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duodecimal
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{{distinguish|Dewey Decimal Classification|Duodecimo}}{{Numeral systems}}The duodecimal system (also known as base 12, dozenal, or rarely uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the base ten numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (i.e. the same number that in decimal is written as "14"). Similarly, in duodecimal "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").The number twelve, 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. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, and not 3, 4, or 6), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system.WEB,weblink Why We Should Switch To A Base-12 Counting System, George Dvorsky, 2013-01-18,weblink" title="web.archive.org/web/20130121100313weblink">weblink 2013-01-21, no, 2013-12-21, Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...) have a terminating representation in duodecimal. In particular, the five most elementary fractions ({{frac||1|2}}, {{frac||1|3}}, {{frac||2|3}}, {{frac||1|4}} and {{frac||3|4}}) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (because it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems. Although the trigesimal and sexagesimal systems (where the reciprocals of all 5-smooth numbers terminate) do even better in this respect, this is at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.

Origin

In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve.
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;CONFERENCE, Decimal vs. Duodecimal: An interaction between two systems of numeration, Matsushita, Shuji, 2nd Meeting of the AFLANG, October 1998, Tokyo, 1998,weblinkweblink" title="web.archive.org/web/20081005230737weblink">weblink 2008-10-05, 2011-05-29, the Chepang language of NepalBOOK, Les principes de construction du nombre dans les langues tibéto-birmanes, Martine, Mazaudon, La Pluralité, Jacques, François, 2002, 91–119, Peeters, Leuven, 90-429-1295-2,weblink and the Maldivian language (Dhivehi) of the people of the Maldives and Minicoy Island in India are known to use duodecimal numerals. Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.BOOK, von Mengden, Ferdinand, 2006, The peculiarities of the Old English numeral system, Medieval English and its Heritage: Structure Meaning and Mechanisms of Change, Nikolaus Ritt, Herbert Schendl, Christiane Dalton-Puffer, Dieter Kastovsky, Peter Lang, Studies in English Medieval Language and Literature, 16, Frankfurt, 125–145, BOOK, von Mengden, Ferdinand, 2010, Cardinal Numerals: Old English from a Cross-Linguistic Perspective, Topics in English Linguistics, 67, Berlin; New York, De Gruyter Mouton, 159–161, Historically, units of time in many civilizations are duodecimal. There are twelve signs of 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. 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 a day, and many other items 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 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.{| class="wikitable" style=text-align:center! colspan="6"|Table of units from a base of 12! Relativevalue! French unitof length! English unitof length! English(Troy) unitof weight! Roman unitof weight! English unitof mass
|120
Units of measurement in France before the French Revolution#Length>piedFoot (unit)>footTroy weight#Units of measurement>poundAncient Roman units of measurement#Weight>libra|
|12−1
Units of measurement in France before the French Revolution#Length>pouce|inchTroy weight#Units of measurement>ounceAncient Roman units of measurement#Weight>unciaslug (unit)#Similar units>slinch
|12−2
Units of measurement in France before the French Revolution#Length>ligneLine (unit)>lineApothecaries' system#English-speaking countries>scruplesscrupulum>scrupulaslug (unit)>slug
|12−3
Point (typography)#Truchet>pointPoint (typography)#Truchet>pointsiliqua>seed|siliqua |
The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers).JOURNAL, Pittman, Richard, 1990, Origin of Mesopotamian duodecimal and sexagesimal counting systems, Philippine Journal of Linguistics, 21, 1, 97, WEB, ja, ヒマラヤの満月と十二進法, The Full Moon in the Himalayas and the Duodecimal System, Nishikawa, Yoshiaki, 2002,weblink 2008-03-24, yes,weblink" title="web.archive.org/web/20080329150110weblink">weblink March 29, 2008, It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.BOOK
, Ifrah
, Georges
, Georges Ifrah
, The Universal History of Numbers: From prehistory to the invention of the computer
, John Wiley and Sons
, 2000
,
, 0-471-39340-1
, Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.BOOK, Macey, Samuel L., The Dynamics of Progress: Time, Method, and Measure, 1989, University of Georgia Press, Atlanta, Georgia, 978-0-8203-3796-8, 92,weblink

Notations and pronunciations

Transdecimal symbols

In a duodecimal place system twelve is written as 10, but there are numerous proposals for how to write ten and eleven.JOURNAL, De Vlieger, Michael, Symbology Overview, The Duodecimal Bulletin, 4X [58], 2, 2010,weblink To allow entry on typewriters, letters such as A and B (as in hexadecimal), T and E (initials of Ten and Eleven), X and E (X from the Roman numeral for ten), or X and Z are used. Some employ Greek letters such as δ (standing for Greek δέκα 'ten') and ε (for Greek ένδεκα 'eleven'), or τ and ε. Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book New Numbers an X and ℰ (script E, {{U+|2130}}).BOOK, Frank Emerson, Andrews, New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics, 1935, 52,weblink Edna Kramer in her 1951 book The Main Stream of Mathematics used a six-pointed asterisk (sextile) ⚹ and a hash (or octothorpe) #. The symbols were chosen because they are available in typewriters, they also are on push-button telephones. This notation was used in publications of the Dozenal Society of America (DSA) in the period 1974–2008.JOURNAL, Annual Meeting of 1973 and Meeting of the Board, The Duodecimal Bulletin, 25 [29], 1, 1974,weblink JOURNAL, De Vlieger, Michael, Going Classic, The Duodecimal Bulletin, 49 [57], 2, 2008,weblink From 2008 to 2015, the DSA used (File:Dozenal us 10.svg|15x15px) and (File:Dozenal us 11.svg|15x15px), the symbols devised by William Addison Dwiggins.JOURNAL, Mo for Megro, The Duodecimal Bulletin, 1, 1, 1945,weblink {{Symb|(File:Dozenal gb 10.svg) (File:Dozenal gb 11.svg)(File:Duodecimal-digit-ten-Dozenal-Society-of-America.svg) (File:Dozenal us 11.svg)}}The Dozenal Society of Great Britain proposed a rotated digit two ({{d2}}) and a reversed or rotated digit three ({{d3}}). This notation was introduced by Sir Isaac Pitman.Pitman, Isaac (ed.): A triple (twelve gross) Gems of Wisdom. London 1860JOURNAL, Pitman, Isaac, A Reckoning Reform [reprint from 1857], The Duodecimal Bulletin, 3, 2, 1947,weblink 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.WEB,weblink Proposal to encode Duodecimal Digit Forms in the UCS, 2013-03-30, ISO/IEC JTC1/SC2/WG2, Document N4399, 2016-05-30, Karl Pentzlin, Of these, the British/Pitman forms were accepted for encoding as characters at code points {{unichar|218A|TURNED DIGIT TWO}} and {{unichar|218B|TURNED DIGIT THREE}}. They were included in the Unicode 8.0 release in June 2015.WEB,weblink The Unicode Standard, Version 8.0: Number Forms, Unicode Consortium, 2016-05-30, WEB
,weblink
, The Unicode Standard 8.0
, 2014-07-18
, They are available in LaTeX as textturntwo and textturnthree.WEB,weblink The Comprehensive LATEX Symbol List, 2009, 2016-05-30, Scott Pakin, After the Pitman digits were added to Unicode the DSA took a vote and then began publishing content using the Pitman digits instead.WEB,weblink What should the DSA do about transdecimal characters?, The Dozenal Society of America, 2018-01-01, They still use the letters X and E in ASCII text.Other proposals are more creative or aesthetic; for example, many do not use any Arabic numerals under the principle of "separate identity."As these Unicode symbols are not rendering in many browsers, this article uses physically rotated '2' and '3' glyphs.

{{anchor|Humphrey point}}Base notation

There are also varying proposals of how to distinguish a duodecimal number from a decimal one.JOURNAL, Volan, John, July 2015, Base Annotation Schemes,weblink Duodecomal Bulletin, 62, They include italicizing duodecimal numbers "54 = 64", adding a "Humphrey point" (a semicolon ";" instead of a decimal point ".") to duodecimal numbers "54;6 = 64.5", or some combination of the two. Others use subscript or affixed labels to indicate the base, allowing for more than decimal and duodecimal to be represented (for single letters 'z' from "dozenal" is used as 'd' would mean decimal) such as "54z = 64d," "5412 = 6410" or "doz 54 = dec 64." Programming languages limited to ASCII could use a prefix, similar to 0x used for hexadecimal, perhaps 0z but there is no standard.

Pronunciation

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 famous systems.

The do-gro-mo system

In this system, each order has its own name and the prefix e- is added for fractions.JOURNAL, Zirkel, Gene, How Do You Pronounce Dozenals?volume=4E [59]date=2010,weblink {| class="wikitable"! Duodecimal||Name||Decimal||Duodecimal fraction||Name
1;|one1
10;do|pronounced 'doh'}}120;1edo
100;gro|pronounced 'groh'}}1440;01|egro
1,000;mo|pronounced 'moh'}}1,7280;001|emo
10,000;|do-mo20,7360;000,1|edo-mo
100,000;|gro-mo248,8320;000,01|egro-mo
1,000,000;|bi-mo2,985,9840;000,001|ebi-mo
1,000,000,000;|tri-mo5,159,780,3520;000,000,001|etri-mo
Multiple digits in this are pronounced differently. 12; is "one do two", 30; is "three do", 100; is "one gro",

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