View source for Undecimal

{{Short description|Base-11 numeral system}}
'''Undecimal''' (also known as '''unodecimal''', '''undenary''', and the '''base 11''' numeral system) is a [[Positional notation|positional numeral system]] that uses [[11 (number)|eleven]] as its [[Radix|base]]. While no known society counts by elevens, two are purported to have done so: the [[Māori people|Māori]] (one of the two [[Polynesians|Polynesian]] peoples of [[New Zealand]]) and the [[Pangwa|Pañgwa]] (a [[Bantu languages|Bantu]]-speaking people of [[Tanzania]]). The idea of counting by elevens remains of interest for its relation to a traditional method of tally-counting practiced in Polynesia.<ref name="OVERJPS2020">{{cite journal |last=Overmann |first=Karenleigh A |date=2020 |title=The curious idea that Māori once counted by elevens, and the insights it still holds for cross-cultural numerical research |url=http://www.thepolynesiansociety.org/jps/index.php/JPS/article/view/458 |journal=Journal of the Polynesian Society |volume=129 |issue=1 |pages=59–84 |doi=10.15286/jps.129.1.59-84 |access-date=24 July 2020|doi-access=free }}</ref><ref name="OVEROA2020">{{Citation |last=Overmann |first=Karenleigh A |title=Oceanian counting algorithms: Analytical data for EU project 785793 |publisher=Unpublished |date=2021 |url=https://osf.io/z9ux5/ |access-date=March 21, 2022}}</ref> During the [[French Revolution]], undecimal was briefly considered as a possible basis for the reformed system of measurement.<ref name="Delambre1816">{{cite book |last1=Delambre |first1=Jean Baptiste Joseph |title=Mémoires de la classe des Sciences mathématiques et physiques de l'Institut de France, Année 1812, Seconde Partie |date=1816 |publisher=Firmin Didot |location=Paris |pages=xxvii–lxxx |language=fr |chapter=Notice sur la vie et les ouvrages de M. Malus, et de M. le Comte Lagrange |chapter-url=https://books.google.com/books?id=BbxeAAAAcAAJ&pg=PR1}}</ref> Undecimal numerals have applications in computer science,<ref name="Saurabh2013">{{cite journal|first1=Saurabh |last1=Rawat |first2=Anushree |last2=Sah |date=2013 |title=Subtraction in Traditional and Strange Number System by r's and r-1's Compliments |journal=International Journal of Computer Applications |volume=70 |issue=23 |pages=13–17 |doi=10.5120/12206-7640|doi-access=free }}</ref> technology,<ref name="Werner1957">{{cite journal |first=Werner |last= Ulrich |date=November 1957 |title=Non-binary error correction codes |journal=Bell System Technical Journal |volume=36 |issue=6 |pages=1364–1365 |doi= 10.1002/j.1538-7305.1957.tb01514.x |url=https://archive.org/details/bstj36-6-1341/page/n23/mode/2up?q=unodecimal}}</ref> and the [[International Standard Book Number]] system.<ref name="ISBN2017">{{cite book |date=2017 |title=ISBN Users' Manual, International Edition, 7th ed. |location=London |publisher=International ISBN Agency |isbn=978-92-95055-12-4}}</ref> They also occasionally feature in works of popular fiction.<ref>{{cite journal |first1=Debasis |last1=Das |first2=U.A. |last2=Lanjewar |date=January 2012 |title=Realistic Approach of Strange Number System from Unodecimal to Vigesimal |journal=International Journal of Computer Science and Telecommunications |volume=3 |issue=1 |url=https://www.ijcst.org/Volume3/Issue1/p2_3_1.pdf |page=13}}</ref> In undecimal, a capital letter (often [[A]], the symbol for 10 in [[hexadecimal]]; [[T]], the first letter of the English word "ten"; or [[X]], the [[Roman numeral]] 10) or the digit ↊ (called "dek") is typically used as a [[duodecimal#Transdecimal symbols|transdecimal symbol]] to represent the number 10.

==Alleged use of undecimal in cultural number systems==

===Use by the Māori===

====Conant and Williams====

For about a century, the idea that Māori counted by elevens was best known from its mention in the writing of the American mathematician [[Levi Leonard Conant]]. He identified it as a "mistake" originating with a 19th-century dictionary of the New Zealand language published by the Rev. [[William Williams (bishop)|William Williams]], at the time Archdeacon of [[Diocese of Waiapu|Waiapu]].<ref name="CONANT">{{cite book |last=Conant |first=Levi Leonard |date=1896 |title=The number concept: Its origin and development |location=New York |publisher=Macmillan and Co. |pages=122–123 |oclc=990771340}}</ref>{{rp|p. 123}}

<blockquote>"Many years ago a statement appeared which at once attracted attention and awakened curiosity. It was to the effect that the Maoris, the aboriginal inhabitants of New Zealand, used as the basis of their numeral system the number 11; and that the system was quite extensively developed, having simple words for 121 and 1331, i.e. for the square and cube of 11."<ref name="CONANT"/>{{rp|pp. 122–123}}</blockquote>

As published by Williams in the first two editions of the dictionary series, this statement read:

<blockquote>"The Native mode of counting is by elevens, till they arrive at the tenth eleven, which is their hundred; then onwards to the tenth hundred, which is their thousand:* but those Natives who hold intercourse with Europeans have, for the most part, abandoned this method, and, leaving out ''ngahuru'', reckon ''tekau'' or ''tahi tekau'' as 10, ''rua tekau'' as 20, &c. *This seems to be on the principle of putting aside one to every ten as a tally. A parallel to this obtains among the English, as in the case of the baker's dozen."<ref>{{cite book |last=Williams |first=William |title=A dictionary of the New-Zealand language, and a concise grammar; to which are added a selection of colloquial sentences |location=Paihia, NZ |publisher=The Press of the Church Mission Society |date=1844 |oclc=504512293}}</ref>{{rp|p. xv}}</blockquote>

====Lesson and Blosseville====

In 2020, an earlier, Continental origin of the idea the Māori counted by elevens was traced to the published writings of two 19th-century scientific explorers, [[René Primevère Lesson]] and [[Jules de Blosseville]].<ref name="OVERJPS2020"/> They had visited New Zealand in 1824 as part of the 1822–1825 circumnavigational voyage of the ''[[French ship Astrolabe (1811)|Coquille]]'',<ref>{{cite book |last=Duperrey |first=Louis Isidore |url=https://books.google.com/books?id=pWK39IqIYKwC |title=Voyage autour du monde, exécuté par ordre du roi, sur la corvette de sa majesté, la Coquille, pendant les années 1822, 1823, 1824 et 1825, ... et publié sous les auspices de son excellence mgr le cte de Chabrol ... par M.L.I. Duperrey... |date=1829 |publisher=Arthus Bertrand |volume=7 |location=Paris |pages=84–87 |language=fr |trans-title=Voyage around the world, executed by order of the king, on his majesty's corvette, the Coquille, during the years 1822, 1823, 1824 and 1825, ... and published under the auspices of his Excellency the Count of Chabrol ... by M.L.I. Duperry... |chapter=Tableaux des routes parcourues par la Corvette de Sa Majesté, la Coquille, et des observations météorologiques faites a bord du batiment, pendant les années 1822, 1823, 1824 et 1825 |trans-chapter=Tables of the Routes Traveled by Her Majesty's Corvette, the Coquille, and Meteorological Observations Made on Board the Vessel, During the Years 1822, 1823, 1824 and 1825 |oclc=257721098}}</ref> a French [[corvette]] commanded by [[Louis Isidore Duperrey]] and seconded by [[Jules Dumont d'Urville]]. On his return to France in 1825, Lesson published his French translation of an article written by the German botanist [[Adelbert von Chamisso]].<ref name="AVC1825">{{cite book |last=Von Chamisso |first=Adelbert |url=https://books.google.com/books?id=780OAAAAYAAJ |title=Annales maritimes et coloniales |date=1825 |publisher=L'imprimerie Royale |editor-last=Bajot |editor-first=Louis-Marie |volume=2, Part 2 |location=Paris |pages=1–41 |language=fr |trans-title=Maritime and colonial annals |chapter=Du Grand Océan, de ses îles et de ses côtes: par A. de Chamisso, Docteur en philosophie, &c. &c.; traduit sur l'édition anglaise par R. P. Lesson, Médecin de la corvette ''la Coquille'', Pharmacien de la marine, Membre de plusieurs sociétés savantes, &c. |trans-chapter=Of the Great Ocean, its islands and its coasts: by A. de Chamisso, Doctor of Philosophy, &c. &vs.; translated into the English edition by R. P. Lesson, Doctor of the corvette ''La Coquille'', Navy Pharmacist, Member of several learned societies, &c.}}</ref> At von Chamisso's claim that the New Zealand number system was based on twenty ([[vigesimal]]), Lesson inserted a footnote to mark an error:

<blockquote>Von Chamisso's text, as translated by Lesson: "...de l'E. de la mer du Sud ... c'est là qu'on trouve premierement le système arithmétique fondé sur un échelle de vingt, comme dans la Nouvelle-Zélande (2)..."<ref name="AVC1825"/>{{rp|p. 27}} [...east of the South Sea ... is where we first find the arithmetic system based on a scale of twenty, as in New Zealand (2)...]</blockquote>

<blockquote>Lesson's footnote on von Chamisso's text: "(2) Erreur. Le système arithmétique des Zélandais est undécimal, et les Anglais sont les premiers qui ont propagé cette fausse idée. (L.)"<ref name="AVC1825"/>{{rp|p. 27}} [(2) Error. The Zealander arithmetic system is undecimal, and the English are the first to propagate this false idea. (L).]</blockquote>

Von Chamisso had mentioned his error himself in 1821, tracing the source of his confusion and its clarification to [[Thomas Kendall]], the English missionary to New Zealand who provided the material on the [[Māori language]] that was the basis for a grammar published in 1820 by the English linguist [[Samuel Lee (linguist)|Samuel Lee]].<ref name="AVC1821">{{cite book |last=Von Chamisso |first=Adelbert |editor-last=Von Kotzebue |editor-first=Otto |title=A voyage of discovery, into the South Sea and Beering's Straits, for the purpose of exploring a north-east passage, undertaken in the years 1815–1818, at the expense of his highness the Chancellor of the Empire, Count Romanzoff, in the ship Rurick: Vol. III |url=https://books.google.com/books?id=00YbAAAAYAAJ |location=London |publisher=Longman, Hurst, Rees, Orme, and Brown |date=1821 |pages=439–442 |chapter=Corrections and remarks}}</ref><ref name="LEE1820">{{cite book |last=Lee |first=Samuel |date=1820 |title=A grammar and vocabulary of the language of New Zealand |location=London |publisher=Church Missionary Society |oclc=561056725}}</ref> In the same 1821 publication, von Chamisso also identified the Māori number system as decimal, noting the source of the confusion was the Polynesian practice of counting things by pairs, where each pair was counted as a single unit, so that ten units were numerically equivalent to twenty:<ref name="AVC1821"/><ref name="LEE1820"/>

<blockquote>"We have before us a Grammar and Vocabulary of the Language of New Zealand, published by the Church Missionary Society. London, 1820. 8vo. The author of this grammar is the same Mr. Kendall who has communicated to us the Vocabulary in Nicolas's voyage.<ref>{{cite book |editor-last=Nicholas |editor-first=John Liddiard |title=Narrative of a voyage to New Zealand, performed in the years 1814 and 1816, in company with the Rev. Samuel Marsden, principal chaplain of New South Wales |location=London |publisher=James Black and Son |date=1817 |chapter=A vocabulary of English and New Zealand words, the latter being compared with some of those in the Tonga language |last=Kendall |first=Thomas |volume=II |pages=327–352 |oclc=831296147 }}</ref> The language has now been opened to us, and we correct our opinion."<ref name="AVC1821"/>{{rp|p. 13}}</blockquote>

And,

<blockquote>"It is very far from easy to find out the arithmetical system of a people. It is at New Zealand, as at Tonga, the decimal system. What may, perhaps, have deceived Mr. Kendall, at the beginning, in his first attempt in Nicholas's voyage, and which we followed, is the custom of the New Zealanders to count things by pairs. The natives of Tonga count the bananas and fish likewise by pairs and by twenties (''Tecow'', English score)."<ref name="AVC1821"/>{{rp|pp. 441–442}}</blockquote>

Lesson's use of the term "undécimal" in 1825 was possibly a printer's error that conjoined the intended phrase "un décimal," which would have correctly identified New Zealand numeration as decimal.<ref name="OVERJPS2020"/> Lesson knew Polynesian numbers were decimal and highly similar throughout the region, as he had learned a lot about Pacific number systems during his 2.5 years on the ''Coquille'', collecting numerical vocabularies and ultimately publishing or commenting on more than a dozen of them.<ref name="OVERJPS2020"/> He was also familiar with the work of Thomas Kendall and Samuel Lee through his translation of von Chamisso's work.<ref name="AVC1825"/> These circumstances suggest Lesson was unlikely to have misunderstood New Zealand counting as proceeding by elevens.<ref name="OVERJPS2020"/>

Lesson and his shipmate and friend, Blosseville,<ref>{{cite journal |last=Rallet |first=Louis |date=1 May 1953 |title=Un naturaliste saintongeais: René-Primevère Lesson (1794–1849) |trans-title=A naturalist from Saintonge: René-Primevère Lesson (1794–1849) |url=https://gallica.bnf.fr/ark:/12148/bpt6k96021734 |journal=Annales de la Société des Sciences Naturelles de la Charente-Maritime |language=French |volume=3 |issue=1 |pages=77–131 |access-date=25 July 2020 |via=Gallica}}</ref> sent accounts of their alleged discovery of elevens-based counting in New Zealand to their contemporaries. At least two of these correspondents published these reports, including the Italian geographer [[Adriano Balbi]], who detailed a letter he received from Lesson in 1826,<ref name="BALBI">{{cite book |last=Balbi |first=Adriano |url=https://books.google.com/books?id=oV38C96IOBgC |title=Atlas ethnographique du globe, ou classification des peuples anciens et modernes d'aprés leur langue. Vol. 1, Discours préliminaire et introduction |date=1826 |publisher=Paul Renouard |location=Paris |pages=230–278 |language=fr |chapter=Observations sur la classification des langues Océaniennes}}</ref> and the Hungarian astronomer [[Franz Xaver von Zach]], who briefly mentioned the alleged discovery as part of a letter from Blosseville he had received through a third party.<ref>{{cite book |last=Von Zach |first=Franz Xaver |url=https://books.google.com/books?id=2FU7AQAAIAAJ |title=Bulletin des sciences mathématiques, astronomiques, physiques et chimiques, 1ère section, tome V |date=1826 |publisher=Bureau du Bulletin |editor-last=Saigey |editor-first=Emilé |location=Paris |pages=120–123 |language=fr |chapter=Correspondance astronomique, géographique, hydrographique et statistique}}</ref> De Blosseville also mentioned it to the Scottish author [[George Lillie Craik]], who reported this letter in his 1830 book ''The New Zealanders''.<ref>{{cite book |last=Craik |first=George Lillie |title=The New Zealanders |location=Boston |publisher=Wells and Lilly |date=1830 |OCLC=877372740}}</ref> Lesson was also likely the author of an undated essay, written by a Frenchman but otherwise anonymous, found among and published with the papers of the Prussian linguist [[Wilhelm von Humboldt]] in 1839.<ref name="WVH1839">{{cite book |last=Von Humboldt |first=Wilhelm |title=Über die Kawi-Sprache aus der Insel Java, nebst einer Einleitung über die Verschiedenheit des menschlichen Sprachbaues und ihren Einsluss aus die geistige Entwickelung des Menschengeschlechts. Band III. Südsee-Sprachen, als östlicher Zweig des Malayischen |date=1839 |publisher=F. Dümmler |location=Berlin |language=de |oclc=889950161}}</ref><ref>{{cite book |last=Pott |first=August Friedrich |title=Die Quinare Und Vigesimale Zählmethode Bei Völkern Aller Welttheile: Nebst Ausführlicheren Bemerkungen Über die Zahlwörter Indogermanischen Stammes und Einem Anhange Über Fingernamen |location=Halle, Germany |publisher=C. A. Schwetschke und Sohn |date=1847 |oclc= 669402349}}</ref>

The story expanded in its retelling:<ref name="OVERJPS2020"/> The 1826 letter published by Balbi added an alleged numerical vocabulary with terms for eleven squared (''Karaou'') and eleven cubed (''Kamano''), as well as an account of how the number-words and counting procedure were supposedly elicited from local informants.<ref name="BALBI" /> In an interesting twist, it also changed the mistaken classification needing correction from vigesimal to decimal.<ref name="AVC1825"/><ref name="BALBI" /> The 1839 essay published with von Humboldt's papers named [[Thomas Kendall]], the English missionary whose confusion over the effects of pair-counting on Māori numbers had caused von Chamisso to misidentify them as [[vigesimal]].<ref name="AVC1825"/><ref name="AVC1821"/><ref name="WVH1839"/> It also listed places the alleged local informants were supposedly from.<ref name="WVH1839"/>

====Relation to traditional counting====

The idea that Māori counted by elevens highlights an ingenious and pragmatic form of counting once practiced throughout Polynesia.<ref name="OVERJPS2020"/><ref>{{cite journal |last=Overmann |first=Karenleigh A |date=2021 |title=Counting by 'elevens' and why ''nine and two'' make ''twenty'': The material roots of Polynesian numbers |url=https://osf.io/preprints/socarxiv/jbr9g/ |journal=Journal of Mathematics and Culture |volume=15 |issue=3 |pages=1–32 |access-date=2 November 2021}}</ref><ref>{{cite book |last=Overmann |first=Karenleigh A. |title=The Materiality of Numbers: Emergence and Elaboration from Prehistory to Present |publisher=Cambridge University Press |year=2023 |isbn=9781009361248 |url=https://www.cambridge.org/core/books/materiality-of-numbers/0225B896107A5E4BCC6657473ACD1FE8}}</ref> This method of counting set aside every tenth item to mark ten of the counted items; the items set aside were subsequently counted in the same way, with every tenth item now marking a hundred (second round), thousand (third round), ten thousand items (fourth round), and so on.<ref name="OVERJPS2020"/> The counting method worked the same regardless of whether the base unit was a single item, pair, or group of four &mdash; base counting units used throughout the region &mdash; and it was the basis for the unique [[Mangareva#Binary counting system|binary counting]] found in [[Mangareva]], where counting could also proceed by groups of eight.<ref name="OVERJPS2020"/><ref>{{cite journal |last1=Bender |first1=Andrea |last2=Beller |first2=Sieghard |date=2013 |title=Mangarevan invention of binary steps for easier calculation |journal=Proc. Natl. Acad. Sci. USA |volume=111 |issue=4 |pages=1322–1327 |doi=10.1073/pnas.1309160110 |pmid=24344278 |pmc=3910603 |doi-access=free }}</ref>

The method of counting also solves another mystery: why the Hawaiian word for ''twenty'', ''iwakalua'', means "nine and two." When the counting method was used with pairs, nine pairs were counted (18) and the last pair (2) was set aside for the next round.<ref name="OVERJPS2020"/><ref name="OVEROA2020"/>

===Use by the Pañgwa===

Less is known about the idea the [[Pangwa language|Pañgwa]] people of [[Tanzania]] counted by elevens. It was mentioned in 1920 by the British anthropologist [[Northcote W. Thomas]]:
<blockquote>"Another abnormal numeral system is that of the Pangwa, north-east of Lake Nyassa, who use a base of eleven."<ref name="THOMAS">{{cite journal |last=Thomas |first=Northcote W |date=1920 |title=Duodecimal base of numeration |journal=Man |volume=20 |issue=1 |pages=56–60 |doi=10.2307/2840036 |jstor=2840036 |url=http://www.jstor.com/stable/2840036 |access-date=25 July 2020}}</ref>{{rp|p. 59}}</blockquote>

And,

<blockquote>"If we could be certain that ''ki dzigo'' originally bore the meaning of eleven, not ten, in Pangwa, it would be tempting to correlate the ''dzi'' or ''či'' with the same word in Walegga-Lendu, where it means twelve, and thus bring into a relation, albeit of the flimsiest and most remote kind, all three areas in which abnormal systems are in use."<ref name="THOMAS"/>{{rp|p. 59}}</blockquote>

The claim was repeated by the British explorer and colonial administrator [[Harry H. Johnston]] in Vol. II of his 1922 study of the [[Bantu languages|Bantu]] and [[Semi-Bantu]] languages. He too noted suggestive similarities between the Pañgwa term for eleven and terms for ten in related languages:<ref name="BANTU">{{cite book |last=Johnston |first=Harry H |title=A comparative study of the Bantu and Semi-Bantu languages, Vol. II |location=Oxford |publisher=Oxford University Press |date=1922 |pages=463–482 |chapter=The Bantu and Semi-Bantu numerals |oclc=872099614}}</ref>

<blockquote>"Occasionally there are special terms for 'eleven'. So far as my information goes they are the following:

'''Ki-dzigꞷ''' 36 (in this language, the Pangwa of North-east Nyasaland, counting actually goes by elevens. '''Ki-dzigꞷ-kavili''' = 'twenty-two', '''Ki-dzigꞷ-kadatu''' = 'thirty-three'). Yet the root '''-dzigꞷ''' is obviously the same as the '''-tsigꞷ''', which stands for 'ten' in No. 38. It may also be related to the '''-digi''' ('ten') of 148, '''-tuku''' or '''-dugu''' of the Ababua and Congo tongues, '''-dikꞷ''' of 130, '''-liku''' of 175 ('eight'), and the '''Tiag''' of 249."<ref name="BANTU"/>{{rp|p. 477}}
</blockquote>

In Johnston's classification of the [[Bantu languages|Bantu]] and [[Semi-Bantu]] languages,<ref name="BANTU"/>

*36 is Pañgwa, Bantu Group J, N. Ruvuma, NE Nyasaland
*38 is Kiñga, Bantu Group K, Ukiñga
*130 is Ba-ñkutu (Ba-ñkpfutu), Bantu Group DD, Central Congꞷland
*148 is Li-huku, Bantu Group HH, Upper Ituri
*175 is Ifumu or Ifuru (E. Teke), Bantu Group LL, Kwa-Kasai-Upper Ꞷgꞷwe (Teke)
*249 is Afudu, Semi-Bantu Group D, S. Benue

Today, Pañgwa is understood to have decimal numbers, with the numbers six and higher borrowed from [[Swahili language|Swahili]].<ref>{{cite web |url=https://lingweb.eva.mpg.de/channumerals/Pangwa.htm |title=Pangwa, Tanzania |website=Numeral systems of the world's languages |publisher=Max Planck Institute |date=1994 |access-date=24 January 2022}}</ref>

==Undecimal in the history of measurement==
In June 1789, mere weeks before the [[French Revolution]] began with the storming of the [[Bastille]], the [[Academy of Sciences]] established a committee (''[[General Conference on Weights and Measures|la Commission des Poids et Mesures]]'') to standardize systems of weights and measures, a popular reform that was an early step toward creating the international [[metric system]].<ref>{{cite journal |last=Hellman |first=C. Doris |date=January 1936 |title=Legendre and the French Reform of Weights and Measures |url=https://www.jstor.org/stable/301613 |journal=Osiris |publisher=[[University of Chicago Press]] |volume=1 |pages=314–340 |doi=10.1086/368429 |jstor=301613 |s2cid=144499554 |access-date=July 13, 2021 |via=JSTOR}}</ref><ref>{{cite journal |last=Smeaton |first=William A |date=2000 |title=The foundation of the metric system in France in the 1790s |journal=Platinum Metals Rev. |volume=44 |issue=3 |pages=125–134 |url=https://technology.matthey.com/documents/496120/626258/pmr-v44-i3-125-134.pdf/ |access-date=29 October 2023}}</ref> On 27 October 1790, the committee reported they had considered using [[duodecimal]] (base 12) as the basis for weights, lengths/distances, and money because of its greater divisibility, relative to [[decimal]] (base 10).<ref name="Borda1791">{{cite book |last1=Borda |first1=Jean-Charles de |title=Histoire de l'Académie Royale des Sciences. Année M.DCC.LXXXVIII. Avec les Mémoires de Mathématique & de Physique, pour la même Année, tirés des Registres de cette Académie |last2=Lagrange |first2=Joseph-Louis |last3=Lavoisier |first3=Antoine Laurent |last4=Tillet |first4=Mathieu |last5=Condorcet |first5=Jean-Antoine-Nicolas de Caritat |date=1791 |publisher=L'Imprimerie Royale |location=Paris |pages=1–6 |language=fr |chapter=Rapport fait à l'Académie des Sciences, par MM. Borda, Lagrange, Lavoisier, Tillet & Condorcet, le 27 Octobre 1790 |chapter-url=https://books.google.com/books?id=0L5eAAAAcAAJ&pg=PA64}}</ref> However, they ultimately rejected the initiative, deciding a common scale based on spoken numbers would simplify calculations and conversions and make the new system easier to implement.<ref name="Borda1791"/> Mathematician [[Joseph-Louis Lagrange]], a member of the committee, was credited with influencing the committee to select decimal.<ref name="Delambre1816"/> The debate over which one to use seems to have been lively, if not contentious, as at one point, Lagrange suggested adopting 11 as the base number, on the grounds indivisibility was actually advantageous; because 11 was a [[Prime number|prime]] number, no fraction with it as its denominator would be reducible:<ref name="Delambre1816" /><ref>{{cite book |last=Glaser |first=Anton |title=History of Binary and Other Nondecimal Numeration |location=Los Angeles |publisher=Tomash Publishers |date=1981 |oclc=923223696}}</ref>
<blockquote>Delambre wrote: "Il était peu frappé de l'objection que l'on tirait contre ce système du petit nombre des diviseurs de sa base. Il regrettait presque qu'elle ne fut pas un nombre premier, tel que 11, qui nécessairement eût donné un même dénominateur à toutes les fractions. On regardera, si l'on veut, cette idée comme une de ces exagérations qui échappent aux meilleurs esprits dans le feu de la dispute; mais il n'employait ce nombre 11 que pour écarter le nombre 12, que des novateurs plus intrépides auraient voulu substituer à celui de 10, qui fait partout la base de la numération."<ref name="Delambre1816"/>{{rp|p. lxvi}}</blockquote>
<blockquote>As translated: "He [Lagrange] almost regretted [the base] was not a prime number, such as 11, which necessarily would give all fractions the same denominator. This idea will be regarded, if you will, as one of those exaggerations that escape the best minds in the heat of argument; but he only used the number 11 to rule out the number 12, which the more intrepid innovators wanted to substitute for 10, which is the basis of numeration everywhere."</blockquote>
In 1795, in the published public lectures at the [[Joseph-Louis Lagrange#École Normale|École Normale]], Lagrange observed that fractions with varying denominators (e.g., {{frac|1|2}}, {{frac|1|3}}, {{frac|1|4}}, {{frac|1|5}}, {{frac|1|7}}), though simple in themselves, were inconvenient, as their different denominators made them difficult to compare.<ref name="L&L1795">{{cite book |last1=Lagrange |first1=Joseph-Louis |title=Séances des écoles normales, recueillies par des sténographes, et revues par les professeurs. Seconde partie. Débats. Tome premier |last2=Laplace |first2=Pierre-Simon |date=1795 |publisher=L. Reynier |location=Paris |pages=3–23 |language=fr |chapter=Mathématiques |oclc=780161317}}</ref> That is, fractions aren't difficult to compare if the numerator is 1 (e.g., {{frac|1|2}} is larger than {{frac|1|3}}, which in turn is larger than {{frac|1|4}}). However, comparisons become more difficult when both numerators and denominators are mixed: {{frac|3|4}} is larger than {{frac|5|7}}, which in turn is larger than {{frac|2|3}}, though this cannot be determined by simple inspection of the denominators in the way possible if the numerator is 1. He noted the difficulty was resolved if all the fractions had the same denominator:
<blockquote>Lagrange wrote: "On voit aussi par-là, qu'il est indifférent que le nombre qui suit la base du système, comme le nombre 10 dans notre système décimal, ait des diviseurs ou non; peut-être même y aurait-il, à quelques égards, de l'avantage à ce que ce nombre n'eût point de diviseurs, comme le nombre 11, ce qui aurait lieu dans le système undécimal, parce qu'on en serait moins porté à employer les fractions {{frac|1|2}}, {{frac|1|3}}, etc."<ref name="L&L1795"/>{{rp|p. 23}}</blockquote>
<blockquote>As translated: "We also see by this [argument about divisibility], it does not matter whether the number that is the base of the system, like the number 10 in our decimal system, has divisors or not; perhaps there would even be, in some respects, an advantage if this number did not have divisors, like the number 11, which would happen in the undecimal system, because one would be less inclined to use the fractions {{frac|1|2}}, {{frac|1|3}}, etc."</blockquote>

In recounting the story, Ralph H. Beard (in 1947, president of the then-named Duodecimal Society of America) noted that base 11 numbers have the disadvantage that for prime numbers higher than 11, "we are unable to tell, without actually testing them, not only whether or not they are prime, but, surprisingly, whether or not they are odd or even."<ref>{{cite journal |last=Beard |first=Ralph H |date=1947 |title=Disingenuous dissuasions |journal=The Duodecimal Bulletin |volume=3 |issue=2 |pages=6–9}}</ref>{{rp|p. 9}}

==Undecimal in computer science and technology==
Undecimal (often referred to as unodecimal in this context) is useful in computer science and technology for understanding complement (subtracting by negative addition)<ref name="Saurabh2013"/> and performing digit checks on a decimal channel.<ref name="Werner1957"/>

===Transdecimal symbols===
Any numerical system with a base greater than ten requires one or more new digits; "in an undenary system (base eleven) there should be a character for ten."<ref>{{cite book |last1=O'Sullivan |first=D. |title=The principles of arithmetic: A comprehensive text-book for the use of teachers and advanced pupils |location=Dublin |publisher=Alexander Thom |date=1875 |edition=3rd |oclc=443088675}}</ref>{{rp|p. 345}} To allow entry on typewriters, letters such as {{angbr|{{mono|1=A}}}} (as in [[hexadecimal]]), {{angbr|{{mono|1=T}}}} (the initial of "ten"), or {{angbr|{{mono|1=X}}}} (the [[Roman numeral]] 10) are used for the number 10 in base 11. It is also possible to use the digit ↊ ("dek"), the so-called Pitman numeral for 10 proposed in 1947 by [[Isaac Pitman]] as one of the two transdecimal symbols needed to represent base 12 ([[duodecimal]]).<ref>{{cite journal |last=Pitman |first=Isaac |date=1947 |title=A reckoning reform |journal=The Duodecimal Bulletin |volume=3 |issue=2 |pages=1–5}}</ref>

==Undecimal in International Standard Book Numbers (ISBN)==
The 10-digit numbers in the system of [[International Standard Book Number]]s (ISBN) used undecimal as a [[check digit]].<ref name="ISBN2017"/> A check digit is the final digit of an ISBN that is related mathematically to all the other digits it contains that is used to verify their accuracy.<ref>{{cite web |publisher=International ISBN Agency |url=https://www.isbn-international.org/content/what-isbn |title=What is an ISBN? |date=2014 |access-date=July 14, 2021}}</ref> It represents the answer to a mathematical calculation, in this case, one that multiplies the ten digits of the ISBN by the integers ten (leftmost digit) through two (second-to-last rightmost digit, the last being the check digit itself) and then sums them.<ref name="Pearce2015">{{cite web |url=https://isbn-information.com/the-10-digit-isbn.html |title=ISBN Information: Anatomy of a 10-digit ISBN |date=2015 |access-date=July 14, 2021}}</ref> The calculation should yield a multiple of eleven, with its final digit, represented by the digits 0 through 9 or an X (for ten), being equal to the tenth digit of the ISBN.<ref name="Pearce2015"/> {{As of|2007|January|1}}, thirteen-digit ISBNs are the standard.<ref name="ISBN2017"/> The International ISBN Agency provides an online calculator that will convert ten-digit ISBNs into thirteen digits.<ref>{{cite web |publisher=International ISBN Agency |url=https://www.isbn-international.org/content/isbn-calculator |title=ISBN Calculator |date=2014 |access-date=July 14, 2021}}</ref>

==Undecimal in popular fiction==
In the novel ''[[Contact (novel)|Contact]]'' by [[Carl Sagan]], a message left by an unknown advanced intelligence lies hidden inside the number [[pi]]; the message is best revealed when pi is computed in undecimal.<ref>{{Cite web |url=https://math.stackexchange.com/q/1104660 |title=Patterns in pi in "Contact" |website=Mathematics Stack Exchange }}</ref><ref>{{cite book |last=Sagan |first=Carl |title=Contact |location=New York |publisher=Gallery Books |date=1985 |isbn=9781501197987}}</ref>{{rp|p. 317}} In the television series ''[[Babylon 5]]'', the advanced race known as [[Minbari]] use undecimal numbers they realize by counting ten fingers and the head, according to series creator [[J. Michael Straczynski]].<ref>"[[The Quality of Mercy (Babylon 5)|The Quality of Mercy]]", ''[[Babylon 5]]'', orig. airdate 17 August 1994</ref><ref>{{Cite web|url=http://jmsnews.com/messages/message?id=17719|title=JMSNews Minbari base eleven includes [ten] fingers and head...|website=jmsnews.com}}</ref>

== Undecimal powers of 2 ==
{| class="wikitable sortable" style="text-align:center"
|+ Doubles
! Decimal
! Undecimal
|-
| 1
| 1
|-
| 2
| 2
|-
| 4
| 4
|-
| 8
| 8
|-
| 16
| 15
|-
| 32
| 2A
|-
| 64
| 59
|-
| 128
| 107
|-
| 256
| 213
|-
| 512
| 426
|-
| 1024
| 851
|-
| 2048
| 15A2
|}

== Undecimal multiplication chart ==
{| class="wikitable"
|-
! × || '''1''' || '''2''' || '''3''' || '''4''' || '''5'''|| '''6''' || '''7''' || '''8''' || '''9''' || '''A''' || '''10''' || '''11'''
|-
! '''1'''
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || A || 10 || 11
|-
! '''2'''
| 2 || 4 || 6 || 8 || A || 11 || 13 || 15 || 17 || 19 || 20 || 22
|-
! '''3'''
| 3 || 6 || 9 || 11 || 14 || 17 || 1A || 22 || 25 || 28 || 30 || 33
|-
! '''4'''
| 4 || 8 || 11 || 15 || 19 || 22 || 26 || 2A || 33 || 37 || 40 || 44
|-
! '''5'''
| 5 || A || 14 || 19 || 23 || 28 || 32 || 37 || 41 || 46 || 50 || 55
|-
! '''6'''
| 6 || 11 || 17 || 22 || 28 || 33 || 39 || 44 || 4A || 55 || 60 || 66
|-
! '''7'''
| 7 || 13 || 1A || 26 || 32 || 39 || 45 || 51 || 58 || 64 || 70 || 77
|-
! '''8'''
| 8 || 15 || 22 || 2A || 37 || 44 || 51 || 59 || 66 || 73 || 80 || 88
|-
! '''9'''
| 9 || 17 || 25 || 33 || 41 || 4A || 58 || 66 || 74 || 82 || 90 || 99
|-
! '''A'''
| A || 19 || 28 || 37 || 46 || 55 || 64 || 73 || 82 || 91 || A0 || AA
|-
! '''10'''
| 10 || 20 || 30 || 40 || 50 || 60 || 70 || 80 || 90 || A0 || 100 || 110
|-
! '''11'''
| 11 || 22 || 33 || 44 || 55 || 66 || 77 || 88 || 99 || AA || 110 || 121
|}

==See also==
* [[International Standard Book Number#Check digits|Undecimal check digit for ten-digit ISBNs]]

==References==
{{reflist}}

[[Category:Positional numeral systems]]