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eleven as its base while no known society counts by elevens two are purported to have done so the māori one of the two polynesian peoples of new zealand and the pañgwa a 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 1 2 during the french revolution undecimal was briefly considered as a possible basis for the reformed system of measurement 3 today undecimal numerals have applications in computer science 4 technology 5 and the international standard book number system 6 they also occasionally feature in works of popular fiction 7 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 8 p 345 to allow entry on typewriters letters such as a as in hexadecimal t the initial of ten or 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 9 alleged use by the māori edit conant and williams edit 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 at the time archdeacon of waiapu 10 p 123 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 10 pp 122 123 as published by williams in the first two editions of the dictionary series this statement read 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 11 p xv lesson and blosseville edit 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 1 12 they had visited new zealand in 1824 as part of the 1822 1825 circumnavigational voyage of the coquille 13 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 14 at von chamisso s claim that the new zealand number system was based on twenty vigesimal lesson inserted a footnote to mark an error 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 14 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 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 14 p 27 2 error the zealander arithmetic system is undecimal and the english are the first to propagate this false idea l 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 15 16 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 15 16 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 17 the language has now been opened to us and we correct our opinion 15 p 13 and 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 15 pp 441 442 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 1 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 1 he was also familiar with the work of thomas kendall and samuel lee through his translation of von chamisso s work 14 these circumstances suggest lesson was unlikely to have misunderstood new zealand counting as proceeding by elevens 1 lesson and his shipmate and friend blosseville 18 sent accounts of their alleged discovery of elevens based counting in new zealand to their contemporaries 1 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 19 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 20 de blosseville also mentioned it to the scottish author george lillie craik who reported this letter in his 1830 book the new zealanders 21 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 22 23 the story expanded in its retelling 1 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 19 in an interesting twist it also changed the mistaken classification needing correction from vigesimal to decimal 14 19 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 14 15 22 it also listed places the alleged local informants were supposedly from 22 relation to traditional counting edit the idea that māori counted by elevens highlights an ingenious and pragmatic form of counting once practiced throughout polynesia 1 24 25 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 1 the counting method worked the same regardless of whether the base unit was a single item pair or group of four base counting units used throughout the region and it was the basis for the unique binary counting found in mangareva where counting could also proceed by groups of eight 1 26 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 1 2 alleged use by the pañgwa edit less is known about the idea the pañgwa people of tanzania counted by elevens it was mentioned in 1920 by the british anthropologist northcote w thomas another abnormal numeral system is that of the pangwa north east of lake nyassa who use a base of eleven 27 p 59 and 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 27 p 59 the claim was repeated by the british explorer and colonial administrator harry h johnston in vol ii of his 1922 study of the bantu and semi bantu languages he too noted suggestive similarities between the pañgwa term for eleven and terms for ten in related languages 28 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 28 p 477 in johnston s classification of the bantu and semi bantu languages 28 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 pañgwa undecimal counting was not confirmed by later research and the idea is now believed to have been an error 29 today pañgwa is understood to have decimal numbers the numbers ten and higher have been influenced by or borrowed from swahili 30 in the history of measurement edit in june 1789 mere weeks before the french revolution began with the storming of the bastille the academy of sciences established a committee 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 31 32 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 33 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 33 mathematician joseph louis lagrange a member of the committee was credited with influencing the committee to select decimal 3 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 no fraction with it as its denominator would be reducible 3 34 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 3 p lxvi 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 in 1795 in the published public lectures at the école normale lagrange observed that fractions with varying denominators e g 1 2 1 3 1 4 1 5 1 7 though simple in themselves were inconvenient as their different denominators made them difficult to compare 35 that is fractions aren t difficult to compare if the numerator is 1 e g 1 2 is larger than 1 3 which in turn is larger than 1 4 however comparisons become more difficult when both numerators and denominators are mixed 3 4 is larger than 5 7 which in turn is larger than 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 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 1 2 1 3 etc 35 p 23 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 1 2 1 3 etc 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 36 p 9 in science and technology edit undecimal often referred to as unodecimal in this context is useful in computer science and technology for understanding complement subtracting by negative addition 4 and performing digit checks on a decimal channel 5 the 10 digit numbers in the system of international standard book numbers isbn used undecimal as a check digit 6 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 ...