Hexadecimal (also known asbase-16 or simplyhex) is apositional numeral system that represents numbers using aradix (base) of sixteen. Unlike thedecimal 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 ofbinary-coded values. Each hexadecimal digit represents fourbits (binary digits), also known as anibble (or nybble).[1] For example, an 8-bitbyte is two hexadecimal digits and its value can be written as00 toFF in hexadecimal.
In mathematics, a subscript is typically used to specify the base. For example, the decimal value711 would be expressed in hexadecimal as 2C716. In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix0x is used inC, which would denote this value as0x2C7.
Hexadecimal is used in the transfer encodingBase 16, in which each byte of theplain text is broken into two 4-bit values and represented by two hexadecimal digits.
In most current use cases, the letters A–F or a–f represent the values 10–15, while thenumerals 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. Someseven-segment displays 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 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.
In contexts where thebase 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: 15910 is decimal 159; 15916 is hexadecimal 159, which equals 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h.
Donald Knuth introduced the use of a particular typeface to represent a particular radix in his bookThe TeXbook.[2] Hexadecimal representations are written there in atypewriter typeface:5A3,C1F27ED
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
Although best known from theC programming language (and the many languages influenced by C), the prefix0x to indicate a hex constant may have had origins in theIBM Stretch systems. It is derived from the0 prefix already in use foroctal constants. Byte values can be expressed in hexadecimal with the prefix\x followed by two hex digits:'\x1B' represents theEsc control character;"\x1B[0m\x1B[25;1H" is a string containing 11 characters with two embedded Esc characters.[3] To output an integer as hexadecimal with theprintf function family, the format conversion code%X or%x is used.
InXML andXHTML, characters can be expressed as hexadecimalnumeric character references 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).[4]
In Intel-derivedassembly languages and Modula-2,[5] hexadecimal is denoted with a suffixedH orh:FFh or05A3H. Some implementations require a leading zero when the first hexadecimal 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 hexadecimal numerals in based "numeric quotes":16#5A3#,16#C1F27ED#. For bit vector constantsVHDL uses the notationx"5A3",x"C1F27ED".[6]
Verilog represents hexadecimal constants in the form8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal 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*[7] and *print-base*[8] 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.
TI-89 and 92 series uses a0h prefix:0h5A3,0hC1F27ED
ALGOL 68 uses the prefix16r to denote hexadecimal numbers:16r5a3,16rC1F27ED. 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 (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.
InURIs (includingURLs),character codes are written as hexadecimal 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.
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 hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with#:magenta, for example, is represented as#FF00FF.[11] 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 hexadecimal pairs prefixed with=:Espa=F1a is "España" (F1hex is the code forñ in the ISO/IEC 8859-1 character set).[12])
PostScript binary data (such as imagepixels) can be expressed as unprefixed consecutive hexadecimal pairs:AA213FD51B3801043FBC ...
AnyIPv6 address can be written as eight groups of four hexadecimal 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 hexadecimal digits, often in unequal hyphen-separated groupings, for example3F2504E0-4F89-41D3-9A0C-0305E82C3301.
Other symbols for 10–15 and mostly different symbol sets
The use of the lettersA throughF 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 anoverline to denote the values10–15 as0,1,2,3,4 and5.
TheSWAC (1950)[13] andBendix G-15 (1956)[14][13] 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.[15][13]
The LibrascopeLGP-30 (1956) used the lettersF,G,J,K,Q andW for the values 10 to 15.[16][13]
On thePERM (1956) computer, hexadecimal 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.[17]
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.[13]
TheMonrobot XI (1960) used the lettersS,T,U,V,W andX for the values 10 to 15.[13]
TheNECparametron computer NEAC 1103 (1960) used the lettersD,G,H,J,K (and possiblyV) for values 10–15.[18]
New numeric symbols and names were introduced in theBibi-binary notation byBoby Lapointe in 1968.
Bruce Alan Martin's hexadecimal notation proposal[19]
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.[19]
Ronald O. Whitaker's hexadecimal notation proposal.[20][21]
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."[20][21]
Someseven-segment display decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.[22]
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 theNATO phonetic alphabet, theJoint Army/Navy Phonetic Alphabet, or a similarad-hoc system. In the wake of the adoption of hexadecimal amongIBM System/360 programmers, Magnuson (1968)[23] suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A" was pronounced "ann", B "bet", C "chris", etc.[23] Another naming-system was published online by Rogers (2007)[24] 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.[25] The system proposed by Babb was further improved by Atkins-Bittner in 2015-2016.[26]
Others have proposed using the verbal Morse Code conventions to express four-bit hexadecimal digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hexadecimal digit for decimal 15.
Systems of counting ondigits have been devised for both binary and hexadecimal.Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 102310 on ten fingers.[27] Another system for counting up to FF16 (25510) is illustrated on the right.
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210, −B01D9 to represent −72136910 and so on.
Hexadecimal can also be used to express the exact bit patterns used in theprocessor, so a sequence of hexadecimal 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 hexadecimal 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 hexadecimal 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.[28]Using the%a or%A conversion specifiers, this notation can be produced by implementations of theprintf family of functions following the C99 specification[29] andSingle Unix Specification (IEEE Std 1003.1)POSIX standard.[30]
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 hexadecimal than to decimal because each hexadecimal 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 (see table inwritten 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.[31]
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 hexadecimal, 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 hexadecimal to binary is equally direct.[31]
Althoughquaternary (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 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 hexadecimal 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 hexadecimal by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans, only decimal and for most computers, only binary (which can be converted by far more efficient methods) can be easily handled with this method.
Let d be the number to represent in hexadecimal, and the series hihi−1...h2h1 be the hexadecimal digits representing the number.
i ← 1
hi ← d mod 16
d ← (d − hi) / 16
If d = 0 (return series hi) else increment i and go to step 2
"16" may be replaced with any other base that may be desired.
The following is aJavaScript 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 withbitwise operators.
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value — before carrying out multiplication and addition to get the final representation.For example, to convert the number B3AD to decimal, one can split the hexadecimal number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 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 hexadecimal 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 hexadecimal, both of these fractions are written as0.1. Because the radix 16 is aperfect square (42), fractions expressed in hexadecimal 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 hexadecimal 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 hexadecimal (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 hexadecimal are also finitely representable in decimal,duodecimal andsexagesimal: 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 0.19 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.062510 (one-sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560.
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 hexadecimal calculations such as additions and subtractions.[32]
As with theduodecimal 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.[33] Some proposals unify standard measures so that they are multiples of 16.[34][35]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.[36]Nystrom among other things suggestedhexadecimal time, which subdivides a day by 16, so that there are 16 "hours" (or "10tims", pronouncedtontim) in a day.[37]
Look uphexadecimal in Wiktionary, the free dictionary.
The wordhexadecimal is first recorded in 1952.[38] 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.[39]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.[40]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.[41] 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".[42][43]
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 theFortran IV manual forIBM System/360, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants.[44]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".[19]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.
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 theASCII 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 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.
^The string"\x1B[0m\x1B[25;1H" specifies the character sequenceEsc [ 0 m Esc [ 2 5; 1 H. These are the escape sequences used on anANSI terminal that reset the character set and color, and then move the cursor to line 25.
^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 hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.
^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.
^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)
^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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