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69-447: Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses A and B, as in hexadecimal , which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published material: 2 (a turned 2) for ten and 3 (a turned 3) for eleven. The number twelve,

138-426: A great gross , a great-great-gross , and a great-great-great-gross , respectively. In this system, the prefix e - is added for fractions. As numbers get larger (or fractions smaller), the last two morphemes are successively replaced with tri-mo, quad-mo, penta-mo, and so on. Multiple digits in this series are pronounced differently: 12 is "do two"; 30 is "three do"; 100 is "gro"; BA9 is "el gro dek do nine"; B86

207-650: A superior highly composite number , is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range, and the smallest abundant number . All multiples of reciprocals of 3-smooth numbers ( ⁠ a / 2·3 ⁠ where a,b,c are integers) have a terminating representation in duodecimal. In particular, ⁠ + 1 / 4 ⁠  (0.3), ⁠ + 1 / 3 ⁠  (0.4), ⁠ + 1 / 2 ⁠  (0.6), ⁠ + 2 / 3 ⁠  (0.8), and ⁠ + 3 / 4 ⁠  (0.9) all have

276-492: A 32-bit CPU register (in two's complement ), as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard ). Just as decimal numbers can be represented in exponential notation , so too can hexadecimal numbers. P notation uses the letter P (or p , for "power"), whereas E (or e ) serves a similar purpose in decimal E notation . The number after

345-523: A Duodecimal Base Would Simplify Mathematics . Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system. Both the Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of

414-472: A Humphrey point for other duodecimal numbers. The Dozenal Society of America suggested the pronunciation of ten and eleven as "dek" and "el". For the names of powers of twelve, there are two prominent systems. In spite of the efficiency of these newer systems, terms for powers of twelve either already exist or remain easily reconstructed in English using words and affixes. Another nominal for twelve (12 10 )

483-409: A convenient representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). For example, a 6-bit byte can have values ranging from 000000 to 111111 (0 to 63 decimal) in binary form, which can be written as 00 to 3F in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example,

552-491: A day; many other items are counted by the dozen , gross ( 144 , square of 12), or great gross ( 1728 , cube of 12). The Romans used a fraction system based on 12, including the uncia , which became both the English words ounce and inch . Pre- decimalisation , Ireland and the United Kingdom used a mixed duodecimal- vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to

621-509: A decimal rather than duodecimal origin. However, Old Norse used a hybrid decimal–duodecimal counting system, with its words for "one hundred and eighty" meaning 200 and "two hundred" meaning 240. In the British Isles, this style of counting survived well into the Middle Ages as the long hundred . Historically, units of time in many civilizations are duodecimal. There are twelve signs of

690-440: A greater positional-notation value. 20,736 10 or 10,000 12 may be rendered a dozen-great-gross ; so 248,832 10 or 100,000 12 is a gross-great-gross , with 2,985,984 10 or 1,000,000 12 being known as a great-great-gross . It should be made plain that the indice's being a multiple of three, e.g. 10 12 [1,000 12 ], 10 12 [1,000,000 12 ], 10 12 [1,000,000,000 12 ] results, in these examples, in

759-403: A larger proportion lies outside its range of finite representation. All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal : that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in

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828-482: A pair of quaternary digits, and each quaternary digit corresponds to a pair of binary digits. In the above example 2 5 C 16 = 02 11 30 4 . The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping

897-402: 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 10 ). This example converts 1111 2 to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from

966-485: A short terminating representation in duodecimal. There is also higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system. In these respects, duodecimal is considered superior to decimal, which has only 2 and 5 as factors, and other proposed bases like octal or hexadecimal . Sexagesimal (base sixty) does even better in this respect (the reciprocals of all 5-smooth numbers terminate), but at

1035-510: A single hexadecimal digit. This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results. Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently and converted directly: The conversion from hexadecimal to binary is equally direct. Although quaternary (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to

1104-414: A variety of methods have arisen: Sometimes the numbers are known to be Hex. The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers. 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

1173-484: Is "el gro eight do six"; 8BB,15A is "eight gro el do el, one gro five do dek"; ABA is "dek gro el do dek"; BBB is "el gro el do el"; 0.06 is "six egro"; and so on. This system uses "-qua" ending for the positive powers of 12 and "-cia" ending for the negative powers of 12, and an extension of the IUPAC systematic element names (with syllables dec and lev for the two extra digits needed for duodecimal) to express which power

1242-435: Is a dozen (10 12 or 1•10 12 ). One hundred and forty-four (144 10 ) is also known as a gross (100 12 or 1•10 12 ). One thousand, seven hundred and twenty-eight is (1728 10 ) also known as a great-gross (1,000 12 or 1•10 12 ). For the next powers of twelve that follow those aforementioned, the affixes (dozen-, gross-, great-) are used to produce names for these powers of twelve that have

1311-408: Is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen. Software developers and system designers widely use hexadecimal numbers because they provide

1380-454: Is below the DSA's stated threshold. Eight and Sixteen only have 2 as a prime factor. Therefore, in octal and hexadecimal , the only terminating fractions are those whose denominator is a power of two . Thirty is the smallest number that has three different prime factors (2, 3, and 5, the first three primes), and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). Sexagesimal

1449-444: Is illustrated on the right. The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42 10 , −B01D9 to represent −721369 10 and so on. Hexadecimal can also be used to express the exact bit patterns used in the processor , so a sequence of hexadecimal digits may represent a signed or even a floating-point value. This way, the negative number −42 10 can be written as FFFF FFD6 in

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1518-470: Is meant. After hex-, further prefixes continue sept-, oct-, enn-, dec-, lev-, unnil-, unun-. William James Sidis used 12 as the base for his constructed language Vendergood in 1906, noting it being the smallest number with four factors and its prevalence in commerce. The case for the duodecimal system was put forth at length in Frank Emerson Andrews ' 1935 book New Numbers: How Acceptance of

1587-454: Is much more advisable to work with bitwise 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 (11 10 ), 3 (3 10 ), A (10 10 ) and D (13 10 ), and then get

1656-562: Is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following hex dump , each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number. In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give

1725-497: Is the natural number following 143 and preceding 145 . It is coincidentally both the square of twelve (a dozen dozens , or one gross .) and the twelfth Fibonacci number , and the only nontrivial number in the sequence that is square. 144 is a highly totient number . 144 is the smallest number whose fifth power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture . It

1794-401: Is too small, significantly longer expansions are needed for numbers; if a base is too large, one must memorise a large multiplication table to perform arithmetic. Thus, it presumes that "a number base will need to be between about 7 or 8 through about 16, possibly including 18 and 20". The number 12 has six factors, which are 1 , 2 , 3 , 4 , 6 , and 12 , of which 2 and 3 are prime . It is

1863-496: The C99 edition of the C programming language . Using the %a or %A conversion specifiers, this notation can be produced by implementations of the printf family of functions following the C99 specification and Single Unix Specification (IEEE Std 1003.1) POSIX standard. Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even

1932-653: The Calculator utility can be set to Programmer mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 ( octal ), and 2 ( binary ), the bases most commonly used by programmers. In Programmer Mode, the on-screen numeric keypad includes the hexadecimal digits A through F, which are active when "Hex" is selected. In hex mode, however, the Windows Calculator supports only integers. Elementary operations such as division can be carried out indirectly through conversion to an alternate numeral system , such as

2001-454: The P is decimal and represents the binary 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 with 1. (zero is usually 0 with no P ). Example: 1.3DEp42 represents 1.3DE 16  × 2 . P notation is required by the IEEE 754-2008 binary floating-point standard and can be used for floating-point literals in

2070-509: The Roman numeral for ten and a rounded italic capital E similar to open E ), along with italic numerals 0 – 9 . Edna Kramer in her 1951 book The Main Stream of Mathematics used a ⟨ *, # ⟩ ( sextile or six-pointed asterisk, hash or octothorpe). The symbols were chosen because they were available on some typewriters; they are also on push-button telephones . This notation

2139-486: The duodecimal system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals. Some proposals unify standard measures so that they are multiples of 16. An early such proposal was put forward by John W. Nystrom in Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called

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2208-406: The pound sterling or Irish pound ), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places. In a positional numeral system of base n (twelve for duodecimal), each of the first n natural numbers is given a distinct numeral symbol, and then n is denoted "10", meaning 1 times n plus 0 units. For duodecimal,

2277-469: The zodiac , twelve months in a year, and the Babylonians had twelve hours in a day (although at some point, this was changed to 24). Traditional Chinese calendars , clocks, and compasses are based on the twelve Earthly Branches or 24 (12×2) Solar terms . There are 12 inches in an imperial foot, 12  troy ounces in a troy pound, 12  old British pence in a shilling , 24 (12×2) hours in

2346-511: The Pitman digits were added to Unicode, the DSA took a vote and then began publishing PDF content using the Pitman digits instead, but continues to use the letters X and E on its webpage. There are also varying proposals of how to distinguish a duodecimal number from a decimal one. The most common method used in mainstream mathematics sources comparing various number bases uses a subscript "10" or "12", e.g. "54 12 = 64 10 ". To avoid ambiguity about

2415-540: The Tonal System, with Sixteen to the Base , published in 1862. Nystrom among other things suggested hexadecimal time , which subdivides a day by 16, so that there are 16 "hours" (or "10 tims ", pronounced tontim ) in a day. The word hexadecimal is first recorded in 1952. It is macaronic in the sense that it combines Greek ἕξ (hex) "six" with Latinate -decimal . The all-Latin alternative sexadecimal (compare

2484-611: The alphabet for the transdecimal symbols. Latin letters such as ⟨ A, B ⟩ (as in hexadecimal ) or ⟨ T, E ⟩ (initials of Ten and Eleven ) are convenient because they are widely accessible, and for instance can be typed on typewriters. However, when mixed with ordinary prose, they might be confused for letters. As an alternative, Greek letters such as ⟨ τ, ε ⟩ could be used instead. Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his 1935 book New Numbers ⟨ X , Ɛ ⟩ (italic capital X from

2553-507: The base explicitly: 159 10 is decimal 159; 159 16 is hexadecimal 159, which equals 345 10 . Some authors prefer a text subscript, such as 159 decimal and 159 hex , or 159 d and 159 h . Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook . Hexadecimal representations are written there in a typewriter typeface : 5A3 , C1F27ED In linear text systems, such as those used in most computer programming environments,

2622-451: The binary digits in groups of either three or four. As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans, only decimal and for most computers, only binary (which can be converted by far more efficient methods) can be easily handled with this method. Let d be

2691-400: The commonly used decimal system or the binary system where each hex digit corresponds to four binary digits. Alternatively, one can also perform elementary operations directly within the hex system itself — by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm. As with other numeral systems,

2760-470: The cost of unwieldy multiplication tables and a much larger number of symbols to memorize. Georges Ifrah speculatively traced the origin of the duodecimal system to a system of finger counting based on the knuckle bones of the four larger fingers. Using the thumb as a pointer, it is possible to count to 12 by touching each finger bone, starting with the farthest bone on the fifth finger, and counting on. In this system, one hand counts repeatedly to 12, while

2829-440: The decimal value 711 would be expressed in hexadecimal as 2C7 16 . In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C , which would denote this value as 0x2C7 . Hexadecimal is used in the transfer encoding Base 16 , in which each byte of the plain text is broken into two 4-bit values and represented by two hexadecimal digits. In most current use cases,

Duodecimal - Misplaced Pages Continue

2898-411: The decimal. This is my experience; I am certain that even more so it would be the experience of others. But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to

2967-493: The digit decomposition (7,080.9 = 7,000 + 80 + 0.9). Then, the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get: Because the summands are already converted to decimal, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result: That is, (duodecimal) 12,345;6 equals (decimal) 24,677.5 If

3036-491: The duodecimal system. They use the word "dozenal" instead of "duodecimal" to avoid the more overtly decimal terminology. However, the etymology of "dozenal" itself is also an expression based on decimal terminology since "dozen" is a direct derivation of the French word douzaine , which is a derivative of the French word for twelve, douze , descended from Latin duodecim . Mathematician and mental calculator Alexander Craig Aitken

3105-402: The duodecimal. In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien being with twelve fingers and twelve toes using duodecimal arithmetic, using "dek" and "el" as names for ten and eleven, and Andrews' script-X and script-E for the digit symbols. Systems of measurement proposed by dozenalists include: The Dozenal Society of America argues that if a base

3174-606: The end of the 1960s. In 1969, Donald Knuth argued that the etymologically correct term would be senidenary , or possibly sedenary , a Latinate term intended to convey "grouped by 16" modelled on binary , ternary , quaternary , etc. According to Knuth's argument, the correct terms for decimal and octal arithmetic would be denary and octonary , respectively. 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". 144 (number) 144 ( one hundred [and] forty-four )

3243-450: The expansions of some common irrational numbers in decimal and hexadecimal. Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below. The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels . The suanpan (Chinese abacus ) can be used to perform hexadecimal calculations such as additions and subtractions. As with

3312-455: The final result by multiplying each decimal representation by 16 ( p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that: B3AD = (11 × 16 ) + (3 × 16 ) + (10 × 16 ) + (13 × 16 ) which is 45997 in base 10. Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hexadecimal. In Microsoft Windows ,

3381-520: The given number is in decimal and the target base is duodecimal, the method is same. Using the digit conversion tables: (decimal) 10,000 + 2,000 + 300 + 40 + 5 + 0.6 = (duodecimal) 5,954 + 1,1A8 + 210 + 34 + 5 + 0; 7249 To sum these partial products and recompose the number, the addition must be done with duodecimal rather than decimal arithmetic: That is, (decimal) 12,345.6 equals (duodecimal) 7,189; 7249 Hexadecimal Hexadecimal (also known as base-16 or simply hex )

3450-415: The given number must first be decomposed into a sum of numbers with only one significant digit each. For example: This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 7,080.9), these are left out in

3519-454: The hexadecimal system can be used to represent rational numbers , although repeating expansions are common since sixteen (10 16 ) has only a single prime factor: two. For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1 . Because

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3588-421: The latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.1 9 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.0625 10 (one-sixteenth) is equivalent to 0.1 16 , 0.09 12 , and 0;3,45 60 . The table below gives

3657-437: The letters A–F or a–f represent the values 10–15, while the numerals 0–9 are used to represent their decimal values. There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Some seven-segment displays use mixed-case 'A b C d E F' to distinguish the digits A–F from one another and from 0–9. There

3726-625: The meaning of the subscript 10, the subscripts might be spelled out, "54 twelve = 64 ten ". In 2015 the Dozenal Society of America adopted the more compact single-letter abbreviation "z" for "do z enal" and "d" for " d ecimal", "54 z = 64 d ". Other proposed methods include italicizing duodecimal numbers " 54 = 64", adding a "Humphrey point" (a semicolon instead of a decimal point ) to duodecimal numbers "54;6 = 64.5", prefixing duodecimal numbers by an asterisk "*54 = 64", or some combination of these. The Dozenal Society of Great Britain uses an asterisk prefix for duodecimal whole numbers, and

3795-480: The number to represent in hexadecimal, and the series h i h i−1 ...h 2 h 1 be the hexadecimal digits representing the number. "16" may be replaced with any other base that may be desired. The following is a JavaScript implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it

3864-536: The numerals eleven to fifteen. Some people read hexadecimal numbers digit by digit, like a phone number, or using the NATO phonetic alphabet , the Joint Army/Navy Phonetic Alphabet , or a similar ad-hoc system. In the wake of the adoption of hexadecimal among IBM System/360 programmers, Magnuson (1968) suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A"

3933-835: The other displays the number of iterations, until five dozens, i.e. the 60, are full. This system is still in use in many regions of Asia. Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji , Gbiri-Niragu (Gure-Kahugu), Piti , and the Nimbia dialect of Gwandara ; and the Chepang language of Nepal are known to use duodecimal numerals. Germanic languages have special words for 11 and 12, such as eleven and twelve in English . They come from Proto-Germanic * ainlif and * twalif (meaning, respectively, one left and two left ), suggesting

4002-598: The radix 16 is a perfect square (4 ), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using 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

4071-597: The representation of multiples of numbers that are one less than or one more than the base. In the following multiplication table, numerals are written in duodecimal. For example, "10" means twelve, and "12" means fourteen. To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation ). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0;1 and BB,BBB;B to decimal, or any decimal number between 0.1 and 99,999.9 to duodecimal. To use them,

4140-428: The right: Therefore: With little practice, mapping 1111 2 to F 16 in one step becomes easy (see table in 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

4209-488: The smallest number to have six factors, the largest number to have at least half of the numbers below it as divisors, and is only slightly larger than 10. (The numbers 18 and 20 also have six factors but are much larger.) Ten, in contrast, only has four factors, which are 1 , 2 , 5 , and 10 , of which 2 and 5 are prime. Six shares the prime factors 2 and 3 with twelve; however, like ten, six only has four factors (1, 2, 3, and 6) instead of six. Its corresponding base, senary ,

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4278-428: The standard numeral symbols for 0–9 are typically preserved for zero through nine, but there are numerous proposals for how to write the numerals representing "ten" and "eleven". More radical proposals do not use any Arabic numerals under the principle of "separate identity." Pronunciation of duodecimal numbers also has no standard, but various systems have been proposed. Several authors have proposed using letters of

4347-596: 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 on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023 10 on ten fingers. Another system for counting up to FF 16 (255 10 )

4416-770: The word sexagesimal for base 60) is older, and sees at least occasional use from the late 19th century. It is still in use in the 1950s in Bendix documentation. Schwartzman (1994) argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex . Many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal (e.g. French hexadécimal , Italian esadecimale , Romanian hexazecimal , Serbian хексадецимални , etc.) but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic sextándakerfi , Russian шестнадцатеричной etc.) Terminology and notation did not become settled until

4485-438: Was actually used by the ancient Sumerians and Babylonians , among others; its base, sixty , adds the four convenient factors 4, 12, 20, and 60 to 30 but no new prime factors. The smallest number that has four different prime factors is 210 ; the pattern follows the primorials . However, these numbers are quite large to use as bases, and are far beyond the DSA's stated threshold. In all base systems, there are similarities to

4554-419: Was an outspoken advocate of duodecimal: The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in

4623-658: Was introduced by Isaac Pitman in 1857. In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies in the Unicode Standard . Of these, the British/Pitman forms were accepted for encoding as characters at code points U+218A ↊ TURNED DIGIT TWO and U+218B ↋ TURNED DIGIT THREE . They were included in Unicode 8.0 (2015). After

4692-463: Was pronounced "ann", B "bet", C "chris", etc. Another naming-system was published online by Rogers (2007) that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below. Yet another naming system was elaborated by Babb (2015), based on a joke in Silicon Valley . Others have proposed using

4761-403: Was used in publications of the Dozenal Society of America (DSA) from 1974 to 2008. From 2008 to 2015, the DSA used ⟨  [REDACTED] , [REDACTED]  ⟩ , the symbols devised by William Addison Dwiggins . The Dozenal Society of Great Britain (DSGB) proposed symbols ⟨  2 , 3  ⟩ . This notation, derived from Arabic digits by 180° rotation,

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