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			{{Short description|Base-12 numeral system}}
	{{distinguish|Dewey Decimal Classification|duodecimo}}
			{{distinguish|Dewey Decimal Classification|Duodecimo}}
	{{Special characters}}
	{{Numeral systems}}		{{Numeral systems}}
	The '''duodecimal''' system (also known as '''] 12''' or '''dozenal''') is a ] ] using ] as its ]. In this system, the number ] may be written by a rotated "2" (<span style="display:inline-block; {{transform|rotate(180deg)}}">2</span>) and the number ] by a rotated "3" (<span style="display:inline-block; {{transform|rotate(180deg)}}">3</span>). This notation was introduced by Sir ].<ref>Pitman, Isaac (ed.): A triple (twelve gross) Gems of Wisdom. London 1860</ref> These digit forms are available as ] characters on computerized systems since June 2015<ref name="Unicode8">{{cite web|url=http://unicode.org/versions/Unicode8.0.0/|title=Unicode 8.0.0|publisher=Unicode Consortium|accessdate=2016-05-30}}</ref> as ↊ (] 218A) and ↋ (Code point 218B), respectively.<ref>{{cite web		The '''duodecimal''' system, also known as '''base twelve''' or '''dozenal''', is a ] ] using ] as its ]. In duodecimal, the number twelve is denoted "10", meaning 1&nbsp;twelve and 0 ]; in the ] 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&nbsp;], "1000" means twelve&nbsp;], and "0.1" means a twelfth.
	| url = http://www.unicode.org/charts/PDF/U2150.pdf
	| title = The Unicode Standard 8.0
	| accessdate = 2014-07-18
	}}</ref> Other notations use "A", "T", or "X" for ten and "B" or "E" for eleven. The number twelve (that is, the number written as "12" in the ] numerical system) is instead written as "10" in duodecimal (meaning "1 ] 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 ]", "1000" means "1 ]", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").
			Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses {{d2}} and {{d3}}, as in ], which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, {{d2}}, {{d3}}, 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published material: <span style="display: inline-block; transform: scale(-1);">2</span> (a turned 2) for ten and <span style="display: inline-block; transform: scale(-1);">3</span> (a turned 3) for eleven.
	The number twelve, a ], is the smallest number with four non-trivial ] (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the ] range. As a result of this increased factorability of the ] 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.<ref name=io9>{{cite web
	|url = http://io9.com/5977095/why-we-should-switch-to-a-base+12-counting-system
			The number twelve, a ], is the smallest number with four non-trivial ] (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the ] range, and the smallest ]. All multiples of ] of ] numbers ({{math|{{sfrac|''a''|2<sup>''b''</sup>·3<sup>''c''</sup>}}}} where {{mvar|a,b,c}} are integers) have a ] representation in duodecimal. In particular, {{sfrac||1|4}}&nbsp;(0.3), {{sfrac||1|3}}&nbsp;(0.4), {{sfrac||1|2}}&nbsp;(0.6), {{sfrac||2|3}}&nbsp;(0.8), and {{sfrac||3|4}}&nbsp;(0.9) all have 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.<ref name="io9">{{cite web |author=Dvorsky |first=George |date=January 18, 2013 |title=Why We Should Switch To A Base-12 Counting System |url=http://io9.com/5977095/why-we-should-switch-to-a-base+12-counting-system |access-date=December 21, 2013 |website=]}}</ref>
	|title = Why We Should Switch To A Base-12 Counting System
	|author = George Dvorsky
			In these respects, duodecimal is considered superior to decimal, which has only 2 and 5 as factors, and other proposed bases like ] or ]. ] (base sixty) does even better in this respect (the reciprocals of all ] numbers terminate), but at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.
	|date = 2013-01-18
	|accessdate = 2013-12-21
	|deadurl = bot: unknown
	|archiveurl = https://web.archive.org/web/20130121100313/http://io9.com/5977095/why-we-should-switch-to-a-base+12-counting-system
	|archivedate = 2013-01-21
	|df =
	}}</ref> Of its factors, 2 and 3 are ], which means the ] of all ] numbers (such as 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...) have a ] 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 ] 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 ], ], ], ] and ] systems. Although the ] and ] systems (where the reciprocals of all ] 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 ==		== Origin ==
	:''In this section, numerals are based on decimal ]<ref name="Unicode8" />. For example, 10 means ], 12 means ].''		:''In this section, numerals are in ]. For example, "10" means 9+1, and "12" means 9+3.''
			] speculatively traced the origin of the duodecimal system to a system of ] 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 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.<ref>{{cite journal |last=Pittman |first=Richard |date=1990 |title=Origin of Mesopotamian duodecimal and sexagesimal counting systems |journal=Philippine Journal of Linguistics |volume=21 |issue=1 |page=97}}</ref><ref name="Ifrah 2000">{{Cite book| last = Ifrah| first = Georges| author-link = Georges Ifrah| title = The Universal History of Numbers: From prehistory to the invention of the computer| publisher = Wiley | year=2000 |orig-year=1st French ed. 1981 | isbn = 0-471-39340-1}} Translated from the French by David Bellos, E. F. Harding, Sophie Wood and Ian Monk.</ref>
	Languages using duodecimal number systems are uncommon. Languages in the ]n Middle Belt such as ], ] (Gure-Kahugu), ], and the Nimbia dialect of ];<ref>{{Cite conference		Languages using duodecimal number systems are uncommon. Languages in the ]n Middle Belt such as ], ] (Gure-Kahugu), ], and the Nimbia dialect of ];<ref>{{Cite web |last=Matsushita |first=Shuji |date=October 1998 |title=Decimal vs. Duodecimal: An interaction between two systems of numeration |url=http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html |url-status=dead |archive-url=https://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html |archive-date=October 5, 2008 |access-date=May 29, 2011 |website=www3.aa.tufs.ac.jp}}</ref> and the ] of ]<ref>{{Cite book
	| title=Decimal vs. Duodecimal: An interaction between two systems of numeration
	| last=Matsushita
	| first=Shuji
	| conference=2nd Meeting of the AFLANG, October 1998, Tokyo
	| year=1998
	| url=http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
	| archiveurl=https://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
	| archivedate=2008-10-05
	| accessdate=2011-05-29
	| postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
	}}</ref> the ] of ]<ref>{{Cite book
	| contribution=Les principes de construction du nombre dans les langues tibéto-birmanes		| contribution=Les principes de construction du nombre dans les langues tibéto-birmanes
	| first=Martine		| first=Martine
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	| editor-first=Jacques		| editor-first=Jacques
	| editor-last=François		| editor-last=François
	| year=2002		| date=2002
	| pages=91–119		| pages=91–119
	| publisher=Peeters		| publisher=Peeters
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	| isbn=90-429-1295-2		| isbn=90-429-1295-2
	| url=http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf		| url=http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf
			| access-date=2014-03-27
	| postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
			| archive-date=2016-03-28
	}}</ref> and the ] of ] in ] are known to use duodecimal numerals.
			| archive-url=https://web.archive.org/web/20160328145817/http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf
			| url-status=dead
			}}</ref> are known to use duodecimal numerals.
	] have special words for 11 and 12, such as ''eleven'' and ''twelve'' in ]. However, they are considered to come from ] *''ainlif'' and *''twalif'' (respectively ''one left'' and ''two left''), both of which were decimal.<ref>{{cite book|last=von Mengden| first=Ferdinand| year=2006|chapter=The peculiarities of the Old English numeral system|title=Medieval English and its Heritage: Structure Meaning and Mechanisms of Change|editors=Nikolaus Ritt, Herbert Schendl, Christiane Dalton-Puffer, Dieter Kastovsky|publisher=Peter Lang Pub|series=Studies in English Medieval Language and Literature |volume=16 |location=Frankfurt|pages= 125–45}}<br>{{cite book|last=von Mengden |first=Ferdinand |year=2010| title=Cardinal Numerals: Old English from a Cross-Linguistic Perspective|series=Topics in English Linguistics| volume=67|location= Berlin; New York|publisher=De Gruyter Mouton| pages=159–161}}</ref>		] have special words for 11 and 12, such as ''eleven'' and ''twelve'' in ]. They come from ] *''ainlif'' and *''twalif'' (meaning, respectively, ''one left'' and ''two left''), suggesting a decimal rather than duodecimal origin.<ref>{{cite book|last=von Mengden| first=Ferdinand| date=2006|chapter=The peculiarities of the Old English numeral system |title=Medieval English and its Heritage: Structure Meaning and Mechanisms of Change|editor1=Nikolaus Ritt |editor2=Herbert Schendl |editor3=Christiane Dalton-Puffer |editor4=Dieter Kastovsky|publisher=Peter Lang |series=Studies in English Medieval Language and Literature |volume=16 |location=Frankfurt |pages= 125–145}}</ref><ref>{{cite book|last=von Mengden |first=Ferdinand |date=2010| title=Cardinal Numerals: Old English from a Cross-Linguistic Perspective |series=Topics in English Linguistics | volume=67|location= Berlin; New York|publisher=De Gruyter Mouton| pages=159–161}}</ref> However, ] used a hybrid decimal–duodecimal counting system, with its words for "one hundred and eighty" meaning 200 and "two hundred" meaning 240.<ref>{{cite book|last=Gordon|first=E V|title=Introduction to Old Norse|year=1957|publisher=Clarendon Press|location=Oxford|pages=292–293}}</ref> In the British Isles, this style of counting survived well into the Middle Ages as the ].
	Historically, ] of ] in many ]s are duodecimal. There are twelve signs of the ], twelve months in a year, and the ] had twelve hours in a day (although at some point this was changed to 24). Traditional ]s, clocks, and compasses are based on the twelve ]. There are 12 inches in an imperial foot, 12 ] ounces in a troy pound, 12 ] in a ], 24 (12×2) hours in a day, and many other items counted by the ], ] (], ] of 12) or ] (], ] of 12). The Romans used a fraction system based on 12, including the ] which became both the English words '']'' and ''inch''. Pre-], ] and the ] used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the ] or ]), and ] established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.		Historically, ] in many ]s are duodecimal. There are twelve signs of the ], twelve months in a year, and the ] had twelve hours in a day (although at some point, this was changed to 24). Traditional ]s, clocks, and compasses are based on the twelve ] or 24 (12×2) ]s. There are 12&nbsp;inches in an imperial foot, 12&nbsp;] ounces in a troy pound, 12&nbsp;] in a ], 24&nbsp;(12×2) hours in a day; many other items are counted by the ], ] (], ] of 12), or ] (], ] of 12). The Romans used a fraction system based on 12, including the ], which became both the English words '']'' and ''inch''. Pre-], ] and the ] used a mixed duodecimal-] currency system (12&nbsp;pence = 1&nbsp;shilling, 20&nbsp;shillings or 240&nbsp;pence to the ] or ]), and ] 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		{| class="wikitable" style=text-align:center
			|+ Duodecimally divided units
	! colspan="6"|<big>Table of units from a base of 12</big>
	|-		|-
	! Relative<br>value		! rowspan=2 | Relative<br>value
			! colspan=2 | Length
	! French unit<br>of length
			! colspan=2 | Weight
	! English unit<br>of length
			|-
	! English unit<br>of weight
			! French
	! Roman unit<br>of weight
	! English unit<br>of mass		! English
			! English (Troy)
			! Roman
	|-		|-
	|12<sup>0</sup>		|12<sup>0</sup>
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	|]		|]
	|]		|]
	|
	|-		|-
	|12<sup>−1</sup>		|12<sup>−1</sup>
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	|]		|]
	|]		|]
	|]
	|-		|-
	|12<sup>−2</sup>		|12<sup>−2</sup>
Line 83:		Line 66:
	|]		|]
	|2 ]		|2 ]
	|2 ]		|2 ]
	|]
	|-		|-
	|12<sup>−3</sup>		|12<sup>−3</sup>
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	|]		|]
	|]		|]
	|]		|]
			|}
			== Notations and pronunciations ==
			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, 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".<ref name="Symbology Overview">{{cite journal|last=De Vlieger|first=Michael|title=Symbology Overview|journal=The Duodecimal Bulletin|volume=4X |issue=2|date=2010|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue4a2_0.pdf }}</ref> More radical proposals do not use any ] under the principle of "separate identity."<ref name="Symbology Overview" />
			Pronunciation of duodecimal numbers also has no standard, but various systems have been proposed.
			=== Transdecimal symbols ===
			{{infobox symbol
			|mark={{mono|1=<span style="display: inline-block; transform: scale(-1);">2</span>{{NBSP}}<span style="display: inline-block; transform: scale(-1);">3</span>}}
			|name = duodecimal&nbsp;{{angbr|1=ten, eleven}}
			|unicode={{ubli
			| {{unichar|1=218A|2=TURNED DIGIT TWO}}
			| {{unichar|1=218B|2=TURNED DIGIT THREE}}
			}}
			|unicode note=Block ]
			|note={{ubli
			|Arabic digits with 180° rotation, by&nbsp;]
			|In ], using the TIPA package:<ref name="LATEX">{{cite web |url=https://www.ctan.org/pkg/comprehensive |title=The Comprehensive LATEX Symbol List |year=2021 |edition=14.0 |orig-year=2007 |last=Pakin |first=Scott |website=Comprehensive TEX Archive Network }} {{pb}} {{cite web |url=https://www.ctan.org/pkg/tipa |title=tipa – Fonts and macros for IPA phonetics characters |last=Rei |first=Fukui |year=2004 |orig-year=2002 |website=Comprehensive TEX Archive Network |edition=1.3 }} {{pb}} The turned digits 2 and 3 employed in the TIPA package originated in ''The Principles of the International Phonetic Association'', University College London, 1949.</ref><br/>{{angbr|1={{code|\textturntwo}}, {{code|1=\textturnthree}}}}}}
			}}
			Several authors have proposed using letters of the alphabet for the transdecimal symbols. Latin letters such as {{angbr|{{mono|1=A, B}}}} (as in ]) or {{angbr|{{mono|1=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 {{angbr|{{mono|1=τ, ε}}}} could be used instead.<ref name="Symbology Overview"/> Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his 1935 book ''New Numbers'' {{angbr|{{mono|''X'', ''Ɛ''}}}} (italic capital X from the ] for ten and a rounded ] capital E similar to ]), along with italic numerals {{mono|''0''}}–{{mono|''9''}}.<ref name="New Numbers 1935"/>
			Edna Kramer in her 1951 book ''The Main Stream of Mathematics'' used a {{angbr|{{mono|1=*, <nowiki>#</nowiki>}}}} (] or six-pointed asterisk, ] or octothorpe).<ref name="Symbology Overview"/> The symbols were chosen because they were available on some typewriters; they are also on ]s.<ref name="Symbology Overview"/> This notation was used in publications of the Dozenal Society of America (DSA) from 1974 to 2008.<ref>{{cite journal|title=Annual Meeting of 1973 and Meeting of the Board|journal=The Duodecimal Bulletin|volume=25 |issue=1|date=1974|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue251-web_0.pdf}}</ref><ref>{{cite journal|last=De Vlieger|first=Michael|title=Going Classic|journal=The Duodecimal Bulletin|volume=49 |issue=2|date=2008|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue492_0.pdf}}</ref>
			From 2008 to 2015, the DSA used {{angbr|1={{NNBSP}}], ]{{NNBSP}}}}, the symbols devised by ].<ref name="Symbology Overview"/><ref name="DB01">{{cite journal|title=Mo for Megro|journal=The Duodecimal Bulletin|volume=1|issue=1|date=1945|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue011-web.pdf}}</ref>
			The Dozenal Society of Great Britain (DSGB) proposed symbols {{angbr|1={{NNBSP}}<span style="display: inline-block; transform: scale(-1);">{{math|2}}</span>, <span style="display: inline-block; transform: scale(-1);">{{math|3}}</span>{{NNBSP}}}}.<ref name="Symbology Overview"/> This notation, derived from Arabic digits by 180° rotation, was introduced by ] in 1857.<ref name="Symbology Overview"/><ref name="Pitman1857">{{cite news |last=Pitman |first=Isaac |author-link=Isaac Pitman |title=A Reckoning Reform |newspaper=Bedfordshire Independent |date=24 November 1857 }} Reprinted as {{cite journal |last=Pitman |first=Isaac |display-authors=0 |title=Sir Isaac Pitman on the Dozen System: A Reckoning Reform |journal=The Duodecimal Bulletin |volume=3 |issue=2 |pages=1–5 |year=1947 |url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue032-web_0.pdf }}</ref> In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies in the ].<ref name="N4399">{{cite web |author=Pentzlin |first=Karl |date=March 30, 2013 |title=Proposal to encode Duodecimal Digit Forms in the UCS |url=https://www.unicode.org/wg2/docs/n4399.pdf |publisher=ISO/IEC JTC1/SC2/WG2 |access-date=2024-06-25}}</ref> 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 ] (2015).<ref name="Unicode8">{{cite web|url=https://www.unicode.org/charts/PDF/Unicode-8.0/U80-2150.pdf|title=The Unicode Standard, Version 8.0: Number Forms|publisher=Unicode Consortium|access-date=2016-05-30}}</ref><ref>{{cite web
			| url = https://www.unicode.org/charts/PDF/U2150.pdf
			| title = The Unicode Standard 8.0
			| access-date = 2014-07-18
			}}</ref>
			After 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.<ref>{{Cite web |last=The Dozenal Society of America |date=n.d. |title=What should the DSA do about transdecimal characters? |url=https://dozenal.org/drupal/content/what-should-dsa-do-about-transdecimal-characters.html |access-date=January 1, 2018 |website=Dozenal Society of America |publisher=The Dozenal Society of America}}</ref>
			{| class="wikitable" id="transdecimal-symbols-table" style="font-size:90%; max-width:50em;"
			! colspan=2 | Symbols
			! style="width:20em" | Background
			! Note
			|-
			| <big>A</big>
			| <big>B</big>
			| As in ]
			| Allows entry on typewriters.
			|-
			| <big>T</big>
			| <big>E</big>
			| Initials of ''Ten'' and ''Eleven''
			| Used (in lower case) in ]<ref>], ''The Cambridge Introduction to Serialism'' (New York: Cambridge University Press, 2008): 276. {{ISBN|978-0-521-68200-8}} (pbk).</ref>
			|-
			| <big>X</big>
			| <big>E</big>
			| X from the ]; <br> E from ''Eleven''.
	|		|
			|-
			| <big>X</big>
			| <big>Z</big>
			| Origin of Z unknown
			| Attributed to ] & ] by the DSA.<ref name="Symbology Overview" />
			|-
			| <big>δ</big>
			| <big>ε</big>
			| Greek ] from {{lang|grc|δέκα}} "ten"; <br> ] from {{lang|grc|ένδεκα}} "eleven"<ref name="Symbology Overview"/>
			|
			|-
			| <big>τ</big>
			| <big>ε</big>
			| Greek ], ]<ref name="Symbology Overview"/>
			|
			|-
			| <big>W</big>
			| <big>∂</big>
			| W from doubling the Roman numeral V; <br> ∂ based on a pendulum
			| Silvio Ferrari in ''Calcolo Decidozzinale'' (1854).<ref name="Ferrari 1854">{{cite book|first=Silvio |last=Ferrari|title=Calcolo Decidozzinale|date=1854|page=2}}</ref>
			|-
			| <big>''X''</big>
			| <big>''Ɛ''</big>
			| italic ''X'' pronounced "dec"; <br> rounded ] ''Ɛ'', pronounced "elf"
			| Frank Andrews in ''New Numbers'' (1935), with italic ''0''–''9'' for other duodecimal numerals.<ref name="New Numbers 1935">{{cite book|first=Frank Emerson |last=Andrews |title=New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics |date=1935 |page=52 |publisher=Harcourt, Brace and company |url=https://archive.org/details/newnumbershowacc0000fran/page/52/mode/1up?q=%22quantity+eleven%22 |url-access=limited}}</ref>
			|-
			| <big>{{mono|1=*}}</big>
			| <big>{{mono|1=<nowiki>#</nowiki>}}</big>
			| ] or six-pointed asterisk,<br/>] or octothorpe
			| On ]s; used by Edna Kramer in ''The Main Stream of Mathematics'' (1951); used by the DSA {{nobr|1974–2008}}<ref name="bellchange">{{cite journal|title=Annual Meeting of 1973 and Meeting of the Board|journal=The Duodecimal Bulletin|volume=25 |issue=1|date=1974|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue251-web_0.pdf}}</ref><ref name="classic">{{cite journal|last=De Vlieger|first=Michael|title=Going Classic|journal=The Duodecimal Bulletin|volume=49 |issue=2|date=2008|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue492_0.pdf}}</ref><ref name="Symbology Overview"/>
			|-
			| <big><span style="display: inline-block; transform: scale(-1);">2</span></big>
			| <big><span style="display: inline-block; transform: scale(-1);">3</span></big>
			| {{ubli
			| Digits 2 and 3 rotated 180°
			}}
			| ] (1857);<ref name="Pitman1857"/> used by the DSGB; used by the DSA since 2015; included in ] (2015)<ref name="Unicode8">{{cite web|url=https://www.unicode.org/charts/PDF/Unicode-8.0/U80-2150.pdf|title=The Unicode Standard, Version 8.0: Number Forms|publisher=Unicode Consortium|access-date=2016-05-30}}</ref><ref>{{cite web
			| url = https://www.unicode.org/charts/PDF/U2150.pdf
			| title = The Unicode Standard 8.0
			| access-date = 2014-07-18 }}</ref>
			|-
			| <big>]</big>
			| <big>]</big>
			| Pronounced "dek", "el"
			| {{ubl
			|] (1945/1932?).<ref name="Symbology Overview"/><ref name="DB01">{{cite journal|title=Mo for Megro|journal=The Duodecimal Bulletin|volume=1|issue=1|date=1945|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue011-web.pdf}}</ref>
			|Used by the DSA 1945&ndash;1974 and 2008&ndash;2015<ref name="bellchange"/><ref name="classic" />}}
	|}		|}
			==={{anchor|Humphrey point}}Base notation===
	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 (]) on one hand (three on each of four fingers).<ref>Pittman, Richard. 1990. Origin of Mesopotamian duodecimal and sexagesimal counting systems. ''Philippine Journal of Linguistics'' 21(1)"97.</ref><ref>{{Cite web |title=ヒマラヤの満月と十二進法 (The Full Moon in the Himalayas and the Duodecimal System) |last=Nishikawa |first=Yoshiaki |year=2002 |url=http://www.kankyok.co.jp/nue/nue11/nue11_01.html |accessdate=2008-03-24 |postscript={{inconsistent citations}} |deadurl=yes |archiveurl=https://web.archive.org/web/20080329150110/http://www.kankyok.co.jp/nue/nue11/nue11_01.html |archivedate=March 29, 2008 |df= }}</ref> It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional ] 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.<ref name=Ifrah>{{Cite book
			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<sub>12</sub> = 64<sub>10</sub>". To avoid ambiguity about the meaning of the subscript 10, the subscripts might be spelled out, "54<sub>twelve</sub> = 64<sub>ten</sub>". 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<sub>z</sub> = 64<sub>d</sub>".<ref name="Volan 2015">{{Cite journal |last=Volan |first=John |date=July 2015 |title=Base Annotation Schemes |url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue521.pdf |journal=The Duodecimal Bulletin |volume=62}}</ref>
	| last = Ifrah
	| first = Georges
	| author-link = Georges Ifrah
	| title = The Universal History of Numbers: From prehistory to the invention of the computer
	| publisher = ]
	| year= 2000
	| page =
	| isbn = 0-471-39340-1
	| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
	}}. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.</ref><ref name=Macey>{{Cite book|last=Macey|first=Samuel L.|title=The Dynamics of Progress: Time, Method, and Measure|year=1989|publisher=University of Georgia Press|location=Atlanta, Georgia|isbn=978-0-8203-3796-8|page=92|url=https://books.google.com/books?id=xlzCWmXguwsC&pg=PA92&lpg=PA92|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
			Other proposed methods include italicizing duodecimal numbers "''54'' = 64", adding a "Humphrey point" (a ] instead of a ]) to duodecimal numbers "54;6 = 64.5", prefixing duodecimal numbers by an asterisk "*54&nbsp;= 64", or some combination of these. The Dozenal Society of Great Britain uses an asterisk prefix for duodecimal whole numbers, and a Humphrey point for other duodecimal numbers.<ref name="Volan 2015" />
	== Notations and pronunciations ==
	===Transdecimal symbols===
	In a duodecimal place system twelve is written as 10, but there are numerous proposals for how to write ] and ].<ref name="Symbology Overview">{{cite journal|last=De Vlieger|first=Michael|title=Symbology Overview|journal=The Duodecimal Bulletin|volume=4X |issue=2|date=2010|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue4a2_0.pdf}}</ref> The simplified notations use only basic and easy to access letters such as ''T'' and ''E'' (for ''ten'' and ''eleven''), ''X'' and ''Z'', ''t'' and ''e'', ''d'' and ''k'', others use ''A'' and ''B'' or ''a'' and ''b'' as in the ] system. Some employ Greek letters such as δ (standing for Greek δέκα 'ten') and ε (for Greek ένδεκα 'eleven'), or τ and ε.<ref name="Symbology Overview"/> Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book ''New Numbers'' an X (from the ] for ten) and a script E (ℰ, {{U+|2130}}).<ref name="New Numbers 1935">{{cite book|first=Frank Emerson |last=Andrews|title=New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics|date=1935|page=52|url=https://books.google.com/books?id=zfXuAAAAMAAJ&q=eleven}}</ref>
			=== Pronunciation ===
	{{Symb|] ]<br>] ]}}
			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.
	The '''Dozenal Society of Great Britain''' proposes a rotated digit two <span style="display:inline-block; {{transform|rotate(180deg)}}">2</span> (↊, {{U+|218A}}) for ] and a reversed or rotated digit three <span style="display:inline-block; {{transform|rotate(180deg)}}">3</span> (↋, {{U+|218B}}) for ].<ref name="Symbology Overview"/> This notation was introduced by Sir ].<ref name="Symbology Overview"/><ref name="Pitman1857">{{cite journal|last=Pitman|first=Isaac|title=A Reckoning Reform |journal=The Duodecimal Bulletin|volume=3|issue=2|date=1947|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue032-web_0.pdf}}</ref>
			==== Base-12 nomenclature in English ====
	Until 2015, the '''Dozenal Society of America''' ('''DSA''') used ] and ], the symbols devised by ].<ref name="Symbology Overview"/><ref name="DB01">{{cite journal|title=Mo for Megro|journal=The Duodecimal Bulletin|volume=1|issue=1|date=1945|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue011-web.pdf}}</ref> After the Pitman digits (↊, {{U+|218A}} and ↋, {{U+|218B}}) were added to Unicode in 2015<ref name="Unicode8" /><ref name=":0" />, the DSA took a vote and then began publishing content using the Pitman digits instead.<ref>{{Cite web|url=http://www.dozenal.org/drupal/content/what-should-dsa-do-about-transdecimal-characters.html|title=What should the DSA do about transdecimal characters? {{!}} The Dozenal Society of America|website=www.dozenal.org|language=en|access-date=2018-01-01}}</ref><ref name=":1">{{Cite journal|last=Volan|first=John|date=July 2015|title=Base Annotation Schemes|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue521.pdf|journal=Duodecomal Bulletin|volume=62|pages=|via=}}</ref> They still use the letters X and E as the equivalent in ASCII text.
			{{Unreferenced section|date=November 2024}}
			Another nominal for ] (12<sub>]</sub>) is a '']'' (10<sub>12</sub> ''or'' 1•10<sup>1</sup><sub>12</sub>).
			] (144<sub>10</sub>) is also known as a ] (100<sub>12</sub> ''or'' 1•10<sup>2</sup><sub>12</sub>).
	Other proposals are more creative or aesthetic, for example, Edna Kramer in her 1951 book ''The Main Stream of Mathematics'' used a six-pointed asterisk (]) ⚹ for ten and a ] (or octothorpe) # for eleven.<ref name="Symbology Overview"/> The symbols were chosen because they are available in typewriters and already present in ] ].<ref name="Symbology Overview"/> This notation was used in publications of the Dozenal Society of America in the period 1974–2008.<ref>{{cite journal|title=Annual Meeting of 1973 and Meeting of the Board|journal=The Duodecimal Bulletin|volume=25 |issue=1|date=1974|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue251-web_0.pdf}}</ref><ref>{{cite journal|last=De Vlieger|first=Michael|title=Going Classic|journal=The Duodecimal Bulletin|volume=49 |issue=2|date=2008|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue492_0.pdf}}</ref> Many don't use any of the Hindu-Arabic symbols, under the principle of "separate identity."<ref name="Symbology Overview" />
			] is (1728<sub>10</sub>) also known as a ''great-gross'' (1,000<sub>12</sub> or 1•10<sup>3</sup><sub>12</sub>).
			For the next ] of twelve that follow those aforementioned, the affixes (dozen-, gross-, great-) are used to produce names for these powers of twelve that have a greater ] value. ]<sub>10</sub> or 10,000<sub>12</sub> may be rendered a ''dozen-great-gross''; so ]<sub>10</sub> or 100,000<sub>12</sub> is a ''gross-great-gross'', with ]<sub>10</sub> or 1,000,000<sub>12</sub> being known as a ''great-great-gross''.
			It should be made plain that the indice's being a multiple of three, e.g. 10<sup>3</sup><sub>12</sub> , 10<sup>6</sup><sub>12</sub> , 10<sup>9</sup><sub>12</sub> results, in these examples, in a ''great gross'', a ''great-great-gross'', and a ''great-great-great-gross'', respectively.
	===Base notation===
	There are also varying proposals of how to distinguish a duodecimal number from a decimal one, or one in a different base. They include italicizing duodecimal numbers ( ''54'' = 64 ), adding a "Humphrey point" (a semicolon ";" instead of a decimal point "." ) to duodecimal numbers ( 54; = 64. ) ( 54;0 = 64.0 ), or some combination of the two. More also add extra marking to one or more bases. Others use subscript or affixed labels to indicate the base, allowing for more than decimal and duodecimal to be represented:<ref name=":1" />
	{| class="wikitable"		{| class="wikitable"
			! Scientific notation||Positional notation||Name||Decimal
	!Common Base
	!Abb.
	!Letter
	!Cardinal
	!Decimal
	!Duodecimal
	|-		|-
			|style="text-align:center"|1•10<sup>0</sup>
	|'''<u>b</u>in'''ary
			|1
	|<u>b</u>in
			|style="text-align:center"|One
	|b
	|two		|1
	|2
	|2
	|-		|-
			|style="text-align:center" |A•10<sup>0</sup>
	|'''<u>o</u>ct'''al
			|A
	|<u>o</u>ct
	|o		|Ten
	|eight
	|8
	|8
	|-
	|'''<u>d</u>ec'''imal
	|<u>d</u>ec
	|d
	|ten
	|10		|10
	|↊
	|-		|-
			|style="text-align:center" |B•10<sup>0</sup>
	|'''do<u>z</u>'''enal (duodecimal)
			|B
	|do<u>z</u>
			|Eleven
	|z
			|11
	|twelve
	|12		|-
			| style="text-align:center" |1•10<sup>1</sup>
	|10		|10
			| style="text-align:center" |Twelve
			|12
	|-		|-
			| style="text-align:center" |5•10<sup>1</sup>
	|'''he<u>x</u>'''adecimal
			|50
	|he<u>x</u>
			| style="text-align:center" |Five dozen
	|x
			|60
	|sixteen
	|16		|-
			| style="text-align:center" |1•10<sup>2</sup>
	|14
			|100
			| style="text-align:center" |One gross
			|144
			|-
			| style="text-align:center" |2;6•10<sup>2</sup>
			|260
			| style="text-align:center" |Two gross, six dozen
			|360
			|-
			| style="text-align:center" |1•10<sup>3</sup>
			|1,000
			| style="text-align:center" |One great-gross
			|1,728
			|-
			| style="text-align:center" |1•10<sup>4</sup>
			|10,000
			| style="text-align:center" |One dozen-great-gross
			|20,736
			|-
			| style="text-align:center" |1•10<sup>5</sup>
			|100,000
			| style="text-align:center" |One gross-great-gross
			|248,832
			|-
			| style="text-align:center" |1•10<sup>6</sup>
			|1,000,000
			| style="text-align:center" |One great-great-gross
			|2,985,984
	|}		|}
	This allows one to write "54<sub>z</sub> = 64<sub>d</sub>," "54<sub>twelve</sub> = 64<sub>ten</sub>" or "doz 54 = dec 64." In programming, binary, octal, and hexadecimal often use a similar scheme: a binary number starts with <code>0b</code>, octal with <code>0o</code>, and hexadecimal with <code>0x</code>.
	=== Pronunciation ===		==== Duodecimal numbers ====
	The Dozenal Society of America suggests the pronunciation of ten and eleven as "dek" and "el", each ] has its own name and the prefix ''e''- is added for fractions.<ref name="DB01"/><ref name="Zirkel2010">{{cite journal|last=Zirkel|first=Gene|title=How Do You Pronounce Dozenals?		In this system, the prefix ''e''- is added for fractions.<ref name="DB01"/><ref name="Zirkel2010">{{cite journal|last=Zirkel|first=Gene|title=How Do You Pronounce Dozenals?
	|journal=The Duodecimal Bulletin|volume=4E |issue=2|date=2010|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue4b2_0.pdf}}</ref> The symbol corresponding to the decimal point or decimal comma, separating the whole number part from the fractional part, is the semicolon ";". The overall system is:<ref name="DB01"/>		|journal=The Duodecimal Bulletin|volume=4E |issue=2|date=2010|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue4b2_0.pdf }}</ref>
	{| class="wikitable"		{| class="wikitable"
			! Duodecimal <br> number||Number <br> name||Decimal <br> number||Duodecimal <br> fraction||Fraction <br> name
	|-		|-
			|style="text-align:right"|1;
	! Duodecimal||Name||Decimal||Duodecimal fraction||Name
	|-
	|style="text-align:right"|1
	|one		|one
	|style="text-align:right"|1		|style="text-align:right"|1
	|colspan="2"|		|colspan="2"|
	|-		|-
	|style="text-align:right"|10		| style="text-align:right" |10;
	|{{abbr|do|pronounced 'doh'}}		|{{abbr|do|pronounced 'doʊ'}}
	|style="text-align:right"|12		|style="text-align:right"|12
	|style="text-align:right"|0;1||edo		|0;1||edo
	|-		|-
	|style="text-align:right"|100		| style="text-align:right" |100;
	|{{abbr|gro|pronounced 'groh'}}		|{{abbr|gro|pronounced 'ɡroʊ'}}
	|style="text-align:right"|144		|style="text-align:right"|144
			|0;01
	|style="text-align:right"|0;01
	|egro		|egro
	|-		|-
	|style="text-align:right"|1,000		| style="text-align:right" |1,000;
	|{{abbr|mo|pronounced 'moh'}}		|{{abbr|mo|pronounced 'moʊ'}}
	|style="text-align:right"|1,728		|style="text-align:right"|1,728
			|0;001
	|style="text-align:right"|0;001
	|emo		|emo
	|-		|-
	|style="text-align:right"|10,000		| style="text-align:right" |10,000;
	|do-mo		|do-mo
	|style="text-align:right"|20,736		|style="text-align:right"|20,736
	|style="text-align:right"|0;000,1		|0;000,1
	|edo-mo		|edo-mo
	|-		|-
	|style="text-align:right"|100,000		| style="text-align:right" |100,000;
	|gro-mo		|gro-mo
	|style="text-align:right"|248,832		|style="text-align:right"|248,832
	|style="text-align:right"|0;000,01		|0;000,01
	|egro-mo		|egro-mo
	|-		|-
	|style="text-align:right"|1,000,000		| style="text-align:right" |1,000,000;
	|bi-mo		|bi-mo
	|style="text-align:right"|2,985,984		|style="text-align:right" |2,985,984
	|style="text-align:right"|0;000,001		|0;000,001
	|ebi-mo		|ebi-mo
	|-		|-
	|style="text-align:right"|1,000,000,000		| style="text-align:right" |10,000,000;
	|tri-mo		|do-bi-mo
	|style="text-align:right"|5,159,780,352		|style="text-align:right"|35,831,808
	|style="text-align:right"|0;000,000,001		|0;000,000,1
	|etri-mo		|edo-bi-mo
			|-
			| style="text-align:right" |100,000,000;
			|gro-bi-mo
			|style="text-align:right"|429,981,696
			|0;000,000,01
			|egro-bi-mo
	|}		|}
			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 are pronounced differently. 12 is "one do two", 30 is "three do", 100 is "one gro", BA9 (ET9) is "el gro dek do nine", B8,65A,300 (E8,65T,300) is "el do eight bi-mo, six gro five do dek mo, three gro", and so on.<ref name="Zirkel2010"/>
			Multiple digits in this series are pronounced differently: 12 is "do two"; 30 is "three do"; 100 is "gro"; {{d3}}{{d2}}9 is "el gro dek do nine"; {{d3}}86 is "el gro eight do six"; 8{{d3}}{{d3}},15{{d2}} 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.<ref name="Zirkel2010"/>
	==Advocacy and "dozenalism"==
	The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book ''New Numbers: How Acceptance of 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.
			==== Systematic Dozenal Nomenclature (SDN) ====
	]]]
			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 ]s (with syllables '''dec''' and '''lev''' for the two extra digits needed for duodecimal) to express which power is meant.<ref name="DBX1">{{cite journal |url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue511a_0.pdf |title=Systematic Dozenal Nomenclature and other nomenclature system. |journal=The Duodecimal Bulletin |volume=61 |number=1 }}</ref><ref name="DSAGoodman2016"/>
	The Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word "dozenal" instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology.
			{| class="wikitable"
	It should be noted that the etymology of 'dozenal' is itself also an expression based on base-ten terminology since 'dozen' is a direct derivation of the French word 'douzaine' which is a derivative of the French word for twelve, '']'' which is related to the old French word 'doze' from Latin 'duodecim'. It has been suggested by some members of the Dozenal Society of America and Duodecimal Society of Great Britain that a more apt word would be 'uncial'. This is a derivation of the Latin word 'uncia' which means one-twelfth and is the base-twelve analogue of the Latin word 'decima' meaning one-tenth. In the same manner as ''decimal'' comes from the Latin word for one-tenth decima, (Latin for ten was decem), the direct analogue for a base-twelve system is ''uncial''. An early use of this word can be found in Vol 1 Issue 2 of of the DSA dated June 1945 in which a submission on page 9 by a Pvt William S. Crosby titled "The Uncial Jottings of a Harried Infantryman", he includes the same argument for the word 'uncial'. Although not accepted by either of these two 'Uncial' societies, the use is beginning to grow.
			! Duodecimal <br> number||Number <br> name||Decimal <br> number||Duodecimal <br> fraction||Fraction <br> name
			|-
			|style="text-align:right"|1;
			|one
			|style="text-align:right"|1
			|colspan="2"|
			|-
			|style="text-align:right"|10;
			|unqua
			|style="text-align:right"|12
			|0;1
			|uncia
			|-
			|style="text-align:right"|100;
			|biqua
			|style="text-align:right"|144
			|0;01
			|bicia
			|-
			|style="text-align:right"|1,000;
			|triqua
			|style="text-align:right"|1,728
			|0;001
			|tricia
			|-
			|style="text-align:right"|10,000;
			|quadqua
			|style="text-align:right"|20,736
			|0;000,1
			|quadcia
			|-
			|style="text-align:right"|100,000;
			|pentqua
			|style="text-align:right"|248,832
			|0;000,01
			|pentcia
			|-
			|style="text-align:right"|1,000,000;
			|hexqua
			|style="text-align:right"|2,985,984
			|0;000,001
			|hexcia
			|}
			After hex-, further prefixes continue sept-, oct-, enn-, dec-, lev-, unnil-, unun-.
	The renowned mathematician and mental calculator ] was an outspoken advocate of the advantages and superiority of duodecimal over decimal:
	{{quote|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 the decimal. This is my experience; I am certain that even more so it would be the experience of others.|A. C. Aitken||in ''The Listener'', January 25, 1962}}
			== Advocacy and "dozenalism" ==
	{{quote|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 the duodecimal.|A. C. Aitken|'''' (Edinburgh / London: Oliver & Boyd, 1962)}}
			] used 12 as the base for his constructed language ] in 1906, noting it being the smallest number with four factors and its prevalence in commerce.<ref>The Prodigy (Biography of WJS) pg </ref>
			The case for the duodecimal system was put forth at length in ]' 1935 book ''New Numbers: How Acceptance of 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.<ref name="New Numbers 1935"/>
	In ]' short story ] Herbert Ashe, a melancholy English engineer, working for the Southern Argentine Railway company, is converting a duodecimal number system to a ] system. He leaves behind on his death in 1937 a manuscript Orbis Tertius that posthumously identifies him as one of the anonymous authors of the encyclopaedia of Tlön.
			]]]
	In ]'s ] novels, Conrad introduces a duodecimal system of arithmetic at the suggestion of a merchant, who is accustomed to buying and selling goods in dozens and grosses, rather than tens or hundreds. He then invents an entire system of weights and measures in base twelve, including a clock with twelve hours in a day, rather than twenty-four hours.{{citation needed|date=May 2016}}
			Both the Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of 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, '']'', descended from Latin ''duodecim''.
			Mathematician and mental calculator ] was an outspoken advocate of duodecimal:
	In ]'s ''Kryon: Alchemy of the Human Spirit'', a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by ] (a fictional entity believed in by ] circles) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the ], as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.<ref>{{cite book|first=Lee|last=Carroll|title=Kryon—Alchemy of the Human Spirit|url=https://www.kryon.com/k_13.html|date=1995|publisher=The Kryon Writings, Inc.|isbn=0-9636304-8-2}}</ref>
			{{blockquote|text=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 the decimal. This is my experience; I am certain that even more so it would be the experience of others.|author=A. C. Aitken|source="Twelves and Tens" in ''The Listener'' (January 25, 1962)<ref>A. C. Aitken (January 25, 1962) ''The Listener''.</ref>}}
			{{Blockquote|text=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 the duodecimal.|author=A. C. Aitken|source=''The Case Against Decimalisation'' (1962)<ref>A. C. Aitken (1962) . Edinburgh / London: Oliver & Boyd.</ref>}}
	In "Little Twelvetoes", American television series '']'' portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols.<ref></ref>
	=== In computing ===		=== In media ===
			In "Little Twelvetoes," an episode of the American educational television series '']'', a farmer encounters an alien being with twelve fingers on each hand and twelve toes on each foot who uses duodecimal arithmetic. The alien uses "dek" and "el" as names for ten and eleven, and Andrews' script-X and script-E for the digit symbols.<ref>{{cite web|url=http://www.schoolhouserock.tv/Little.html|archive-url=https://web.archive.org/web/20100206052053/http://www.schoolhouserock.tv/Little.html|url-status=dead|archive-date=6 February 2010|title=SchoolhouseRock - Little Twelvetoes|date=6 February 2010}}</ref><ref>{{Cite book|last=Bellos|first=Alex|url=https://books.google.com/books?id=FA_HwoEzSQUC|title=Alex's Adventures in Numberland|date=2011-04-04|publisher=A&C Black|isbn=978-1-4088-0959-4|pages=50|language=en|author-link=Alex Bellos}}</ref>
	In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies of Great Britain and America in the ].<ref name="N4399">{{cite web|url=http://std.dkuug.dk/jtc1/sc2/wg2/docs/n4399.pdf|title=Proposal to encode Duodecimal Digit Forms in the UCS|date=2013-03-30|publisher=ISO/IEC JTC1/SC2/WG2, Document N4399|access-date=2016-05-30|author=Karl Pentzlin}}</ref> Of these, the British forms were accepted for encoding as characters at code points {{U+|218A}} {{smallcaps|turned digit two}} (↊) and {{U+|218B}} {{smallcaps|turned digit three}} (↋) They have been included in the ] release in June 2015.<ref name="Unicode8" /><ref name=":0">{{cite web|url=http://www.unicode.org/charts/PDF/Unicode-8.0/U80-2150.pdf|title=The Unicode Standard, Version 8.0: Number Forms|publisher=Unicode Consortium|accessdate=2016-05-30}}</ref>
			=== Duodecimal systems of measurements ===
	Unicode points {{U+|218C}} and {{U+|218D}} seem to be reserved for the Dwiggins digits (stylized X and E).<ref>{{Cite web|url=http://www.fileformat.info/info/unicode/char/218c/index.htm|title=U+218C|website=www.fileformat.info|access-date=2018-01-02}}</ref>
			] proposed by dozenalists include:
			* Tom Pendlebury's TGM system<ref>{{cite web|last1=Pendlebury|first1=Tom|last2=Goodman|first2=Donald|title=TGM: A Coherent Dozenal Metrology|url=http://www.dozenal.org/drupal/sites_bck/default/files/tgm_0.pdf |date=2012|publisher=The Dozenal Society of Great Britain}}</ref><ref name="DSAGoodman2016">{{cite web |last=Goodman |first=Donald |title=Manual of the Dozenal System |date=2016 |url=http://www.dozenal.org/drupal/sites_bck/default/files/DSA_mods_rev.pdf |publisher=Dozenal Society of America|access-date=27 April 2018}}</ref>
			* Takashi Suga's Universal Unit System<ref>{{cite web|last=Suga|first=Takashi|title=Proposal for the Universal Unit System|url=http://www.asahi-net.or.jp/~dd6t-sg/univunit-e/revised.pdf |date=22 May 2019}}</ref><ref name="DSAGoodman2016" />
			* John Volan's Primel system<ref>{{cite journal |last1=Volan |first1=John |date= |title=The Primel Metrology |url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue531.pdf |journal=The Duodecimal Bulletin |volume=63 |issue=1 |pages=38–60<!--be careful; the Duodecimal Bulletin, as one might expect, numbers its pages in duodecimal--> }}</ref>
			== Comparison to other number systems ==
	Few fonts support these new characters, but , ], ], ], and Symbola do.
			:''In this section, numerals are in decimal. For example, "10" means 9+1, and "12" means 9+3.''
			The Dozenal Society of America argues that if a base 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".<ref name="dsafaq" />
	Also, the turned digits two and three are available in ] as <code>\textturntwo</code> and <code>\textturnthree</code>.<ref name="LATEX">{{cite web|url=http://library.caltech.edu/etd/symbols-a4.pdf|title=The Comprehensive LATEX Symbol List|date=2009|access-date=2016-05-30|author=Scott Pakin}}</ref>
			The number 12 has six factors, which are ], ], ], ], ], and ], of which 2 and 3 are ]. It 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 ], ], ], and ], of which 2 and 5 are prime.<ref name="dsafaq" /> 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, ], is below the DSA's stated threshold.
	=== Duodecimal clock ===
	*
	*
			] and ] only have 2 as a prime factor. Therefore, in ] and ], the only ] are those whose ] is a ].
	=== Duodecimal metric systems ===
	] proposed by dozenalists include:
	* Tom Pendlebury's TGM system<ref>{{cite web|last=Pendlebury|first=Tom|title=TGM. A coherent dozenal metrology based on Time, Gravity and Mass|url=http://www.dozenalsociety.org.uk/pdfs/TGMbooklet.pdf|date=May 2011|publisher=The Dozenal Society of Great Britain}}</ref>
	* Takashi Suga's Universal Unit System<ref>{{cite web|last=Suga|first=Takashi|title=Proposal for the Universal Unit System|url=http://www.asahi-net.or.jp/~dd6t-sg/univunit-e/|date=2002}}</ref>
			] 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). ] was actually used by the ancient ]ians and ]ns, among others; its base, ], 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 ]; the pattern follows the ]s. However, these numbers are quite large to use as bases, and are far beyond the DSA's stated threshold.
	== Comparison to other numeral systems ==
	]
			In all base systems, there are similarities to the representation of multiples of numbers that are one less than or one more than the base.{{Clear}}''In the following multiplication table, numerals are written in duodecimal. For example, "10" means twelve, and "12" means fourteen.''<!-- the {{-}} template keeps the multiplication table from squeezing the heading for the next section-->
	The number 12 has six factors, which are ], ], ], ], ], and ], of which 2 and 3 are ]. The decimal system has only four factors, which are ], ], ], and ], of which 2 and 5 are prime. ] (base 20) adds two factors to those of ten, namely ] and ], but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base, and so the digit set and the multiplication table are much larger. Binary has only two factors, 1 and 2, the latter being prime. ] (base 16) has five factors, adding 4, ] and ] to those of 2, but no additional prime. Trigesimal (base 30) is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). ]—which the ancient ] and ]ns among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors. The smallest system that has four different prime factors is base 210 and the pattern follows the ]s. In all base systems, there are similarities to the representation of multiples of numbers which are one less than the base.{{Clear}}<!-- the {{-}} template keeps the multiplication table from squeezing the heading for the next section-->
			{| class="wikitable" style="text-align: right;"
			|+ Duodecimal multiplication table
			!style="width:7.69%"|×
			!style="text-align: right; width:7.69%"|1
			!style="text-align: right; width:7.69%"|2
			!style="text-align: right; width:7.69%"|3
			!style="text-align: right; width:7.69%"|4
			!style="text-align: right; width:7.69%"|5
			!style="text-align: right; width:7.69%"|6
			!style="text-align: right; width:7.69%"|7
			!style="text-align: right; width:7.69%"|8
			!style="text-align: right; width:7.69%"|9
			!style="text-align: right; width:7.69%"|{{d2}}
			!style="text-align: right; width:7.69%"|{{d3}}
			!style="text-align: right; width:7.69%"|10
			|-
			!style="text-align: right; width:7.69%"|1
			|1||2||3||4||5||6||7||8||9||{{d2}}||{{d3}}||10
			|-
			!style="text-align: right;"|2
			|2||4||6||8||{{d2}}||10||12||14||16||18||1{{d2}}||20
			|-
			!style="text-align: right;|3
			|3||6||9||10||13||16||19||20||23||26||29||30
			|-
			!style="text-align: right;"|4
			|4||8||10||14||18||20||24||28||30||34||38||40
			|-
			!style="text-align: right;"|5
			|5||{{d2}}||13||18||21||26||2{{d3}}||34||39||42||47||50
			|-
			!style="text-align: right;"|6
			|6||10||16||20||26||30||36||40||46||50||56||60
			|-
			!style="text-align: right;"|7
			|7||12||19||24||2{{d3}}||36||41||48||53||5{{d2}}||65||70
			|-
			!style="text-align: right;"|8
			|8||14||20||28||34||40||48||54||60||68||74||80
			|-
			!style="text-align: right;"|9
			|9||16||23||30||39||46||53||60||69||76||83||90
			|-
			!style="text-align: right;"|{{d2}}
			|{{d2}}||18||26||34||42||50||5{{d2}}||68||76||84||92||A0
			|-
			!style="text-align: right;"|{{d3}}
			|{{d3}}||1{{d2}}||29||38||47||56||65||74||83||92||{{d2}}1||B0
			|-
			!style="text-align: right;"|10
			|10||20||30||40||50||60||70||80||90||A0||B0||100
			|}
	== Conversion tables to and from decimal ==		== Conversion tables to and from decimal ==
	To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under ]). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0.01 and ƐƐƐ,ƐƐƐ.ƐƐ to decimal, or any decimal number between 0.01 and 999,999.99 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example:		To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under ]). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0;1 and {{d3}}{{d3}},{{d3}}{{d3}}{{d3}};{{d3}} to decimal, or any decimal number between 0.1 and 99,999.9 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example:
	:123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08		:12,345.6 = 10,000 + 2,000 + 300 + 40 + 5 + 0.6
	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, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). 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:		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 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:
	:<small>(duodecimal)</small> 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = <small>(decimal)</small> 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58{{overline|3}}333333333... + 0.0{{overline|5}}5555555555...		:<small>(duodecimal)</small> 10,000 + 2,000 + 300 + 40 + 5 + 0;6 <br> = <small>(decimal)</small> 20,736 + 3,456 + 432 + 48 + 5 + 0.5
	Now, because the summands are already converted to base ten, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result:		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:
	Duodecimal -----> Decimal		Duodecimal ---> Decimal
	100,000 = 248,832		10,000 = 20,736
	20,000 = 41,472		2,000 = 3,456
	3,000 = 5,184		300 = 432
	400 = 576		40 = 48
	50 = 60		5 = 5
	+ 6 = + 6		+ 0;6 = + 0.5
			-----------------------------
	0.7 = 0.58{{overline|3}}333333333...
			12,345;6 = 24,677.5
	0.08 = 0.0{{overline|5}}5555555555...
	--------------------------------------------
	123,456.78 = 296,130.63{{overline|8}}888888888...
	That is, <small>(duodecimal)</small> 123,456.78 equals <small>(decimal)</small> 296,130.63{{overline|8}} ≈ 296,130.64		That is, <small>(duodecimal)</small> 12,345;6 equals <small>(decimal)</small> 24,677.5
	If the given number is in decimal and the target base is duodecimal, the method is basically same. Using the digit conversion tables:		If the given number is in decimal and the target base is duodecimal, the method is same. Using the digit conversion tables:
	<small>(decimal)</small> 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = <small>(duodecimal)</small> 49,ᘔ54 + Ɛ,6ᘔ8 + 1,8ᘔ0 + 294 + 42 + 6 + 0.8{{overline|4972}}4972497249724972497... + 0.{{overline|0Ɛ62ᘔ68781Ɛ05915343ᘔ}}0Ɛ62...		<small>(decimal)</small> 10,000 + 2,000 + 300 + 40 + 5 + 0.6 <br> = <small>(duodecimal)</small> 5,954 + 1,1{{d2}}8 + 210 + 34 + 5 + 0;{{overline|7249}}
			To sum these partial products and recompose the number, the addition must be done with duodecimal rather than decimal arithmetic:
	However, in order to do this sum and recompose the number, now the addition tables for the duodecimal system have to be used, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in duodecimal as well. In decimal, 6 + 6 equals 12, but in duodecimal it equals 10; so, if using decimal arithmetic with duodecimal numbers one would arrive at an incorrect result. Doing the arithmetic properly in duodecimal, one gets the result:
	Decimal -----> Duodecimal		Decimal --> Duodecimal
	100,000 = 49,ᘔ54		10,000 = 5,954
	20,000 = Ɛ,6ᘔ8		2,000 = 1,1{{d2}}8
	3,000 = 1,8ᘔ0		300 = 210
	400 = 294		40 = 34
	50 = 42		5 = 5
	+ 6 = + 6		+ 0.6 = + 0;{{overline|7249}}
			-------------------------------
	0.7 = 0.8{{overline|4972}}4972497249724972497...
	0.08 = 0.{{overline|0Ɛ62ᘔ68781Ɛ05915343ᘔ}}0Ɛ62...		12,345.6 = 7,189;{{overline|7249}}
	--------------------------------------------------------
	123,456.78 = 5Ɛ,540.9{{overline|43ᘔ0Ɛ62ᘔ68781Ɛ059153}}43ᘔ...
	That is, <small>(decimal)</small> 123,456.78 equals <small>(duodecimal)</small> 5Ɛ,540.9{{overline|43ᘔ0Ɛ62ᘔ68781Ɛ059153}}... ≈ 5Ɛ,540.94		That is, <small>(decimal)</small> 12,345.6 equals <small>(duodecimal)</small> 7,189;{{overline|7249}}
	=== Duodecimal to decimal digit conversion ===		=== Duodecimal to decimal digit conversion ===
	{|class="wikitable"		{| class="wikitable"
			! Duod.
	|-
			! Dec.
	! style="background:silver;" | ''Duod.''
	! ''Dec.''		! Duod.
			! Dec.
	! style="background:silver;" | ''Duod.''
	! ''Dec.''		! Duod.
			! Dec.
	! style="background:silver;" | ''Duod.''
	! ''Dec.''		! Duod.
			! Dec.
	! style="background:silver;" | ''Duod.''
	! ''Dec.''		! Duod.
			! Dec.
	! style="background:silver;" | ''Duod.''
	! ''Dec.''		! Duod.
			! Dec.
	! style="background:silver;" | ''Duod.''
			|- style="text-align:right"
	! ''Dec.''
	! style="background:silver;" | ''Duod.''
	! ''Dec.''
	! style="background:silver;" | ''Duod.''
	! ''Dec.''
	! style="background:silver;" | ''Duod.''
	! ''Dec.''
	|-
	| style="background:silver;"| '''1,000,000'''
	| 2,985,984
	| style="background:silver;"| '''100,000'''
	| 248,832
	| style="background:silver;"| '''10,000'''		| style="background:silver;"| '''10,000'''
	| 20,736		| 20,736
Line 346:		Line 525:
	| style="background:silver;"| '''1'''		| style="background:silver;"| '''1'''
	| 1		| 1
	| style="background:silver;"| '''0.1'''		| style="background:silver;"| '''0;1'''
	| 0.08{{overline|3}}		| style="text-align:left" | 0.08{{overline|3}}
	| style="background:silver;"| '''0.01'''		|- style="text-align:right"
	| 0.0069{{overline|4}}
	|-
	| style="background:silver;"| '''2,000,000'''
	| 5,971,968
	| style="background:silver;"| '''200,000'''
	| 497,664
	| style="background:silver;"| '''20,000'''		| style="background:silver;"| '''20,000'''
	| 41,472		| 41,472
Line 365:		Line 538:
	| style="background:silver;"| '''2'''		| style="background:silver;"| '''2'''
	| 2		| 2
	| style="background:silver;"| '''0.2'''		| style="background:silver;"| '''0;2'''
	| 0.1{{overline|6}}		| style="text-align:left" | 0.1{{overline|6}}
	| style="background:silver;"| '''0.02'''		|- style="text-align:right"
	| 0.013{{overline|8}}
	|-
	| style="background:silver;"| '''3,000,000'''
	| 8,957,952
	| style="background:silver;"| '''300,000'''
	| 746,496
	| style="background:silver;"| '''30,000'''		| style="background:silver;"| '''30,000'''
	| 62,208		| 62,208
Line 384:		Line 551:
	| style="background:silver;"| '''3'''		| style="background:silver;"| '''3'''
	| 3		| 3
	| style="background:silver;"| '''0.3'''		| style="background:silver;"| '''0;3'''
			| style="text-align:left" | 0.25
	| 0.25
	| style="background:silver;"| '''0.03'''		|- style="text-align:right"
	| 0.0208{{overline|3}}
	|-
	| style="background:silver;"| '''4,000,000'''
	| 11,943,936
	| style="background:silver;"| '''400,000'''
	| 995,328
	| style="background:silver;"| '''40,000'''		| style="background:silver;"| '''40,000'''
	| 82,944		| 82,944
Line 403:		Line 564:
	| style="background:silver;"| '''4'''		| style="background:silver;"| '''4'''
	| 4		| 4
	| style="background:silver;"| '''0.4'''		| style="background:silver;"| '''0;4'''
	| 0.{{overline|3}}		| style="text-align:left" | 0.{{overline|3}}
	| style="background:silver;"| '''0.04'''		|- style="text-align:right"
	| 0.02{{overline|7}}
	|-
	| style="background:silver;"| '''5,000,000'''
	| 14,929,920
	| style="background:silver;"| '''500,000'''
	| 1,244,160
	| style="background:silver;"| '''50,000'''		| style="background:silver;"| '''50,000'''
	| 103,680		| 103,680
Line 422:		Line 577:
	| style="background:silver;"| '''5'''		| style="background:silver;"| '''5'''
	| 5		| 5
	| style="background:silver;"| '''0.5'''		| style="background:silver;"| '''0;5'''
	| 0.41{{overline|6}}		| style="text-align:left" | 0.41{{overline|6}}
	| style="background:silver;"| '''0.05'''		|- style="text-align:right"
	| 0.0347{{overline|2}}
	|-
	| style="background:silver;"| '''6,000,000'''
	| 17,915,904
	| style="background:silver;"| '''600,000'''
	| 1,492,992
	| style="background:silver;"| '''60,000'''		| style="background:silver;"| '''60,000'''
	| 124,416		| 124,416
Line 441:		Line 590:
	| style="background:silver;"| '''6'''		| style="background:silver;"| '''6'''
	| 6		| 6
	| style="background:silver;"| '''0.6'''		| style="background:silver;"| '''0;6'''
			| style="text-align:left" | 0.5
	| 0.5
	| style="background:silver;"| '''0.06'''		|- style="text-align:right"
	| 0.041{{overline|6}}
	|-
	| style="background:silver;"| '''7,000,000'''
	| 20,901,888
	| style="background:silver;"| '''700,000'''
	| 1,741,824
	| style="background:silver;"| '''70,000'''		| style="background:silver;"| '''70,000'''
	| 145,152		| 145,152
Line 460:		Line 603:
	| style="background:silver;"| '''7'''		| style="background:silver;"| '''7'''
	| 7		| 7
	| style="background:silver;"| '''0.7'''		| style="background:silver;"| '''0;7'''
	| 0.58{{overline|3}}		| style="text-align:left" | 0.58{{overline|3}}
	| style="background:silver;"| '''0.07'''		|- style="text-align:right"
	| 0.0486{{overline|1}}
	|-
	| style="background:silver;"| '''8,000,000'''
	| 23,887,872
	| style="background:silver;"| '''800,000'''
	| 1,990,656
	| style="background:silver;"| '''80,000'''		| style="background:silver;"| '''80,000'''
	| 165,888		| 165,888
Line 479:		Line 616:
	| style="background:silver;"| '''8'''		| style="background:silver;"| '''8'''
	| 8		| 8
	| style="background:silver;"| '''0.8'''		| style="background:silver;"| '''0;8'''
	| 0.{{overline|6}}		| style="text-align:left" | 0.{{overline|6}}
	| style="background:silver;"| '''0.08'''		|- style="text-align:right"
	| 0.0{{overline|5}}
	|-
	| style="background:silver;"| '''9,000,000'''
	| 26,873,856
	| style="background:silver;"| '''900,000'''
	| 2,239,488
	| style="background:silver;"| '''90,000'''		| style="background:silver;"| '''90,000'''
	| 186,624		| 186,624
Line 498:		Line 629:
	| style="background:silver;"| '''9'''		| style="background:silver;"| '''9'''
	| 9		| 9
	| style="background:silver;"| '''0.9'''		| style="background:silver;"| '''0;9'''
			| style="text-align:left" | 0.75
	| 0.75
	| style="background:silver;"| '''0.09'''		|- style="text-align:right"
			| style="background:silver;"| '''{{nowrap|{{d2}}0,000}}'''
	| 0.0625
	|-
	| style="background:silver;"| '''ᘔ,000,000'''
	| 29,859,840
	| style="background:silver;"| '''ᘔ00,000'''
	| 2,488,320
	| style="background:silver;"| '''ᘔ0,000'''
	| 207,360		| 207,360
	| style="background:silver;"| '''ᘔ,000'''		| style="background:silver;"| '''{{d2}},000'''
	| 17,280		| 17,280
	| style="background:silver;"| '''ᘔ00'''		| style="background:silver;"| '''{{d2}}00'''
	| 1,440		| 1,440
	| style="background:silver;"| '''ᘔ0'''		| style="background:silver;"| '''{{d2}}0'''
	| 120		| 120
	| style="background:silver;"| '''ᘔ'''		| style="background:silver;"| '''{{d2}}'''
	| 10		| 10
	| style="background:silver;"| '''0.ᘔ'''		| style="background:silver;"| '''0;{{d2}}'''
	| 0.8{{overline|3}}		| style="text-align:left" | 0.8{{overline|3}}
	| style="background:silver;"| '''0.0ᘔ'''		|- style="text-align:right"
			| style="background:silver;"| '''{{d3}}0,000'''
	| 0.069{{overline|4}}
	|-
	| style="background:silver;"| '''Ɛ,000,000'''
	| 32,845,824
	| style="background:silver;"| '''Ɛ00,000'''
	| 2,737,152
	| style="background:silver;"| '''Ɛ0,000'''
	| 228,096		| 228,096
	| style="background:silver;"| '''Ɛ,000'''		| style="background:silver;"| '''{{d3}},000'''
	| 19,008		| 19,008
	| style="background:silver;"| '''Ɛ00'''		| style="background:silver;"| '''{{d3}}00'''
	| 1,584		| 1,584
	| style="background:silver;"| '''Ɛ0'''		| style="background:silver;"| '''{{d3}}0'''
	| 132		| 132
	| style="background:silver;"| '''Ɛ'''		| style="background:silver;"| '''{{d3}}'''
	| 11		| 11
	| style="background:silver;"| '''0.Ɛ'''		| style="background:silver;"| '''0;{{d3}}'''
	| 0.91{{overline|6}}		| style="text-align:left" | 0.91{{overline|6}}
	| style="background:silver;"| '''0.0Ɛ'''
	| 0.0763{{overline|8}}
	|}		|}
	=== Decimal to duodecimal digit conversion ===		=== Decimal to duodecimal digit conversion ===
	{|class="wikitable"		{| class="wikitable"
			! Dec.
	|-
			! Duod.
	! style="background:silver;"| ''Dec.''
	! ''Duod.''		! Dec.
			! Duod.
	! style="background:silver;"| ''Dec.''
	! ''Duod.''		! Dec.
			! Duod.
	! style="background:silver;"| ''Dec.''
	! ''Duod.''		! Dec.
			! Duod.
	! style="background:silver;"| ''Dec.''
	! ''Duod.''		! Dec.
			! Duod.
	! style="background:silver;"| ''Dec.''
	! ''Duod.''		! Dec.
			! Duodecimal
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	|-		|-
	| style="background:silver;"| '''100,000'''
	| 49,ᘔ54
	| style="background:silver;"| '''10,000'''		| style="background:silver;"| '''10,000'''
	| 5,954		| 5,954
	| style="background:silver;"| '''1,000'''		| style="background:silver;"| '''1,000'''
			| 6{{d3}}4
	| 6Ɛ4
	| style="background:silver;"| '''100'''		| style="background:silver;"| '''100'''
	| 84		| 84
	| style="background:silver;"| '''10'''		| style="background:silver;"| '''10'''
	| ᘔ		| {{d2}}
	| style="background:silver;"| '''1'''		| style="background:silver;"| '''1'''
	| 1		| 1
	| style="background:silver;"| '''0.1'''		| style="background:silver;"| '''0.1'''
	| 0.1{{overline|2497}}		| 0;1{{overline|2497}}
	| style="background:silver;"| '''0.01'''
	| 0.0{{overline|15343ᘔ0Ɛ62ᘔ68781Ɛ059}}
	|-		|-
	| style="background:silver;"| '''200,000'''
	| 97,8ᘔ8
	| style="background:silver;"| '''20,000'''		| style="background:silver;"| '''20,000'''
			| {{d3}},6{{d2}}8
	| Ɛ,6ᘔ8
	| style="background:silver;"| '''2,000'''		| style="background:silver;"| '''2,000'''
	| 1,1ᘔ8		| 1,1{{d2}}8
	| style="background:silver;"| '''200'''		| style="background:silver;"| '''200'''
	| 148		| 148
Line 592:		Line 698:
	| 2		| 2
	| style="background:silver;"| '''0.2'''		| style="background:silver;"| '''0.2'''
	| 0.{{overline|2497}}		| 0;{{overline|2497}}
	| style="background:silver;"| '''0.02'''
	| 0.0{{overline|2ᘔ68781Ɛ05915343ᘔ0Ɛ6}}
	|-		|-
	| style="background:silver;"| '''300,000'''
	| 125,740
	| style="background:silver;"| '''30,000'''		| style="background:silver;"| '''30,000'''
	| 15,440		| 15,440
	| style="background:silver;"| '''3,000'''		| style="background:silver;"| '''3,000'''
	| 1,8ᘔ0		| 1,8{{d2}}0
	| style="background:silver;"| '''300'''		| style="background:silver;"| '''300'''
	| 210		| 210
Line 609:		Line 711:
	| 3		| 3
	| style="background:silver;"| '''0.3'''		| style="background:silver;"| '''0.3'''
	| 0.3{{overline|7249}}		| 0;3{{overline|7249}}
	| style="background:silver;"| '''0.03'''
	| 0.0{{overline|43ᘔ0Ɛ62ᘔ68781Ɛ059153}}
	|-		|-
	| style="background:silver;"| '''400,000'''
	| 173,594
	| style="background:silver;"| '''40,000'''		| style="background:silver;"| '''40,000'''
	| 1Ɛ,194		| 1{{d3}},194
	| style="background:silver;"| '''4,000'''		| style="background:silver;"| '''4,000'''
	| 2,394		| 2,394
Line 626:		Line 724:
	| 4		| 4
	| style="background:silver;"| '''0.4'''		| style="background:silver;"| '''0.4'''
	| 0.{{overline|4972}}		| 0;{{overline|4972}}
	| style="background:silver;"| '''0.04'''
	| 0.{{overline|05915343ᘔ0Ɛ62ᘔ68781Ɛ}}
	|-		|-
	| style="background:silver;"| '''500,000'''
	| 201,428
	| style="background:silver;"| '''50,000'''		| style="background:silver;"| '''50,000'''
	| 24,Ɛ28		| 24,{{d3}}28
	| style="background:silver;"| '''5,000'''		| style="background:silver;"| '''5,000'''
	| 2,ᘔ88		| 2,{{d2}}88
	| style="background:silver;"| '''500'''		| style="background:silver;"| '''500'''
	| 358		| 358
Line 643:		Line 737:
	| 5		| 5
	| style="background:silver;"| '''0.5'''		| style="background:silver;"| '''0.5'''
	| 0.6		| 0;6
	| style="background:silver;"| '''0.05'''
	| 0.0{{overline|7249}}
	|-		|-
	| style="background:silver;"| '''600,000'''
	| 24Ɛ,280
	| style="background:silver;"| '''60,000'''		| style="background:silver;"| '''60,000'''
	| 2ᘔ,880		| 2{{d2}},880
	| style="background:silver;"| '''6,000'''		| style="background:silver;"| '''6,000'''
	| 3,580		| 3,580
Line 660:		Line 750:
	| 6		| 6
	| style="background:silver;"| '''0.6'''		| style="background:silver;"| '''0.6'''
	| 0.{{overline|7249}}		| 0;{{overline|7249}}
	| style="background:silver;"| '''0.06'''
	| 0.0{{overline|8781Ɛ05915343ᘔ0Ɛ62ᘔ6}}
	|-		|-
	| style="background:silver;"| '''700,000'''
	| 299,114
	| style="background:silver;"| '''70,000'''		| style="background:silver;"| '''70,000'''
	| 34,614		| 34,614
Line 671:		Line 757:
	| 4,074		| 4,074
	| style="background:silver;"| '''700'''		| style="background:silver;"| '''700'''
			| 4{{d2}}4
	| 4ᘔ4
	| style="background:silver;"| '''70'''		| style="background:silver;"| '''70'''
	| 5ᘔ		| 5{{d2}}
	| style="background:silver;"| '''7'''		| style="background:silver;"| '''7'''
	| 7		| 7
	| style="background:silver;"| '''0.7'''		| style="background:silver;"| '''0.7'''
	| 0.8{{overline|4972}}		| 0;8{{overline|4972}}
	| style="background:silver;"| '''0.07'''
	| 0.0{{overline|ᘔ0Ɛ62ᘔ68781Ɛ05915343}}
	|-		|-
	| style="background:silver;"| '''800,000'''
	| 326,Ɛ68
	| style="background:silver;"| '''80,000'''		| style="background:silver;"| '''80,000'''
	| 3ᘔ,368		| 3{{d2}},368
	| style="background:silver;"| '''8,000'''		| style="background:silver;"| '''8,000'''
	| 4,768		| 4,768
Line 694:		Line 776:
	| 8		| 8
	| style="background:silver;"| '''0.8'''		| style="background:silver;"| '''0.8'''
	| 0.{{overline|9724}}		| 0;{{overline|9724}}
	| style="background:silver;"| '''0.08'''
	| 0.{{overline|0Ɛ62ᘔ68781Ɛ05915343ᘔ}}
	|-		|-
	| style="background:silver;"| '''900,000'''
	| 374,ᘔ00
	| style="background:silver;"| '''90,000'''		| style="background:silver;"| '''90,000'''
	| 44,100		| 44,100
Line 711:		Line 789:
	| 9		| 9
	| style="background:silver;"| '''0.9'''		| style="background:silver;"| '''0.9'''
	| 0.ᘔ{{overline|9724}}		| 0;{{d2}}{{overline|9724}}
	| style="background:silver;"| '''0.09'''
	| 0.1{{overline|0Ɛ62ᘔ68781Ɛ05915343ᘔ}}
	|}		|}
			== Fractions and irrational numbers ==
	=== Conversion of powers ===
			=== Fractions ===
	{|class="wikitable"
			Duodecimal ] for rational numbers with ] denominators terminate:
	|-
			* {{sfrac|2}} = 0;6
	! rowspan="2" | ''Exponent''
			* {{sfrac|3}} = 0;4
	! colspan="2" | b=2
			* {{sfrac|4}} = 0;3
	! colspan="2" | b=3
			* {{sfrac|6}} = 0;2
	! colspan="2" | b=4
			* {{sfrac|8}} = 0;16
	! colspan="2" | b=5
			* {{sfrac|9}} = 0;14
	! colspan="2" | b=6
			* {{sfrac|10}} = 0;1 (this is one twelfth, {{sfrac|{{d2}}}} is one tenth)
	! colspan="2" | b=7
			* {{sfrac|14}} = 0;09 (this is one sixteenth, {{sfrac|12}} is one fourteenth)
	|-
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	|-
	| b<sup>6</sup>
	| style="background:silver;"| '''64'''
	| 54
	| style="background:silver;"| '''729'''
	| 509
	| style="background:silver;"| '''4,096'''
	| 2,454
	| style="background:silver;"| '''15,625'''
	| 9,061
	| style="background:silver;"| '''46,656'''
	| 23,000
	| style="background:silver;"| '''117,649'''
	| 58,101
	|-
	| b<sup>5</sup>
	| style="background:silver;"| '''32'''
	| 28
	| style="background:silver;"| '''243'''
	| 183
	| style="background:silver;"| '''1,024'''
	| 714
	| style="background:silver;"| '''3,125'''
	| 1,985
	| style="background:silver;"| '''7,776'''
	| 4,600
	| style="background:silver;"| '''16,807'''
	| 9,887
	|-
	| b<sup>4</sup>
	| style="background:silver;"| '''16'''
	| 14
	| style="background:silver;"| '''81'''
	| 69
	| style="background:silver;"| '''256'''
	| 194
	| style="background:silver;"| '''625'''
	| 441
	| style="background:silver;"| '''1,296'''
	| 900
	| style="background:silver;"| '''2,401'''
	| 1,481
	|-
	| b<sup>3</sup>
	| style="background:silver;"| '''8'''
	| 8
	| style="background:silver;"| '''27'''
	| 23
	| style="background:silver;"| '''64'''
	| 54
	| style="background:silver;"| '''125'''
	| ᘔ5
	| style="background:silver;"| '''216'''
	| 160
	| style="background:silver;"| '''343'''
	| 247
	|-
	| b<sup>2</sup>
	| style="background:silver;"| '''4'''
	| 4
	| style="background:silver;"| '''9'''
	| 9
	| style="background:silver;"| '''16'''
	| 14
	| style="background:silver;"| '''25'''
	| 21
	| style="background:silver;"| '''36'''
	| 30
	| style="background:silver;"| '''49'''
	| 41
	|-
	| b<sup>1</sup>
	| style="background:silver;"| '''2'''
	| 2
	| style="background:silver;"| '''3'''
	| 3
	| style="background:silver;"| '''4'''
	| 4
	| style="background:silver;"| '''5'''
	| 5
	| style="background:silver;"| '''6'''
	| 6
	| style="background:silver;"| '''7'''
	| 7
	|-
	| b<sup>−1</sup>
	| style="background:silver;"| '''0.5'''
	| 0.6
	| style="background:silver;"| '''0.{{overline|3}}'''
	| 0.4
	| style="background:silver;"| '''0.25'''
	| 0.3
	| style="background:silver;"| '''0.2'''
	| 0.{{overline|2497}}
	| style="background:silver;"| '''0.1{{overline|6}}'''
	| 0.2
	| style="background:silver;"| '''0.{{overline|142857}}'''
	| 0.{{overline|186ᘔ35}}
	|-
	| b<sup>−2</sup>
	| style="background:silver;"| '''0.25'''
	| 0.3
	| style="background:silver;"| '''0.{{overline|1}}'''
	| 0.14
	| style="background:silver;"| '''0.0625'''
	| 0.09
	| style="background:silver;"| '''0.04'''
	| 0.{{overline|05915343ᘔ0<br>Ɛ62ᘔ68781Ɛ}}
	| style="background:silver;"| '''0.02{{overline|7}}'''
	| 0.04
	| style="background:silver;"| '''0.{{overline|0204081632653<br>06122448979591<br>836734693877551}}'''
	| 0.{{overline|02Ɛ322547ᘔ05ᘔ<br>644ᘔ9380Ɛ908996<br>741Ɛ615771283Ɛ}}
	|}
			while other rational numbers have ] duodecimal fractions:
	{|class="wikitable"
			* {{sfrac|5}} = 0;{{Overline|2497}}
	|-
			* {{sfrac|7}} = 0;{{Overline|186{{D2}}35}}
	! rowspan="2" | ''Exponent''
			* {{sfrac|{{d2}}}} = 0;1{{Overline|2497}} (one tenth)
	! colspan="2" | b=8
			* {{sfrac|{{d3}}}} = 0;{{Overline|1}} (one eleventh)
	! colspan="2" | b=9
			* {{sfrac|11}} = 0;{{Overline|0{{D3}}}} (one thirteenth)
	! colspan="2" | '''b=10'''
			* {{sfrac|12}} = 0;0{{Overline|{{D2}}35186}} (one fourteenth)
	! colspan="2" | b=11
			* {{sfrac|13}} = 0;0{{Overline|9724}} (one fifteenth)
	! colspan="2" | '''b=12'''
	|-
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	! style="background:silver;"| ''Dec.''
	! ''Duod.''
	|-
	| b<sup>6</sup>
	| style="background:silver;"| '''262,144'''
	| 107,854
	| style="background:silver;"| '''531,441'''
	| 217,669
	| style="background:silver;"| '''1,000,000'''
	| 402,854
	| style="background:silver;"| '''1,771,561'''
	| 715,261
	| style="background:silver;"| '''2,985,984'''
	| 1,000,000
	|-
	| b<sup>5</sup>
	| style="background:silver;"| '''32,768'''
	| 16,Ɛ68
	| style="background:silver;"| '''59,049'''
	| 2ᘔ,209
	| style="background:silver;"| '''100,000'''
	| 49,ᘔ54
	| style="background:silver;"| '''161,051'''
	| 79,24Ɛ
	| style="background:silver;"| '''248,832'''
	| 100,000
	|-
	| b<sup>4</sup>
	| style="background:silver;"| '''4,096'''
	| 2,454
	| style="background:silver;"| '''6,561'''
	| 3,969
	| style="background:silver;"| '''10,000'''
	| 5,954
	| style="background:silver;"| '''14,641'''
	| 8,581
	| style="background:silver;"| '''20,736'''
	| 10,000
	|-
	| b<sup>3</sup>
	| style="background:silver;"| '''512'''
	| 368
	| style="background:silver;"| '''729'''
	| 509
	| style="background:silver;"| '''1,000'''
	| 6Ɛ4
	| style="background:silver;"| '''1,331'''
	| 92Ɛ
	| style="background:silver;"| '''1,728'''
	| 1,000
	|-
	| b<sup>2</sup>
	| style="background:silver;"| '''64'''
	| 54
	| style="background:silver;"| '''81'''
	| 69
	| style="background:silver;"| '''100'''
	| 84
	| style="background:silver;"| '''121'''
	| ᘔ1
	| style="background:silver;"| '''144'''
	| 100
	|-
	| b<sup>1</sup>
	| style="background:silver;"| '''8'''
	| 8
	| style="background:silver;"| '''9'''
	| 9
	| style="background:silver;"| '''10'''
	| ᘔ
	| style="background:silver;"| '''11'''
	| Ɛ
	| style="background:silver;"| '''12'''
	| 10
	|-
	| b<sup>−1</sup>
	| style="background:silver;"| '''0.125'''
	| 0.16
	| style="background:silver;"| '''0.{{overline|1}}'''
	| 0.14
	| style="background:silver;"| '''0.1'''
	| 0.1{{overline|2497}}
	| style="background:silver;"| '''0.{{overline|09}}'''
	| 0.{{overline|1}}
	| style="background:silver;"| '''0.08{{overline|3}}'''
	| 0.1
	|-
	| b<sup>−2</sup>
	| style="background:silver;"| '''0.015625'''
	| 0.023
	| style="background:silver;"| '''0.{{overline|012345679}}'''
	| 0.0194
	| style="background:silver;"| '''0.01'''
	| 0.0{{overline|15343ᘔ0Ɛ6<br>2ᘔ68781Ɛ059}}
	| style="background:silver;"| '''0.{{overline|00826446280<br>99173553719}}'''
	| 0.{{overline|0123456789Ɛ}}
	| style="background:silver;"| '''0.0069{{overline|4}}'''
	| 0.01
	|}
			{| class="wikitable"
	== Some properties ==
	(In this section, all numbers are written with duodecimal)
	All ] end with square digits (i.e. end with 0, 1, 4 or 9), if ''n'' is divisible by both 2 and 3, then ''n''<sup>2</sup> ends with 0, if ''n'' is not divisible by 2 or 3, then ''n''<sup>2</sup> ends with 1, if ''n'' is divisible by 2 but not by 3, then ''n''<sup>2</sup> ends with 4, if ''n'' is not divisible by 2 but by 3, then ''n''<sup>2</sup> ends with 9. If the unit digit of ''n''<sup>2</sup> is 0, then the dozens digit of ''n''<sup>2</sup> is either 0 or 3, if the unit digit of ''n''<sup>2</sup> is 1, then the dozens digit of ''n''<sup>2</sup> is even, if the unit digit of ''n''<sup>2</sup> is 4, then the dozen digit of ''n''<sup>2</sup> is 0, 1, 4, 5, 8 or 9, if the unit digit of ''n''<sup>2</sup> is 9, then the dozen digit of ''n''<sup>2</sup> is either 0 or 6. (More specially, all squares of (primes ≥ 5) end with 1)
	The ] of a square is 1, 3, 4, 5 or Ɛ.
	No ]s with more than one digit are squares, in fact, a square cannot end with three same digits except 000.
	No four-digit ]s are squares.
	A ] can end with all digits except 2, 6 and ᘔ (in fact, no ]s end with 2, 6 or ᘔ), if ''n'' is not congruent to 2 mod 4, then ''n''<sup>3</sup> ends with the same digit as ''n''; if ''n'' is congruent to 2 mod 4, then ''n''<sup>3</sup> ends with the digit (the last digit of ''n'' +- 6).
	The ] of a cube can be any number.
	If ''k''≥2, then ''n''<sup>''k''+2</sup> ends with the same digit as ''n''<sup>''k''</sup>.
	Except for 6 and 24, all even ]s end with 54.
	The ] of an even perfect number is 1, 4, 6 or ᘔ.
	The unit digits of powers of 2 are 1, 2, 4, 8, 4, 8, 4, 8, 4, ... (periodic with period 2)
	The unit digits of powers of 3 are 1, 3, 9, 3, 9, 3, 9, 3, 9, ... (periodic with period 2)
	The unit digits of powers of 5 are 1, 5, 1, 5, 1, 5, 1, 5, 1, ... (periodic with period 2)
	The unit digits of powers of 7 are 1, 7, 1, 7, 1, 7, 1, 7, 1, ... (periodic with period 2)
	The unit digits of powers of Ɛ are 1, Ɛ, 1, Ɛ, 1, Ɛ, 1, Ɛ, 1, ... (periodic with period 2)
	The final two digits of powers of 2 are 01, 02, 04, 08, 14, 28, 54, ᘔ8, 94, 68, 14, 28, 54, ᘔ8, 94, 68, 14, ... (periodic with period 6)
	The final two digits of powers of 3 are 01, 03, 09, 23, 69, 83, 09, 23, 69, 83, 09, 23, 69, ... (periodic with period 4)
	The final two digits of powers of 5 are 01, 05, 21, ᘔ5, 41, 85, 61, 65, 81, 45, ᘔ1, 25, 01, 05, 21, ᘔ5, 41, ... (periodic with period 10)
	The final two digits of powers of 7 are 01, 07, 41, 47, 81, 87, 01, 07, 41, 47, 81, 87, 01, ... (periodic with period 6)
	The final two digits of powers of Ɛ are 01, 0Ɛ, ᘔ1, 2Ɛ, 81, 4Ɛ, 61, 6Ɛ, 41, 8Ɛ, 21, ᘔƐ, 01, 0Ɛ, ᘔ1, 2Ɛ, 81, ... (periodic with period 10)
	The digital roots of powers of 2 are 1, 2, 4, 8, 5, ᘔ, 9, 7, 3, 6, 1, 2, 4, 8, 5, ᘔ, 9, 7, 3, 6, 1, ... (periodic with period ᘔ)
	The digital roots of powers of 3 are 1, 3, 9, 5, 4, 1, 3, 9, 5, 4, 1, ... (periodic with period 5)
	The digital roots of powers of 5 are 1, 5, 3, 4, 9, 1, 5, 3, 4, 9, 1, ... (periodic with period 5)
	The digital roots of powers of 7 are 1, 7, 5, 2, 3, ᘔ, 4, 6, 9, 8, 1, 7, 5, 2, 3, ᘔ, 4, 6, 9, 8, 1, ... (periodic with period ᘔ)
	The digital roots of powers of Ɛ are 1, Ɛ, Ɛ, Ɛ, Ɛ, Ɛ, ... (periodic with period 1)
	The period of the unit digit of powers of a number coprime to 10 is at most 2, the final two digit is at most 20, the final three digits is 200, the final four digits is 2000, ..., the final ''n'' digits is 2×10<sup>''n''−1</sup> (''n''≥2).
	The unit digit of a ] can be any digit except 6 (interestingly, if the unit digit of a Fibonacci number is 0, then the dozens digit of this number must also be 0, thus, all Fibonacci numbers divisible by 6 are also divisible by 100), and the unit digit of a ] cannot be 0 or 9 (thus, no Lucas number is divisible by 10).
	The unit digits of Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 1, 9, ᘔ, 7, 5, 0, 5, 5, ᘔ, 3, 1, 4, 5, 9, 2, Ɛ, 1, 0, 1, 1, 2, 3, 5, ... (periodic with period 20)
	The unit digits of Lucas numbers are 2, 1, 3, 4, 7, Ɛ, 6, 5, Ɛ, 4, 3, 7, ᘔ, 5, 3, 8, Ɛ, 7, 6, 1, 7, 8, 3, Ɛ, 2, 1, 3, 4, 7, Ɛ, ... (periodic with period 20)
	The final two digits of Fibonacci numbers are 00, 01, 01, 02, 03, 05, 08, 11, 19, 2ᘔ, 47, 75, 00, 75, 75, 2ᘔ, ᘔ3, 11, Ɛ4, 05, Ɛ9, 02, ƐƐ, 01, 00, 01, 01, 02, 03, 05, ... (periodic with period 20)
	The final two digits of Lucas numbers are 02, 01, 03, 04, 07, 0Ɛ, 16, 25, 3Ɛ, 64, ᘔ3, 47, 2ᘔ, 75, ᘔ3, 58, 3Ɛ, 97, 16, Ɛ1, 07, Ɛ8, 03, ƐƐ, 02, 01, 03, 04, 07, 0Ɛ, ... (periodic with period 20)
	The digital roots of Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 2, ᘔ, 1, Ɛ, 1, 1, 2, 3, 5, ... (periodic with period ᘔ)
	The digital roots of Lucas numbers are 2, 1, 3, 4, 7, Ɛ, 7, 7, 3, ᘔ, 2, 1, 3, 4, 7, Ɛ, ... (periodic with period ᘔ)
	The period of the unit digit of Fibonacci numbers is 20, the final two digits is also 20, the final three digits is 200, the final four digits is 2000, ..., the final ''n'' digits is 2×10<sup>''n''−1</sup> (''n''≥2).
	All ]s end with prime digits or 1 (i.e. end with 1, 2, 3, 5, 7 or Ɛ), more generally, except for 2 and 3, all prime numbers end with 1, 5, 7 or Ɛ (1 and all prime digits that do not divide 10).
	The density of primes end with 1 is a relatively low, but the density of primes end with 5, 7 and Ɛ are nearly equal. (since all prime squares except 4 and 9 end with 1, no prime squares end with 5, 7 or Ɛ)
	Except (3, 5), all ]s end with (5, 7) or (Ɛ, 1).
	All ]s except 11 has an odd number of digits, since all even-digit palindrome numbers are divisible by 11.
	All ]s end with digit 1, 3, 7 or 9.
	Except for 3, all ]s end with 5.
	Except for 3, all ]s end with 7.
	Except for 2 and 3, all ]s end with 5 or Ɛ.
	Except for 5 and 7, all ]s end with Ɛ.
	A prime ''p'' is ] if and only if ''p'' ends with 7 or Ɛ (or ''p''=3).
	A prime ''p'' is ] if and only if ''p'' ends with 5 or Ɛ (or ''p''=2).
	All ]s end with 5 or 7. (in fact, for all primes ''p'' ≥ 5, (''p''-1)/(the period length of 1/''p'') is odd ] ''p'' is end with 5 or 7, since 10 is a ] mod ''p'' if and only if ''p'' is end with 5 or 7)
	If ''p'' is a safe prime other than 5, 7 and Ɛ, then the ] of 1/''p'' is (''p''-1)/2. (this is not true for all primes ends with Ɛ (other than Ɛ itself), the first counterexample is ''p'' = 2ƐƐ, where the period length of 1/''p'' is only 37)
	By ], if ''p'' is a prime other than 2, 3 and Ɛ, then ''p'' divides the ] with length ''p''−1. (The converse is not true, the first counterexample is 55, which is composite (equals 5×11) but divides the repunit with length 54) Thus, we can prove that every positive integer coprime to 10 has a repunit multiple, and every positive integer has a multiple uses only 0's and 1's. (the smallest multiple of ''n'' uses only 0's and 1's are 1, 10, 10, 10, 101, 10, 1001, 100, 100, 1010, 11111111111, 10, 11, 10010, 1010, 100, ...)
	The only two ]s < 10<sup>10</sup> are 1685 and 5Ɛ685, note that both of the numbers end with 685, and it is conjectured that all base 10 Wieferich primes end with 685.
	Most numbers that end with 2 are ] (in fact, all nontotients < 58 except 2ᘔ end with 2), except 2 itself, the first counterexample is 92, which equals φ(ᘔ1) = φ(Ɛ<sup>2</sup>).
	All orders of non-] ] end with 0, however, we can prove that no groups with order 10, 20, 30 or 40 are simple, thus 50 is the smallest order of non-cyclic simple group, (50 is the order of the ] <math>A_5</math>, which is a non-cyclic simple group) next three orders of non-cyclic simple group are 120, 260 and 360. (Edit: I found that this is not true, the smallest counterexample is 14ᘔ28, however, all such numbers are divisible by 4 and either 3 or 5)
	== Some special numbers ==
	(In this section, all numbers are written with duodecimal, using X for ten and E for eleven)
	In duodecimal, as in decimal, there are special numbers that exhibit digit patterns when, for example, multiplied by small integers:
	2497 * 1 = 2497
	2497 * 2 = 4972
	2497 * 3 = 7249
	2497 * 4 = 9724
	2497 * 5 = EEEE
	186X35 * 1 = 186X35
	186X35 * 2 = 35186X
	186X35 * 3 = 5186X3
	186X35 * 4 = 6X3518
	186X35 * 5 = 86X351
	186X35 * 6 = X35186
	186X35 * 7 = EEEEEE
	8579214E36429X7 * 1 = 8579214E36429X7
	8579214E36429X7 * 2 = 14E36429X7085792
	8579214E36429X7 * 3 = 214E36429X708579
	8579214E36429X7 * 4 = 29X708579214E364
	8579214E36429X7 * 5 = 36429X708579214E
	8579214E36429X7 * 6 = 429X708579214E36
	8579214E36429X7 * 7 = 4E36429X70857921
	8579214E36429X7 * 8 = 579214E36429X708
	8579214E36429X7 * 9 = 6429X708579214E3
	8579214E36429X7 * X = 708579214E36429X
	8579214E36429X7 * E = 79214E36429X7085
	8579214E36429X7 * 10 = 8579214E36429X70
	8579214E36429X7 * 11 = 9214E36429X70857
	8579214E36429X7 * 12 = 9X708579214E3642
	8579214E36429X7 * 13 = X708579214E36429
	8579214E36429X7 * 14 = E36429X708579214
	8579214E36429X7 * 15 = EEEEEEEEEEEEEEEE
	123456789XE * 1 = 123456789XE
	123456789XE * 2 = 2468E13579X
	123456789XE * 3 = 36X147E2589
	123456789XE * 4 = 4915X26E378
	123456789XE * 5 = 5E4X3928167
	123456789XE * 6 = 718293X4E56
	123456789XE * 7 = 83E72X61945
	123456789XE * 8 = 962E851X734
	123456789XE * 9 = X8641E97523
	123456789XE * X = EX987654312
	123456789XE * E = 111111111101
	123456789E * 1 = 123456789E (miss digit X)
	123456789E * 2 = 2468E1357X (miss digit 9)
	123456789E * 3 = 36X147E259 (miss digit 8)
	123456789E * 4 = 4915X26E38 (miss digit 7)
	123456789E * 5 = 5E4X392817 (miss digit 6)
	123456789E * 6 = 718293X4E6 (miss digit 5)
	123456789E * 7 = 83E72X6195 (miss digit 4)
	123456789E * 8 = 962E851X74 (miss digit 3)
	123456789E * 9 = X8641E9753 (miss digit 2)
	123456789E * X = EX98765432 (miss digit 1)
	123456789E * E = 11111111111
	275 * 5 = 1111
	275 * X = 2222
	275 * 13 = 3333
	275 * 18 = 4444
	275 * 21 = 5555
	275 * 26 = 6666
	275 * 2E = 7777
	275 * 34 = 8888
	275 * 39 = 9999
	275 * 42 = XXXX
	275 * 47 = EEEE
	1X537 * 7 = 111111
	1X537 * 12 = 222222
	1X537 * 19 = 333333
	1X537 * 24 = 444444
	1X537 * 2E = 555555
	1X537 * 36 = 666666
	1X537 * 41 = 777777
	1X537 * 48 = 888888
	1X537 * 53 = 999999
	1X537 * 5X = XXXXXX
	1X537 * 65 = EEEEEE
	123456789E * E = 11111111111
	123456789E * 1X = 22222222222
	123456789E * 29 = 33333333333
	123456789E * 38 = 44444444444
	123456789E * 47 = 55555555555
	123456789E * 56 = 66666666666
	123456789E * 65 = 77777777777
	123456789E * 74 = 88888888888
	123456789E * 83 = 99999999999
	123456789E * 92 = XXXXXXXXXXX
	123456789E * X1 = EEEEEEEEEEE
	92X79E43715865 * 15 = 1111111111111111
	92X79E43715865 * 2X = 2222222222222222
	92X79E43715865 * 43 = 3333333333333333
	92X79E43715865 * 58 = 4444444444444444
	92X79E43715865 * 71 = 5555555555555555
	92X79E43715865 * 86 = 6666666666666666
	92X79E43715865 * 9E = 7777777777777777
	92X79E43715865 * E4 = 8888888888888888
	92X79E43715865 * 109 = 9999999999999999
	92X79E43715865 * 122 = XXXXXXXXXXXXXXXX
	92X79E43715865 * 137 = EEEEEEEEEEEEEEEE
	8327 * 17 = 111111
	8327 * 32 = 222222
	8327 * 49 = 333333
	8327 * 64 = 444444
	8327 * 7E = 555555
	8327 * 96 = 666666
	8327 * E1 = 777777
	8327 * 108 = 888888
	8327 * 123 = 999999
	8327 * 13X = XXXXXX
	8327 * 155 = EEEEEE
	112233445566778899XE * EE = 1111111111111111111111
	112233445566778899XE * 1EX = 2222222222222222222222
	112233445566778899XE * 2E9 = 3333333333333333333333
	112233445566778899XE * 3E8 = 4444444444444444444444
	112233445566778899XE * 4E7 = 5555555555555555555555
	112233445566778899XE * 5E6 = 6666666666666666666666
	112233445566778899XE * 6E5 = 7777777777777777777777
	112233445566778899XE * 7E4 = 8888888888888888888888
	112233445566778899XE * 8E3 = 9999999999999999999999
	112233445566778899XE * 9E2 = XXXXXXXXXXXXXXXXXXXXXX
	112233445566778899XE * XE1 = EEEEEEEEEEEEEEEEEEEEEE
	1^2 = 01, 0 + 1 = 1
	E^2 = X1, X + 1 = E
	56^2 = 2630, 26 + 30 = 56
	66^2 = 3630, 36 + 30 = 66
	EE^2 = EX01, EX + 01 = EE
	444^2 = 170294, 170 + 294 = 444
	778^2 = 4X4294, 4X4 + 294 = 778
	EEE^2 = EEX001, EEX + 001 = EEE
	12XX^2 = 1661144, 166 + 1144 = 12XX
	1640^2 = 2401400, 240 + 1400 = 1640
	2046^2 = 4161830, 416 + 1830 = 2046
	2929^2 = 7802169, 780 + 2169 = 2929
	3333^2 = X862469, X86 + 2469 = 3333
	4973^2 = 1E062X69, 1E06 + 2X69 = 4973
	5E60^2 = 2E603000, 2E60 + 3000 = 5E60
	2^2 + 5^2 = 25
	X^2 + 5^2 = X5
	5^3 + 7^3 + 7^3 = 577
	6^3 + 6^3 + 8^3 = 668
	X^3 + 8^3 + 3^3 = X83
	1^5 + 4^5 + 7^5 + 6^5 + 5^5 = 14765
	9^5 + 3^5 + 8^5 + X^5 + 4^5 = 938X4
	3^6 + 6^6 + 9^6 + 8^6 + 6^6 + 2^6 = 369862
	X^6 + 2^6 + 3^6 + 9^6 + 4^6 + X^6 = X2394X
	14 = (1 * 4)^2
	20 = (2^2 + 0^2)!
	21 = (2^2 + 1^2)^2
	24 = 2^2 + 4!
	30 = ((3 + 0)!)^2
	43 = 4! + 3^3
	48 = 4^3 - 8
	84 = (8 + sqrt(4))^2
	X1 = (X + 1)^2
	X5 = (X - 5)^3
	X8 = (X - 8)^7
	XE = X + E^2
	121 = (12 - 1)^2
	169 = (1 * (6 + 9))^2
	230 = (30 / 2)^2
	1 * X + 1 = E
	12 * X + 2 = EX
	123 * X + 3 = EX9
	1234 * X + 4 = EX98
	12345 * X + 5 = EX987
	123456 * X + 6 = EX9876
	1234567 * X + 7 = EX98765
	12345678 * X + 8 = EX987654
	123456789 * X + 9 = EX9876543
	123456789X * X + X = EX98765432
	123456789XE * X + E = EX987654321
	0 * E + 1 = 1
	1 * E + 2 = 11
	12 * E + 3 = 111
	123 * E + 4 = 1111
	1234 * E + 5 = 11111
	12345 * E + 6 = 111111
	123456 * E + 7 = 1111111
	1234567 * E + 8 = 11111111
	12345678 * E + 9 = 111111111
	123456789 * E + X = 1111111111
	123456789X * E + E = 11111111111
	123456789XE * E + 10 = 111111111111
	1^2 = 1
	11^2 = 121
	111^2 = 12321
	1111^2 = 1234321
	11111^2 = 123454321
	111111^2 = 12345654321
	1111111^2 = 1234567654321
	11111111^2 = 123456787654321
	111111111^2 = 12345678987654321
	1111111111^2 = 123456789X987654321
	11111111111^2 = 123456789XEX987654321
	111111111111^2 = 123456789E00EX987654321 ((1_''n'')^2 is not ] if ''n'' >= 10)
	E^2 = X1
	EE^2 = EX01
	EEE^2 = EEX001
	EEEE^2 = EEEX0001
	EEEEE^2 = EEEEX00001
	EEEEEE^2 = EEEEEX000001
	EEEEEEE^2 = EEEEEEX0000001
	EEEEEEEE^2 = EEEEEEEX00000001
	EEEEEEEEE^2 = EEEEEEEEX000000001
	EEEEEEEEEE^2 = EEEEEEEEEX0000000001
	EEEEEEEEEEE^2 = EEEEEEEEEEX00000000001
	EEEEEEEEEEEE^2 = EEEEEEEEEEEX000000000001
	10XE * 1 = 10XE
	10XE * 2 = 219X
	10XE * 3 = 3289
	10XE * 4 = 4378
	10XE * 5 = 5467
	10XE * 6 = 6556
	10XE * 7 = 7645
	10XE * 8 = 8734
	10XE * 9 = 9823
	10XE * X = X912
	10XE * E = EX01
	10EXE * 1 = 10EXE
	10EXE * 2 = 21E9X
	10EXE * 3 = 32E89
	10EXE * 4 = 43E78
	10EXE * 5 = 54E67
	10EXE * 6 = 65E56
	10EXE * 7 = 76E45
	10EXE * 8 = 87E34
	10EXE * 9 = 98E23
	10EXE * X = X9E12
	10EXE * E = EXE01
	1 / E = 0.111111111111 ...
	1 / 11 = 0.0E 0E 0E 0E 0E 0E ...
	1 / EE = 0.01 01 01 01 01 01 ...
	1 / 101 = 0.00EE 00EE 00EE ...
	1 / E1 = 0.010EXE 010EXE ...
	1 / 111 = 0.00E 00E 00E 00E ...
	1 / X1 = 0.0123456789E 0123456789E ...
	1 / 121 = 0.00X28466482X0 EE1937557391E ...
	1 / XE1 = 0.00112233445566778899XE 00112233445566778899XE ...
	1 / XEE1 = 0.000111222333444555666777888999XXE 000111222333444555666777888999XXE ...
	1 / XEEE1 = 0.0000111122223333444455556666777788889999XXXE 0000111122223333444455556666777788889999XXXE ...
	1 / EXEE = 0.00 01 01 02 03 05 08 11 19 2X 47 ...
	1 / EEXEEE = 0.000 001 001 002 003 005 008 011 019 02X 047 075 100 175 275 42X 6X3 ...
	1 / EEEXEEEE = 0.0000 0001 0001 0002 0003 0005 0008 0011 0019 002X 0047 0075 0100 0175 0275 042X 06X3 0E11 15E4 ...
	1 / EXE01 = 0.0000101111212222323333434444545555656666767777878888989999X9XXXXEE 0000101111212222323333434444545555656666767777878888989999X9XXXXEE ...
	1 / EX01 = 0.00 01 02 03 04 05 06 07 08 09 0X 0E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22 23 24 25 26 27 28 29 2X 2E 30 31 32 33 34 35 36 37 38 39 3X 3E 40 41 42 43 44 45 46 47 48 49 4X 4E 50 51 52 53 54 55 56 57 58 59 5X 5E 60 61 62 63 64 65 66 67 68 69 6X 6E 70 71 72 73 74 75 76 77 78 79 7X 7E 80 81 82 83 84 85 86 87 88 89 8X 8E 90 91 92 93 94 95 96 97 98 99 9X 9E X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 XX XE E0 E1 E2 E3 E4 E5 E6 E7 E8 E9 EE 00 01 02 03 04 05 06 07 08 09 0X 0E 10 11 12 13 14 15 16 17 18 19 1X 1E 20 21 22 23 24 25 26 27 28 29 2X 2E 30 31 32 33 34 35 36 37 38 39 3X 3E 40 41 42 43 44 45 46 47 48 49 4X 4E 50 51 52 53 54 55 56 57 58 59 5X 5E 60 61 62 63 64 65 66 67 68 69 6X 6E 70 71 72 73 74 75 76 77 78 79 7X 7E 80 81 82 83 84 85 86 87 88 89 8X 8E 90 91 92 93 94 95 96 97 98 99 9X 9E X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 XX XE E0 E1 E2 E3 E4 E5 E6 E7 E8 E9 EE ...
	1 / EEX001 = 0.000 001 002 003 004 005 006 007 008 009 00X 00E 010 011 012 013 014 ...
	1 / EEEX0001 = 0.0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 000X 000E 0010 0011 0012 0013 0014 ...
	1 / EX = 0.01 02 04 08 14 28 54 X8
	194
	368
	714
	1228
	2454 ... (])
	1 / E9 = 0.01 03 09 23 69
	183
	509
	1323
	3969 ... (powers of 3)
	1 / E8 = 0.01 04 14 54
	194
	714
	2454 ... (powers of 4)
	1 / XE = 0.0112358
	11
	19
	2X
	47
	75
	100
	175
	275
	42X
	6X3
	E11
	15E4 ... (]s)
	1 / 9E = 0.0125
	10
	25
	5X
	121
	2X0
	6X1
	1462 ... (]s)
	1 / 5E = 0.02 04 08 14 28 54 X8
	194
	368
	714
	1228
	2454 ... (])
	1 / 3E = 0.03 09 23 69
	183
	509
	1323
	3969 ... (powers of 3)
	1 / 2E = 0.04 14 54
	194
	714
	2454 ... (powers of 4)
	148421
	28
	54
	X8
	194
	368
	714
	1228
	1 / 1E = (]) ... 2454
	23931
	69
	183
	509
	1323
	1 / 2E = (power of 3) ... 3969
	1441
	54
	194
	714
	1 / 3E = (power of 4) ... 2454
	11853211
	19
	2X
	47
	75
	100
	175
	275
	42X
	6X3
	E11
	1 / 10E = (]s) ... 15E4
	10521
	25
	5X
	121
	2X0
	6X1
	1 / 11E = (]s) ... 1462
	ab * 5 * 25 = abab
	ab * 7 * 17 * 111 = ababab
	ab * 5 * 25 * 75 * 175 = abababab
	abc * 7 * 11 * 17 = abcabc
	abcd * 75 * 175 = abcdabcd
	== Prime numbers and divisibility rules==
	(In this section, all numbers are written with duodecimal)
	A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself. Natural numbers greater than 1 that are not prime are called composite.
	The first 1ᘔ5 prime numbers (all the prime numbers less than 1000) are:
	:2, 3, 5, 7, Ɛ, 11, 15, 17, 1Ɛ, 25, 27, 31, 35, 37, 3Ɛ, 45, 4Ɛ, 51, 57, 5Ɛ, 61, 67, 6Ɛ, 75, 81, 85, 87, 8Ɛ, 91, 95, ᘔ7, ᘔƐ, Ɛ5, Ɛ7, 105, 107, 111, 117, 11Ɛ, 125, 12Ɛ, 131, 13Ɛ, 141, 145, 147, 157, 167, 16Ɛ, 171, 175, 17Ɛ, 181, 18Ɛ, 195, 19Ɛ, 1ᘔ5, 1ᘔ7, 1Ɛ1, 1Ɛ5, 1Ɛ7, 205, 217, 21Ɛ, 221, 225, 237, 241, 24Ɛ, 251, 255, 25Ɛ, 267, 271, 277, 27Ɛ, 285, 291, 295, 2ᘔ1, 2ᘔƐ, 2Ɛ1, 2ƐƐ, 301, 307, 30Ɛ, 315, 321, 325, 327, 32Ɛ, 33Ɛ, 347, 34Ɛ, 357, 35Ɛ, 365, 375, 377, 391, 397, 3ᘔ5, 3ᘔƐ, 3Ɛ5, 3Ɛ7, 401, 40Ɛ, 415, 41Ɛ, 421, 427, 431, 435, 437, 447, 455, 457, 45Ɛ, 465, 46Ɛ, 471, 481, 485, 48Ɛ, 497, 4ᘔ5, 4Ɛ1, 4ƐƐ, 507, 511, 517, 51Ɛ, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58Ɛ, 591, 59Ɛ, 5Ɛ1, 5Ɛ5, 5Ɛ7, 5ƐƐ, 611, 615, 617, 61Ɛ, 637, 63Ɛ, 647, 655, 661, 665, 66Ɛ, 675, 687, 68Ɛ, 695, 69Ɛ, 6ᘔ7, 6Ɛ1, 701, 705, 70Ɛ, 711, 71Ɛ, 721, 727, 735, 737, 745, 747, 751, 767, 76Ɛ, 771, 775, 77Ɛ, 785, 791, 797, 7ᘔ1, 7ƐƐ, 801, 80Ɛ, 817, 825, 82Ɛ, 835, 841, 851, 855, 85Ɛ, 865, 867, 871, 881, 88Ɛ, 8ᘔ5, 8ᘔ7, 8ᘔƐ, 8Ɛ5, 8Ɛ7, 901, 905, 907, 90Ɛ, 91Ɛ, 921, 927, 955, 95Ɛ, 965, 971, 987, 995, 9ᘔ7, 9ᘔƐ, 9Ɛ1, 9Ɛ5, 9ƐƐ, ᘔ07, ᘔ0Ɛ, ᘔ11, ᘔ17, ᘔ27, ᘔ35, ᘔ37, ᘔ3Ɛ, ᘔ41, ᘔ45, ᘔ4Ɛ, ᘔ5Ɛ, ᘔ6Ɛ, ᘔ77, ᘔ87, ᘔ91, ᘔ95, ᘔ9Ɛ, ᘔᘔ7, ᘔᘔƐ, ᘔƐ7, ᘔƐƐ, Ɛ11, Ɛ15, Ɛ1Ɛ, Ɛ21, Ɛ25, Ɛ2Ɛ, Ɛ31, Ɛ37, Ɛ45, Ɛ61, Ɛ67, Ɛ6Ɛ, Ɛ71, Ɛ91, Ɛ95, Ɛ97, Ɛᘔ5, ƐƐ5, ƐƐ7
	Except 2 and 3, all primes end in 1, 5, 7 or Ɛ. The first ''k'' such that all of 10''k'', 10''k'' + 1, 10''k'' + 2, ..., 10''k'' + Ɛ are all composite is 38, i.e. all of 380, 381, 382, ..., 38Ɛ are composite.
	The density of primes end with 1 is a relatively low, but the density of primes end with 5, 7 and Ɛ are nearly equal. (i.e. for a given natural number ''N'', the numbers of primes end with 1 less than ''N'' is usually smaller than the number of primes end with 5 (or 7, or Ɛ) less than ''N'')
	There are 2ᘔ primes between 1 and 100, 23 primes between 101 and 200, 1ᘔ primes between 201 and 300, 1ᘔ primes between 301 and 400, 1Ɛ primes between 401 and 500, 1ᘔ primes between 501 and 600, 16 primes between 601 and 700, 1ᘔ primes between 701 and 800, 18 primes between 801 and 900, 16 primes between 901 and ᘔ00, 1ᘔ primes between ᘔ01 and Ɛ00, 17 primes between Ɛ01 and 1000.
	:''This section focuses on duodecimal ]s.''
	;1
	Any integer is divisible by '''1'''.
	;2
	If a number is divisible by '''2''' then the unit digit of that number will be 0, 2, 4, 6, 8 or ᘔ.
	;3
	If a number is divisible by '''3''' then the unit digit of that number will be 0, 3, 6 or 9.
	;4
	If a number is divisible by '''4''' then the unit digit of that number will be 0, 4 or 8.
	;5
	To test for divisibility by 5, double the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by '''5''' then the given number is divisible by 5.
	This rule comes from 21(5*5)
	Examples: <br>
	'''13''' &nbsp;&nbsp;&nbsp; rule => |1-2*3| = 5 which is divisible by 5.<br>
	'''2Ɛᘔ5''' &nbsp; rule => |2Ɛᘔ-2*5| = 2Ɛ0(5*70) which is divisible by 5(or apply the rule on 2Ɛ0).
	'''OR'''
	To test for divisibility by 5, subtract the units digit and triple of the result to the number formed by the rest of the digits. If the result is divisible by '''5''' then the given number is divisible by 5.
	This rule comes from 13(5*3)
	Examples: <br>
	'''13''' &nbsp;&nbsp;&nbsp; rule => |3-3*1| = 0 which is divisible by 5.<br>
	'''2Ɛᘔ5''' &nbsp; rule => |5-3*2Ɛᘔ| = 8Ɛ1(5*195) which is divisible by 5(or apply the rule on 8Ɛ1).
	'''OR'''
	Form the alternating sum of blocks of two from right to left. If the result is divisible by '''5''' then the given number is divisible by 5.
	This rule comes from 101, since 101 = 5*25, thus this rule can be also tested for the divisibility by 25.
	Example:<br>
	'''97,374,627''' => 27-46+37-97 = -7Ɛ which is divisible by 5.
	;6
	If a number is divisible by '''6''' then the unit digit of that number will be 0 or 6.
	;7
	To test for divisibility by 7, triple the units digit and add the result to the number formed by the rest of the digits. If the result is divisible by '''7''' then the given number is divisible by 7.
	This rule comes from 2Ɛ(7*5)
	Examples:<br>
	'''12'''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;rule => |3*2+1| = 7 which is divisible by 7.<br>
	'''271Ɛ'''&nbsp;&nbsp;&nbsp;&nbsp;rule => |3*Ɛ+271| = 29ᘔ(7*4ᘔ) which is divisible by 7(or apply the rule on 29ᘔ).<br>
	'''OR'''
	To test for divisibility by 7, subtract the units digit and double the result from the number formed by the rest of the digits. If the result is divisible by '''7''' then the given number is divisible by 7.
	This rule comes from 12(7*2)
	Examples:<br>
	'''12'''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;rule => |2-2*1| = 0 which is divisible by 7.<br>
	'''271Ɛ'''&nbsp;&nbsp;&nbsp;&nbsp;rule => |Ɛ-2*271| = 513(7*89) which is divisible by 7(or apply the rule on 513).<br>
	'''OR'''
	To test for divisibility by 7, 4 times the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by '''7''' then the given number is divisible by 7.
	This rule comes from 41(7*7)
	Examples:<br>
	'''12'''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;rule => |4*2-1| = 7 which is divisible by 7.<br>
	'''271Ɛ'''&nbsp;&nbsp;&nbsp;&nbsp;rule => |4*Ɛ-271| = 235(7*3Ɛ) which is divisible by 7(or apply the rule on 235).<br>
	'''OR'''
	Form the alternating sum of blocks of three from right to left. If the result is divisible by '''7''' then the given number is divisible by 7.
	This rule comes from 1001, since 1001 = 7*11*17, thus this rule can be also tested for the divisibility by 11 and 17.
	Example:<br>
	'''386,967,443''' => 443-967+386 = -168 which is divisible by 7.
	;8
	If the 2-digit number formed by the last 2 digits of the given number is divisible by '''8''' then the given number is divisible by 8.
	Example: 1Ɛ48, 4120
	rule => since 48(8*7) divisible by 8, then 1Ɛ48 is divisible by 8.
	rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8.
	;9
	If the 2-digit number formed by the last 2 digits of the given number is divisible by '''9''' then the given number is divisible by 9.
	Example: 7423, 8330
	rule => since 23(9*3) divisible by 9, then 7423 is divisible by 9.
	rule => since 30(9*4) divisible by 9, then 8330 is divisible by 9.
	;ᘔ
	If the number is divisible by 2 and 5 then the number is divisible by '''ᘔ'''.
	;Ɛ
	If the sum of the digits of a number is divisible by '''Ɛ''' then the number is divisible by Ɛ (the equivalent of ] in decimal).
	Example: 29, 61Ɛ13
	rule => 2+9 = Ɛ which is divisible by Ɛ, then 29 is divisible by Ɛ.
	rule => 6+1+Ɛ+1+3 = 1ᘔ which is divisible by Ɛ, then 61Ɛ13 is divisible by Ɛ.
	;10
	If a number is divisible by '''10''' then the unit digit of that number will be 0.
	;11
	Sum the alternate digits and subtract the sums. If the result is divisible by '''11''' the number is divisible by 11 (the equivalent of divisibility by eleven in decimal).
	Example: 66, 9427
	rule => |6-6| = 0 which is divisible by 11, then 66 is divisible by 11.
	rule => |(9+2)-(4+7)| = |ᘔ-ᘔ| = 0 which is divisible by 11, then 9427 is divisible by 11.
	;12
	If the number is divisible by 2 and 7 then the number is divisible by '''12'''.
	;13
	If the number is divisible by 3 and 5 then the number is divisible by '''13'''.
	;14
	If the 2-digit number formed by the last 2 digits of the given number is divisible by '''14''' then the given number is divisible by 14.
	==Fractions and irrational numbers==
	===Fractions===
	Duodecimal ] may be simple:
	* {{sfrac|2}} = 0.6
	* {{sfrac|3}} = 0.4
	* {{sfrac|4}} = 0.3
	* {{sfrac|6}} = 0.2
	* {{sfrac|8}} = 0.16
	* {{sfrac|9}} = 0.14
	* {{sfrac|10}} = 0.1 (note that this is a twelfth, {{sfrac|ᘔ}} is a tenth)
	* {{sfrac|14}} = 0.09 (note that this is a sixteenth, {{sfrac|12}} is a fourteenth)
	or complicated:
	* {{sfrac|5}} = 0.249724972497... recurring (rounded to 0.24ᘔ)
	* {{sfrac|7}} = 0.186ᘔ35186ᘔ35... recurring (rounded to 0.187)
	* {{sfrac|ᘔ}} = 0.1249724972497... recurring (rounded to 0.125)
	* {{sfrac|Ɛ}} = 0.111111111111... recurring (rounded to 0.111)
	* {{sfrac|11}} = 0.0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1)
	* {{sfrac|12}} = 0.0ᘔ35186ᘔ35186... recurring (rounded to 0.0ᘔ3)
	* {{sfrac|13}} = 0.0972497249724... recurring (rounded to 0.097)
	{|class="wikitable"
	|-
	! Examples in duodecimal		! Examples in duodecimal
	! Decimal equivalent		! Decimal equivalent
	|-		|-
	| 1 × ({{sfrac|5|8}}) = 0.76		| 1 × {{sfrac|5|8}} = 0.76
	| 1 × ({{sfrac|5|8}}) = 0.625		| 1 × {{sfrac|5|8}} = 0.625
	|-		|-
	| 100 × ({{sfrac|5|8}}) = 76		| 100 × {{sfrac|5|8}} = 76
	| 144 × ({{sfrac|5|8}}) = 90		| 144 × {{sfrac|5|8}} = 90
	|-		|-
	| {{sfrac|576|9}} = 76		| {{sfrac|576|9}} = 76
Line 1,638:		Line 829:
	| {{sfrac|576|9}} = 64		| {{sfrac|576|9}} = 64
	|-		|-
	| 1ᘔ.6 + 7.6 = 26		| 1{{d2}}.6 + 7.6 = 26
	| 22.5 + 7.5 = 30		| 22.5 + 7.5 = 30
	|}		|}
	As explained in ]s, whenever an ] is written in ] notation in any base, the fraction can be expressed exactly (terminates) if and only if all the ]s of its denominator are also prime factors of the base. Thus, in base-ten (=&nbsp;2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: {{sfrac|8}}&nbsp;=&nbsp;{{sfrac|(2×2×2)}}, {{sfrac|20}}&nbsp;=&nbsp;{{sfrac|(2×2×5)}} and {{sfrac|500}}&nbsp;=&nbsp;{{sfrac|(2×2×5×5×5)}} can be expressed exactly as 0.125, 0.05 and 0.002 respectively. {{sfrac|3}} and {{sfrac|7}}, however, recur (0.333... and 0.142857142857...). In the duodecimal (=&nbsp;2×2×3) system, {{sfrac|8}} is exact; {{sfrac|20}} and {{sfrac|500}} recur because they include 5 as a factor; {{sfrac|3}} is exact; and {{sfrac|7}} recurs, just as it does in decimal.		As explained in ]s, whenever an ] is written in ] notation in any base, the fraction can be expressed exactly (terminates) if and only if all the ]s of its denominator are also prime factors of the base.
			Because <math>2\times5=10</math> in the decimal system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: {{sfrac|8}}&nbsp;=&nbsp;{{sfrac|(2×2×2)}}, {{sfrac|20}}&nbsp;=&nbsp;{{sfrac|(2×2×5)}}, and {{sfrac|500}}&nbsp;=&nbsp;{{sfrac|(2×2×5×5×5)}} can be expressed exactly as 0.125, 0.05, and 0.002 respectively. {{sfrac|3}} and {{sfrac|7}}, however, recur (0.333... and 0.142857142857...).
	The number of denominators which give terminating fractions within a given number of digits, say ''n'', in a base ''b'' is the number of factors (divisors) of ''b<sup>n</sup>'', the ''n''th power of the base ''b'' (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of ''b<sup>n</sup>'' is given using its prime factorization.
			Because <math>2\times2\times3=12</math> in the duodecimal system, {{sfrac|8}} is exact; {{sfrac|20}} and {{sfrac|500}} recur because they include 5 as a factor; {{sfrac|3}} is exact, and {{sfrac|7}} recurs, just as it does in decimal.
	For decimal, 10<sup>''n''</sup> = 2<sup>''n''</sup> * 5<sup>''n''</sup>. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together.
	Factors of 10<sup>''n''</sup> = (''n''+1)(''n''+1) = (''n''+1)<sup>2</sup>.
			The number of denominators that give terminating fractions within a given number of digits, {{math|''n''}}, in a base {{math|''b''}} is the number of factors (divisors) of <math>b^n</math>, the {{math|''n''}}th power of the base {{math|''b''}} (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of ''<math>b^n</math>'' is given using its prime factorization.
	For example, the number 8 is a factor of 10<sup>3</sup> (1000), so 1/8 and other fractions with a denominator of 8 can not require more than 3 fractional decimal digits to terminate. 5/8 = 0.625<sub>ten</sub>
			For decimal, <math>10^n=2^n\times 5^n</math>. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together, so the number of factors of ''<math>10^n</math>'' is <math>(n+1)(n+1)=(n+1)^2</math>.
	For duodecimal, 12<sup>''n''</sup> = 2<sup>2''n''</sup> * 3<sup>''n''</sup>. This has (2''n''+1)(''n''+1) divisors. The sample denominator of 8 is a factor of a gross (12<sup>2</sup> = 144), so eighths can not need more than two duodecimal fractional places to terminate. 5/8 = 0.76<sub>twelve</sub>
			For example, the number 8 is a factor of 10<sup>3</sup> (1000), so <math display="inline">\frac{1}{8}</math> and other fractions with a denominator of 8 cannot require more than three fractional decimal digits to terminate. <math display="inline">\frac{5}{8}=0.625_{10}.</math>
	Because both ten and twelve have two unique prime factors, the number of divisors of ''b<sup>n</sup>'' for ''b'' = 10 or 12 grows quadratically with the exponent ''n'' (in other words, of the order of ''n''<sup>2</sup>).
			For duodecimal, <math>10^n=2^{2n}\times 3^n</math>. This has <math>(2n+1)(n+1)</math> divisors. The sample denominator of 8 is a factor of a gross <math display="inline">12^2=144</math> (in decimal), so eighths cannot need more than two duodecimal fractional places to terminate. <math display="inline">\frac{5}{8}=0.76_{12}.</math>
			Because both ten and twelve have two unique prime factors, the number of divisors of ''<math>b^n</math>'' for {{math|''b'' {{=}} 10 or 12}} grows quadratically with the exponent {{math|''n''}} (in other words, of the order of <math>n^2</math>).
	=== Recurring digits ===		=== Recurring digits ===
	] argues that factors of 3 are more commonly encountered in real-life ] problems than factors of 5.<ref>http://www.dozenal.org/articles/DSA-DozenalFAQs.pdf</ref> Thus, in practical applications, the nuisance of ] is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.		The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life ] problems than factors of 5.<ref name="dsafaq">{{cite web |author=De Vlieger |first=Michael Thomas |date=30 November 2011 |title=Dozenal FAQs |url=https://dozenal.org/articles/DSA-DozenalFAQs.pdf |access-date=November 20, 2022 |website=dozenal.org |publisher=The Dozenal Society of America }}</ref> Thus, in practical applications, the nuisance of ] is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.
	However, when recurring fractions ''do'' occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because ] (twelve) is between two ]s, ] (eleven) and ] (thirteen), whereas ten is adjacent to the ] ]. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so ], which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor ] in its factorization, whereas only one out of every five contains the prime factor ]. All other prime factors, except 2, are not shared by either ten or twelve, so they do not		However, when recurring fractions ''do'' occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because ] (twelve) is between two ]s, ] (eleven) and ] (thirteen), whereas ten is adjacent to the ] ]. Nonetheless, having a shorter or longer period does not help the main inconvenience that one does not get a finite representation for such fractions in the given base (so ], which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor ] in its factorization, whereas only one out of every five contains the prime factor ]. All other prime factors, except 2, are not shared by either ten or twelve, so they do not
	influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor ] appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are ] will have a shorter, more convenient terminating representation in duodecimal than in decimal representation (e.g. 1/(2<sup>2</sup>) = 0.25 <sub>ten</sub> = 0.3 <sub>twelve</sub>; 1/(2<sup>3</sup>) = 0.125 <sub>ten</sub> = 0.16 <sub>twelve</sub>; 1/(2<sup>4</sup>) = 0.0625 <sub>ten</sub> = 0.09 <sub>twelve</sub>; 1/(2<sup>5</sup>) = 0.03125 <sub>ten</sub> = 0.046 <sub>twelve</sub>; etc.).		influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base).
			Also, the prime factor ] appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are ] will have a shorter, more convenient terminating representation in duodecimal than in decimal:
	Values in '''bold''' indicate that value is exact.
			* 1/(2<sup>2</sup>) = 0.25<sub>10</sub> = 0.3<sub>12</sub>
	{|class="wikitable"
			* 1/(2<sup>3</sup>) = 0.125<sub>10</sub> = 0.16<sub>12</sub>
	|- style="text-align:center;"
			* 1/(2<sup>4</sup>) = 0.0625<sub>10</sub> = 0.09<sub>12</sub>
	| colspan="3"| '''Decimal base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Orange">'''11'''</span></SMALL><br><SMALL>All other primes: <span style="color:Red">'''7 13 17 19 23 29 31'''</span></SMALL>
			* 1/(2<sup>5</sup>) = 0.03125<sub>10</sub> = 0.046<sub>12</sub>
	| colspan="3"| '''Duodecimal base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''Ɛ'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Orange">'''11'''</span></SMALL><br><SMALL>All other primes: <span style="color:Red">'''5 7 15 17 1Ɛ 25 27'''</span></SMALL>
	|-
			{| class="wikitable"
			|- style="text-align:center;"
			| colspan="3"| '''Decimal base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''11'''</span></SMALL><br><SMALL>All other primes: <span style="color:Red">'''7'''</span>, <span style="color:Red">'''13'''</span>, <span style="color:Red">'''17'''</span>, <span style="color:Red">'''19'''</span>, <span style="color:Red">'''23'''</span>, <span style="color:Red">'''29'''</span>, <span style="color:Red">'''31'''</span></SMALL>
			| colspan="3"| '''Duodecimal base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''{{d3}}'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''11 (=13<sub>10</sub>)'''</span></SMALL><br><SMALL>All other primes: <span style="color:Red">'''5'''</span>, <span style="color:Red">'''7'''</span>, <span style="color:Red">'''15 (=17<sub>10</sub>)'''</span>, <span style="color:Red">'''17 (=19<sub>10</sub>)'''</span>, <span style="color:Red">'''1{{d3}} (=23<sub>10</sub>)'''</span>, <span style="color:Red">'''25 (=29<sub>10</sub>)'''</span>, <span style="color:Red">'''27 (=31<sub>10</sub>)'''</span></SMALL>
			|-
	! Fraction		! Fraction
	! <SMALL>Prime factors<br>of the denominator</SMALL>		! <SMALL>Prime factors<br>of the denominator</SMALL>
Line 1,677:		Line 876:
	| style="text-align:center;"| 1/2		| style="text-align:center;"| 1/2
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| '''0.5'''		| 0.5
	| '''0.6'''		| 0;6
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| style="text-align:center;"| 1/2		| style="text-align:center;"| 1/2
Line 1,684:		Line 883:
	| style="text-align:center;"| 1/3		| style="text-align:center;"| 1/3
	| style="text-align:center;"| <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.'''{{overline|3}}		| 0.{{overline|3}}
	| '''0.4'''		| 0;4
	| style="text-align:center;"| <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/3		| style="text-align:center;"| 1/3
Line 1,691:		Line 890:
	| style="text-align:center;"| 1/4		| style="text-align:center;"| 1/4
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| '''0.25'''		| 0.25
	| '''0.3'''		| 0;3
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| style="text-align:center;"| 1/4		| style="text-align:center;"| 1/4
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	| style="text-align:center;"| 1/5		| style="text-align:center;"| 1/5
	| style="text-align:center;"| <span style="color:Green">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''5'''</span>
	| '''0.2'''		| 0.2
	| style="background:silver;"| '''0.'''{{overline|2497}}		| 0;{{overline|2497}}
	| style="text-align:center;"| <span style="color:Red">'''5'''</span>		| style="text-align:center;"| <span style="color:Red">'''5'''</span>
	| style="text-align:center;"| 1/5		| style="text-align:center;"| 1/5
Line 1,705:		Line 904:
	| style="text-align:center;"| 1/6		| style="text-align:center;"| 1/6
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.1'''{{overline|6}}		| 0.1{{overline|6}}
	| '''0.2'''		| 0;2
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/6		| style="text-align:center;"| 1/6
Line 1,712:		Line 911:
	| style="text-align:center;"| 1/7		| style="text-align:center;"| 1/7
	| style="text-align:center;"| <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Red">'''7'''</span>
	| style="background:silver;"| '''0.'''{{overline|142857}}		| 0.{{overline|142857}}
	| style="background:silver;"| '''0.'''{{overline|186ᘔ35}}		| 0;{{overline|186{{d2}}35}}
	| style="text-align:center;"| <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Red">'''7'''</span>
	| style="text-align:center;"| 1/7		| style="text-align:center;"| 1/7
Line 1,719:		Line 918:
	| style="text-align:center;"| 1/8		| style="text-align:center;"| 1/8
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| '''0.125'''		| 0.125
	| '''0.16'''		| 0;16
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| style="text-align:center;"| 1/8		| style="text-align:center;"| 1/8
Line 1,726:		Line 925:
	| style="text-align:center;"| 1/9		| style="text-align:center;"| 1/9
	| style="text-align:center;"| <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.'''{{overline|1}}		| 0.{{overline|1}}
	| '''0.14'''		| 0;14
	| style="text-align:center;"| <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/9		| style="text-align:center;"| 1/9
Line 1,733:		Line 932:
	| style="text-align:center;"| 1/10		| style="text-align:center;"| 1/10
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
	| '''0.1'''		| 0.1
	| style="background:silver;"| '''0.1'''{{overline|2497}}		| 0;1{{overline|2497}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>
	| style="text-align:center;"| 1/ᘔ		| style="text-align:center;"| 1/{{d2}}
	|-		|-
	| style="text-align:center;"| 1/11		| style="text-align:center;"| 1/11
	| style="text-align:center;"| <span style="color:Orange">'''11'''</span>		| style="text-align:center;"| <span style="color:Magenta">'''11'''</span>
	| style="background:silver;"| '''0.'''{{overline|09}}		| 0.{{overline|09}}
	| style="background:silver;"| '''0.'''{{overline|1}}		| 0;{{overline|1}}
	| style="text-align:center;"| <span style="color:Blue">'''Ɛ'''</span>		| style="text-align:center;"| <span style="color:Blue">'''{{d3}}'''</span>
	| style="text-align:center;"| 1/Ɛ		| style="text-align:center;"| 1/{{d3}}
	|-		|-
	| style="text-align:center;"| 1/12		| style="text-align:center;"| 1/12
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.08'''{{overline|3}}		| 0.08{{overline|3}}
	| '''0.1'''		| 0;1
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/10		| style="text-align:center;"| 1/10
Line 1,754:		Line 953:
	| style="text-align:center;"| 1/13		| style="text-align:center;"| 1/13
	| style="text-align:center;"| <span style="color:Red">'''13'''</span>		| style="text-align:center;"| <span style="color:Red">'''13'''</span>
	| style="background:silver;"| '''0.'''{{overline|076923}}		| 0.{{overline|076923}}
	| style="background:silver;"| '''0.'''{{overline|0Ɛ}}		| 0;{{overline|0{{d3}}}}
	| style="text-align:center;"| <span style="color:Orange">'''11'''</span>		| style="text-align:center;"| <span style="color:Magenta">'''11'''</span>
	| style="text-align:center;"| 1/11		| style="text-align:center;"| 1/11
	|-		|-
	| style="text-align:center;"| 1/14		| style="text-align:center;"| 1/14
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
	| style="background:silver;"| '''0.0'''{{overline|714285}}		| 0.0{{overline|714285}}
	| style="background:silver;"| '''0.0'''{{overline|ᘔ35186}}		| 0;0{{overline|{{d2}}35186}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
	| style="text-align:center;"| 1/12		| style="text-align:center;"| 1/12
Line 1,768:		Line 967:
	| style="text-align:center;"| 1/15		| style="text-align:center;"| 1/15
	| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>		| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
	| style="background:silver;"| '''0.0'''{{overline|6}}		| 0.0{{overline|6}}
	| style="background:silver;"| '''0.0'''{{overline|9724}}		| 0;0{{overline|9724}}
	| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Red">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Red">'''5'''</span>
	| style="text-align:center;"| 1/13		| style="text-align:center;"| 1/13
Line 1,775:		Line 974:
	| style="text-align:center;"| 1/16		| style="text-align:center;"| 1/16
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| '''0.0625'''		| 0.0625
	| '''0.09'''		| 0;09
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| style="text-align:center;"| 1/14		| style="text-align:center;"| 1/14
Line 1,782:		Line 981:
	| style="text-align:center;"| 1/17		| style="text-align:center;"| 1/17
	| style="text-align:center;"| <span style="color:Red">'''17'''</span>		| style="text-align:center;"| <span style="color:Red">'''17'''</span>
	| style="background:silver;"| '''0.'''{{overline|0588235294117647}}		| 0.{{overline|0588235294117647}}
	| style="background:silver;"| '''0.'''{{overline|08579214Ɛ36429ᘔ7}}		| 0;{{overline|08579214{{d3}}36429{{d2}}7}}
	| style="text-align:center;"| <span style="color:Red">'''15'''</span>		| style="text-align:center;"| <span style="color:Red">'''15'''</span>
	| style="text-align:center;"| 1/15		| style="text-align:center;"| 1/15
Line 1,789:		Line 988:
	| style="text-align:center;"| 1/18		| style="text-align:center;"| 1/18
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.0'''{{overline|5}}		| 0.0{{overline|5}}
	| '''0.08'''		| 0;08
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/16		| style="text-align:center;"| 1/16
Line 1,796:		Line 995:
	| style="text-align:center;"| 1/19		| style="text-align:center;"| 1/19
	| style="text-align:center;"| <span style="color:Red">'''19'''</span>		| style="text-align:center;"| <span style="color:Red">'''19'''</span>
	| style="background:silver;"| '''0.'''{{overline|052631578947368421}}		| 0.{{overline|052631578947368421}}
	| style="background:silver;"| '''0.'''{{overline|076Ɛ45}}		| 0;{{overline|076{{d3}}45}}
	| style="text-align:center;"| <span style="color:Red">'''17'''</span>		| style="text-align:center;"| <span style="color:Red">'''17'''</span>
	| style="text-align:center;"| 1/17		| style="text-align:center;"| 1/17
Line 1,803:		Line 1,002:
	| style="text-align:center;"| 1/20		| style="text-align:center;"| 1/20
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
	| '''0.05'''		| 0.05
	| style="background:silver;"| '''0.0'''{{overline|7249}}		| 0;0{{overline|7249}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>
	| style="text-align:center;"| 1/18		| style="text-align:center;"| 1/18
Line 1,810:		Line 1,009:
	| style="text-align:center;"| 1/21		| style="text-align:center;"| 1/21
	| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span>
	| style="background:silver;"| '''0.'''{{overline|047619}}		| 0.{{overline|047619}}
	| style="background:silver;"| '''0.0'''{{overline|6ᘔ3518}}		| 0;0{{overline|6{{d2}}3518}}
	| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Red">'''7'''</span>
	| style="text-align:center;"| 1/19		| style="text-align:center;"| 1/19
	|-		|-
	| style="text-align:center;"| 1/22		| style="text-align:center;"| 1/22
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Orange">'''11'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
	| style="background:silver;"| '''0.0'''{{overline|45}}		| 0.0{{overline|45}}
	| style="background:silver;"| '''0.0'''{{overline|6}}		| 0;0{{overline|6}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''Ɛ'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''{{d3}}'''</span>
	| style="text-align:center;"| 1/1ᘔ		| style="text-align:center;"| 1/1{{d2}}
	|-		|-
	| style="text-align:center;"| 1/23		| style="text-align:center;"| 1/23
	| style="text-align:center;"| <span style="color:Red">'''23'''</span>		| style="text-align:center;"| <span style="color:Red">'''23'''</span>
	| style="background:silver;"| '''0.'''{{overline|0434782608695652173913}}		| 0.{{overline|0434782608695652173913}}
	| style="background:silver;"| '''0.'''{{overline|06316948421}}		| 0;{{overline|06316948421}}
	| style="text-align:center;"| <span style="color:Red">'''1Ɛ'''</span>		| style="text-align:center;"| <span style="color:Red">'''1{{d3}}'''</span>
	| style="text-align:center;"| 1/1Ɛ		| style="text-align:center;"| 1/1{{d3}}
	|-		|-
	| style="text-align:center;"| 1/24		| style="text-align:center;"| 1/24
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.041'''{{overline|6}}		| 0.041{{overline|6}}
	| '''0.06'''		| 0;06
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/20		| style="text-align:center;"| 1/20
Line 1,838:		Line 1,037:
	| style="text-align:center;"| 1/25		| style="text-align:center;"| 1/25
	| style="text-align:center;"| <span style="color:Green">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''5'''</span>
	| '''0.04'''		| 0.04
			| 0;{{overline|05915343{{d2}}0{{d3}}62{{d2}}68781{{d3}}}}
	| style="background:silver;"| '''0.'''{{overline|05915343ᘔ0Ɛ62ᘔ68781Ɛ}}
	| style="text-align:center;"| <span style="color:Red">'''5'''</span>		| style="text-align:center;"| <span style="color:Red">'''5'''</span>
	| style="text-align:center;"| 1/21		| style="text-align:center;"| 1/21
Line 1,845:		Line 1,044:
	| style="text-align:center;"| 1/26		| style="text-align:center;"| 1/26
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
	| style="background:silver;"| '''0.0'''{{overline|384615}}		| 0.0{{overline|384615}}
	| style="background:silver;"| '''0.0'''{{overline|56}}		| 0;0{{overline|56}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Orange">'''11'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
	| style="text-align:center;"| 1/22		| style="text-align:center;"| 1/22
	|-		|-
	| style="text-align:center;"| 1/27		| style="text-align:center;"| 1/27
	| style="text-align:center;"| <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.'''{{overline|037}}		| 0.{{overline|037}}
	| '''0.054'''		| 0;054
	| style="text-align:center;"| <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/23		| style="text-align:center;"| 1/23
Line 1,859:		Line 1,058:
	| style="text-align:center;"| 1/28		| style="text-align:center;"| 1/28
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
	| style="background:silver;"| '''0.03'''{{overline|571428}}		| 0.03{{overline|571428}}
	| style="background:silver;"| '''0.0'''{{overline|5186ᘔ3}}		| 0;0{{overline|5186{{d2}}3}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
	| style="text-align:center;"| 1/24		| style="text-align:center;"| 1/24
Line 1,866:		Line 1,065:
	| style="text-align:center;"| 1/29		| style="text-align:center;"| 1/29
	| style="text-align:center;"| <span style="color:Red">'''29'''</span>		| style="text-align:center;"| <span style="color:Red">'''29'''</span>
	| style="background:silver;"| '''0.'''{{overline|0344827586206896551724137931}}		| 0.{{overline|0344827586206896551724137931}}
	| style="background:silver;"| '''0.'''{{overline|04Ɛ7}}		| 0;{{overline|04{{d3}}7}}
	| style="text-align:center;"| <span style="color:Red">'''25'''</span>		| style="text-align:center;"| <span style="color:Red">'''25'''</span>
	| style="text-align:center;"| 1/25		| style="text-align:center;"| 1/25
Line 1,873:		Line 1,072:
	| style="text-align:center;"| 1/30		| style="text-align:center;"| 1/30
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
	| style="background:silver;"| '''0.0'''{{overline|3}}		| 0.0{{overline|3}}
	| style="background:silver;"| '''0.0'''{{overline|4972}}		| 0;0{{overline|4972}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>, <span style="color:Red">'''5'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>, <span style="color:Red">'''5'''</span>
	| style="text-align:center;"| 1/26		| style="text-align:center;"| 1/26
Line 1,880:		Line 1,079:
	| style="text-align:center;"| 1/31		| style="text-align:center;"| 1/31
	| style="text-align:center;"| <span style="color:Red">'''31'''</span>		| style="text-align:center;"| <span style="color:Red">'''31'''</span>
	| style="background:silver;"| '''0.'''{{overline|032258064516129}}		| 0.{{overline|032258064516129}}
			| 0;{{overline|0478{{d2}}{{d2}}093598166{{d3}}74311{{d3}}28623{{d2}}55}}
	| style="background:silver;"| '''0.'''{{overline|0478ᘔᘔ093598166Ɛ74311Ɛ28623ᘔ55}}
	| style="text-align:center;"| <span style="color:Red">'''27'''</span>		| style="text-align:center;"| <span style="color:Red">'''27'''</span>
	| style="text-align:center;"| 1/27		| style="text-align:center;"| 1/27
Line 1,887:		Line 1,086:
	| style="text-align:center;"| 1/32		| style="text-align:center;"| 1/32
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| '''0.03125'''		| 0.03125
	| '''0.046'''		| 0;046
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>
	| style="text-align:center;"| 1/28		| style="text-align:center;"| 1/28
	|-		|-
	| style="text-align:center;"| 1/33		| style="text-align:center;"| 1/33
	| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Orange">'''11'''</span>		| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span>
	| style="background:silver;"| '''0.'''{{overline|03}}		| 0.{{overline|03}}
	| style="background:silver;"| '''0.0'''{{overline|4}}		| 0;0{{overline|4}}
	| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Blue">'''Ɛ'''</span>		| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Blue">'''{{d3}}'''</span>
	| style="text-align:center;"| 1/29		| style="text-align:center;"| 1/29
	|-		|-
	| style="text-align:center;"| 1/34		| style="text-align:center;"| 1/34
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''17'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''17'''</span>
	| style="background:silver;"| '''0.0'''{{overline|2941176470588235}}		| 0.0{{overline|2941176470588235}}
	| style="background:silver;"| '''0.0'''{{overline|429ᘔ708579214Ɛ36}}		| 0;0{{overline|429{{d2}}708579214{{d3}}36}}
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''15'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''15'''</span>
	| style="text-align:center;"| 1/2ᘔ		| style="text-align:center;"| 1/2{{d2}}
	|-		|-
	| style="text-align:center;"| 1/35		| style="text-align:center;"| 1/35
	| style="text-align:center;"| <span style="color:Green">'''5'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Green">'''5'''</span>, <span style="color:Red">'''7'''</span>
	| style="background:silver;"| '''0.0'''{{overline|285714}}		| 0.0{{overline|285714}}
	| style="background:silver;"| '''0.'''{{overline|0414559Ɛ3931}}		| 0;{{overline|0414559{{d3}}3931}}
	| style="text-align:center;"| <span style="color:Red">'''5'''</span>, <span style="color:Red">'''7'''</span>		| style="text-align:center;"| <span style="color:Red">'''5'''</span>, <span style="color:Red">'''7'''</span>
	| style="text-align:center;"| 1/2Ɛ		| style="text-align:center;"| 1/2{{d3}}
	|-		|-
	| style="text-align:center;"| 1/36		| style="text-align:center;"| 1/36
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
	| style="background:silver;"| '''0.02'''{{overline|7}}		| 0.02{{overline|7}}
	| '''0.04'''		| 0;04
	| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>		| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
	| style="text-align:center;"| 1/30		| style="text-align:center;"| 1/30
	|}		|}
	The duodecimal period length of 1/''n'' are		The duodecimal period length of 1/''n'' are (in decimal)
	:0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... {{OEIS|id=A246004}}		:0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... {{OEIS|id=A246004}}
	The duodecimal period length of 1/(''n''th prime) are		The duodecimal period length of 1/(''n''th prime) are (in decimal)
	:0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... {{OEIS|id=A246489}}		:0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... {{OEIS|id=A246489}}
	Smallest prime with duodecimal period ''n'' are		Smallest prime with duodecimal period ''n'' are (in decimal)
	:11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... {{OEIS|id=A252170}}		:11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... {{OEIS|id=A252170}}
	=== Irrational numbers ===		=== Irrational numbers ===
			The representations of ]s in any positional number system (including decimal and duodecimal) neither terminate nor ]. The following table gives the first digits for some important ] and ] numbers in both decimal and duodecimal.
	As for ]s, none of them have a finite representation in ''any'' of the ]-based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no ''finite'' sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 10<sup>2</sup> + 2 × 10<sup>1</sup> + 3 × 10<sup>0</sup> + 4 × 1/10<sup>1</sup> + 5 × 1/10<sup>2</sup> + 6 × 1/10<sup>3</sup> (this is also the reason why fractions that contain prime factors in their denominator not in common with those of the base do not have a terminating representation in that base). Moreover, the infinite series of digits of an irrational number does not exhibit ]; instead, the different digits often succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important ] and ] irrational numbers. Some of these numbers may be perceived as having fortuitous patterns, making them easier to memorize, when represented in one base or the other.
	{|class="wikitable"		{| class="wikitable"
			! Algebraic irrational number
	|-
	! ''Algebraic irrational number''
	! In decimal		! In decimal
	! '''In duodecimal'''		! In duodecimal
	|-		|-
	| style="text-align:center;"| ] <SMALL>(the length of the ] of a unit ])</SMALL>		| style="text-align:center;"| ], the square root of 2
	| 1.41421356237309... (≈ 1.4142)		| 1.414213562373...
			| 1;4{{d3}}79170{{d2}}07{{d3}}8...
	| 1.4Ɛ79170ᘔ07Ɛ857... (≈ 1.5)
	|-		|-
	| style="text-align:center;"| ] <SMALL>(the length of the diagonal of a unit ], or twice the ] of an ] of unit side)</SMALL>		| style="text-align:center;"| {{mvar|]}} (phi), the golden ratio = <math>\tfrac{1+\sqrt{5}}{2}</math>
	| 1.73205080756887... (≈ 1.732)		| 1.618033988749...
			| 1;74{{d3}}{{d3}}6772802{{d2}}...
	| 1.894Ɛ97ƐƐ968704... (≈ 1.895)
	|-		|-
			! Transcendental number
	| style="text-align:center;"| ] <SMALL>(the length of the ] of a 1×2 ])</SMALL>
	| 2.2360679774997... (≈ 2.236)
	| 2.29ƐƐ132540589... (≈ 2.2ᘔ)
	|-
	| style="text-align:center;"| ] <SMALL>(phi, the golden ratio = <math>\scriptstyle \frac{1+\sqrt{5}}{2}</math>)</SMALL>
	| 1.6180339887498... (≈ 1.618)
	| 1.74ƐƐ6772802ᘔ4... (≈ 1.75)
	|-
	! ''Transcendental irrational number''
	! In decimal		! In decimal
	! '''In duodecimal'''		! In duodecimal
	|-		|-
	| style="text-align:center;"| '']'' <SMALL>(pi, the ratio of ] to ])</SMALL>		| style="text-align:center;"| {{mvar|]}} (pi), the ratio of a circle's ] to its ]
			| 3.141592653589...
	| 3.1415926535897932384626433<br>8327950288419716939937510...<br>(≈ 3.1416)
			| 3;184809493{{d3}}91...
	| 3.184809493Ɛ918664573ᘔ6211Ɛ<br>Ɛ151551ᘔ05729290ᘔ7809ᘔ492...<br>(≈ 3.1848)
	|-
	| style="text-align:center;"| ] <SMALL>(the base of the ])</SMALL>
	| 2.718281828459045... (≈ 2.718)
	| 2.8752360698219Ɛ8... (≈ 2.875)
	|}
	The first few digits of the decimal and duodecimal representation of another important number, the ] (the status of which as a rational or irrational number is not yet known), are:
	{|class="wikitable"
	|-
	! ''Number''
	! In decimal
	! '''In duodecimal'''
	|-		|-
	| style="text-align:center;"| ] <SMALL>(the limiting difference between the ] and the natural logarithm)</SMALL>		| style="text-align:center;"| {{mvar|]}}, the base of the ]
			| 2.718281828459...
	| 0.57721566490153... (≈ 0.577)
			| 2;875236069821...
	| 0.6Ɛ15188ᘔ6760Ɛ3... (≈ 0.7)
	|}		|}
	== See also ==		== See also ==
	* ] (base 6)		* ] (base 20)
	* ] (base 10)
	* ] (base 60)		* ] (base 60)
Line 1,991:		Line 1,167:
	== External links ==		== External links ==
	*		*
			**
	*
			** , the DSA website's page of external links to third-party tools
	*
	*		*
			* {{cite web |last=Lauritzen |first=Bill |year=1994 |title=Nature's Numbers |work=Earth360 |url=http://www.earth360.com/new_number.html }}
	*
			* {{cite web |last=Savard |first=John J. G. |year=2018 |title=Changing the Base |orig-year=2016 |work=quadibloc |url=http://www.quadibloc.com/math/baseint.htm |access-date=2018-07-17 }}
	*
	* {{cite web|last=Grime|first=James|title=Base 12: Dozenal or Duodecimal|url=http://www.numberphile.com/videos/base_12.html|work=Numberphile|publisher=]}}
	]		]
			]

---

Latest revision as of 06:51, 11 January 2025

Base-12 numeral system Not to be confused with Dewey Decimal Classification or Duodecimo.

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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 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 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 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 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 the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.

Origin

In this section, numerals are in decimal. For example, "10" means 9+1, and "12" means 9+3.

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 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 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 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 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 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.

Duodecimally divided units

Relative value	Length		Weight
	French	English	English (Troy)	Roman
12	pied	foot	pound	libra
12	pouce	inch	ounce	uncia
12	ligne	line	2 scruples	2 scrupula
12	point	point	seed	siliqua

Notations and pronunciations

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, 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.

Transdecimal symbols

2 3
duodecimal ⟨ten, eleven⟩
In Unicode	U+218A ↊ TURNED DIGIT TWO U+218B ↋ TURNED DIGIT THREE
Block Number Forms
Note
Arabic digits with 180° rotation, by Isaac Pitman In LaTeX, using the TIPA package: ⟨\textturntwo, \textturnthree⟩

Several authors have proposed using letters of 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 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 was used in publications of the Dozenal Society of America (DSA) from 1974 to 2008.

From 2008 to 2015, the DSA used ⟨ ,  ⟩, 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, 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 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.

Symbols		Background	Note
A	B	As in hexadecimal	Allows entry on typewriters.
T	E	Initials of Ten and Eleven	Used (in lower case) in music set theory
X	E	X from the Roman numeral; E from Eleven.
X	Z	Origin of Z unknown	Attributed to D'Alembert & Buffon by the DSA.
δ	ε	Greek delta from δέκα "ten"; epsilon from ένδεκα "eleven"
τ	ε	Greek tau, epsilon
W	∂	W from doubling the Roman numeral V; ∂ based on a pendulum	Silvio Ferrari in Calcolo Decidozzinale (1854).
X	Ɛ	italic X pronounced "dec"; rounded italic Ɛ, pronounced "elf"	Frank Andrews in New Numbers (1935), with italic 0–9 for other duodecimal numerals.
*	#	sextile or six-pointed asterisk, hash or octothorpe	On push-button telephones; used by Edna Kramer in The Main Stream of Mathematics (1951); used by the DSA 1974–2008
2	3	Digits 2 and 3 rotated 180°	Isaac Pitman (1857); used by the DSGB; used by the DSA since 2015; included in Unicode 8.0 (2015)
		Pronounced "dek", "el"	William Dwiggins (1945/1932?). Used by the DSA 1945–1974 and 2008–2015

Base notation

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. "54₁₂ = 64₁₀". To avoid ambiguity about 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 "dozenal" and "d" for "decimal", "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 a Humphrey point for other duodecimal numbers.

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 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.

Base-12 nomenclature in English

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Another nominal for twelve (12₁₀) is a dozen (10₁₂ or 1•10₁₂).

One hundred and forty-four (144₁₀) is also known as a gross (100₁₂ or 1•10₁₂).

One thousand, seven hundred and twenty-eight is (1728₁₀) also known as a great-gross (1,000₁₂ or 1•10₁₂).

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 a greater positional-notation value. 20,736₁₀ or 10,000₁₂ may be rendered a dozen-great-gross; so 248,832₁₀ or 100,000₁₂ is a gross-great-gross, with 2,985,984₁₀ or 1,000,000₁₂ being known as a great-great-gross.

It should be made plain that the indice's being a multiple of three, e.g. 10₁₂ , 10₁₂ , 10₁₂ results, in these examples, in a great gross, a great-great-gross, and a great-great-great-gross, respectively.

Scientific notation	Positional notation	Name	Decimal
1•10	1	One	1
A•10	A	Ten	10
B•10	B	Eleven	11
1•10	10	Twelve	12
5•10	50	Five dozen	60
1•10	100	One gross	144
2;6•10	260	Two gross, six dozen	360
1•10	1,000	One great-gross	1,728
1•10	10,000	One dozen-great-gross	20,736
1•10	100,000	One gross-great-gross	248,832
1•10	1,000,000	One great-great-gross	2,985,984

Duodecimal numbers

In this system, the prefix e- is added for fractions.

Duodecimal number	Number name	Decimal number	Duodecimal fraction	Fraction name
1;	one	1
10;	do	12	0;1	edo
100;	gro	144	0;01	egro
1,000;	mo	1,728	0;001	emo
10,000;	do-mo	20,736	0;000,1	edo-mo
100,000;	gro-mo	248,832	0;000,01	egro-mo
1,000,000;	bi-mo	2,985,984	0;000,001	ebi-mo
10,000,000;	do-bi-mo	35,831,808	0;000,000,1	edo-bi-mo
100,000,000;	gro-bi-mo	429,981,696	0;000,000,01	egro-bi-mo

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 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.

Systematic Dozenal Nomenclature (SDN)

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 is meant.

Duodecimal number	Number name	Decimal number	Duodecimal fraction	Fraction name
1;	one	1
10;	unqua	12	0;1	uncia
100;	biqua	144	0;01	bicia
1,000;	triqua	1,728	0;001	tricia
10,000;	quadqua	20,736	0;000,1	quadcia
100,000;	pentqua	248,832	0;000,01	pentcia
1,000,000;	hexqua	2,985,984	0;000,001	hexcia

After hex-, further prefixes continue sept-, oct-, enn-, dec-, lev-, unnil-, unun-.

Advocacy and "dozenalism"

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 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 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 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 the decimal. This is my experience; I am certain that even more so it would be the experience of others.

— A. C. Aitken, "Twelves and Tens" in The Listener (January 25, 1962)

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 the duodecimal.

— A. C. Aitken, The Case Against Decimalisation (1962)

In media

In "Little Twelvetoes," an episode of the American educational television series Schoolhouse Rock!, a farmer encounters an alien being with twelve fingers on each hand and twelve toes on each foot who uses duodecimal arithmetic. The alien uses "dek" and "el" as names for ten and eleven, and Andrews' script-X and script-E for the digit symbols.

Duodecimal systems of measurements

Systems of measurement proposed by dozenalists include:

• Tom Pendlebury's TGM system
• Takashi Suga's Universal Unit System
• John Volan's Primel system

Comparison to other number systems

In this section, numerals are in decimal. For example, "10" means 9+1, and "12" means 9+3.

The Dozenal Society of America argues that if a base 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 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, 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 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 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.

Duodecimal multiplication table

×	1	2	3	4	5	6	7	8	9	A	B	10
1	1	2	3	4	5	6	7	8	9	A	B	10
2	2	4	6	8	A	10	12	14	16	18	1A	20
3	3	6	9	10	13	16	19	20	23	26	29	30
4	4	8	10	14	18	20	24	28	30	34	38	40
5	5	A	13	18	21	26	2B	34	39	42	47	50
6	6	10	16	20	26	30	36	40	46	50	56	60
7	7	12	19	24	2B	36	41	48	53	5A	65	70
8	8	14	20	28	34	40	48	54	60	68	74	80
9	9	16	23	30	39	46	53	60	69	76	83	90
A	A	18	26	34	42	50	5A	68	76	84	92	A0
B	B	1A	29	38	47	56	65	74	83	92	A1	B0
10	10	20	30	40	50	60	70	80	90	A0	B0	100

Conversion tables to and from decimal

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, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example:

12,345.6 = 10,000 + 2,000 + 300 + 40 + 5 + 0.6

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 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:

(duodecimal) 10,000 + 2,000 + 300 + 40 + 5 + 0;6
= (decimal) 20,736 + 3,456 + 432 + 48 + 5 + 0.5

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:

Duodecimal --->  Decimal
  10,000    =   20,736
   2,000    =    3,456 
     300    =      432
      40    =       48
       5    =        5
 +     0;6  =  +     0.5
-----------------------------
  12,345;6  =   24,677.5

That is, (duodecimal) 12,345;6 equals (decimal) 24,677.5

If 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:

  Decimal --> Duodecimal
  10,000    =   5,954
   2,000    =   1,1A8
     300    =     210
      40    =      34
       5    =       5
 +     0.6  =  +    0;7249
-------------------------------
  12,345.6  =   7,189;7249

That is, (decimal) 12,345.6 equals (duodecimal) 7,189;7249

Duodecimal to decimal digit conversion

Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.
10,000	20,736	1,000	1,728	100	144	10	12	1	1	0;1	0.083
20,000	41,472	2,000	3,456	200	288	20	24	2	2	0;2	0.16
30,000	62,208	3,000	5,184	300	432	30	36	3	3	0;3	0.25
40,000	82,944	4,000	6,912	400	576	40	48	4	4	0;4	0.3
50,000	103,680	5,000	8,640	500	720	50	60	5	5	0;5	0.416
60,000	124,416	6,000	10,368	600	864	60	72	6	6	0;6	0.5
70,000	145,152	7,000	12,096	700	1,008	70	84	7	7	0;7	0.583
80,000	165,888	8,000	13,824	800	1,152	80	96	8	8	0;8	0.6
90,000	186,624	9,000	15,552	900	1,296	90	108	9	9	0;9	0.75
A0,000	207,360	A,000	17,280	A00	1,440	A0	120	A	10	0;A	0.83
B0,000	228,096	B,000	19,008	B00	1,584	B0	132	B	11	0;B	0.916

Decimal to duodecimal digit conversion

Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duod.	Dec.	Duodecimal
10,000	5,954	1,000	6B4	100	84	10	A	1	1	0.1	0;12497
20,000	B,6A8	2,000	1,1A8	200	148	20	18	2	2	0.2	0;2497
30,000	15,440	3,000	1,8A0	300	210	30	26	3	3	0.3	0;37249
40,000	1B,194	4,000	2,394	400	294	40	34	4	4	0.4	0;4972
50,000	24,B28	5,000	2,A88	500	358	50	42	5	5	0.5	0;6
60,000	2A,880	6,000	3,580	600	420	60	50	6	6	0.6	0;7249
70,000	34,614	7,000	4,074	700	4A4	70	5A	7	7	0.7	0;84972
80,000	3A,368	8,000	4,768	800	568	80	68	8	8	0.8	0;9724
90,000	44,100	9,000	5,260	900	630	90	76	9	9	0.9	0;A9724

Fractions and irrational numbers

Fractions

Duodecimal fractions for rational numbers with 3-smooth denominators terminate:

• ⁠1/2⁠ = 0;6
• ⁠1/3⁠ = 0;4
• ⁠1/4⁠ = 0;3
• ⁠1/6⁠ = 0;2
• ⁠1/8⁠ = 0;16
• ⁠1/9⁠ = 0;14
• ⁠1/10⁠ = 0;1 (this is one twelfth, ⁠1/A⁠ is one tenth)
• ⁠1/14⁠ = 0;09 (this is one sixteenth, ⁠1/12⁠ is one fourteenth)

while other rational numbers have recurring duodecimal fractions:

• ⁠1/5⁠ = 0;2497
• ⁠1/7⁠ = 0;186A35
• ⁠1/A⁠ = 0;12497 (one tenth)
• ⁠1/B⁠ = 0;1 (one eleventh)
• ⁠1/11⁠ = 0;0B (one thirteenth)
• ⁠1/12⁠ = 0;0A35186 (one fourteenth)
• ⁠1/13⁠ = 0;09724 (one fifteenth)

Examples in duodecimal	Decimal equivalent
1 × ⁠5/8⁠ = 0.76	1 × ⁠5/8⁠ = 0.625
100 × ⁠5/8⁠ = 76	144 × ⁠5/8⁠ = 90
⁠576/9⁠ = 76	⁠810/9⁠ = 90
⁠400/9⁠ = 54	⁠576/9⁠ = 64
1A.6 + 7.6 = 26	22.5 + 7.5 = 30

As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base.

Because ${\displaystyle 2\times 5=10}$ in the decimal system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: ⁠1/8⁠ = ⁠1/(2×2×2)⁠, ⁠1/20⁠ = ⁠1/(2×2×5)⁠, and ⁠1/500⁠ = ⁠1/(2×2×5×5×5)⁠ can be expressed exactly as 0.125, 0.05, and 0.002 respectively. ⁠1/3⁠ and ⁠1/7⁠, however, recur (0.333... and 0.142857142857...).

Because ${\displaystyle 2\times 2\times 3=12}$ in the duodecimal system, ⁠1/8⁠ is exact; ⁠1/20⁠ and ⁠1/500⁠ recur because they include 5 as a factor; ⁠1/3⁠ is exact, and ⁠1/7⁠ recurs, just as it does in decimal.

The number of denominators that give terminating fractions within a given number of digits, n, in a base b is the number of factors (divisors) of ${\displaystyle b^{n}}$, the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of ${\displaystyle b^{n}}$ is given using its prime factorization.

For decimal, ${\displaystyle 10^{n}=2^{n}\times 5^{n}}$. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together, so the number of factors of ${\displaystyle 10^{n}}$ is ${\displaystyle (n+1)(n+1)=(n+1)^{2}}$.

For example, the number 8 is a factor of 10 (1000), so ${\textstyle {\frac {1}{8}}}$ and other fractions with a denominator of 8 cannot require more than three fractional decimal digits to terminate. ${\textstyle {\frac {5}{8}}=0.625_{10}.}$

For duodecimal, ${\displaystyle 10^{n}=2^{2n}\times 3^{n}}$. This has ${\displaystyle (2n+1)(n+1)}$ divisors. The sample denominator of 8 is a factor of a gross ${\textstyle 12^{2}=144}$ (in decimal), so eighths cannot need more than two duodecimal fractional places to terminate. ${\textstyle {\frac {5}{8}}=0.76_{12}.}$

Because both ten and twelve have two unique prime factors, the number of divisors of ${\displaystyle b^{n}}$ for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of ${\displaystyle n^{2}}$).

Recurring digits

The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life division problems than factors of 5. Thus, in practical applications, the nuisance of repeating decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.

However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two prime numbers, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the composite number 9. Nonetheless, having a shorter or longer period does not help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, whereas only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base).

Also, the prime factor 2 appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are powers of two will have a shorter, more convenient terminating representation in duodecimal than in decimal:

• 1/(2) = 0.25₁₀ = 0.3₁₂
• 1/(2) = 0.125₁₀ = 0.16₁₂
• 1/(2) = 0.0625₁₀ = 0.09₁₂
• 1/(2) = 0.03125₁₀ = 0.046₁₂

Decimal base Prime factors of the base: 2, 5 Prime factors of one below the base: 3 Prime factors of one above the base: 11 All other primes: 7, 13, 17, 19, 23, 29, 31			Duodecimal base Prime factors of the base: 2, 3 Prime factors of one below the base: B Prime factors of one above the base: 11 (=1310) All other primes: 5, 7, 15 (=1710), 17 (=1910), 1B (=2310), 25 (=2910), 27 (=3110)
Fraction	Prime factors of the denominator	Positional representation	Positional representation	Prime factors of the denominator	Fraction
1/2	2	0.5	0;6	2	1/2
1/3	3	0.3	0;4	3	1/3
1/4	2	0.25	0;3	2	1/4
1/5	5	0.2	0;2497	5	1/5
1/6	2, 3	0.16	0;2	2, 3	1/6
1/7	7	0.142857	0;186A35	7	1/7
1/8	2	0.125	0;16	2	1/8
1/9	3	0.1	0;14	3	1/9
1/10	2, 5	0.1	0;12497	2, 5	1/A
1/11	11	0.09	0;1	B	1/B
1/12	2, 3	0.083	0;1	2, 3	1/10
1/13	13	0.076923	0;0B	11	1/11
1/14	2, 7	0.0714285	0;0A35186	2, 7	1/12
1/15	3, 5	0.06	0;09724	3, 5	1/13
1/16	2	0.0625	0;09	2	1/14
1/17	17	0.0588235294117647	0;08579214B36429A7	15	1/15
1/18	2, 3	0.05	0;08	2, 3	1/16
1/19	19	0.052631578947368421	0;076B45	17	1/17
1/20	2, 5	0.05	0;07249	2, 5	1/18
1/21	3, 7	0.047619	0;06A3518	3, 7	1/19
1/22	2, 11	0.045	0;06	2, B	1/1A
1/23	23	0.0434782608695652173913	0;06316948421	1B	1/1B
1/24	2, 3	0.0416	0;06	2, 3	1/20
1/25	5	0.04	0;05915343A0B62A68781B	5	1/21
1/26	2, 13	0.0384615	0;056	2, 11	1/22
1/27	3	0.037	0;054	3	1/23
1/28	2, 7	0.03571428	0;05186A3	2, 7	1/24
1/29	29	0.0344827586206896551724137931	0;04B7	25	1/25
1/30	2, 3, 5	0.03	0;04972	2, 3, 5	1/26
1/31	31	0.032258064516129	0;0478AA093598166B74311B28623A55	27	1/27
1/32	2	0.03125	0;046	2	1/28
1/33	3, 11	0.03	0;04	3, B	1/29
1/34	2, 17	0.02941176470588235	0;0429A708579214B36	2, 15	1/2A
1/35	5, 7	0.0285714	0;0414559B3931	5, 7	1/2B
1/36	2, 3	0.027	0;04	2, 3	1/30

The duodecimal period length of 1/n are (in decimal)

0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... (sequence A246004 in the OEIS)

The duodecimal period length of 1/(nth prime) are (in decimal)

0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... (sequence A246489 in the OEIS)

Smallest prime with duodecimal period n are (in decimal)

11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... (sequence A252170 in the OEIS)

Irrational numbers

The representations of irrational numbers in any positional number system (including decimal and duodecimal) neither terminate nor repeat. The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal.

Algebraic irrational number	In decimal	In duodecimal
√2, the square root of 2	1.414213562373...	1;4B79170A07B8...
φ (phi), the golden ratio = ${\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}$	1.618033988749...	1;74BB6772802A...
Transcendental number	In decimal	In duodecimal
π (pi), the ratio of a circle's circumference to its diameter	3.141592653589...	3;184809493B91...
e, the base of the natural logarithm	2.718281828459...	2;875236069821...

See also

• Vigesimal (base 20)
• Sexagesimal (base 60)

References

1. Dvorsky, George (January 18, 2013). "Why We Should Switch To A Base-12 Counting System". Gizmodo. Retrieved December 21, 2013.
2. Pittman, Richard (1990). "Origin of Mesopotamian duodecimal and sexagesimal counting systems". Philippine Journal of Linguistics. 21 (1): 97.
3. Ifrah, Georges (2000) . The Universal History of Numbers: From prehistory to the invention of the computer. Wiley. ISBN 0-471-39340-1. Translated from the French by David Bellos, E. F. Harding, Sophie Wood and Ian Monk.
4. Matsushita, Shuji (October 1998). "Decimal vs. Duodecimal: An interaction between two systems of numeration". www3.aa.tufs.ac.jp. Archived from the original on October 5, 2008. Retrieved May 29, 2011.
5. Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibéto-birmanes". In François, Jacques (ed.). La Pluralité (PDF). Leuven: Peeters. pp. 91–119. ISBN 90-429-1295-2. Archived from the original (PDF) on 2016-03-28. Retrieved 2014-03-27.
6. von Mengden, Ferdinand (2006). "The peculiarities of the Old English numeral system". In Nikolaus Ritt; Herbert Schendl; Christiane Dalton-Puffer; Dieter Kastovsky (eds.). Medieval English and its Heritage: Structure Meaning and Mechanisms of Change. Studies in English Medieval Language and Literature. Vol. 16. Frankfurt: Peter Lang. pp. 125–145.
7. von Mengden, Ferdinand (2010). Cardinal Numerals: Old English from a Cross-Linguistic Perspective. Topics in English Linguistics. Vol. 67. Berlin; New York: De Gruyter Mouton. pp. 159–161.
8. Gordon, E V (1957). Introduction to Old Norse. Oxford: Clarendon Press. pp. 292–293.
9. ^ De Vlieger, Michael (2010). "Symbology Overview" (PDF). The Duodecimal Bulletin. 4X (2).
10. Pakin, Scott (2021) . "The Comprehensive LATEX Symbol List". Comprehensive TEX Archive Network (14.0 ed.). Rei, Fukui (2004) . "tipa – Fonts and macros for IPA phonetics characters". Comprehensive TEX Archive Network (1.3 ed.). The turned digits 2 and 3 employed in the TIPA package originated in The Principles of the International Phonetic Association, University College London, 1949.
11. ^ Andrews, Frank Emerson (1935). New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics. Harcourt, Brace and company. p. 52.
12. "Annual Meeting of 1973 and Meeting of the Board" (PDF). The Duodecimal Bulletin. 25 (1). 1974.
13. De Vlieger, Michael (2008). "Going Classic" (PDF). The Duodecimal Bulletin. 49 (2).
14. ^ "Mo for Megro" (PDF). The Duodecimal Bulletin. 1 (1). 1945.
15. ^ Pitman, Isaac (24 November 1857). "A Reckoning Reform". Bedfordshire Independent. Reprinted as "Sir Isaac Pitman on the Dozen System: A Reckoning Reform" (PDF). The Duodecimal Bulletin. 3 (2): 1–5. 1947.
16. Pentzlin, Karl (March 30, 2013). "Proposal to encode Duodecimal Digit Forms in the UCS" (PDF). ISO/IEC JTC1/SC2/WG2. Retrieved 2024-06-25.
17. ^ "The Unicode Standard, Version 8.0: Number Forms" (PDF). Unicode Consortium. Retrieved 2016-05-30.
18. "The Unicode Standard 8.0" (PDF). Retrieved 2014-07-18.
19. The Dozenal Society of America (n.d.). "What should the DSA do about transdecimal characters?". Dozenal Society of America. The Dozenal Society of America. Retrieved January 1, 2018.
20. Arnold Whittall, The Cambridge Introduction to Serialism (New York: Cambridge University Press, 2008): 276. ISBN 978-0-521-68200-8 (pbk).
21. Ferrari, Silvio (1854). Calcolo Decidozzinale. p. 2.
22. ^ "Annual Meeting of 1973 and Meeting of the Board" (PDF). The Duodecimal Bulletin. 25 (1). 1974.
23. ^ De Vlieger, Michael (2008). "Going Classic" (PDF). The Duodecimal Bulletin. 49 (2).
24. "The Unicode Standard 8.0" (PDF). Retrieved 2014-07-18.
25. ^ Volan, John (July 2015). "Base Annotation Schemes" (PDF). The Duodecimal Bulletin. 62.
26. ^ Zirkel, Gene (2010). "How Do You Pronounce Dozenals?" (PDF). The Duodecimal Bulletin. 4E (2).
27. "Systematic Dozenal Nomenclature and other nomenclature system" (PDF). The Duodecimal Bulletin. 61 (1).
28. ^ Goodman, Donald (2016). "Manual of the Dozenal System" (PDF). Dozenal Society of America. Retrieved 27 April 2018.
29. The Prodigy (Biography of WJS) pg
30. A. C. Aitken (January 25, 1962) "Twelves and Tens" The Listener.
31. A. C. Aitken (1962) The Case Against Decimalisation. Edinburgh / London: Oliver & Boyd.
32. "SchoolhouseRock - Little Twelvetoes". 6 February 2010. Archived from the original on 6 February 2010.
33. Bellos, Alex (2011-04-04). Alex's Adventures in Numberland. A&C Black. p. 50. ISBN 978-1-4088-0959-4.
34. Pendlebury, Tom; Goodman, Donald (2012). "TGM: A Coherent Dozenal Metrology" (PDF). The Dozenal Society of Great Britain.
35. Suga, Takashi (22 May 2019). "Proposal for the Universal Unit System" (PDF).
36. Volan, John. "The Primel Metrology" (PDF). The Duodecimal Bulletin. 63 (1): 38–60.
37. ^ De Vlieger, Michael Thomas (30 November 2011). "Dozenal FAQs" (PDF). dozenal.org. The Dozenal Society of America. Retrieved November 20, 2022.

External links

• Dozenal Society of America
  • "The DSA Symbology Synopsis"
  • "Resources", the DSA website's page of external links to third-party tools
• Dozenal Society of Great Britain
• Lauritzen, Bill (1994). "Nature's Numbers". Earth360.
• Savard, John J. G. (2018) . "Changing the Base". quadibloc. Retrieved 2018-07-17.

Categories:

• Positional numeral systems
• 12 (number)