"Sexadecimal", "Hex digit", and "Hex format" redirect here. For base 60, seeSexagesimal. For binary-coded hexadecimals, seeNibble. For hexadecimal file formats, seeHex file (disambiguation).
Hexadecimal (hex for short) is apositional numeral system for representing a numeric value asbase 16. For the most common convention, a digit is represented as "0" to "9" like fordecimal and as a letter of the alphabet from "A" to "F" (either upper or lower case) for the digits with decimal value 10 to 15.
Special notation is often used to indicate that a number is hex. Inmathematics, a subscript is typically used to specify the base. For example, the decimal value491 would be expressed in hex as 1EB16. Incomputer programming, various notations are used. InC and many related languages, the prefix0x is used. For example,0x1EB.
Typically, a hex representation convention allows either lower or upper case letters and treats the letter the same regardless of its case.
Often when rendering non-textual data, a value stored in memory is displayed as a sequence of hex digits with spaces that between values. For instance, in the followinghex dump, each 8-bitbyte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.
There are several conventions for expressing that a number is represented as hex.
A decimal subscript can give the base explicitly. For example 15910 indicates decimal 159, 15916 indicates hex 159. Some prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h
In C and many languages influenced by it, the prefix0x indicates that the numeric literal after it is in hex, a character of a string or character literal can be expressed as hex with the prefix\x (for example'\x1B' represents theEsc control character) and to output an integer as hex viaprintf-like function, the format conversion code%X or%x is used
InURIs (includingURLs),character codes are written as hex pairs prefixed with%:http://www.example.com/name%20with%20spaces where%20 is the code for thespace (blank) character,ASCII code point 20 in hex, 32 in decimal.
InXML andXHTML, a character can be expressed as a hexnumeric character reference using the notationode;, for instanceT represents the character U+0054 (the uppercase letter "T"). If there is nox the number is decimal (thusT is the same character).[2]
In Intel-derivedassembly languages and Modula-2,[3] hex is denoted with a suffixedH orh:FFh or05A3H. Some implementations require a leading zero when the first hex digit character is not a decimal digit, so one would write0FFh instead ofFFh. Some other implementations (such as NASM) allow C-style numbers (0x42)
Some assembly languages (Microchip) use the notationH'ABCD' (for ABCD16); similarly,Fortran 95 uses Z'ABCD'
Ada andVHDL enclose hex numerals in based "numeric quotes":16#5A3#,16#C1F27ED#. For bit vector constantsVHDL uses the notationx"5A3",x"C1F27ED".[4]
Verilog represents hex constants in the form8'hFF, where 8 is the number of bits in the value and FF is the hex constant
TheIcon andSmalltalk languages use the prefix16r:16r5A3
PostScript and theBourne shell and its derivatives denote hex with prefix16#:16#5A3,16#C1F27ED
Common Lisp uses the prefixes#x and#16r. Setting the variables *read-base*[5] and *print-base*[6] to 16 can also be used to switch the reader and printer of a Common Lisp system to hex representation for reading and printing numbers. Thus hex numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.
TI-89 and 92 series uses a0h prefix:0h5A3,0hC1F27ED
ALGOL 68 uses the prefix16r to denote hex numbers:16r5a3,16rC1F27ED. Binary, quaternary (base-4), and octal numbers can be specified similarly.
The most common format for hex on IBM mainframes (zSeries) and midrange computers (IBM i) running the traditional OS's (zOS,zVSE,zVM,TPF,IBM i) isX'5A3' orX'C1F27ED', and is used in Assembler,PL/I,COBOL,JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.
In some contexts, a number is always written as hex, and therefore, needs no identification notation.
In theUnicode standard, a character value is represented withU+ followed by the hex value, e.g.U+00A1 is theinverted exclamation point (¡).
Color references in HTML,CSS andX Window can be expressed with six hex digits (two each for the red, green and blue components, in that order) prefixed with#:magenta, for example, is represented as#FF00FF.[10] CSS also allows 3-hexdigit abbreviations with one hexdigit per component:#FA3 abbreviates#FFAA33 (a golden orange:).
InMIME (e-mail extensions)quoted-printable encoding, character codes are written as hex pairs prefixed with=:Espa=F1a is "España" (F1hex is the code forñ in the ISO/IEC 8859-1 character set).[11])
PostScript binary data (such as imagepixels) can be expressed as unprefixed consecutive hex pairs:AA213FD51B3801043FBC ...
AnyIPv6 address can be written as eight groups of four hex digits (sometimes calledhextets), where each group is separated by a colon (:). This, for example, is a valid IPv6 address:2001:0db8:85a3:0000:0000:8a2e:0370:7334 or abbreviated by removing leading zeros as2001:db8:85a3::8a2e:370:7334 (IPv4 addresses are usually written in decimal).
Globally unique identifiers are written as thirty-two hex digits, often in unequal hyphen-separated groupings, for example3F2504E0-4F89-41D3-9A0C-0305E82C3301.
Bruce Alan Martin's hex notation proposal[12]Ronald O. Whitaker's hex notation proposal.[13][14]
Notable other hexadecimal representations that use symbols other than letters "A" through "F" to represent the digits above 9 include:
During the 1950s, some installations, such as Bendix-14, favored using the digits 0 through 5 with anoverline to denote the values10–15 as0,1,2,3,4 and5.
TheSWAC (1950)[15] andBendix G-15 (1956)[16][15] computers used the lowercase lettersu,v,w,x,y andz for the values 10 to 15.
TheORDVAC andILLIAC I (1952) computers (and some derived designs, e.g.BRLESC) used the uppercase lettersK,S,N,J,F andL for the values 10 to 15.[17][15]
The LibrascopeLGP-30 (1956) used the lettersF,G,J,K,Q andW for the values 10 to 15.[18][15]
On thePERM (1956) computer, hex numbers were written as lettersO for zero,A toN andP 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.[19]
TheHoneywellDatamatic D-1000 (1957) used the lowercase lettersb,c,d,e,f, andg whereas theElbit 100 (1967) used the uppercase lettersB,C,D,E,F andG for the values 10 to 15.[15]
TheMonrobot XI (1960) used the lettersS,T,U,V,W andX for the values 10 to 15.[15]
TheNECparametron computer NEAC 1103 (1960) used the lettersD,G,H,J,K (and possiblyV) for values 10–15.[20]
New numeric symbols and names were introduced in theBibi-binary notation byBoby Lapointe in 1968.
Bruce Alan Martin ofBrookhaven National Laboratory considered the choice of A–F "ridiculous". In a 1968 letter to the editor of theCACM, he proposed an entirely new set of symbols based on the bit locations.[12]
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."[13][14]
Someseven-segment display decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.[21]
Hex can express the bit pattern in aprocessor, so a sequence of hex digits may represent asigned or even afloating-point value. This way, the negative number −4210 can be written as FFFF FFD6 in a 32-bitCPU register (intwo's complement), as C228 0000 in a 32-bitFPU register or C045 0000 0000 0000 in a 64-bit FPU register (in theIEEE floating-point standard).
Just as decimal numbers can be represented inexponential notation, so too can hex numbers.P notation uses the letterP (orp, for "power"), whereasE (ore) serves a similar purpose in decimalE notation. The number after theP isdecimal and represents thebinary exponent. Increasing the exponent by 1 multiplies by 2, not 16:20p0 = 10p1 = 8p2 = 4p3 = 2p4 = 1p5. Usually, the number is normalized so that the hex digits start with1. (zero is usually0 with noP).
Example:1.3DEp42 represents1.3DE16 × 24210.
P notation is required by theIEEE 754-2008 binary floating-point standard and can be used for floating-point literals in theC99 edition of theC programming language.[22]Using the%a or%A conversion specifiers, this notation can be produced by implementations of theprintf family of functions following the C99 specification[23] andSingle Unix Specification (IEEE Std 1003.1)POSIX standard.[24]
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 hex numbers digit by digit, like a phone number, or using theNATO phonetic alphabet, theJoint Army/Navy Phonetic Alphabet, or a similarad hoc system. In the wake of the adoption of hex amongIBM System/360 programmers, Magnuson (1968)[25] suggested a pronunciation guide that gave short names to the letters of hex – for instance, "A" was pronounced "ann", B "bet", C "chris", etc.[25] Another naming-system was published online by Rogers (2007)[26] 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 inSilicon Valley.[27] The system proposed by Babb was further improved by Atkins-Bittner in 2015-2016.[28]
Others have proposed using the verbalMorse code conventions to express four-bit hex digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hex digit for decimal 15.
Systems of counting ondigits have been devised for both binary and hex.Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 102310 on ten fingers.[29] Another system for counting up to FF16 (25510) is illustrated on the right.
The programmableRPN-calculatorHP-16C Computer Scientist from 1982 was designed for programmers. One of its key features was the conversion between different numeral systems (note hex number in display).
Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hex than to decimal because each hex digit maps to a whole number of bits (410).This example converts 11112 to base ten. Since eachposition in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:
00012 = 110
00102 = 210
01002 = 410
10002 = 810
Therefore:
11112
= 810 + 410 + 210 + 110
= 1510
With little practice, mapping 11112 to F16 in one step becomes easy. The advantage of using hex rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hex, it is trivial to regard the binary string as 4-digit groups and map each to a single hex digit.[30]
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
(1001011100)2
= 51210 + 6410 + 1610 + 810 + 410
= 60410
Compare this to the conversion to hex, where each group of four digits can be considered independently and converted directly:
(1001011100)2
=
0010
0101
11002
=
2
5
C16
=
25C16
The conversion from hex to binary is equally direct.[30]
Althoughquaternary (base 4) is little used, it can easily be converted to and from hex or binary. Each hex digit corresponds to a pair of quaternary digits, and each quaternary digit corresponds to a pair of binary digits. In the above example 2 5 C16 = 02 11 304.
Theoctal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hex via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.
As with all bases there is a simplealgorithm for converting a representation of a number to hex by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans, only decimal and for most computers, only binary (which can be converted by far more efficient methods) can be easily handled with this method.
Let d be the number to represent in hex, and the series hihi−1...h2h1 be the hex digits representing the number.
i ← 1
hi ← d mod 16
d ← (d − hi) / 16
If d = 0 (return series hi) else increment i and go to step 2
"16" may be replaced with any other base that may be desired.
The following is aJavaScript implementation of the above algorithm for converting any number to a hex in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work withbitwise operators.
It is also possible to make the conversion by assigning each place in the source base the hex 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 hex number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16p (p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:
Elementary operations such as division can be carried out indirectly through conversion to an alternatenumeral 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 aslong division and the traditional subtraction algorithm.
As with other numeral systems, the hex system can be used to representrational numbers, althoughrepeating expansions are common since sixteen (1016) has only a single prime factor: two.
For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two forbinary or dividing one by sixteen for hex, both of these fractions are written as0.1. Because the radix 16 is aperfect square (42), fractions expressed in hex have an odd period much more often than decimal ones, and there are nocyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has aprime factor not found in the radix; thus, when using hex notation, all fractions with denominators that are not apower of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hex (and binary) less convenient thandecimal for representing rational numbers since a larger proportion lies outside its range of finite representation.
All rational numbers finitely representable in hex are also finitely representable in decimal,duodecimal andsexagesimal: that is, any hex number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hex. For example, decimal 0.1 corresponds to the infinite recurring representation 0.19 in hex. However, hex is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.062510 (one-sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560.
n
Decimal Prime factors of: base, b = 10:2,5; b − 1 = 9:3; b + 1 = 11:11
Hexadecimal Prime factors of: base, b = 1610 = 10:2; b − 1 = 1510 = F:3, 5; b + 1 = 1710 = 11:11
The traditionalChinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteentaels. Thesuanpan (Chineseabacus) can be used to perform hex calculations such as additions and subtractions.[31]
As with theduodecimal system, there have been occasional attempts to promote hex as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals.[32] Some proposals unify standard measures so that they are multiples of 16.[33][34]An early such proposal was put forward byJohn W. Nystrom inProject 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.[35]Nystrom among other things suggestedhexadecimal time, which subdivides a day by 16,so that there are 16 "hours" (or "10tims", pronouncedtontim) in a day.[36]
Look uphexadecimal in Wiktionary, the free dictionary.
The wordhexadecimal is first recorded in 1952.[37] It ismacaronic in the sense that it combinesGreek ἕξ (hex) "six" withLatinate-decimal.The all-Latin alternativesexadecimal (compare the wordsexagesimal for base 60) is older, and sees at least occasional use from the late 19th century.[38]It is still in use in the 1950s inBendix documentation.Schwartzman (1994) argues that use ofsexadecimal may have been avoided because of its suggestive abbreviation tosex.[39]Many western languages since the 1960s have adopted terms equivalent in formation tohexadecimal (e.g. Frenchhexadécimal, Italianesadecimale, Romanianhexazecimal, Serbianхексадецимални, etc.)but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandicsextándakerfi, Russianшестнадцатеричной 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 besenidenary, or possiblysedenary, a Latinate term intended to convey "grouped by 16" modelled onbinary,ternary,quaternary, etc.According to Knuth's argument, the correct terms fordecimal andoctal arithmetic would bedenary andoctonary, respectively.[40]Alfred B. Taylor usedsenidenary in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".[41][42]
The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of thepublication of theFortran IV manual forIBM System/360, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants.[43]As noted above, alternative notations were used byNEC (1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martinin his letter to the editor of theCACM complains that
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".[12]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" (asBrahmi numerals, and later in aHindu–Arabic numeral system),and that the recentASCII standards (ASA X3.4-1963 and USAS X3.4-1968)"should have preserved six code table positions following the ten decimal digits-- rather than needlessly filling these with punctuation characters"(":;<=>?") that might have been placed elsewhere among the 128 available positions.
Base16 is abinary to text encoding in the family that also containsBase32,Base58, andBase64. Data is broken into 4-bit sequences, and each value (0-15) is encoded as a character. Although any 16 characters could be used, in practice, theASCII digits "0"–"9" and letters "A"–"F" (or "a"–"f") are used to align with the typical notation for hex numbers.
Support for Base16 encoding is ubiquitous in modern computing. It is the basis for theW3C standard forURL percent encoding, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers.
Advantages of Base16 encoding include:
Most programming languages have facilities to parse ASCII-encoded hex
Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64, respectively
The notation is well-known; easily understood without needing a symbol lookup table
Many CPU architectures have dedicated instructions that allow access to a half-byte (aka nibble), making it more efficient in hardware than Base32 and Base64
Disadvantages include:
Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte; in contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively
Complexity of accepting both upper and lower case letters
^BBC BASIC programs are not fully portable toMicrosoft BASIC (without modification) since the latter takes& to prefixoctal values. (Microsoft BASIC primarily uses&O to prefix octal, and it uses&H to prefix hex, but the ampersand alone yields a default interpretation as an octal prefix.
^Knuth, Donald Ervin (1986).The TeXbook(PDF). Duane Bibby. Reading, Mass.: American Mathematical Society and Addison-Wesley Publishing Company.ISBN0-201-13448-9.OCLC12973034. Archived fromthe original(PDF) on 2025-11-19.
^abWhitaker, Ronald O. (January 1972). Written at Indianapolis, Indiana, US."More on man/machine"(PDF). Letters.Datamation. Vol. 18, no. 1. Barrington, Illinois, US:Technical Publishing Company. p. 103.Archived(PDF) from the original on 2022-12-05. Retrieved2022-12-24. (1 page)
^abcdefgSavard, John J. G. (2018) [2005]."Computer Arithmetic".quadibloc. The Early Days of Hexadecimal.Archived from the original on 2018-07-16. Retrieved2018-07-16.
^"2.1.3 Sexadecimal notation".G15D Programmer's Reference Manual(PDF). Los Angeles, CA, US:Bendix Computer, Division ofBendix Aviation Corporation. p. 4.Archived(PDF) from the original on 2017-06-01. Retrieved2017-06-01.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.
^Manthey, Steffen; Leibrandt, Klaus (2002-07-02)."Die PERM und ALGOL"(PDF) (in German).Archived(PDF) from the original on 2018-10-03. Retrieved2018-05-19.
^abMano, M. Morris; Ciletti, Michael D. (2013).Digital Design – With an Introduction to the Verilog HDL (Fifth ed.).Pearson Education. pp. 6,8–10.ISBN978-0-13-277420-8.
^Nystrom (1862), p. 33:"In expressing time, angle of a circle, or points on the compass, the unittim should be noted as integer, and parts thereof astonal fractions, as 5·86tims is five times andmetonby [*"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.unifoundry.comArchived 2021-05-19 at theWayback Machine ]."
^C. E. Fröberg,Hexadecimal Conversion Tables, Lund (1952).
^The Century Dictionary of 1895 hassexadecimal in the more general sense of "relating to sixteen".An early explicit use ofsexadecimal in the sense of "using base 16" is found also in 1895, in theJournal of the American Geographical Society of New York, vols. 27–28, p. 197.
^Schwartzman, Steven (1994).The Words of Mathematics: An etymological dictionary of mathematical terms used in English. The Mathematical Association of America. p. 105.ISBN0-88385-511-9. s.v. hexadecimal
^Alfred B. Taylor,Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, 15 September 1859. See pages and 33 and 41.
^Alfred B. Taylor, "Octonary numeration and its application to a system of weights and measures",Proc Amer. Phil. Soc. Vol XXIVArchived 2016-06-24 at theWayback Machine, Philadelphia, 1887; pages 296–366. See pages 317 and 322.
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