View source for Duodecimal

{{Short description|Base-12 numeral system}}
{{distinguish|Dewey Decimal Classification|Duodecimo}}
{{Numeral systems}}
The '''duodecimal''' system, also known as '''base twelve''' or '''dozenal''', is a [[positional notation|positional]] [[numeral system]] using [[12 (number)|twelve]] as its [[radix|base]]. In duodecimal, the number twelve is denoted "10", meaning 1&nbsp;twelve and 0 [[1|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&nbsp;[[square (algebra)|squared]] (144), "1,000" means twelve&nbsp;[[cube (algebra)|cubed]] (1,728), and "0.1" means a twelfth (.8333...).

Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses {{d2}} and {{d3}}, as in [[hexadecimal]], which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, {{d2}}, {{d3}}, and finally 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 (dek, pronounced dɛk) and <span style="display: inline-block; transform: scale(-1);">3</span> (a turned 3) for eleven (el, pronounced ɛl).

The number twelve, a [[superior highly composite number]], is the smallest number with four non-trivial [[integer factorization|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 [[multiplicative inverse|reciprocals]] of [[smooth number|3-smooth]] numbers ({{math|{{sfrac|''a''|2<sup>''b''</sup>·3<sup>''c''</sup>}}}} where {{mvar|a,b,c}} are integers) have a [[terminating decimal|terminating]] 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=[[Gizmodo]]}}</ref>

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 [[regular number|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.<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 [[Nigeria]]n Middle Belt such as [[Janji language|Janji]], [[Gbiri-Niragu language|Gbiri-Niragu]] (Gure-Kahugu), [[Piti language|Piti]], and the Nimbia dialect of [[Gwandara language|Gwandara]];<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 [[Chepang language]] of [[Nepal]]<ref>{{Cite book
| contribution=Les principes de construction du nombre dans les langues tibéto-birmanes
| first=Martine
| last=Mazaudon
| title=La Pluralité
| editor-first=Jacques
| editor-last=François
| date=2002
| pages=91–119
| publisher=Peeters
| place=Leuven
| isbn=90-429-1295-2
| url=http://lacito.vjf.cnrs.fr/documents/publi/num_WEB.pdf
| access-date=2014-03-27
| archive-date=2016-03-28
| 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.

[[Germanic languages]] have special words for 11 and 12, such as ''eleven'' and ''twelve'' in [[English language|English]]. They come from [[Proto-Germanic]] *''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, [[Old Norse]] 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 [[long hundred]].

Historically, [[Unit of time|units of time]] in many [[civilization]]s 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 calendar]]s, clocks, and compasses are based on the twelve [[Earthly Branches]] or 24 (12×2) [[Solar term]]s. There are 12&nbsp;inches in an imperial foot, 12&nbsp;[[troy weight|troy]] ounces in a troy pound, 12&nbsp;[[British One Penny coin (pre-decimal)|old British pence]] in a [[shilling]], 24&nbsp;(12×2) hours in a day; many other items are counted by the [[dozen]], [[gross (unit)|gross]] ([[144 (number)|144]], [[square number|square]] of 12), or [[great gross]] ([[1728 (number)|1728]], [[cube (arithmetic)|cube]] of 12). The Romans used a fraction system based on 12, including the [[uncia (length)|uncia]], which became both the English words ''[[ounce]]'' and ''inch''. Pre-[[Decimal Day|decimalisation]], [[Republic of Ireland|Ireland]] and the [[United Kingdom]] used a mixed duodecimal-[[vigesimal]] currency system (12&nbsp;pence = 1&nbsp;shilling, 20&nbsp;shillings or 240&nbsp;pence to the [[pound sterling]] or [[Irish pound]]), and [[Charlemagne]] established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.

{| class="wikitable" style=text-align:center
|+ Duodecimally divided units
|-
! rowspan=2 | Relative<br>value
! colspan=2 | Length
! colspan=2 | Weight
|-
! French
! English
! English (Troy)
! Roman
|-
|12<sup>0</sup>
|[[Units of measurement in France before the French Revolution#Length|pied]]
|[[Foot (unit)|foot]]
|[[Troy weight#Units of measurement|pound]]
|[[Ancient Roman units of measurement#Weight|libra]]
|-
|12<sup>−1</sup>
|[[Units of measurement in France before the French Revolution#Length|pouce]]
|[[inch]]
|[[Troy weight#Units of measurement|ounce]]
|[[Ancient Roman units of measurement#Weight|uncia]]
|-
|12<sup>−2</sup>
|[[Units of measurement in France before the French Revolution#Length|ligne]]
|[[Line (unit)|line]]
|2 [[Apothecaries' system#English-speaking countries|scruples]]
|2 [[scrupulum|scrupula]]
|-
|12<sup>−3</sup>
|[[Point (typography)#Truchet|point]]
|[[Point (typography)#Truchet|point]]
|[[siliqua|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".<ref name="Symbology Overview">{{cite journal|last=De Vlieger|first=Michael|title=Symbology Overview|journal=The Duodecimal Bulletin|volume=4X [58]|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 [[Arabic numerals]] 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 [[Number Forms]]
|note={{ubli
|Arabic digits with 180° rotation, by&nbsp;[[Isaac Pitman|Isaac&nbsp;Pitman]]
|In [[LaTeX]], 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 [[hexadecimal]]) 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 [[Roman numeral]] for ten and a rounded [[Italic script|italic]] capital E similar to [[Latin epsilon|open E]]), 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>}}}} ([[Astrological aspect#Sextile|sextile]] or six-pointed asterisk,<ref>Note that the symbol displayed is a standard asterisk; for technical reasons, this page cannot display the sextile inline.</ref> [[Number sign|hash]] or octothorpe).<ref name="Symbology Overview"/> The symbols were chosen because they were available on some typewriters; they are also on [[push-button telephone]]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 [29]|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 [57]|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}}[[File:Dozenal us 10.svg|8px]], [[File:Dozenal us 11.svg|8px]]{{NNBSP}}}}, the symbols devised by [[William Addison Dwiggins]].<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 [[Isaac Pitman]] 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 [[Unicode|Unicode Standard]].<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 [[Unicode 8.0]] (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 [[hexadecimal]]
| Allows entry on typewriters.
|-
| <big>T</big>
| <big>E</big>
| Initials of ''Ten'' and ''Eleven''
| Used (in lower case) in [[Pitch class|music set theory]]<ref>[[Arnold Whittall]], ''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 [[Roman numeral]]; <br> E from ''Eleven''.
|
|-
| <big>X</big>
| <big>Z</big>
| Origin of Z unknown
| Attributed to [[Jean le Rond d'Alembert|D'Alembert]] & [[Georges-Louis Leclerc, Comte de Buffon|Buffon]] by the DSA.<ref name="Symbology Overview" />
|-
| <big>δ</big>
| <big>ε</big>
| Greek [[Delta (letter)|delta]] from {{lang|grc|δέκα}} "ten"; <br> [[epsilon]] from {{lang|grc|ένδεκα}} "eleven"<ref name="Symbology Overview"/>
|
|-
| <big>τ</big>
| <big>ε</big>
| Greek [[tau]], [[epsilon]]<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 [[italic script|italic]] ''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>
| [[Astrological aspect#Sextile|sextile]] or six-pointed asterisk,<br/>[[Number sign|hash]] or octothorpe
| On [[push-button telephone]]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 [29]|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 [57]|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°
}} 
| [[Isaac Pitman]] (1857);<ref name="Pitman1857"/> used by the DSGB; used by the DSA since 2015; included in [[Unicode 8.0]] (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>[[File:Dozenal us 10.svg|8px]]</big>
| <big>[[File:Dozenal us 11.svg|8px]]</big>
| Pronounced "dek", "el"
| {{ubl
|[[William Addison Dwiggins|William Dwiggins]] (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===
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>

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&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" />

=== Pronunciation ===
The Dozenal Society of America suggested the pronunciation of ten and eleven as "dek" and "el". Terms for some powers of twelve already exist in English.

==== Base-12 nomenclature in English ====
Another nominal for [[12 (number)|twelve]] (12<sub>[[decimal|10]]</sub>) is a ''[[dozen]]'' (10<sub>12</sub> ''or'' 1•10<sup>1</sup><sub>12</sub>).

[[144 (number)|One hundred and forty-four]] (144<sub>10</sub>) is also known as a [[Gross (unit)|''gross'']] (100<sub>12</sub> ''or'' 1•10<sup>2</sup><sub>12</sub>).<ref>{{Cite web |title=Definition of GROSS |url=https://www.merriam-webster.com/dictionary/gross |access-date=2025-02-17 |website=www.merriam-webster.com |language=en}}</ref>

[[1728 (number)|One thousand, seven hundred and twenty-eight]] 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>).<ref>{{Cite web |title=Definition of GREAT GROSS |url=https://www.merriam-webster.com/dictionary/great%20gross |access-date=2025-02-17 |website=www.merriam-webster.com |language=en}}</ref>

==== Duodecimal numbers ====
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 [59]|issue=2|date=2010|url=http://www.dozenal.org/drupal/sites_bck/default/files/DuodecimalBulletinIssue4b2_0.pdf }}</ref>

{| class="wikitable"
! 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;
|{{abbr|do|pronounced 'doʊ'}}
|style="text-align:right"|12
|0;1||edo
|-
| style="text-align:right" |100;
|{{abbr|gro|pronounced 'ɡroʊ'}}
|style="text-align:right"|144
|0;01
|egro
|-
| style="text-align:right" |1,000;
|{{abbr|mo|pronounced 'moʊ'}}
|style="text-align:right"|1,728
|0;001
|emo
|-
| style="text-align:right" |10,000;
|do-mo
|style="text-align:right"|20,736
|0;000,1
|edo-mo
|-
| style="text-align:right" |100,000;
|gro-mo
|style="text-align:right"|248,832
|0;000,01
|egro-mo
|-
| style="text-align:right" |1,000,000;
|bi-mo
|style="text-align:right" |2,985,984
|0;000,001
|ebi-mo
|-
| style="text-align:right" |10,000,000;
|do-bi-mo
|style="text-align:right"|35,831,808
|0;000,000,1
|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 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"/>

==== 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 name]]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"/>

{| class="wikitable"
! 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-.

== Advocacy and "dozenalism" ==
[[William James Sidis]] used 12 as the base for his constructed language [[William James Sidis#Vendergood language|Vendergood]] in 1906, noting it being the smallest number with four factors and its prevalence in commerce.<ref>The Prodigy (Biography of WJS) pg [42]</ref>

The case for the duodecimal system was put forth at length in [[F. Emerson Andrews|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.<ref name="New Numbers 1935"/>

[[File:Keys in dozenal clock.svg|thumb|A duodecimal clockface as in the logo of the Dozenal Society of America, here used to denote [[Key (music)|musical keys]]]]
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, ''[[:wikt:douze|douze]]'', descended from Latin ''duodecim''.

Mathematician and mental calculator [[Alexander Aitken|Alexander Craig Aitken]] was an outspoken advocate of duodecimal:
{{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) [http://www.dozenalsociety.org.uk/archives/aitken.html "Twelves and Tens"] ''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) [http://www.dozenalsociety.org.uk/pdfs/aitken.pdf ''The Case Against Decimalisation'']. Edinburgh / London: Oliver & Boyd.</ref>}}

=== 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.<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>

=== Duodecimal systems of measurements ===
[[Systems of measurement]] 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 ==
:''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" />

The number 12 has six factors, which are [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], and [[12 (number)|12]], of which 2 and 3 are [[prime number|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 (number)|1]], [[2 (number)|2]], [[5 (number)|5]], and [[10 (number)|10]], 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, [[senary]], is below the DSA's stated threshold.

[[8|Eight]] and [[16 (number)|sixteen]] only have 2 as a prime factor. Therefore, in [[octal]] and [[hexadecimal]], the only [[Repeating decimal|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 [[Sumer]]ians and [[Babylonia]]ns, 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 (number)|210]]; the pattern follows the [[primorial]]s. 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.{{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-->

{| 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 ==
To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under [[Base conversion|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 {{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:

: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:

:<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

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, <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 same. Using the digit conversion tables:

<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:

   Decimal --> Duodecimal
   
   10,000    =   5,954
    2,000    =   1,1{{d2}}8
      300    =     210
       40    =      34
        5    =       5
  +     0.6  =  +    0;{{overline|7249}}
 -------------------------------
   12,345.6  =   7,189;{{overline|7249}}

That is, <small>(decimal)</small> 12,345.6 equals <small>(duodecimal)</small> 7,189;{{overline|7249}}

=== Duodecimal to decimal digit conversion ===
{| class="wikitable"
! Duod.
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
|- style="text-align:right"
| style="background:silver;"| '''10,000'''
| 20,736
| style="background:silver;"| '''1,000'''
| 1,728
| style="background:silver;"| '''100'''
| 144
| style="background:silver;"| '''10'''
| 12
| style="background:silver;"| '''1'''
| 1
| style="background:silver;"| '''0;1'''
| style="text-align:left" | 0.08{{overline|3}}
|- style="text-align:right"
| style="background:silver;"| '''20,000'''
| 41,472
| style="background:silver;"| '''2,000'''
| 3,456
| style="background:silver;"| '''200'''
| 288
| style="background:silver;"| '''20'''
| 24
| style="background:silver;"| '''2'''
| 2
| style="background:silver;"| '''0;2'''
| style="text-align:left" | 0.1{{overline|6}}
|- style="text-align:right"
| style="background:silver;"| '''30,000'''
| 62,208
| style="background:silver;"| '''3,000'''
| 5,184
| style="background:silver;"| '''300'''
| 432
| style="background:silver;"| '''30'''
| 36
| style="background:silver;"| '''3'''
| 3
| style="background:silver;"| '''0;3'''
| style="text-align:left" | 0.25
|- style="text-align:right"
| style="background:silver;"| '''40,000'''
| 82,944
| style="background:silver;"| '''4,000'''
| 6,912
| style="background:silver;"| '''400'''
| 576
| style="background:silver;"| '''40'''
| 48
| style="background:silver;"| '''4'''
| 4
| style="background:silver;"| '''0;4'''
| style="text-align:left" | 0.{{overline|3}}
|- style="text-align:right"
| style="background:silver;"| '''50,000'''
| 103,680
| style="background:silver;"| '''5,000'''
| 8,640
| style="background:silver;"| '''500'''
| 720
| style="background:silver;"| '''50'''
| 60
| style="background:silver;"| '''5'''
| 5
| style="background:silver;"| '''0;5'''
| style="text-align:left" | 0.41{{overline|6}}
|- style="text-align:right"
| style="background:silver;"| '''60,000'''
| 124,416
| style="background:silver;"| '''6,000'''
| 10,368
| style="background:silver;"| '''600'''
| 864
| style="background:silver;"| '''60'''
| 72
| style="background:silver;"| '''6'''
| 6
| style="background:silver;"| '''0;6'''
| style="text-align:left" | 0.5
|- style="text-align:right"
| style="background:silver;"| '''70,000'''
| 145,152
| style="background:silver;"| '''7,000'''
| 12,096
| style="background:silver;"| '''700'''
| 1,008
| style="background:silver;"| '''70'''
| 84
| style="background:silver;"| '''7'''
| 7
| style="background:silver;"| '''0;7'''
| style="text-align:left" | 0.58{{overline|3}}
|- style="text-align:right"
| style="background:silver;"| '''80,000'''
| 165,888
| style="background:silver;"| '''8,000'''
| 13,824
| style="background:silver;"| '''800'''
| 1,152
| style="background:silver;"| '''80'''
| 96
| style="background:silver;"| '''8'''
| 8
| style="background:silver;"| '''0;8'''
| style="text-align:left" | 0.{{overline|6}}
|- style="text-align:right"
| style="background:silver;"| '''90,000'''
| 186,624
| style="background:silver;"| '''9,000'''
| 15,552
| style="background:silver;"| '''900'''
| 1,296
| style="background:silver;"| '''90'''
| 108
| style="background:silver;"| '''9'''
| 9
| style="background:silver;"| '''0;9'''
| style="text-align:left" | 0.75
|- style="text-align:right"
| style="background:silver;"| '''{{nowrap|{{d2}}0,000}}'''
| 207,360
| style="background:silver;"| '''{{d2}},000'''
| 17,280
| style="background:silver;"| '''{{d2}}00'''
| 1,440
| style="background:silver;"| '''{{d2}}0'''
| 120
| style="background:silver;"| '''{{d2}}'''
| 10
| style="background:silver;"| '''0;{{d2}}'''
| style="text-align:left" | 0.8{{overline|3}}
|- style="text-align:right"
| style="background:silver;"| '''{{d3}}0,000'''
| 228,096
| style="background:silver;"| '''{{d3}},000'''
| 19,008
| style="background:silver;"| '''{{d3}}00'''
| 1,584
| style="background:silver;"| '''{{d3}}0'''
| 132
| style="background:silver;"| '''{{d3}}'''
| 11
| style="background:silver;"| '''0;{{d3}}'''
| style="text-align:left" | 0.91{{overline|6}}
|}

=== Decimal to duodecimal digit conversion ===
{| class="wikitable"
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
! Duod.
! Dec.
! Duodecimal
|-
| style="background:silver;"| '''10,000'''
| 5,954
| style="background:silver;"| '''1,000'''
| 6{{d3}}4
| style="background:silver;"| '''100'''
| 84
| style="background:silver;"| '''10'''
| {{d2}}
| style="background:silver;"| '''1'''
| 1
| style="background:silver;"| '''0.1'''
| 0;1{{overline|2497}}
|-
| style="background:silver;"| '''20,000'''
| {{d3}},6{{d2}}8
| style="background:silver;"| '''2,000'''
| 1,1{{d2}}8
| style="background:silver;"| '''200'''
| 148
| style="background:silver;"| '''20'''
| 18
| style="background:silver;"| '''2'''
| 2
| style="background:silver;"| '''0.2'''
| 0;{{overline|2497}}
|-
| style="background:silver;"| '''30,000'''
| 15,440
| style="background:silver;"| '''3,000'''
| 1,8{{d2}}0
| style="background:silver;"| '''300'''
| 210
| style="background:silver;"| '''30'''
| 26
| style="background:silver;"| '''3'''
| 3
| style="background:silver;"| '''0.3'''
| 0;3{{overline|7249}}
|-
| style="background:silver;"| '''40,000'''
| 1{{d3}},194
| style="background:silver;"| '''4,000'''
| 2,394
| style="background:silver;"| '''400'''
| 294
| style="background:silver;"| '''40'''
| 34
| style="background:silver;"| '''4'''
| 4
| style="background:silver;"| '''0.4'''
| 0;{{overline|4972}}
|-
| style="background:silver;"| '''50,000'''
| 24,{{d3}}28
| style="background:silver;"| '''5,000'''
| 2,{{d2}}88
| style="background:silver;"| '''500'''
| 358
| style="background:silver;"| '''50'''
| 42
| style="background:silver;"| '''5'''
| 5
| style="background:silver;"| '''0.5'''
| 0;6
|-
| style="background:silver;"| '''60,000'''
| 2{{d2}},880
| style="background:silver;"| '''6,000'''
| 3,580
| style="background:silver;"| '''600'''
| 420
| style="background:silver;"| '''60'''
| 50
| style="background:silver;"| '''6'''
| 6
| style="background:silver;"| '''0.6'''
| 0;{{overline|7249}}
|-
| style="background:silver;"| '''70,000'''
| 34,614
| style="background:silver;"| '''7,000'''
| 4,074
| style="background:silver;"| '''700'''
| 4{{d2}}4
| style="background:silver;"| '''70'''
| 5{{d2}}
| style="background:silver;"| '''7'''
| 7
| style="background:silver;"| '''0.7'''
| 0;8{{overline|4972}}
|-
| style="background:silver;"| '''80,000'''
| 3{{d2}},368
| style="background:silver;"| '''8,000'''
| 4,768
| style="background:silver;"| '''800'''
| 568
| style="background:silver;"| '''80'''
| 68
| style="background:silver;"| '''8'''
| 8
| style="background:silver;"| '''0.8'''
| 0;{{overline|9724}}
|-
| style="background:silver;"| '''90,000'''
| 44,100
| style="background:silver;"| '''9,000'''
| 5,260
| style="background:silver;"| '''900'''
| 630
| style="background:silver;"| '''90'''
| 76
| style="background:silver;"| '''9'''
| 9
| style="background:silver;"| '''0.9'''
| 0;{{d2}}{{overline|9724}}
|}

== Fractions and irrational numbers ==
=== Fractions ===
Duodecimal [[Fraction (mathematics)|fractions]] for rational numbers with [[Smooth number|3-smooth]] denominators terminate:
* {{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 (this is one twelfth, {{sfrac|{{d2}}}} is one tenth)
* {{sfrac|14}} = 0;09 (this is one sixteenth, {{sfrac|12}} is one fourteenth)

while other rational numbers have [[recurring decimal|recurring]] duodecimal fractions:
* {{sfrac|5}} = 0;{{Overline|2497}}
* {{sfrac|7}} = 0;{{Overline|186{{D2}}35}}
* {{sfrac|{{d2}}}} = 0;1{{Overline|2497}} (one tenth)
* {{sfrac|{{d3}}}} = 0;{{Overline|1}} (one eleventh)
* {{sfrac|11}} = 0;{{Overline|0{{D3}}}} (one thirteenth)
* {{sfrac|12}} = 0;0{{Overline|{{D2}}35186}} (one fourteenth)
* {{sfrac|13}} = 0;0{{Overline|9724}} (one fifteenth)

{| class="wikitable"
! Examples in duodecimal
! Decimal equivalent
|-
| 1 × {{sfrac|5|8}} = 0.76
| 1 × {{sfrac|5|8}} = 0.625
|-
| 100 × {{sfrac|5|8}} = 76
| 144 × {{sfrac|5|8}} = 90
|-
| {{sfrac|576|9}} = 76
| {{sfrac|810|9}} = 90
|-
| {{sfrac|400|9}} = 54
| {{sfrac|576|9}} = 64
|-
| 1{{d2}};6 + 7;6 = 26
| 22.5 + 7.5 = 30
|}

As explained in [[recurring decimal]]s, 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 factor]]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...).

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.

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 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 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>

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 ===
The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life [[division (mathematics)|division]] 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 [[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 (number)|12]] (twelve) is between two [[prime number]]s, [[11 (number)|11]] (eleven) and [[13 (number)|13]] (thirteen), whereas ten is adjacent to the [[composite number]] [[9 (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 (number)|3]] in its factorization, whereas only one out of every five contains the prime factor [[5 (number)|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 (number)|2]] appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are [[power of two|powers of two]] will have a shorter, more convenient terminating representation in duodecimal than in decimal:

* 1/(2<sup>2</sup>) = 0.25<sub>10</sub> = 0.3<sub>12</sub>
* 1/(2<sup>3</sup>) = 0.125<sub>10</sub> = 0.16<sub>12</sub>
* 1/(2<sup>4</sup>) = 0.0625<sub>10</sub> = 0.09<sub>12</sub>
* 1/(2<sup>5</sup>) = 0.03125<sub>10</sub> = 0.046<sub>12</sub>

{| 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
! <SMALL>Prime factors<br>of the denominator</SMALL>
! Positional representation
! Positional representation
! <SMALL>Prime factors<br>of the denominator</SMALL>
! Fraction
|-
| style="text-align:center;"| 1/2
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.5
| 0;6
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/2
|-
| style="text-align:center;"| 1/3
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
| 0.{{overline|3}}
| 0;4
| style="text-align:center;"| <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/3
|-
| style="text-align:center;"| 1/4
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.25
| 0;3
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/4
|-
| style="text-align:center;"| 1/5
| style="text-align:center;"| <span style="color:Green">'''5'''</span>
| 0.2
| 0;{{overline|2497}}
| style="text-align:center;"| <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/5
|-
| style="text-align:center;"| 1/6
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.1{{overline|6}}
| 0;2
| 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;"| 1/7
| style="text-align:center;"| <span style="color:Red">'''7'''</span>
| 0.{{overline|142857}}
| 0;{{overline|186{{d2}}35}}
| style="text-align:center;"| <span style="color:Red">'''7'''</span>
| style="text-align:center;"| 1/7
|-
| style="text-align:center;"| 1/8
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.125
| 0;16
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/8
|-
| style="text-align:center;"| 1/9
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
| 0.{{overline|1}}
| 0;14
| style="text-align:center;"| <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/9
|-
| style="text-align:center;"| 1/10
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| 0.1
| 0;1{{overline|2497}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/{{d2}}
|-
| style="text-align:center;"| 1/11
| style="text-align:center;"| <span style="color:Magenta">'''11'''</span>
| 0.{{overline|09}}
| 0;{{overline|1}}
| style="text-align:center;"| <span style="color:Blue">'''{{d3}}'''</span>
| style="text-align:center;"| 1/{{d3}}
|-
| style="text-align:center;"| 1/12
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.08{{overline|3}}
| 0;1
| 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;"| 1/13
| style="text-align:center;"| <span style="color:Red">'''13'''</span>
| 0.{{overline|076923}}
| 0;{{overline|0{{d3}}}}
| style="text-align:center;"| <span style="color:Magenta">'''11'''</span>
| style="text-align:center;"| 1/11
|-
| style="text-align:center;"| 1/14
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| 0.0{{overline|714285}}
| 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;"| 1/12
|-
| style="text-align:center;"| 1/15
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
| 0.0{{overline|6}}
| 0;0{{overline|9724}}
| 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;"| 1/16
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.0625
| 0;09
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/14
|-
| style="text-align:center;"| 1/17
| style="text-align:center;"| <span style="color:Red">'''17'''</span>
| 0.{{overline|0588235294117647}}
| 0;{{overline|08579214{{d3}}36429{{d2}}7}}
| style="text-align:center;"| <span style="color:Red">'''15'''</span>
| style="text-align:center;"| 1/15
|-
| style="text-align:center;"| 1/18
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.0{{overline|5}}
| 0;08
| 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;"| 1/19
| style="text-align:center;"| <span style="color:Red">'''19'''</span>
| 0.{{overline|052631578947368421}}
| 0;{{overline|076{{d3}}45}}
| style="text-align:center;"| <span style="color:Red">'''17'''</span>
| style="text-align:center;"| 1/17
|-
| style="text-align:center;"| 1/20
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| 0.05
| 0;0{{overline|7249}}
| 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;"| 1/21
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span>
| 0.{{overline|047619}}
| 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;"| 1/19
|-
| style="text-align:center;"| 1/22
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
| 0.0{{overline|45}}
| 0;0{{overline|6}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''{{d3}}'''</span>
| style="text-align:center;"| 1/1{{d2}}
|-
| style="text-align:center;"| 1/23
| style="text-align:center;"| <span style="color:Red">'''23'''</span>
| 0.{{overline|0434782608695652173913}}
| 0;{{overline|06316948421}}
| style="text-align:center;"| <span style="color:Red">'''1{{d3}}'''</span>
| style="text-align:center;"| 1/1{{d3}}
|-
| style="text-align:center;"| 1/24
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.041{{overline|6}}
| 0;06
| 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;"| 1/25
| style="text-align:center;"| <span style="color:Green">'''5'''</span>
| 0.04
| 0;{{overline|05915343{{d2}}0{{d3}}62{{d2}}68781{{d3}}}}
| style="text-align:center;"| <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/21
|-
| style="text-align:center;"| 1/26
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
| 0.0{{overline|384615}}
| 0;0{{overline|56}}
| 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;"| 1/27
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
| 0.{{overline|037}}
| 0;054
| style="text-align:center;"| <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/23
|-
| style="text-align:center;"| 1/28
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| 0.03{{overline|571428}}
| 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;"| 1/24
|-
| style="text-align:center;"| 1/29
| style="text-align:center;"| <span style="color:Red">'''29'''</span>
| 0.{{overline|0344827586206896551724137931}}
| 0;{{overline|04{{d3}}7}}
| style="text-align:center;"| <span style="color:Red">'''25'''</span>
| style="text-align:center;"| 1/25
|-
| 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>
| 0.0{{overline|3}}
| 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;"| 1/26
|-
| style="text-align:center;"| 1/31
| style="text-align:center;"| <span style="color:Red">'''31'''</span>
| 0.{{overline|032258064516129}}
| 0;{{overline|0478{{d2}}{{d2}}093598166{{d3}}74311{{d3}}28623{{d2}}55}}
| style="text-align:center;"| <span style="color:Red">'''27'''</span>
| style="text-align:center;"| 1/27
|-
| style="text-align:center;"| 1/32
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.03125
| 0;046
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/28
|-
| style="text-align:center;"| 1/33
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span>
| 0.{{overline|03}}
| 0;0{{overline|4}}
| 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;"| 1/34
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''17'''</span>
| 0.0{{overline|2941176470588235}}
| 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;"| 1/2{{d2}}
|-
| style="text-align:center;"| 1/35
| style="text-align:center;"| <span style="color:Green">'''5'''</span>, <span style="color:Red">'''7'''</span>
| 0.0{{overline|285714}}
| 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;"| 1/2{{d3}}
|-
| style="text-align:center;"| 1/36
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.02{{overline|7}}
| 0;04
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 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, ... {{OEIS|id=A246004}}

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}}

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}}

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

{| class="wikitable"
! Algebraic irrational number
! In decimal
! In duodecimal
|-
| style="text-align:center;"| [[Square root of 2|{{sqrt|2}}]], the square root of 2
| 1.414213562373...
| 1;4{{d3}}79170{{d2}}07{{d3}}8...
|-
| style="text-align:center;"| {{mvar|[[Golden ratio|φ]]}} (phi), the golden ratio = <math>\tfrac{1+\sqrt{5}}{2}</math>
| 1.618033988749...
| 1;74{{d3}}{{d3}}6772802{{d2}}...
|-
! Transcendental number
! In decimal
! In duodecimal
|-
| style="text-align:center;"| {{mvar|[[Pi|π]]}} (pi), the ratio of a circle's [[circumference]] to its [[diameter]]
| 3.141592653589...
| 3;184809493{{d3}}91...
|-
| style="text-align:center;"| {{mvar|[[E (mathematical constant)|e]]}}, the base of the [[natural logarithm]]
| 2.718281828459...
| 2;875236069821...
|}

== See also ==
* [[Vigesimal]] (base 20)
* [[Sexagesimal]] (base 60)

== References ==
{{reflist}}

== External links ==
* [http://www.dozenal.org/ Dozenal Society of America]
** [https://dozenal.org/drupal/content/dsa-symbology-synopsis.html "The DSA Symbology Synopsis"]
** [https://dozenal.org/drupal/content/resources.html "Resources"], the DSA website's page of external links to third-party tools
* [http://www.dozenalsociety.org.uk/ Dozenal Society of Great Britain]
* {{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 }}

[[Category:Positional numeral systems]]
[[Category:12 (number)]]