journal of the American Geographical Society of New York'', vols. 27–28, p. 197.</ref> 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''.<ref>{{cite book|first=Steven|last=Schwartzman|title=The Words of Mathematics: An etymological dictionary of mathematical terms used in English|date=1994|publisher=The Mathematical Association of America|page=105|isbn=0-88385-511-9}} s.v. hexadecimal</ref> 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. In 1969, Donald Knuth 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'', ''quaternary'', etc. According to Knuth's argument, the correct terms for ''decimal'' and ''octal'' arithmetic would be ''denary'' and ''octonary'', respectively.<ref>Knuth, Donald. (1969). ''The Art of Computer Programming, Volume 2''. {{isbn|0-201-03802-1}}. (Chapter 17.)</ref> 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".<ref>Alfred B. Taylor, [https://archive.org/details/reportonweights00taylgoog Report on Weights and Measures], Pharmaceutical Association, 8th Annual Session, Boston, 15 September 1859. See pages and 33 and 41.</ref><ref>Alfred B. Taylor, "Octonary numeration and its application to a system of weights and measures", [https://books.google.com/books?id=KsAUAAAAYAAJ&pg=PA296 ''Proc Amer. Phil. Soc.'' Vol XXIV] {{Webarchive|url=https://web.archive.org/web/20160624070056/https://books.google.com/books?id=KsAUAAAAYAAJ&pg=PA296 |date=2016-06-24 }}, Philadelphia, 1887; pages 296–366. See pages 317 and 322.</ref>
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, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants.<ref>[http://www.bitsavers.org/pdf/ibm/360/fortran/C28-6515-6_FORTRAN_IV_Language_1966.pdf IBM System/360 FORTRAN IV Language] {{Webarchive|url=https://web.archive.org/web/20210519073220/http://www.bitsavers.org/pdf/ibm/360/fortran/C28-6515-6_FORTRAN_IV_Language_1966.pdf#page=13|date=2021-05-19}} (1966), p. 13.</ref> As noted above, alternative notations were used by NEC (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 {{blockquote|With the ridiculous choice of letters A, B, C, D, E, F as hexadecimal number symbols adding to already troublesome problems of distinguishing octal (or hex) numbers from decimal numbers (or variable names), the time is overripe for reconsideration of our number symbols. This should have been done before poor choices gelled into a de facto standard!}} 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".<ref name="Martin_1968"/> 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, and later in a Hindu–Arabic numeral system), and that the recent ASCII 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. <!--The etymologically proper Greek term would be ''hexadecadic'' / ''ἑξαδεκαδικός'' / ''hexadekadikós'' {{cn}} (although in Modern Greek, ''decahexadic'' / ''δεκαεξαδικός'' / ''dekaexadikos'' is more commonly used). --> <!--please, do not add anything on "drome numbers", nobody outside MathWorld refers to them and it may be a deliberate fake entry-->
==Base16== '''Base16''' is a binary to text encoding in the family that also contains Base32, Base58, and Base64. Data is broken into 4-bit sequences, and each value (0-15) is encoded as a character. Although any 16 characters could be used, in practice, the ASCII digits "0"–"9" and letters "A"–"F" (or "a"–"f") are used to align with the typical notation for hex numbers.
Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the W3C 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.
Advantages of Base16 encoding include: * Most programming languages have facilities to parse ASCII-encoded hex * Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64, respectively * The notation is well-known and easily understood without needing a symbol lookup table * Many CPU architectures have dedicated instructions that allow access to a half-byte (aka nibble), making it more efficient in hardware than Base32 and Base64
Disadvantages include: * 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 * Complexity of accepting both upper and lower case letters
==See also== * {{Annotated link |Bailey–Borwein–Plouffe formula}} * {{Annotated link |Hex dump}} * {{Annotated link |Hex editor}} * {{Annotated link |Hexadecimal time}} * {{Annotated link |Hexspeak}} * {{Annotated link |IBM hexadecimal floating-point}} * {{Annotated link |P notation}} <!-- Please do NOT add links for particular hex encoders or decoders. There are plenty of examples already. See WP:EL (http://en.wikipedia.org/wiki/Wikipedia:EL) for an explanation of why external links are routinely vigorously eliminated from articles.-->
==References== <references> <ref name="Savard_2018_CA">{{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}}</ref> <ref name="Bendix">{{cite book |title=G15D Programmer's Reference Manual |chapter=2.1.3 Sexadecimal notation |publisher=Bendix Computer, Division of Bendix Aviation Corporation |location=Los Angeles, CA, US |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.}}</ref> <ref name="Illiac-I">{{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 |location=Urbana, Illinois, US |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}}</ref> <ref name="RP_1957_LGP-30">{{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's 6-bit character code.)</ref> <ref name="PERM">{{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 |archive-date=2018-10-03 |archive-url=https://web.archive.org/web/20181003172841/http://www.manthey.cc/sites/seminars/src/History.pdf |url-status=live }}</ref> </references>
<!-- ==External links== If you're here, you're probably thinking about adding an external link to an online calculator or some such. Some points to keep in mind (from the policy at WP:EL, http://en.wikipedia.org/w/index.php?title=Wikipedia:External_links): * The "External links" section should be kept to a minimum. A lack of external links ... is not a reason to add them. * Links to be avoided: ** Any site that does not provide a unique resource beyond what the article might contain... ** Links mainly intended to promote a website ** Sites that are only indirectly related to the article's subject
Since the article is about hexadecimal representation and mentions standard tools for conversion only as a minor example, there is little any external link to an online calculator or converter could possibly add to the reader's knowledge. + + But a online calculator and or converter does "provide a unique resource beyond what the article contains" What's the harm in linking to a good page rather than making the user search for one and finding a the poor one? ~~~~ -->
Category:Binary arithmetic Category:Hexadecimal numeral system Category:Power-of-two numeral systems Category:Positional numeral systems