Numeral Systems

In comparative studies, within Hamito-Semitic and between the latter and Indo-European, many scholars have tried to compare numerals as such. In this paper, it is suggested to compare other features in the domain of numerals, such as the order of tens and units in composed numerals, the formation of tens and the number of counted object.

The importance of the prime factorization problem is very well known
(e.g., many security protocols are based on the impossibility of a fast factorization of integers on traditional computers). It is necessary from a number k to establish two primes a and b giving k = a · b. Usually, k is written in a positional numeral system. However, there exists a variety of numeral systems that can be used to represent numbers. Is it true that the prime factorization is difficult in any numeral system? In this paper, a numeral system with partial carrying is described. It is shown that this system contains numerals allowing one to reduce the problem of prime factorization to solving [K/2] − 1 systems of equations, where K is the number of digits in k (the concept of digit in this system is more complex than the traditional one) and [u] is the integer part of u. Thus, it is shown that the difficulty of prime factorization is not in the problem itself but in the fact that the positional numeral system is used traditionally to represent numbers participating in the prime  factorization. Obviously, this does not mean that P=NP since it is not known whether it is possible to re-write a number given in the traditional positional numeral system to the new one in a polynomial time.

El propósito de la presente ponencia consiste -simplemente- en presentar una breve descripción de los numerales de la lengua selknam de Tierra del Fuego, agregando una revisión de las comparaciones hasta el momento efectuadas de estos numerales, tanto dentro como fuera de la familia lingüística chon.

Publicado en: A. Regúnaga, S. Spinelli y M. E. Orden (Comps). IV Encuentro de Lenguas Indígenas Americanas (ELIA). Libro de Actas: 685-703. Santa Rosa: EdUNLPam. (ISBN: 978-950-863-335-4, CDD: 907.2).

http://www.unlpam.edu.ar/images/extension/edunlpam/Actas%20ELIA%20-%20UNLPam.pdf

The article reexamines the reconstruction of the Proto-Indo-Hittite counting system. Using comparative and typological arguments, I attempt to show that the original Indo-Hittite system was structured as ‘1’, ‘2’, ‘3’, ‘>3/several/many’. Words for a new numeral, ‘4’, were introduced in Anatolian and Narrow Indo-European branches separately after their split. Anatolian *meyu- ‘4’ (Hittite meyu- ~ Luwian mauwa-) is unetymologizable within Indo-Euro-pean. I suppose that it is an archaism and propose an external (Proto-Altaic) non-numeral cognate for this Anatolian stem. Anatolian *meyu- is one of the rather numerous Nostratic roots that were retained in Anatolian, but lost in the Narrow Indo-European branch.

Resumen El propósito de este trabajo es reunir y analizar descriptiva y comparativamente los sistemas numerales documentados en las lenguas de la familia lingüística alacalufe. Se presenta un breve panorama de las características morfosintácticas de los numerales en kawésqar, se comparan las formas de las lenguas históricamente conocidas, discutiendo las etimologías propuestas y/o planteando nuevas, se ofrecen reconstrucciones de los sistemas de numerales en proto-alacalufe nuclear y en proto-alacalufe, se discuten algunos presuntos numerales atribuidos lenguas alacalufes, y se evalúan las comparaciones externas efectuadas hasta el momento.

La lengua aonekko 'a'ien (literalmente "lengua sureña"), también conocida como tehuelche, forma parte de la familia lingüística chon, la cual está compuesta por las lenguas de los pueblos cazadores-recolectores pedestres más australes de Sudamérica (Lehmann-Nitsche 1913, Suárez 1988 [1970]). Junto con el extinguido teushen, forma la rama continental de esta familia lingüística, diferenciada de la rama insular compuesta por las lenguas selknam y haush de Tierra del Fuego. El aonekko 'a'ien se habló en la parte meridional de la Patagonia continental argentina y chilena (principalmente en territorio perteneciente a las actuales provincias argentinas de Santa Cruz y Chubut, con expansiones hasta el sur de la provincia de Buenos Aires durante el siglo XVIII). La última hablante falleció a principios del año 2019. A diferencia de las lenguas hermanas haush y selknam, el aonekko 'a'ien posee (y poseía ya en el siglo XIX) un sistema numeral decimal, lo que parece ser poco frecuente en lenguas de pueblos estrictamente cazadores-recolectores (Epps et al. 2012). El propósito del presente trabajo es comparar el sistema registrado en los últimos años por la lingüista Ana Fernández Garay (1998) con el que fuera documentado a mediados del siglo XIX por el misionero Theophilus Schmid (1912a y 1912b). Los numerales de ambos cronolectos muestran ciertas diferencias, que permiten esbozar la evolución del sistema durante el último siglo y medio 2 .

La lengua yagan se habló en la costa sur de la Isla Grande de Tierra del Fuego y territorio insular adyacente, hasta el Cabo de Hornos. La mayor parte de las fuentes disponibles del yagan proporcionan datos sobre una cantidad muy reducida de numerales, no más de tres (o, en los registros más recientes, cuatro) unidades, pero hay autores que han hablado de un sistema mayor, de hasta nueve o diez unidades. El propósito de este trabajo es reunir, analizar y evaluar --desde puntos de vista tanto descriptivo como histórico-comparativo-- los datos sobre numerales de esta lengua dispersos en la bibliografía.

This work defends the strength of the contextual and combinatorial arguments of the hypothesis proposed by Eduardo Orduña about the identification of several Iberian elements as numerals. Additionally, some aspects of the original hypothesis are corrected and new positive arguments are added, the most relevant are those related to the marks of value of the Iberian coins, because they are compatible with some of the proposed numerals.

In this paper, we seek to draw attention to Malayo-Polynesian languages outside of the Oceanic subgroup with innovative bases and complex numerals involving various additive, subtractive, and multiplicative procedures. We highlight the fact that the number of languages showing such innovations is
more than previously recognized in the literature. Finally, we observe that the concentration of complex numeral innovations in the region of eastern Indonesia suggests Papuan influence, either through contact or substrate. However, we also note that sociocultural factors, in the form of numeral taboos and conventionalized counting practices, may have played a role in
driving innovations in numerals.

In this paper, we examine the processes underlying the Yorùbá numeral system and describe a computational system that is capable of converting cardinal numbers to their equivalent Standard Yorùbá number names. First, we studied the mathematical and linguistic basis of the Yorùbá numeral system so as to formalise its arithmetic and syntactic procedures. Next, the process involved in formulating a ContextFree Grammar (CFG) to capture the structure of the Yorùbá numeral system was highlighted. Thereafter, the model was reduced into a set of computer programs to implement the numerical to lexical conversion process. System evaluation was done by ranking the output from the software and comparing the output with the representations given by a group of Yorùbá native speakers. The result showed that the system gave correct representation for numbers and produced a recall of 100% with respect to the collected corpus. Our future study is focused on developing a text normalisation system that will produce number names for other numerical expressions such as ordinal numbers, date, time, money, ratio, etc. in Yorùbá text

This article discusses the earliest attestations and history of various English indefinite hyperbolic numerals such as zillion, jillion, and umpteen, identifying them as a previously unrecognized category of contemporary English words with a peculiarly American emphasis. Umpteen originated in the late 1870s but became normative in the Midwest in the 1890s, not as World War I British soldiers' cant as frequently claimed. Zillion first flourished in African American English in the 1920s, while jillion originated independently in Texas around the same time. These ought to be regarded as a contemporaneous cluster of locally developed words within specific discourse communities rather than random nonce inventions. Since 1985 a massive expansion has occurred in the lexicon, primarily using the emphatic prefixes ka-, ba-, and ga-. Interviews conducted with African American youth in 2009–10 suggest that zillion may be acquiring a fixed numeral meaning (following trillion) among some speakers. Indefinite hyperbolic numerals represent a lexical category newly, promiscuously, and playfully developed in late nineteenth-and twentieth-century Englishes, in a discursive environment where numerical reference had become normative.

Col presente lavoro, a conclusione di un periodo di riflessione sul tema dei numerali e delle istanze di numerazione, si intende offrire un contributo alla costituzione di un Atlante generale dei numerali e delle istanze di numerazione, ambizioso obiettivo della ricerca cofinanziata dal MIUR per il biennio 2000-2001 e svolta sotto il coordinamento di Domenico Silvestri dell'Istituto Universitario Orientale di Napoli.

Number and quantity in Slovak and Hungarian
Jubilees are connected with numbers, the jubilant, Slavomír Ondrejovič translated one of W. Humboldt’s works (Humboldt, 2000) to Slovak, which is a fundament of the theory about the linguistic image of the world. Our jubilant also researches the sociolinguistic aspect of the Hungarian-Slovak relations. That is why we analyse the grammatical category of numerus and the word class of numerals in Hungarian and Slovak from the aspect of linguistic relativity and ethnolinguistics.

La llengua ibèrica era la llengua vernacle dels pobles ibers i la trobem expressada bàsicament en escriptura ibèrica nord-oriental i residualment en escriptura sud-oriental i greco-ibèrica. Les inscripcions ibèriques es poden transcriure, però no es poden traduir, ja que no disposem de cap llengua suficientment propera. Tot i que la llengua ibèrica molt probablement comparteixi família lingüística amb la basca, atès que els numerals són similars, quinze segles separen els darrers textos ibèrics dels primers en basc. Tant l’escriptura ibèrica nord-oriental, com la sud-oriental són escriptures híbrides en les que conviuen signes alfabètics i sil·làbics, però no deriven una de l’altra, sinó que pertanyen a dues branques diferents de la família paleohispànica. L’ancestre comú, que derivaria de l'escriptura fenícia, seria el responsable dels trets comuns, com el dualisme, que mitjançant marques permet reaprofitar el mateix signe per representar dos valors fonètics propers.
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The Iberian language was the vernacular language of the Iberian peoples, and we find it expressed basically in north-eastern Iberian writing and residually in south-eastern and Greco-Iberian writing. Iberian inscriptions can be transcribed, but cannot be translated, as we do not have any language close enough. Although the Iberian language most likely shares a linguistic family with Basque, since the numerals are similar, fifteen centuries separate the last Iberian texts from the first in Basque. Both North-eastern and South-eastern Iberian writing are hybrid writings in which alphabetic and syllabic signs coexist, but they do not derive from each other, but belong to two different branches of the Paleo-Hispanic family. The common ancestor, which would derive from Phoenician writing, would be responsible for common features, such as dualism, which by means of marks allows the same sign to be reused to represent two close phonetic values.
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La langue ibérique était la langue vernaculaire des peuples ibériques et nous la trouvons exprimée essentiellement dans l'écriture ibérique du nord-est et résiduellement dans l'écriture du sud-est et gréco-ibérique. Les inscriptions ibériques peuvent être transcrites, mais ne peuvent pas être traduites, car nous n'avons pas de langue assez proche. Bien que la langue ibérique partage très probablement une famille linguistique avec le basque, puisque les chiffres sont similaires, quinze siècles séparent les derniers textes ibériques des premiers en basque. Les écritures ibériques du nord-est et du sud-est sont des écritures hybrides dans lesquelles coexistent des signes alphabétiques et syllabiques, mais ils ne dérivent pas l'un de l'autre, mais appartiennent à deux branches différentes de la famille paléo-hispanique. L'ancêtre commun, qui dériverait de l'écriture phénicienne, serait responsable de traits communs, tels que le dualisme, qui à travers des marques permet de réutiliser le même signe pour représenter deux valeurs phonétiques proches.
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La lengua ibérica era la lengua vernácula de los pueblos íberos y la encontramos expresada básicamente en escritura ibérica nororiental y residualmente en escritura sudoriental y greco-ibérica. Las inscripciones ibéricas se pueden transcribir, pero no se pueden traducir, ya que no disponemos de ninguna lengua suficientemente cercana. Aunque la lengua ibérica muy probablemente comparta familia lingüística con la vasca, dado que los numerales son similares, quince siglos separan los últimos textos ibéricos de los primeros en vasco. Tanto la escritura ibérica nororiental, como la sudoriental son escrituras híbridas en las que conviven signos alfabéticos y silábicos, pero no derivan una de la otra, sino que pertenecen a dos ramas diferentes de la familia paleohispánica. El ancestro común, que derivaría de la escritura fenicia, sería el responsable de los rasgos comunes, como el dualismo, que mediante marcas permite reaprovechar el mismo signo para representar dos valores fonéticos cercanos.

In this global era, scenarios of language contact are increasing quite rapidly. Minor languages, in general, tend to borrow words from major, or at least larger, languages in the same way that communities import new gadgets and tools. That is why nouns are most often borrowed in many languages. In languages of northern Pakistan, such as Burushaski, Domaaki, and Shina, Urdu numerals are borrowed with high frequency in daily use, though speakers still are able to produce Burushaski (and so forth) numerals during elicitation. In this paper, I argue that the use of Urdu numerals, a decimal (base 10 numeral) system, is more consistent with the demands of the global era, than the traditional vigesimal (base 20 numeral) system that is present in the traditional number system in many languages of north Pakistan. I illustrate this by describing particular situations in which Burushaski first language speakers use Urdu numeral for both large and small numbers.

Ethnomathematics gestures, signs and communication in the mathematics of cultural diversity Etnomatematica gesti, segni e comunicazione nella matematica della diversità culturale Giovanni Giuseppe Nicosia IIS Serpieri, Bologna RSDDM, ISGEm, FLC-CGIL Waza gengo: mente, corpo, conoscenza e relazione educativa. Scambi fra tradizioni d'Oriente ed esperienze innovative d'Occidente -6/11/2013 -Bologna

Данная работа посвящена исследованию системы числительных адыгейского языка, уточнению и проверке выделенных в других работах морфосинтаксических классов (т.е. таких как, например, количественные числительные, кратные числительные и т. д.) и подклассов (т. е. разных группы числительных, образованных по единой морфосинтаксической модели).

In the paper, eight Mande counting systems (Modern and Old Bamana, Boko, Dzuungoo, Mwan, Dan-Gwɛɛtaa, San-Maka, Soninke) are analyzed in detail, followed by the presentation of a summary table for 30 languages representing all the groups of the Mande family. In Mande languages, elements of different models are present: decimal, vigesimal, octogesimal; some elements of a quinary and quindecimal systems are also attested. Certain languages of the family possess subtractive numerals; the subtrahend may be as high as 25. In the old Bamana system, Jogo-Ton and Jogo-Ligbi, division is applied. In money counting, many Mande languages apply the principle of multiplying by five.
It is suggested that the quinary and decimal models should be considered original for Mande languages, while the vigesimal model (as well as rare pentadecimal and octogesimal ones) as a result of recent evolution, quite probably under the influence of Senufo, Gur, and probably also Kru languages.

1) «Sonderbar setzt die germanische Sprache blos die Zahlen 11 und 12 mit -lif zusammen, die weiteren aber mit -zehun; es koennte eben so gut: dreilif (13), vierlif (14) u. s. w. oder auch: einzehn (11) zweizehn (12) wie im Lateinischen heißen. Jenes Abbrechen mit Zwoelf und das Anheben einer neuen Ordnung mit Dreizehn scheint mir daher eine Spur der alten Duodecimalrechnung. Das lif bedeutet seiner Wurzel nach ueber, uebrig; einlif sagt folglich eins über zehn […]»

De acuerdo a su etimología, la mayor parte de  los numerales de las lenguas guaicurúes pueden distribuirse en los siguientes tres grandes grupos:

(1) préstamos recientes del español o el portugués (p. ej. toba do:s-olqa / do:s-ole  ‘dos’, kadiweu no:be ‘nueve’, etc.)

(2) formas analizables como verbos o sintagmas originarios (p. ej. toba antiguo nalotapegat  ‘cuatro’, de una raíz verbal -alot(a)- ‘estar o ir parejos’; abipón keienk ñate ‘cuatro’, literalmente ‘dedos del avestruz’; etc.)

(3) formas compuestas a partir de raíces de clasificadores deícticos de posición y movimiento, como *n:a ‘presente, acercándose’,*n:i ‘tridimensional’, etc.

Las formas del grupo (3) tienen reflejos bien documentados sobre todo en guaicurú del norte (mbayá y kadiweu) y en abipón; pero también en algunas formas antiguas  de numerales bajos del toba (siglos XVII a XIX) y del mocoví (siglo XVIII). En el presente trabajo, se reconstruyen mediante la metodología histórico-comparativa, los numerales del proto-guaicurú a partir de las formas del grupo (3) registradas en las lenguas históricamente registradas.  Las conclusiones indican un sistema originario que habría incluido originalmente sólo dos términos: ‘uno’ y ‘dos’. Ambos numerales eran, en principio, las formas singular y no singular de los clasificadores de posición y movimiento. Tal tipo de sistema de numerales parece ser poco frecuente en las lenguas del mundo, por lo que resulta de trascendencia para el estudio tipológico de los sistemas de numerales y de sus orígenes.

Recent revivals of local cultures and languages, as well as ideologically-loaded debates on the role of dialects at school usually ignore the fact that the scientific study of dialects and of closely related systems can be very interesting for a theory of language and also in the domain of linguistic education. In particular, dialects provide larger sets of variants and of variant combinations that can become relevant in the definition of typological variation. Even when their linguistic "substance" is quite similar, dialects display different forms of linguistic patterning and of fine-grained variation. Focusing on the Romance-Germanic contact area encompassing the alpine space, patterns of variation in the inflectional behaviour of numerals will be considered. This topic provides a good example of dialects exhibiting a higher degree of complexity than most European standard languages.

Most Turkic numerals are etymologically difficult, even ones that in most languages are transparent, several have (very unusually) geminates in them, and of course some have been from time to time etymologized Altaically without explaining ANYTHING.  It is also the case that in Early Turkic the word one followed by the word 20 (bir yegirmi) means 11 not 21 and so on.  That supposedly but has been explained by Clark and adopted by Erdal, so we tested in empirically and we found ourselves taking the elevator to the wrong floor--perhaps the first time any theory in historical lx has been tested against reality.  We propose here how a particular way of counting on fingers would explain this system (called Oberstufenzählung, which we were surprised to discover was not an Altaic term or even Turkic), namely closing the fingers one after another and then opening them again as one counts from 11 to 20, which implies the yegirmi must mean 'finger(s) opened after having been closed'.  But suspecting how the experts are likely to react, we will give the actual etymology in Pt. 2 if ever.  We also point out that the PIE words for 10 and 20 and this time quite obviously mean 'closed hand' and 'open hand'.  But there of course we already know how the experts will react.  So again why waste the effort of writing down etymologies that are (a) obvious and (b) will be either ignored or treated with disdain?

Across multiple disciplines, written numerical notation is a topic of keen interest, yet several unresolved issues in its analysis are either elided or taken as settled. Numerical notation is a complex phenomenon with multiple independent histories—more than 100 distinct systems used over the past 5,500 years, interweaving with, rather than strictly paralleling, the histories of writing systems. Social, semiotic, and cognitive approaches are brought to bear on six incompletely answered questions about numeration in relation to the earliest writing. Is numerical notation a necessary precursor to writing? Does the earliest numerical notation initially serve a bookkeeping function for early states? What is the relationship between tallying and numerical notation? Does the use of numerical notation change human cognition about the domain of number? How does the emergence of numerical notation relate to linguistic representations of number? Finally, among all domains of knowledge, why is number so widely represented using graphic notations? Recognizing that these issues are not resolved, and identifying different possible resolutions, must be preliminary to fully integrating numerical notation within the broader history of writing.

The vast majority of the three hundred or so modern Tibeto-Burman languages have decimal numeral systems, like most contemporary languages, and more specifically like their large influential neighbours, the Sinitic and Indo-Aryan languages. Like most languages again, their favorite way of constructing higher numbers is by multiplication and addition.

Humans seem to be genetically endowed with a mathematical aptitude known as subitizing. It refers to the ability to correctly recognize very limited quantities (up to 3/4 items). Conversely, higher cardinalities require a different, non-innate process: counting. In keeping with the hypothesis that cognition plays a major role in structuring the linguistic complexity (cf. Lakoff and Johnson, 1980; Heine, 1997), here I suggest regarding subitizing as a significant shaping force in language. Our cognitive limitations may be reflected in the organization of linguistic structures, especially in the field of numerals. There are a number of clues which lead us to believe that certain fundamental linguistic features have autonomy and priority compared to others. These include: the diffusion of “1-2-(3)-many” numeral systems, the morpho-syntax of the lowest number words, the suppletion of the first ordinals, etc. Evidence of this in the IE. family was collected in §4. The split between the set of the first few elements and the rest is sometimes regarded as accidental or due to a chronological gap in historical evolution. However, I think that subitizing, as an innate and universal cognitive pattern, can provide a more satisfactory explanation. Since the first quantities are perceived more clearly and immediatly, the structure of some linguistic domains (e.g. numerals) reflects this cognitive hierarchy.