View source for Hexadecimal

{{Short description|Base-16 numerical system}}
{{Redirect|Sexadecimal|base 60|Sexagesimal}}
{{Redirect|Hex digit|binary-coded hexadecimals|Nibble}}
{{Redirect|Hex format|hexadecimal file formats|Hex file (disambiguation)}}
{{Use dmy dates|date=March 2020|cs1-dates=y}}
{{Numeral systems}}
'''Hexadecimal''' (also known as '''base-16''' or simply '''hex''') is a [[Numeral system#Positional systems in detail|positional numeral system]] that represents numbers using a [[radix]] (base) of sixteen. Unlike the [[decimal]] system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen.
Software developers and system designers widely use hexadecimal numbers because they provide a convenient representation of [[binary code|binary-coded]] values. Each hexadecimal digit represents four [[bit]]s (binary digits), also known as a [[nibble]] (or nybble).<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> For example, a 6-bit [[byte]] can have values ranging from {{mono|000000}} to {{mono|111111}} (0 to 63 decimal) in binary form, which can be written as {{mono|00}} to {{mono|3F}} in hexadecimal.
In mathematics, a subscript is typically used to specify the base. For example, the decimal value {{val|711|fmt=commas}} would be expressed in hexadecimal as {{hexadecimal|711}}. In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix <code>0x</code> is used in [[C (programming language)|C]], which would denote this value as <code>0x{{hexadecimal|711|no}}</code>.
Hexadecimal is used in the transfer encoding '''Base 16''', in which each byte of the [[plaintext|plain text]] 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 [[Arabic numerals|numerals]] 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. Some [[seven-segment display]]s use mixed-case 'A b C d E F' to distinguish the digits A–F from one another and from 0–9.
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]], 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 0a
</syntaxhighlight></div>
====Distinguishing from decimal====
In contexts where the [[radix|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<sub>10</sub> is decimal 159; 159<sub>16</sub> is hexadecimal 159, which equals 345<sub>10</sub>. Some authors 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>.
[[Donald Knuth]] introduced the use of a particular typeface to represent a particular radix in his book ''The TeXbook''.<ref>{{Cite book |last=Knuth |first=Donald Ervin |url=https://www.worldcat.org/oclc/12973034 |title=The TeXbook |date=1986 |others=Duane Bibby |isbn=0-201-13447-0 |location=Reading, Mass. |oclc=12973034 |access-date=2022-03-15  |archive-date=2022-01-16  |archive-url=https://web.archive.org/web/20220116012119/http://www.worldcat.org/oclc/12973034 |url-status=live }}</ref> Hexadecimal representations are written there in a [[Monospaced font|typewriter typeface]]: {{mono|5A3}}, {{mono|C1F27ED}}
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:<!--
* * * These are ordered from most to least likely to be encountered by lay people
* * * -->
* {{anchor|_nix}}Although best known from the [[C (programming language)|C programming language]] (and the many languages influenced by C), the prefix <code>0x</code> to indicate a hex constant may have had origins in the [[IBM Stretch]] systems. It is derived from the <code>0</code> prefix already in use for [[octal]] constants. Byte values can be expressed in hexadecimal with the prefix <code>\x</code> followed by two hex digits: <code>'\x1B'</code> represents the [[Escape character|Esc]] control character; <code>"\x1B[0m\x1B[25;1H"</code> is a string containing 11 characters with two embedded Esc characters.<ref>The string <code>"\x1B[0m\x1B[25;1H"</code> specifies the character sequence {{mono|Esc [ 0 m Esc [ 2 5; 1 H}}. These are the escape sequences used on an [[ANSI escape code|ANSI terminal]] that reset the character set and color, and then move the cursor to line 25.</ref> To output an integer as hexadecimal with the [[printf]] function family, the format conversion code <code>%X</code> or <code>%x</code> is used.
* In [[XML]] and [[XHTML]], characters can be expressed as hexadecimal [[numeric character reference]]s using the notation <code>&amp;#x''code'';</code>, for instance <code>&amp;#x0054;</code> represents the character U+0054 (the uppercase letter "T"). If there is no {{code|x}} the number is decimal (thus <code>&amp;#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 language]]s 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> hexadecimal 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 hexadecimal 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 ([[MOS Technology 6502|6502]], [[Motorola]]), [[Pascal (programming language)|Pascal]], [[Delphi (programming language)|Delphi]], some versions of [[BASIC]] ([[Commodore BASIC|Commodore]]), [[GameMaker Studio|GameMaker Language]], [[Godot (game engine)|Godot]] and [[Forth (programming language)|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 language features|Fortran 95]] uses Z'ABCD'.
* [[Ada (programming language)|Ada]] and [[VHDL]] enclose hexadecimal 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 hexadecimal constants in the form <code>8'hFF</code>, where 8 is the number of bits in the value and FF is the hexadecimal constant.
* The [[Icon (programming language)|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 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]],<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 hexadecimal numbers with <code>&amp;H</code>: <code>&amp;H5A3</code>
* [[BBC BASIC]] and [[Locomotive BASIC]] use <code>&amp;</code> for hex.<ref>BBC BASIC programs are not fully portable to [[Microsoft BASIC]] (without modification) since the latter takes <code>&amp;</code> to prefix [[octal]] values. (Microsoft BASIC primarily uses <code>&amp;O</code> to prefix octal, and it uses <code>&amp;H</code> to prefix hexadecimal, 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 hexadecimal numbers: <code>16r5a3</code>, <code>16rC1F27ED</code>. Binary, quaternary (base-4), and octal numbers can be specified similarly.
* The most common format for hexadecimal on IBM mainframes ([[zSeries]]) and midrange computers ([[IBM i]]) running the traditional OS's ([[z/OS|zOS]], [[VSE (operating system)|zVSE]], [[z/VM|zVM]], [[Transaction Processing Facility|TPF]], [[IBM i]]) is <code>X'5A3'</code> or <code>X'C1F27ED'</code>, and is used in Assembler, [[PL/I]], [[COBOL]], [[Job Control Language|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.
====Syntax that is always Hex====
Sometimes the numbers are known to be Hex.
* In [[URI]]s (including [[URL]]s), [[character encoding|character codes]] are written as hexadecimal pairs prefixed with {{code|%}}: {{code|<nowiki>http://www.example.com/name%20with%20spaces</nowiki>}} where {{code|%20}} is the code for the [[Space (punctuation)#Space characters and digital typography|space (blank)]] character, [[ASCII]] code point 20 in hex, 32 in decimal.
* 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]] (¡).
* [[Web colors|Color references]] in HTML, [[Cascading Style Sheets|CSS]] and [[X window system|X&nbsp;Window]] can be expressed with six hexadecimal 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}} (a golden orange: {{color box|#FA3}}).
* In [[MIME]] (e-mail extensions) [[quoted-printable]] encoding, character codes are written as hexadecimal 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&nbsp;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 [[pixel]]s) can be expressed as unprefixed consecutive hexadecimal pairs: {{code|AA213FD51B3801043FBC}}&nbsp;...
* Any [[IPv6 address]] can be written as eight groups of four hexadecimal digits (sometimes called [[hextet (computing)|hextet]]s), 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 address]]es are usually written in decimal).
* [[Globally unique identifier]]s are written as thirty-two hexadecimal digits, often in unequal hyphen-separated groupings, for example {{code|3F2504E0-4F89-41D3-9A0C-0305E82C3301}}.
===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]] to denote the values {{nowrap|10–15}} as {{overline|0}}, {{overline|1}}, {{overline|2}}, {{overline|3}}, {{overline|4}} and {{overline|5}}.
* The [[SWAC (computer)|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 (computer)|PERM]] (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.<ref name="PERM"/>
* The [[Honeywell]] [[Datamatic D-1000]] (1957) used the lowercase letters ''b'', ''c'', ''d'', ''e'', ''f'', and ''g'' whereas the [[Elbit]]&nbsp;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]]&nbsp;1020 (1964) used the letters ''L'', ''C'', ''A'', ''S'', ''M'' and ''D'' for the values 10 to 15.<ref name="Savard_2018_CA"/>
[[File:Table_de_correspondance_entre_le_Bibinaire_et_les_autres_notations.svg|thumb|[[Bibi-binary]]]]
* New numeric symbols and names were introduced in the [[Bibi-binary]] notation by [[Boby Lapointe]] in 1968.
[[Image:Bruce Martin hexadecimal notation proposal.png|thumb|Bruce Alan Martin's hexadecimal notation proposal<ref name="Martin_1968"/>]]
* Bruce Alan Martin of [[Brookhaven National Laboratory]] considered the choice of A–F "ridiculous". In a 1968 letter to the editor of the [[Communications of the ACM|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>
[[File:Base-16 digits.svg|thumb|Ronald O. Whitaker's hexadecimal notation proposal.<ref name="Whitaker_1972"/><ref name="Whitaker_1975"/>]]
* 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>
===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 [[ICAO spelling alphabet|NATO phonetic alphabet]], the [[Joint Army/Navy Phonetic Alphabet]], or a similar ''ad-hoc'' system. In the wake of the adoption of hexadecimal among [[IBM System/360]] programmers, Magnuson (1968)<ref name=Magnuson-1968-01/> suggested a pronunciation guide that gave short names to the letters of hexadecimal – 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 (TV series)|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 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.
[[File:Hexadecimal-counting.jpg|right|thumb|Hexadecimal finger-counting scheme]]
Systems of counting on [[Digit (anatomy)|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 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
! Number !! Pronunciation
|-
| A || ann
|-
| B || bet
|-
| C || chris
|-
| D || dot
|-
| E || ernest
|-
| F || frost
|-
| 1A || annteen
|-
| A0 || annty
|-
| 5B || fifty-bet
|-
| A01C || annty christeen
|-
| 1AD0 || annteen dotty
|-
| 3A7D || thirty-ann seventy-dot
|}
{| class="wikitable" style="display: inline-table; margin-right: 50px;; text-align:right;"
|+ Rogers (2007)<ref name=Rogers-2007/><br /> naming method
! Number !! Pronunciation
|-
| A    || ten
|-
| B    || eleven
|-
| C    || twelve
|-
| D    || draze
|-
| E    || eptwin
|-
| F    || fim
|-
| 10   || tex
|-
| 11   || oneteek
|-
| 1F   || fimteek
|-
| 50   || fiftek
|-
| C0   || twelftek
|-
| 100  || hundrek
|-
| 1000 || thousek
|-
| 3E   || thirtek-eptwin
|-
| E1   || eptek-one
|-
| C4A  || twelve-hundrek-fourtek-ten
|-
| 1743 || one-thousek-seven-<br />-hundrek-fourtek-three
|}
===Signs===
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42<sub>10</sub>, −B01D9 to represent −721369<sub>10</sub> and so on.
Hexadecimal can also be used to express the exact bit patterns used in the [[central processing unit|processor]], so a sequence of hexadecimal digits may represent a [[signedness|signed]] or even a [[floating-point arithmetic|floating-point]] value. This way, the negative number −42<sub>10</sub> can be written as FFFF&nbsp;FFD6 in a 32-bit [[Processor register|CPU register]] (in [[two's complement]]), as C228&nbsp;0000 in a 32-bit [[Floating-point unit|FPU]] register or C045&nbsp;0000&nbsp;0000&nbsp;0000 in a 64-bit FPU register (in the [[IEEE floating-point standard]]).
===Hexadecimal exponential notation===
Just as decimal numbers can be represented in [[exponential notation]], so too can hexadecimal 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 hexadecimal 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)|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|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>
==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 [[Reverse Polish notation|RPN]]-calculator [[HP-16C|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 hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (4<sub>10</sub>).
This example converts 1111<sub>2</sub> to base ten. Since each [[Positional notation|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>
|-
| &nbsp;|| = 15<sub>10</sub>
|}
With little practice, mapping 1111<sub>2</sub> to F<sub>16</sub> in one step becomes easy (see table in [[#Written representation|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.<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>
|-
| &nbsp;|| = 604<sub>10</sub>
|}
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently and converted directly:
{|
|-
| (1001011100)<sub>2</sub>|| = ||0010<sub>&nbsp;</sub>||0101<sub>&nbsp;</sub>||1100<sub>2</sub>
|-
| &nbsp;|| = || align="center" |2|| align="center" |5||align="center" |C<sub>16</sub>
|-
| &nbsp;|| = || colspan="5" |25C<sub>16</sub>
|}
The conversion from hexadecimal 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 numeral system|quaternary]] (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 2&nbsp;5&nbsp;C<sub>16</sub> = 02&nbsp;11&nbsp;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 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]] 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 h<sub>i</sub>h<sub>i−1</sub>...h<sub>2</sub>h<sub>1</sub> be the hexadecimal 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 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]].
<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 hexadecimal [[multiplication table]]]]
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 (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 hexadecimal.
In [[Microsoft Windows]], the [[Calculator (Windows)|Calculator]] utility can be set to Programmer mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 ([[octal]]), and 2 ([[Binary numeral system|binary]]), the bases most commonly used by programmers. In Programmer Mode, the on-screen [[numeric keypad]] 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 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 — 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 hexadecimal system can be used to represent [[rational number]]s, although [[repeating decimal|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 numeral system|binary]] or dividing one by sixteen for hexadecimal, both of these fractions are written as <code>0.1</code>. Because the radix 16 is a [[square number|perfect square]] (4<sup>2</sup>), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no [[cyclic number]]s (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 hexadecimal 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 hexadecimal (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 hexadecimal are also finitely representable in decimal, [[duodecimal]] and [[sexagesimal]]: 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{{overline|9}} in hexadecimal. However, hexadecimal 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 />(hexadecimal)
! 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
| align="center" | '''19'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|052631578947368421}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|0D79435E5}}
| align="center" | '''13'''
| align="center" | 1/13
|-
| 20
| align="center" | 1/20
| align="center" | {{color|#920000|'''2'''}}, {{color|#920000|'''5'''}}
| '''0.05'''
| bgcolor="#e4e4e4" | '''0.0'''{{overline|C}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''5'''}}
| align="center" | 1/14
|-
| 21
| align="center" | 1/21
| align="center" | {{color|#000092|'''3'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|047619}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|0C3}}
| align="center" | {{color|#000092|'''3'''}}, '''7'''
| align="center" | 1/15
|-
| 22
| align="center" | 1/22
| align="center" | {{color|#920000|'''2'''}}, {{color|#004900|'''11'''}}
| bgcolor=#e4e4e4 | '''0.0'''{{overline|45}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|BA2E8}}
| align="center" | {{color|#920000|'''2'''}}, '''B'''
| align="center" | 1/16
|-
| 23
| align="center" | 1/23
| align="center" | '''23'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|0434782608695652173913}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|0B21642C859}}
| align="center" | '''17'''
| align="center" | 1/17
|-
| 24
| align="center" | 1/24
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.041'''{{overline|6}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|A}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/18
|-
| 25
| align="center" | 1/25
| align="center" | {{color|#920000|'''5'''}}
| '''0.04'''
| bgcolor="#e4e4e4" | '''0.'''{{overline|0A3D7}}
| align="center" | {{color|#000092|'''5'''}}
| align="center" | 1/19
|-
| 26
| align="center" | 1/26
| align="center" | {{color|#920000|'''2'''}}, '''13'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|384615}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|9D8}}
| align="center" | {{color|#920000|'''2'''}}, '''D'''
| align="center" | 1/1A
|-
| 27
| align="center" | 1/27
| align="center" | {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.'''{{overline|037}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|097B425ED}}
| align="center" | {{color|#000092|'''3'''}}
| align="center" | 1/1B
|-
| 28
| align="center" | 1/28
| align="center" | {{color|#920000|'''2'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.03'''{{overline|571428}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|924}}
| align="center" | {{color|#920000|'''2'''}}, '''7'''
| align="center" | 1/1C
|-
| 29
| align="center" | 1/29
| align="center" | '''29'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|0344827586206896551724137931}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|08D3DCB}}
| align="center" | '''1D'''
| align="center" | 1/1D
|-
| 30
| align="center" | 1/30
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}, {{color|#920000|'''5'''}}
| bgcolor=#e4e4e4 | '''0.0'''{{overline|3}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|8}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}, {{color|#000092|'''5'''}}
| align="center" | 1/1E
|-
| 31
| align="center" | 1/31
| align="center" | '''31'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|032258064516129}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|08421}}
| align="center" | '''1F'''
| align="center" | 1/1F
|-
| 32
| align="center" | 1/32
| align="center" | {{color|#920000|'''2'''}}
| '''0.03125'''
| '''0.08'''
| align="center" | {{color|#920000|'''2'''}}
| align="center" | 1/20
|-
| 33
| align="center" | 1/33
| align="center" | {{color|#000092|'''3'''}}, {{color|#004900|'''11'''}}
| bgcolor=#e4e4e4 | '''0.'''{{overline|03}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|07C1F}}
| align="center" | {{color|#000092|'''3'''}}, '''B'''
| align="center" | 1/21
|-
| 34
| align="center" | 1/34
| align="center" | {{color|#920000|'''2'''}}, '''17'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|2941176470588235}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|78}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#004900|'''11'''}}
| align="center" | 1/22
|-
| 35
| align="center" | 1/35
| align="center" | {{color|#920000|'''5'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|285714}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|075}}
| align="center" | {{color|#000092|'''5'''}}, '''7'''
| align="center" | 1/23
|-
| 36
| align="center" | 1/36
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.02'''{{overline|7}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|71C}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/24
|-
| 37
| align="center" | 1/37
| align="center" | '''37'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|027}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|06EB3E453}}
| align="center" | '''25'''
| align="center" | 1/25
|}
===Irrational numbers===
The table below gives the expansions of some common [[irrational number]]s in decimal and hexadecimal.
{| class="wikitable"
! rowspan=2 | Number
! colspan=2 | Positional representation
|-
! Decimal
! Hexadecimal
|-
| [[Square root of 2|{{sqrt|2}}]] (the length of the [[diagonal]] of a unit [[Square (geometry)|square]])
| {{val|1.414213562373095048}}...
| 1.6A09E667F3BCD...
|-
| [[Square root of 3|{{sqrt|3}}]] (the length of the diagonal of a unit [[cube]])
| {{val|1.732050807568877293}}...
| 1.BB67AE8584CAA...
|-
| [[Square root of 5|{{sqrt|5}}]] (the length of the [[diagonal]] of a 1×2 [[rectangle]])
| {{val|2.236067977499789696}}...
| 2.3C6EF372FE95...
|-
| {{mvar|[[Golden ratio|φ]]}} (phi, the [[golden ratio]] = {{math|(1+{{radical|5}})/2}})
| {{val|1.618033988749894848}}...
| 1.9E3779B97F4A...
|-
| {{mvar|[[Pi|π]]}} (pi, the ratio of [[circumference]] to [[diameter]] of a circle)
| {{val|3.141592653589793238462643}}<br />{{val|383279502884197169399375105}}...
| 3.243F6A8885A308D313198A2E0<br />3707344A4093822299F31D008...
|-
| {{mvar|[[E (mathematical constant)|e]]}} (the base of the [[natural logarithm]])
| {{val|2.718281828459045235}}...
| 2.B7E151628AED2A6B...
|-
| {{mvar|[[Thue–Morse constant|τ]]}} (the [[Thue–Morse constant]])
| {{val|0.412454033640107597}}...
| 0.6996 9669 9669 6996...
|-
| {{mvar|[[Euler-Mascheroni constant|γ]]}} (the limiting difference between the [[harmonic series (mathematics)|harmonic series]] and the natural logarithm)
| {{val|0.577215664901532860}}...
| 0.93C467E37DB0C7A4D1B...
|}
===Powers===
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
{| class="wikitable"
! 2<sup>''x''</sup> !! Value !! Value (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<sub>hex</sub> || style="text-align:right;" | 16<sub>dec</sub>
|-
| 2<sup>5</sup> || style="text-align:right;" | 20<sub>hex</sub> || style="text-align:right;" | 32<sub>dec</sub>
|-
| 2<sup>6</sup> || style="text-align:right;" | 40<sub>hex</sub> || style="text-align:right;" | 64<sub>dec</sub>
|-
| 2<sup>7</sup> || style="text-align:right;" | 80<sub>hex</sub> || style="text-align:right;" | 128<sub>dec</sub>
|-
| 2<sup>8</sup> || style="text-align:right;" | 100<sub>hex</sub> || style="text-align:right;" | 256<sub>dec</sub>
|-
| 2<sup>9</sup> || style="text-align:right;" | 200<sub>hex</sub> || style="text-align:right;" | 512<sub>dec</sub>
|-
| 2<sup>A</sup> (2{{sup|10{{sub|dec}}}}) || style="text-align:right;" | 400<sub>hex</sub> || style="text-align:right;" | 1024<sub>dec</sub>
|-
| 2<sup>B</sup> (2{{sup|11{{sub|dec}}}}) || style="text-align:right;" | 800<sub>hex</sub> || style="text-align:right;" | 2048<sub>dec</sub>
|-
| 2<sup>C</sup> (2{{sup|12{{sub|dec}}}}) || style="text-align:right;" | 1000<sub>hex</sub> || style="text-align:right;" | 4096<sub>dec</sub>
|-
| 2<sup>D</sup> (2{{sup|13{{sub|dec}}}}) || style="text-align:right;" | 2000<sub>hex</sub> || style="text-align:right;" | 8192<sub>dec</sub>
|-
| 2<sup>E</sup> (2{{sup|14{{sub|dec}}}}) || style="text-align:right;" | 4000<sub>hex</sub> || style="text-align:right;" | 16,384<sub>dec</sub>
|-
| 2<sup>F</sup> (2{{sup|15{{sub|dec}}}}) || style="text-align:right;" | 8000<sub>hex</sub> || style="text-align:right;" | 32,768<sub>dec</sub>
|-
| 2<sup>10</sup> (2{{sup|16{{sub|dec}}}}) || style="text-align:right;" | 10000<sub>hex</sub> || style="text-align:right;" | 65,536<sub>dec</sub>
|}
==Cultural history==
The traditional [[Chinese units of measurement]] were base-16. For example, one jīn&nbsp;(斤) in the old system equals sixteen [[tael]]s. The [[suanpan]] (Chinese [[abacus]]) can be used to perform hexadecimal 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 hexadecimal 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 language|Greek]] ἕξ (hex)  "six" with [[Latin]]ate ''-decimal''.
The  all-Latin alternative ''[[:wikt:sexadecimal|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 Corporation|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 |date=2021-05-19  }} (1966), p. 13.</ref>
As noted above, alternative notations were used by [[NEC]]  (1960)  and The Pacific Data Systems&nbsp;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 [[Communications of the ACM|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 [[Etymology|etymologically]] proper [[Greek language|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 (transfer encoding)==
'''Base16''' (as a proper name without a space) can also refer to a [[binary to text encoding]] belonging to the same family as [[Base32]], [[Base58]], and [[Base64]].
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]] 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]]"), 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 [[World Wide Web Consortium|W3C]] standard for [[Percent-encoding|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]], [[Base64]] (content encoding schemes)
* [[Hexadecimal time]]
* [[IBM hexadecimal floating-point]]
* [[Hex editor]]
* [[Hex dump]]
* [[Bailey–Borwein–Plouffe formula]] (BBP)
* [[Hexspeak]]
* [[P notation]]
<!-- Please do NOT add any 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==
{{reflist|refs=
<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>
}}
<!--
==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]]