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38-477: The Gwandara people are one of the indigenous tribes of FCT Abuja, the capital city of Nigeria. The Nimbia dialect has a duodecimal numeral system (they count in base 12), whereas other dialects, such as Karshi below, have decimal systems: It is thought that Nimbia, which is isolated from the rest of Gwandara, acquired its duodecimal system from neighboring East Kainji languages . It is duodecimal even to powers of base twelve: The Nimbia 12 number set number system

76-399: 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

114-524: 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

152-473: 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 )

190-491: 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

228-512: 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

266-419: 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, a superior highly composite number , is the smallest number with four non-trivial factors (2, 3, 4, 6), and

304-443: 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

342-486: 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

380-447: Is "el gro dek do nine"; B86 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

418-438: 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

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456-454: 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

494-427: Is denoted "10", meaning 1 twelve and 0 units ; in the decimal system, this number is instead written as "12" meaning 1 ten and 2 units, and the string "10" means ten. In duodecimal, "100" means twelve  squared , "1000" means twelve  cubed , and "0.1" means a twelfth. 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

532-415: Is known for making division easier. This article about a West Chadic language is a stub . You can help Misplaced Pages by expanding it . This Nigeria -related article is a stub . You can help Misplaced Pages by expanding it . Duodecimal The duodecimal system, also known as base twelve or dozenal , is a positional numeral system using twelve as its base . In duodecimal, the number twelve

570-492: Is 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 ,

608-399: 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

646-512: 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

684-406: 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,

722-469: 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

760-512: 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

798-612: 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

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836-470: 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

874-411: 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

912-493: 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

950-491: 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

988-404: 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

1026-490: 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 Piti language Piti (Pitti, Bishi, Bisi)

1064-415: 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

1102-626: 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

1140-839: 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

1178-597: 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,

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1216-541: 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

1254-430: 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

1292-540: The two extra digits needed for duodecimal) to express which power 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

1330-438: 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

1368-419: 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

1406-663: 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

1444-408: 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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