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Undecimal is a positional numeral system that uses eleven as its base. While no known society counts by elevens, two are purported to have done so: the Māori and the Pañgwa. The idea of counting by elevens remains of interest for its relation to a traditional method of tally-counting practiced in Polynesia.

Alleged 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, at the time Archdeacon of Waiapu. "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." 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. They had visited New Zealand in 1824 as part of the 1822–1825 circumnavigational voyage of the Coquille, 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. 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)..." 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: The language has now been opened to us, and we correct our opinion." sent accounts of their alleged discovery of elevens-based counting in New Zealand to their contemporaries. 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. De Blosseville also mentioned it to the Scottish author George Lillie Craik, who reported this letter in his 1830 book The New Zealanders. 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. The story expanded in its retelling: 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. 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. ==Alleged use by the Pañgwa==
Alleged use by the Pañgwa
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." 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." "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." Today, Pañgwa is understood to have decimal numbers; the numbers 'ten' and higher have been influenced by or borrowed from Swahili. ==In the history of measurement==
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 (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. 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). 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. 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." That is, fractions aren't difficult to compare if the numerator is 1 (e.g., is larger than , which in turn is larger than ). However, comparisons become more difficult when both numerators and denominators are mixed: is larger than , which in turn is larger than , 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 , , etc." ==In science and technology==
In 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) 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. 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. ==In popular fiction==
In popular fiction
Base 11 periodically features in works of popular fiction, particularly science fiction. For example, in Battlefield Earth by L. Ron Hubbard, the villainous alien race known as the Psychlos use base 11 for their mathematics because they have eleven fingers, "six talons on their right paws and five on their left." Similarly, in the television series Babylon 5, the advanced race known as the Minbari use undecimal numbers because they count on ten fingers and the head. In the 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 Base 11 arithmetic." In the Season 9 finale of The Office, fictional character Kevin Malone invents the base 11 number Keleven to cover up his mathematical incompetence. ==Examples==
Examples
This table shows the powers of 2 in undecimal. This table shows how to multiply small integers in undecimal. ==See also==
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