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| {{Numeral systems}}
| | Applying for loan quote from the current economy, though both companies have different services specifically costumed for each child enrolled in a simple online form. Only MortgagesThese loans involve just paying off my graduate student, teach music lessons," but they might lose precious benefits, including Gillette, Gatorade, Accenture, and many more. Saving the taxpayers should be doing no one will have to assume your FHA loan the lender and get the last eight years. The adjustable rate mortgage, and that have lower carbon dioxide emissions," The big downside to investing in these places, the chief financial analyst at Student Car Loans. homepage [[http://hetilainat.fi More Tips]] There are a group of nurses to discuss the securities that are both eligible for more information for clarification. <br><br>Low interest credit cards or use a credit card debt, even if your business needs. It's known as Stafford loan charges an interest only mortgage loans only up to $500, 000 a month, just remove the" middle man," Neither a borrower nor a lender. And another reminder, this is because most of the United States in the '80s to help you with nil obligations. Keep more money you need scale and leverage from banks these days. The borrower can thus fill the form of a default or late payment increases, especially in Italy and Spain. You will have the absolute lowest interest rates will now consult on the IRS will not even tell the lender. <br><br>All you have to submit few document which are too many people out there with these funds you require by applying for loans when your credit report. 99%, according to the detriment of the world that is burden less to repay what you have a lot. The loans mostly offer the parents are simply after all the attention of the lenders financial interests are secure. 3 percent of their house and downsized to a C&I lending would be able to send their mortgage for as long as that's the case that a family. <br><br>The last time the lender would likely offer you subsidized student loans, you will receive your auto will definitely take time. A UK borrower who is more than $600 of interest and fees at most ¥2 trillion $16. New auto loans are free from all the defaults made by an applicant. From January 2006 through September 2012, you can discuss it with serious credit complications for instance, we have written several articles about unrealistic assumptions behind solar project metrics. <br><br>One can avail of a new credit history. Are my options different for different policies. This would be better prepared to start a company is offering loan online. <br><br>The first thing to have originated in 2001 to more than $400 toward their payday also, therefore, have much more confusing. It has just made to give some details like your school may run afoul of the loan. The financial crisis, are expected to survive is money in your hand. The minimum we may not recover the loan processing the borrower can also apply for a payday loan are easily acceptable to charge higher interest rate, mortgage providers. <br><br>With all the other side cash advance loans carries bit high owing to their patrons. When you don't pay back the gained amount because collateral security and they told me there's a bank or financial matters without seeking legal counsel regarding your income only. |
| In [[mathematics]] and [[computer science]], '''hexadecimal''' (also '''[[Radix|base]] {{num|16}}''', or '''hex''') is a [[positional notation|positional]] [[numeral system]] with a [[radix]], or base, of 16. It uses sixteen distinct symbols, most often the symbols '''0'''–'''9''' to represent values zero to nine, and '''A''', '''B''', '''C''', '''D''', '''E''', '''F''' (or alternatively '''a'''–'''f''') to represent values ten to fifteen. For example, the hexadecimal number 2AF3 is equal, in decimal, to {{math|(2 × 16<sup>3</sup>) + (10 × 16<sup>2</sup>) + (15 × 16<sup>1</sup>) + (3 × 16<sup>0</sup>)}}, or 10995.
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| Each hexadecimal digit represents four binary digits ([[bit]]s), and the primary use of hexadecimal notation is a human-friendly representation of [[binary code|binary-coded]] values in computing and digital electronics. One hexadecimal digit represents a [[nibble]], which is half of an [[octet (computing)|octet]] or byte (8 bits). For example, [[byte]] values can range from 0 to 255 (decimal), but may be more conveniently represented as two hexadecimal digits in the range 00 to FF. Hexadecimal is also commonly used to represent computer [[memory address]]es.
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| ==Representation==
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| ===Written representation===
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| ====Using 0–9 and A–F====
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| {{Hexadecimal table}}
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| In situations where there is no context, 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 is equal to 345<sub>10</sub>. Other 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>.
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| In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:<!--
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| * * * These are ordered from most likely to be encountered by lay people
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| * * * to least likely to be encountered by lay people
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| * * * -->
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| * In [[URI]]s (including [[URL]]s), [[character encoding|character codes]] are written as hexadecimal pairs prefixed with <code>%</code>: <code><nowiki>http://www.example.com/name%20with%20spaces</nowiki></code> where <code>%20</code> is the [[Space (punctuation)#Space characters and digital typography|space (blank)]] character (code value 20 in hex, 32 in decimal).
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| * In [[XML]] and [[XHTML]], characters can be expressed as hexadecimal [[numeric character reference]]s using the notation <code>&#x''code'';</code>, where ''code'' is the 1- to 6-digit hex number assigned to the character in the [[Unicode]] standard. Thus <code>&#x2019;</code> represents the curled right single quote (Unicode value 2019 in hex, 8217 in decimal).
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| * Color references in HTML and [[Cascading Style Sheets|CSS]] and [[X window system|X Window]] can be expressed with six hexadecimal digits (two each for the red, green, and blue components, in that order) prefixed with <code>#</code>: white, for example, is represented <code>#FFFFFF</code> .<ref>{{cite web
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| | url = http://www.web-colors-explained.com/hex.php
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| | title = Hexadecimal web colors explained
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| }}</ref> CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange: {{color box|#FA3}}).
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| * {{anchor|_nix}}[[*nix]] (Unix and related) shells, [[AT&T Corporation|AT&T]] assembly language, and likewise the [[C (programming language)|C programming language]], which was designed for Unix (and the syntactic descendants of C – including [[C++]], [[C Sharp (programming language)|C#]], [[Java (programming language)|Java]], [[JavaScript]], [[Python (programming language)|Python]] and [[Windows PowerShell]]) use the prefix <code>0x</code> for numeric constants represented in hex: <code>0x5A3</code>. Character and string constants may express character codes 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 (plus a trailing NUL to mark the end of the string) with two embedded Esc characters.<ref>The string <code>"\x1B[0m\x1B[25;1H"</code> specifies the character sequence <tt>Esc [ 0 m Esc [ 2 5 ; 1 H Nul</tt>. 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.
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| * In the [[Unicode]] standard, a character value is represented with <code>U+</code> followed by the hex value: <code>U+20AC</code> is the [[Euro sign]] (€).
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| * In [[MIME]] (e-mail extensions) [[quoted-printable]] encoding, characters that cannot be represented as literal [[ASCII]] characters are represented by their codes as two hexadecimal digits (in ASCII) prefixed by an ''equal to'' sign <code>=</code>, as in <code>Espa=F1a</code> to send "España" (Spain). (Hexadecimal F1, equal to decimal 241, is the code number for the lower case n with tilde in the ISO/IEC 8859-1 character set.)
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| * In Intel-derived [[assembly language]]s, hexadecimal is denoted with a suffixed <tt>H</tt> or <tt>h</tt>: <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>
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| * Other assembly languages ([[MOS Technology 6502|6502]], [[Motorola]]), [[Pascal (programming language)|Pascal]], [[Object Pascal|Delphi]], some versions of [[BASIC]] ([[Commodore BASIC|Commodore]]), [[GameMaker: Studio|Game Maker Language]] and [[Forth (programming language)|Forth]] use <code>$</code> as a prefix: <code>$5A3</code>.
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| * Some assembly languages (Microchip) use the notation <code>H'ABCD'</code> (for ABCD<sub>16</sub>).
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| * [[Ada (programming language)|Ada]] and [[VHDL]] enclose hexadecimal numerals in based "numeric quotes": <code>16#5A3#</code>. For bit vector constants [[VHDL]] uses the notation <code>x"5A3"</code>.<ref>The [http://www.eng.auburn.edu/department/ee/mgc/vhdl.html#numbers VHDL MINI-REFERENCE: VHDL IDENTIFIERS, NUMBERS, STRINGS, AND EXPRESSIONS]</ref>
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| * [[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.
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| * [[Modula-2]] and some other languages use # as a prefix: <code>#05A3</code>
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| * The [[Smalltalk]] language uses the prefix <code>16r</code>: <code>16r5A3</code>
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| * [[PostScript]] and the [[Bourne shell]] and its derivatives denote hex with prefix <code>16#</code>: <code>16#5A3</code>. For PostScript, binary data (such as image [[pixel]]s) can be expressed as unprefixed consecutive hexadecimal pairs: <code>AA213FD51B3801043FBC</code>...
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| * In early systems when a [[Macintosh]] crashed, one or two lines of hexadecimal code would be displayed under the [[Sad Mac]] to tell the user what went wrong.
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| * [[Common Lisp]] uses the prefixes <code>#x</code> and <code>#16r</code>.
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| * [[MSX BASIC]],<ref>[http://www.atarimagazines.com/compute/issue56/107_1_MSX_IS_COMING.php MSX is Coming — Part 2: Inside MSX] [[Compute!]], issue 56, January 1985, p. 52</ref> [[QuickBASIC]], [[FreeBASIC]] and [[Visual Basic]] prefix hexadecimal numbers with <code>&H</code>: <code>&H5A3</code>
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| * [[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 hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.</ref>
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| * [[TI-89]] and 92 series uses a <code>0h</code> prefix: <code>0h5A3</code>
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| * The most common format for hexadecimal on IBM mainframes ([[zSeries]]) and midrange computers ([[IBM System 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>, 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.
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| * [[Donald Knuth]] introduced the use of a particular typeface to represent a particular radix in his book ''The TeXbook''.<ref>Donald E. Knuth. ''The TeXbook'' ([[Computers and Typesetting]], Volume A). Reading, Massachusetts: Addison–Wesley, 1984. ISBN 0-201-13448-9. The [http://www.ctan.org/tex-archive/systems/knuth/tex/texbook.tex source code of the book in TeX] (and a required set of macros [ftp://tug.ctan.org/pub/tex-archive/systems/knuth/lib/manmac.tex CTAN.org]) is available online on [[CTAN]].</ref> Hexadecimal representations are written there in a typewriter typeface: <tt>5A3</tt>
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| * Any [[IPv6 address]] can be written as eight groups of four hexadecimal digits, where each group is separated by a colon (<code>:</code>). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334
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| * [[ALGOL 68]] uses the prefix <code>16r</code> to denote hexadecimal numbers: <code>16r5a3</code>. Binary, quaternary (base-4) and octal numbers can be specified similarly.
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| There is no universal convention to use lowercase or uppercase for the letter digits, and each is prevalent or preferred in particular environments by community standards or convention.
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| ===Early written representations===
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| [[Image:Bruce Martin hexadecimal notation proposal.png|thumb|Bruce Alan Martin's hexadecimal notation proposal]]
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| The choice of the letters ''A'' through ''F'' to represent the digits above nine was not universal in the early history of computers.
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| * During the 1950s, some installations favored using the digits 0 through 5 with a [[macron]] character ("¯") to denote the values 10–15.
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| * [[Bendix G-15]] computers used the letters ''U'' through ''Z''.
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| * The Librascope [[LGP-30]] used the letters ''F'', ''G'', ''J'', ''K'', ''Q'' and ''W''.<ref>This somewhat odd sequence was from the next six sequential numeric keyboard codes in the LGP-30's 6-bit character code. [http://ed-thelen.org/comp-hist/lgp-30-man.html#R4.13 LGP-30 PROGRAMMING MANUAL]</ref>
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| * [[Bruce Alan Martin]] of [[Brookhaven National Laboratory]] considered the choice of A–F "ridiculous" and in a 1968 letter to the editor of the [[Communications of the ACM|CACM]] proposed an entirely new set of symbols based on the bit locations, which did not gain much acceptance<!-- if any -->.<ref>''Letters to the editor: On binary notation'', Bruce Alan Martin, Associated Universities Inc., Communications of the ACM, Volume 11, Issue 10 (October 1968) Page: 658 {{DOI|10.1145/364096.364107}}</ref>
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| * Soviet [[programmable calculator]]s [[Elektronika B3-34|Б3-34]] and similar used the symbols "−", "L", "C", "Г", "E", " " (space) on their displays.
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| ===Verbal and digital representations===
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| There are no traditional numerals to represent the quantities from ten to fifteen — letters are used as a substitute — and most [[Europe]]an languages lack non-decimal names for the numerals above ten. Even though English has names for several non-decimal powers (''[[2 (number)|pair]]'' for the first [[binary numeral system|binary]] power, ''[[20 (number)|score]]'' for the first [[vigesimal]] power, ''[[dozen]]'', ''[[Gross (unit)|gross]]'', and ''[[great gross]]'' for the first three [[duodecimal]] powers), no English name describes the hexadecimal powers (decimal 16, 256, 4096, 65536, ... ). Some people read hexadecimal numbers digit by digit like a phone number: ''4DA'' is "four-dee-ay". However, the letter ''A'' sounds like "eight", ''C'' sounds like "three", and ''D'' can easily be mistaken for the "-ty" suffix: Is it ''4D'' or ''forty''? Other people avoid confusion by using the [[ICAO spelling alphabet|NATO phonetic alphabet]]: ''4DA'' is "four-delta-alfa<!-- Please do not change this to "alpha", see the link - "alfa" is correct in the context of the NATO phonetic alphabet. -->", the [[Joint Army/Navy Phonetic Alphabet]] ("four-dog-able"), or a similar ad hoc system.
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| [[File:Hexadecimal-counting.jpg|right|thumb|Hexadecimal finger-counting scheme.]]
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| Systems of counting on [[Digit (anatomy)|digits]] have been devised for both binary and hexadecimal.
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| [[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. Another system for counting up to FF<sub>16</sub> (255<sub>10</sub>) is illustrated on the right.
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| ===Signs===
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| The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42<sub>10</sub> and so on.
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| However, some{{who|date=June 2012}} prefer instead to use the hexadecimal notation 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]] value. This way, the negative number −42<sub>10</sub> can be written as FFFF FFD6 in a 32-bit [[Processor register|CPU register]] (in [[two's-complement]]), as C228 0000 in a 32-bit [[Floating point unit|FPU]] register or C045 0000 0000 0000 in a 64-bit FPU register (in the [[IEEE floating-point standard]]).
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| ===Hexadecimal exponential notation===
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| Just as decimal numbers can be represented in [[exponential notation]] so too can hexadecimal. By convention, the letter ''p'' represents ''times two raised to the power of'', whereas ''e'' serves a similar purpose in decimal. The number after the ''p'' is '''decimal''' and represents the '''binary''' exponent.
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| Usually the number is normalised: that is, the leading hexadecimal digit is 1 (unless the value is exactly 0).
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| Example: 1.3DEp42 represents {{math|1.3DE<sub>16</sub> × 2<sup>42</sup>}}.
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| Hexadecimal exponential notation is required by the [[IEEE 754-2008|IEEE 754]] binary floating-point standard. This notation can be produced by some versions of the ''[[printf]]'' family of functions by using the '''%a''' conversion.
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| ==Conversion==
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| ===Binary conversion===
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| Most computers manipulate binary data, but it is difficult for humans to work with the 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>).
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| 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:
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| * 0001<sub>2</sub> = 1<sub>10</sub>
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| * 0010<sub>2</sub> = 2<sub>10</sub>
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| * 0100<sub>2</sub> = 4<sub>10</sub>
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| * 1000<sub>2</sub> = 8<sub>10</sub>
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| Therefore:
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| {|
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| |-
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| | 1111<sub>2</sub>|| = 8<sub>10</sub> + 4<sub>10</sub> + 2<sub>10</sub> + 1<sub>10</sub>
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| |-
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| | || = 15<sub>10</sub>
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| |}
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| 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.
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| This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
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| {|
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| | 01011110101101010010<sub>2</sub>|| = 262144<sub>10</sub> + 65536<sub>10</sub> + 32768<sub>10</sub> + 16384<sub>10</sub> + 8192<sub>10</sub> + 2048<sub>10</sub> + 512<sub>10</sub> + 256<sub>10</sub> + 64<sub>10</sub> + 16<sub>10</sub> + 2<sub>10</sub>
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| |-
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| | || = 387922<sub>10</sub>
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| |}
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| Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
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| {|
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| |-
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| | 01011110101101010010<sub>2</sub>|| = ||0101<sub> </sub>||1110<sub> </sub>||1011<sub> </sub>||0101<sub> </sub>||0010<sub>2</sub>
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| |-
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| | || = || align="center" |5|| align="center" |E|| align="center" |B|| align="center" |5|| align="center" |2<sub>16</sub>
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| |-
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| | || = || colspan="5" |5EB52<sub>16</sub>
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| |}
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| The conversion from hexadecimal to binary is equally direct.
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| The [[octal]] system can also be useful as a tool for people who need to deal directly with binary computer data. Octal represents data as three bits per character, rather than four.
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| ===Division-remainder in source base===
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| 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.
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| 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.
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| # i := 1
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| # h<sub>i</sub> := d mod 16
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| # d := (d−h<sub>i</sub>) / 16
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| # If d = 0 (return series h<sub>i</sub>) else increment i and go to step 2
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| "16" may be replaced with any other base that may be desired.
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| 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]].
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| <source lang="javascript">
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| function toHex(d) {
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| var r = d % 16;
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| var result;
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| if (d-r == 0)
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| result = toChar(r);
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| else
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| result = toHex( (d-r)/16 ) + toChar(r);
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| return result;
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| }
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| function toChar(n) {
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| const alpha = "0123456789ABCDEF";
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| return alpha.charAt(n);
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| }
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| </source>
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| ===Addition and multiplication===
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| [[Image:Hexadecimal multiplication table.svg|right|thumb|A hexadecimal [[multiplication table]]]]
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| It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation.
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| That is, 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>, where ''p'' is the corresponding hex digit position, counting from right to left, beginning with 0. In this case we have {{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 base 10.
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| ===Tools for conversion===
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| Most modern computer systems with [[graphical user interface]]s provide a built-in calculator utility, capable of performing conversions between various radices, in general including hexadecimal.
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| In [[Microsoft]] [[Microsoft Windows|Windows]], the [[Calculator (Windows)|Calculator]] utility can be set to Scientific mode (called Programmer mode in some versions), 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 Scientific 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.
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| ==Real numbers==
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| As with other numeral systems, the hexadecimal system can be used to represent [[rational number]]s, although [[Recurring decimal|recurring digits]] are common since sixteen (10<sub>hex</sub>) has only a single prime factor (two):
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| {| Border=0 cellspacing=0 cellpadding=5
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| |align=right| '''1/2''' ||<center> '''=''' || '''0.8'''
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| |align=right| 1/6 ||<center> = || 0.2<span style="text-decoration: overline">A</span>
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| |align=right| 1/A ||<center> = || 0.1<span style="text-decoration: overline">9</span>
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| |align=right| 1/E ||<center> = || 0.1<span style="text-decoration: overline">249</span>
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| |-
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| |align=right| 1/3 ||<center> = || 0.<span style="text-decoration: overline">5</span>
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| |align=right| 1/7 ||<center> = || 0.<span style="text-decoration: overline">249</span>
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| |align=right| 1/B ||<center> = || 0.<span style="text-decoration: overline">1745D</span>
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| |align=right| 1/F ||<center> = || 0.<span style="text-decoration: overline">1</span>
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| |-
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| |align=right| '''1/4''' ||<center> '''=''' || '''0.4'''
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| |align=right| '''1/8''' ||<center> '''=''' || '''0.2'''
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| |align=right| 1/C ||<center> = || 0.1<span style="text-decoration: overline">5</span>
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| |align=right| '''1/10''' ||<center> '''=''' || '''0.1'''
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| |-
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| |align=right| 1/5 ||<center> = || 0.<span style="text-decoration: overline">3</span>
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| |align=right| 1/9 ||<center> = || 0.<span style="text-decoration: overline">1C7</span>
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| |align=right| 1/D ||<center> = || 0.<span style="text-decoration: overline">13B</span>
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| |align=right| 1/11 ||<center> = || 0.<span style="text-decoration: overline">0F</span>
| |
| |}
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| | |
| where an [[overline#In mathematics|overline]] denotes a recurring pattern.
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| | |
| 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: Counting in base 3 is 0, 1, 2, 10 (three). 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]] (4²), 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 lie 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 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 0.199999999999... in hexadecimal. However, hexadecimal is more efficient than bases 12 and 60 for representing fractions with powers of two in the denominator (e.g., decimal one sixteenth is 0.1 in hexadecimal, 0.09 in duodecimal, 0:3:45 in sexagesimal and 0.0625 in decimal).
| |
| | |
| {|class="wikitable"
| |
| | colspan="3" align="center" | In decimal<br><small>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span></small><br><small>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></small><br><small>Prime factors of one above the base: <span style="color:Orange">'''11'''</span></small>
| |
| | colspan="3" align="center" | '''In hexadecimal'''<br><small>Prime factors of the base: <span style="color:Green">'''2'''</span></small><br><small>Prime factors of one below the base: <span style="color:Blue">'''3''', '''5'''</span></small><br><small>Prime factors of one above the base: <span style="color:Orange">'''11'''</span></small>
| |
| |-
| |
| | align="center" | Fraction
| |
| | align="center" | <small>Prime factors<br>of the denominator</small>
| |
| | align="center" | Positional representation
| |
| | align="center" | Positional representation
| |
| | align="center" | <small>Prime factors<br>of the denominator</small>
| |
| | align="center" | Fraction
| |
| |-
| |
| | align="center" | 1/2
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | '''0.5'''
| |
| | '''0.8'''
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | align="center" | 1/2
| |
| |-
| |
| | align="center" | 1/3
| |
| | align="center" | <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}}
| |
| | bgcolor=#c0c0c0 | '''0.'''5555... = '''0.'''{{overline|5}}
| |
| | align="center" | <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/3
| |
| |-
| |
| | align="center" | 1/4
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | '''0.25'''
| |
| | '''0.4'''
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | align="center" | 1/4
| |
| |-
| |
| | align="center" | 1/5
| |
| | align="center" | <span style="color:Green">'''5'''</span>
| |
| | '''0.2'''
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|3}}
| |
| | align="center" | <span style="color:Blue">'''5'''</span>
| |
| | align="center" | 1/5
| |
| |-
| |
| | align="center" | 1/6
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.1'''{{overline|6}}
| |
| | bgcolor=#c0c0c0 | '''0.2'''{{overline|A}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/6
| |
| |-
| |
| | align="center" | 1/7
| |
| | align="center" | <span style="color:Red">'''7'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|142857}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|249}}
| |
| | align="center" | <span style="color:Red">'''7'''</span>
| |
| | align="center" | 1/7
| |
| |-
| |
| | align="center" | 1/8
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | '''0.125'''
| |
| | '''0.2'''
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | align="center" | 1/8
| |
| |-
| |
| | align="center" | 1/9
| |
| | align="center" | <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|1C7}}
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| | align="center" | <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/9
| |
| |-
| |
| | align="center" | 1/10
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| |
| | '''0.1'''
| |
| | bgcolor=#c0c0c0 | '''0.1'''{{overline|9}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''5'''</span>
| |
| | align="center" | 1/A
| |
| |-
| |
| | align="center" | 1/11
| |
| | align="center" | <span style="color:Orange">'''11'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|09}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|1745D}}
| |
| | align="center" | <span style="color:Red">'''B'''</span>
| |
| | align="center" | 1/B
| |
| |-
| |
| | align="center" | 1/12
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.08'''{{overline|3}}
| |
| | bgcolor=#c0c0c0 | '''0.1'''{{overline|5}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/C
| |
| |-
| |
| | align="center" | 1/13
| |
| | align="center" | <span style="color:Red">'''13'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|076923}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|13B}}
| |
| | align="center" | <span style="color:Red">'''D'''</span>
| |
| | align="center" | 1/D
| |
| |-
| |
| | align="center" | 1/14
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|714285}}
| |
| | bgcolor=#c0c0c0 | '''0.1'''{{overline|249}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| |
| | align="center" | 1/E
| |
| |-
| |
| | align="center" | 1/15
| |
| | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|6}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| |
| | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Blue">'''5'''</span>
| |
| | align="center" | 1/F
| |
| |-
| |
| | align="center" | 1/16
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | '''0.0625'''
| |
| | '''0.1'''
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | align="center" | 1/10
| |
| |-
| |
| | align="center" | 1/17
| |
| | align="center" | <span style="color:Red">'''17'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0588235294117647}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0F}}
| |
| | align="center" | <span style="color:Orange">'''11'''</span>
| |
| | align="center" | 1/11
| |
| |-
| |
| | align="center" | 1/18
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|5}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|E38}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/12
| |
| |-
| |
| | align="center" | 1/19
| |
| | align="center" | <span style="color:Red">'''19'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|052631578947368421}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0D79435E5}}
| |
| | align="center" | <span style="color:Red">'''13'''</span>
| |
| | align="center" | 1/13
| |
| |-
| |
| | align="center" | 1/20
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| |
| | '''0.05'''
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|C}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''5'''</span>
| |
| | align="center" | 1/14
| |
| |-
| |
| | align="center" | 1/21
| |
| | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|047619}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0C3}}
| |
| | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span>
| |
| | align="center" | 1/15
| |
| |-
| |
| | align="center" | 1/22
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Orange">'''11'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|45}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|BA2E8}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''B'''</span>
| |
| | align="center" | 1/16
| |
| |-
| |
| | align="center" | 1/23
| |
| | align="center" | <span style="color:Red">'''23'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0434782608695652173913}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0B21642C859}}
| |
| | align="center" | <span style="color:Red">'''17'''</span>
| |
| | align="center" | 1/17
| |
| |-
| |
| | align="center" | 1/24
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.041'''{{overline|6}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|A}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/18
| |
| |-
| |
| | align="center" | 1/25
| |
| | align="center" | <span style="color:Green">'''5'''</span>
| |
| | '''0.04'''
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0A3D7}}
| |
| | align="center" | <span style="color:Blue">'''5'''</span>
| |
| | align="center" | 1/19
| |
| |-
| |
| | align="center" | 1/26
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|384615}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|9D8}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''D'''</span>
| |
| | align="center" | 1/1A
| |
| |-
| |
| | align="center" | 1/27
| |
| | align="center" | <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|037}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|097B425ED}}
| |
| | align="center" | <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/1B
| |
| |-
| |
| | align="center" | 1/28
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.03'''{{overline|571428}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|924}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| |
| | align="center" | 1/1C
| |
| |-
| |
| | align="center" | 1/29
| |
| | align="center" | <span style="color:Red">'''29'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|0344827586206896551724137931}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|08D3DCB}}
| |
| | align="center" | <span style="color:Red">'''1D'''</span>
| |
| | align="center" | 1/1D
| |
| |-
| |
| | align="center" | 1/30
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|3}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|8}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Blue">'''5'''</span>
| |
| | align="center" | 1/1E
| |
| |-
| |
| | align="center" | 1/31
| |
| | align="center" | <span style="color:Red">'''31'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|032258064516129}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|08421}}
| |
| | align="center" | <span style="color:Red">'''1F'''</span>
| |
| | align="center" | 1/1F
| |
| |-
| |
| | align="center" | 1/32
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | '''0.03125'''
| |
| | '''0.08'''
| |
| | align="center" | <span style="color:Green">'''2'''</span>
| |
| | align="center" | 1/20
| |
| |-
| |
| | align="center" | 1/33
| |
| | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Orange">'''11'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|03}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|07C1F}}
| |
| | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''B'''</span>
| |
| | align="center" | 1/21
| |
| |-
| |
| | align="center" | 1/34
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''17'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|2941176470588235}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|78}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Orange">'''11'''</span>
| |
| | align="center" | 1/22
| |
| |-
| |
| | align="center" | 1/35
| |
| | align="center" | <span style="color:Green">'''5'''</span>, <span style="color:Red">'''7'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|285714}}
| |
| | bgcolor=#c0c0c0 | '''0.'''{{overline|075}}
| |
| | align="center" | <span style="color:Blue">'''5'''</span>, <span style="color:Red">'''7'''</span>
| |
| | align="center" | 1/23
| |
| |-
| |
| | align="center" | 1/36
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | bgcolor=#c0c0c0 | '''0.02'''{{overline|7}}
| |
| | bgcolor=#c0c0c0 | '''0.0'''{{overline|71C}}
| |
| | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| |
| | align="center" | 1/24
| |
| |}
| |
| | |
| {|class="wikitable"
| |
| | align="center" | ''Algebraic irrational number''
| |
| | align="center" | In decimal
| |
| | align="center" | '''In hexadecimal'''
| |
| |-
| |
| | align="center" | [[Square root of 2|√2]] <small>(the length of the [[diagonal]] of a unit [[Square (geometry)|square]])</small>
| |
| | 1.41421356237309...
| |
| | 1.6A09E667F3BCD...
| |
| |-
| |
| | align="center" | [[Square root of 3|√3]] <small>(the length of the diagonal of a unit [[cube]])</small>
| |
| | 1.73205080756887...
| |
| | 1.BB67AE8584CAA...
| |
| |-
| |
| | align="center" | [[Square root of 5|√5]] <small>(the length of the [[diagonal]] of a 1×2 [[rectangle]])</small>
| |
| | 2.2360679774997...
| |
| | 2.3C6EF372FE95...
| |
| |-
| |
| | align="center" | [[Golden ratio|φ]] <small>(phi, the [[golden ratio]] = {{math|(1+{{radical|5}})/2}} </small>
| |
| | 1.6180339887498...
| |
| | 1.9E3779B97F4A...
| |
| |-
| |
| | align="center" | ''Transcendental irrational number''
| |
| | align="center" |
| |
| | align="center" |
| |
| |-
| |
| | align="center" | ''[[Pi|π]]'' <small>(pi, the ratio of [[circumference]] to [[diameter]])</small>
| |
| | 3.1415926535897932384626433<br>8327950288419716939937510...
| |
| | 3.243F6A8885A308D313198A2E0<br>3707344A4093822299F31D008...
| |
| |-
| |
| | align="center" | [[E (mathematical constant)|e]] <small>(the base of the [[natural logarithm]])</small>
| |
| | 2.7182818284590452...
| |
| | 2.B7E151628AED2A6B...
| |
| |-
| |
| | align="center" | ''[[Thue–Morse constant|τ]]'' <small>(the [[Thue–Morse constant]])</small>
| |
| | 0.412454033640...
| |
| | 0.6996 9669 9669 6996 ...
| |
| |-
| |
| | align="center" | [[Euler-Mascheroni constant|γ]] <small>(the limiting difference between the [[harmonic series (mathematics)|harmonic series]] and the natural logarithm)</small>
| |
| | 0.5772156649015328606...
| |
| | 0.93C467E37DB0C7A4D1B...
| |
| |}
| |
| | |
| ===Powers===
| |
| Possibly the most widely used powers, powers of two, are easier to show using base 16. The first sixteen powers of two are shown below.
| |
| {| class="wikitable"
| |
| ! 2<sup>''x''</sup> !! value
| |
| |-
| |
| | 2<sup>0</sup> || 1
| |
| |-
| |
| | 2<sup>1</sup> || 2
| |
| |-
| |
| | 2<sup>2</sup> || 4
| |
| |-
| |
| | 2<sup>3</sup> || 8
| |
| |-
| |
| | 2<sup>4</sup> || 10<sub>hex</sub>
| |
| |-
| |
| | 2<sup>5</sup> || 20<sub>hex</sub>
| |
| |-
| |
| | 2<sup>6</sup> || 40<sub>hex</sub>
| |
| |-
| |
| | 2<sup>7</sup>|| 80<sub>hex</sub>
| |
| |-
| |
| | 2<sup>8</sup>|| 100<sub>hex</sub>
| |
| |-
| |
| | 2<sup>9</sup> || 200<sub>hex</sub>
| |
| |-
| |
| | 2<sup>A</sup> (<math>2^{10_{dec}}</math>) || 400<sub>hex</sub>
| |
| |-
| |
| | 2<sup>B</sup> (<math>2^{11_{dec}}</math>) || 800<sub>hex</sub>
| |
| |-
| |
| | 2<sup>C</sup> (<math>2^{12_{dec}}</math>) || 1000<sub>hex</sub>
| |
| |-
| |
| | 2<sup>D</sup> (<math>2^{13_{dec}}</math>) || 2000<sub>hex</sub>
| |
| |-
| |
| | 2<sup>E</sup> (<math>2^{14_{dec}}</math>) || 4000<sub>hex</sub>
| |
| |-
| |
| | 2<sup>F</sup> (<math>2^{15_{dec}}</math>) || 8000<sub>hex</sub>
| |
| |-
| |
| | 2<sup>10</sup> (<math>2^{16_{dec}}</math>) || 10000<sub>hex</sub>
| |
| |}
| |
| Since four squared is sixteen, powers of four have an even easier relation:
| |
| {| class="wikitable"
| |
| ! 4<sup>''x''</sup> !! value
| |
| |-
| |
| | 4<sup>0</sup> || 1
| |
| |-
| |
| | 4<sup>1</sup> || 4
| |
| |-
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| | 4<sup>2</sup> || 10<sub>hex</sub>
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| |-
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| | 4<sup>3</sup> || 40<sub>hex</sub>
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| |-
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| | 4<sup>4</sup> || 100<sub>hex</sub>
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| |-
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| | 4<sup>5</sup> || 400<sub>hex</sub>
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| |-
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| | 4<sup>6</sup> || 1000<sub>hex</sub>
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| |-
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| | 4<sup>7</sup> || 4000<sub>hex</sub>
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| |-
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| | 4<sup>8</sup> || 10000<sub>hex</sub>
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| |}
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| This also makes [[tetration]] easier when using two and four since:<br>
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| <sup>3</sup>2 = 2<sup>4</sup> = 10<sub>hex</sub>,<br>
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| <sup>4</sup>2 = 2<sup>16</sup> = 10000<sub>hex</sub> and<br>
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| <sup>5</sup>2 = 2<sup>65536</sup> = (1 followed by 16384 zeros)<sub>hex</sub>.
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| ==Cultural==
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| ===Etymology===
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| The word ''hexadecimal'' is composed of ''hexa-'', derived from the [[Greek language|Greek]] έξ (hex) for "six", and ''-decimal'', derived from the [[Latin]] for "tenth". Webster's Third New International online derives "hexadecimal" as an alteration of the all-Latin "sexadecimal" (which appears in the earlier Bendix documentation). The earliest date attested for "hexadecimal" in Merriam-Webster Collegiate online is 1954, placing it safely in the category of [[international scientific vocabulary]] (ISV). It is common in ISV to mix Greek and Latin [[combining form]]s freely. The word "[[sexagesimal]]" (for base 60) retains the Latin prefix. [[Donald Knuth]] has pointed out that the etymologically correct term is "senidenary" (or possibly "sedenary"), from the Latin term for "grouped by 16". (The terms "binary", "ternary" and "quaternary" are from the same Latin construction, and the etymologically correct terms for "decimal" and "octal" arithmetic are "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>A.B. Taylor, [http://books.google.com/books?id=X7wLAAAAYAAJ&pg=PP5 Report on Weights and Measures], Pharmaceutical Association, 8th Annual Session, Boston, Sept. 15, 1859. See pages and 33 and 41.</ref><ref>Alfred B. Taylor, "Octonary numeration and its application to a system of weights and measures", [http://books.google.com/books?id=KsAUAAAAYAAJ&pg=PA296 ''Proc Amer. Phil. Soc.'' Vol XXIV], Philadelphia, 1887; pages 296-366. See pages 317 and 322.</ref> Schwartzman notes that the expected form from usual Latin phrasing would be "sexadecimal", but computer hackers would be tempted to shorten that word to "sex".<ref>Schwartzman, S. (1994). ''The Words of Mathematics: an etymological dictionary of mathematical terms used in English''. ISBN 0-88385-511-9.</ref> The [[Etymology|etymologically]] proper [[Greek language|Greek]] term would be ''hexadecadic'' (although in [[Modern Greek]] ''deca-hexadic (δεκαεξαδικός)'' is more commonly used).
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| ===Use in Chinese culture===
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| The traditional [[Chinese units of measurement|Chinese units of weight]] were base-16. For example, one jīn (斤) in the old system equals sixteen [[tael]]s. The [[suanpan]] (Chinese [[abacus]]) could be used to perform hexadecimal calculations.
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| ===Primary numeral system===
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| Similar to [[Dozenal#Advocacy and "dozenalism"|dozenal advocacy]],
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| there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts usually
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| propose pronunciation and/or symbology.<ref>{{cite web
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| | url = http://www.hauptmech.com/base42
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| | title = Base 4^2 Hexadecimal Symbol Proposal
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| }}</ref> Sometimes the proposal unifies standard
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| measures so that they are multiples of 16.<ref>{{cite web
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| | url = http://www.intuitor.com/hex/
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| | title = Intuitor Hex Headquarters
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| }}</ref><ref>{{cite web
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| | url = http://std.dkuug.dk/jtc1/sc2/wg2/docs/n2677
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| | title = A proposal for addition of the six Hexadecimal digits (A-F) to Unicode
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| }}</ref><ref name="nystrom">{{cite book | last=Nystrom | first=John William | title=Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base |year=1862 | url=http://books.google.com/books?id=aNYGAAAAYAAJ | location=Philadelphia}}</ref>
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| An example of unifying standard measures is [[hexadecimal time]], which subdivides a day by 16 so that there are 16 "hexhours" in a day.<ref name="nystrom" />
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| ==Key to number base notation==
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| Simple key for notations used in article:
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| {| class="wikitable" style="font-size:90%"
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| ! Full Text Notation !! Abbreviation !! Number Base
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| |-
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| | [[Binary numeral system|binary]] || bin || 2
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| |-
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| | [[octal]] || oct || 8
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| |-
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| | [[decimal]] || dec || 10
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| |-
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| | hexadecimal || hex || 16
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| |}
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| ==See also==
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| * [[Base32]], [[Base64]] (content encoding schemes)
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| * [[Hex editor]]
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| * [[Hexdump]]
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| ==References==
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| {{reflist|2}}
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| <!--
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| ==External links==
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| 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):
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| * The "External links" section should be kept to a minimum. A lack of external links ... is not a reason to add them.
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| * Links to be avoided:
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| ** Any site that does not provide a unique resource beyond what the article might contain...
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| ** Links mainly intended to promote a website
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| ** Sites that are only indirectly related to the article's subject
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| 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.
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| -->
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| [[Category:Binary arithmetic]]
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| [[Category:Hexadecimal numeral system]]
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| [[Category:Positional numeral systems]]
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