Hindu-Arabic Numerals Research Articles

An Analysis of Nonpositional Numeral Systems

These numeration systems are usually broadly classified as being either positional or nonpositional. However, this dichotomous classification is somewhat coarse. To see this, consider the Roman numerals. These are often cited as the standard example of a nonpositional numeral system. However, if the Roman numerals were truly nonpositional, then we would expect IV and VI to represent the same value, which they do not. On the other hand, Roman numerals certainly do not form a positional system; they appear to be a broadly nonpositional system with some positional features. Conversely, there exist positional systems which display some of the features of nonpositional systems—specifically, the Babylonian sexagesimal [9, Chapter 26] and Maya vigesimal [9, Chapter 28] positional systems. Unlike the Hindu-Arabic numerals, each of these systems can have more than one symbol per position. The Babylonian system has dedicated symbols for 1 and for 10 which are then combined additively to make up the value (B59) in any given position. Thus, if we confine our attention to a single position, we find a nonpositional numeral system at work. This is evidence that the Babylonian sexagesimal positional system evolved from an earlier Sumerian decimal nonpositional system (which we will see in the next section). We find a similar phenomenon with the Maya system: there are dedicated symbols for 1 and for 5, which are used to make up the value in each position, hinting at the earlier use of a quinary nonpositional system. Clearly, a finer classification than the traditional positional-versus-nonpositional scheme is called for. The existence of such ‘‘mixed’’ numeral systems is acknowledged in [1], and a simple system of classification appears in [9, p. 429], into positional, nonpositional, and so-called hybrid systems; details can be found in [10, pp. 34–63]. The classification in the present paper will be rather different from that of Ifrah. The goal of this paper is to obtain a finer classification for nonpositional systems. Note that my analysis and classification will be of the mathematical and symbolical structure of numeral systems; a linguistical analysis of numeral systems has already been given in [15], for example. I will use the abbreviation ‘‘NNS’’ throughout to mean ‘‘nonpositional numeral system’’. Throughout this article, the word ‘‘number’’ will refer to a quantity, whilst the word ‘‘numeral’’ will describe a symbol. Thus, for example, ‘‘2’’ is a numeral denoting the number 2. However, since the Hindu-Arabic numerals will not be subject to analysis, there is no danger of confusion in our using these to denote numbers. Nonpositional numerical notations are by far the most common throughout history. They are characterised by the feature that the position of a particular symbol within the representation of a number is irrelevant. We may regard this as a consequence of the ‘‘additive’’ nature of such systems: juxtaposition of symbols denotes addition. For example, if h and j stand for 1 and 10, respectively, in a

A vanished world: medieval Spain's golden age of enlightenment

In a world troubled by religious strife and division, Chris Lowney's vividly written new book offers a hopeful historical reminder: Muslims, Christians, and Jews once lived together in Spain, creating a centuries-long flowering of commerce, culture, art, and architecture. Written with a narrative drive reminiscent of Barbara Tuchman's A Distant Mirror, this new work takes us back to a medieval Iberia that prefigured the Renaissance. In 711, a ragtag army of Muslim North Africans conquered Christian Spain and launched Western Europe's first (and to date only) Islamic state. In 1492, Ferdinand and Isabella vanquished Spain's last Muslim kingdom, forced Jews to convert or emigrate, and dispatched Christopher Columbus to the New World. In the years between, Spain's Muslims, Christians, and Jews forged a golden age for each faith and distanced Spain from a Europe mired in the Dark Ages. Medieval Spain's pioneering innovations touched every dimension of Western life: Spaniards introduced Europeans to paper manufacture and to the Hindu-Arabic numerals that supplanted the Roman numeral system. Spanish scholars translated what stood for centuries as Europe's standard medical handbook. Spain's farmers adopted irrigation technology from the Near East to nurture Europe's first crops of citrus and cotton. Spanish artisans graced luxurious homes with the fountains, gardens, and decorative tile that remain hallmarks of southern Spain's distinctive decor. Spain's religious scholars authored works that still profoundly influence their respective faiths, from the masterpiece of the Jewish kabbalah to the meditations of Sufism's greatest master to the eloquent arguments of Maimonides that humans can successfully marry religious faith and reasoned philosophical inquiry. No less astonishing than medieval Spain's wide-ranging accomplishments was the simple fact its Muslims, Christians, and Jews often managed to live and work side by side, bestowing tolerance and freedom of worship on the religious minorities in their midst. A Vanished World chronicles this impossibly panoramic sweep of human history and achievement, encompassing both the agony of jihad, Crusades, and Inquisition, and the glory of a multireligious, multicultural civilization that forever changed the West. One gnarled root of today's religious animosities stretches back to medieval Spain, but so does a more nourishing root of much modern religious wisdom. In a world torn by religious antagonism, Chris Lowney offers enduring lessons learned from medieval Spanish villages where Muslims, Christians, and Jews rubbed shoulders on a daily basis.

The Measure of Reality: Quantification and Western Society, 1250-1600 (review)

Reviewed by: The Measure of Reality: Quantification and Western Society, 1250–1600* Theodore M. Porter (bio) The Measure of Reality: Quantification and Western Society, 1250–1600. By Alfred W. Crosby. Cambridge: Cambridge University Press, 1996. Pp. xii+245; illustrations, notes, index. $24.95. Alfred Crosby is well known as the author of an academic press best-seller, Ecological Imperialism: The Biological Expansion of Europe, 900–1900 (New York: Cambridge University Press, 1986). There he argued that the European success in conquering and settling other continents involved, in effect, importing large parts of their biological environments with them, especially crops, domestic animals, and diseases. The book provided a grand explanation, a big picture whose details, scope, and implications could be debated and refined by specialists. A readable text written with verve, it makes an excellent work for students and for professors willing to venture outside their bailiwicks. The Measure of Reality has similar ambitions. Crosby introduces it as part of the same project, “the third book I have written in my lifelong search for explanations for the amazing success of European imperialism” (p. ix). Here again we have lively and engaging prose, rich with similes and illustrations, aiming to define a big topic. Even more than in his previous work, he defies all provincialism. This is a book about the late medieval and Renaissance period with no evidence of research in Latin, or indeed any language but English, and very little reliance on primary sources. His use of scholarship is spotty and idiosyncratic. His reading in the history of technology and science shows a marked preference for older studies. He does [End Page 555] not even nod to the outpouring of work in the last two decades on the history of quantification since 1600. His grand argument has yawning gaps. The link between quantification and imperialism he takes largely for granted. Supposing he is right about the medieval origins of a quantitative mentality, he provides no elaboration of how it mattered beyond a nod to the well-worn Weberian tropes of rationalization and bureaucratization. Although the book purports to be about broad shifts of thinking, mentalités, virtually all its evidence relates to elite philosophers and artists. Crosby takes as self-evident that the modern world was made by science and technology. He further presumes that numbers and mathematics provide their natural language. Indeed, he construes a quantitative mentality as their precondition. We find, he writes, little of “science and technology as such” (?) at the “beginnings of western imperialism” (p. x). The vast expansion of European science and of European power alike, he tells us, was triggered by new, quantitative patterns of cognition. His topic, then, is an epochal “acceleration after 1250 or so in the West’s shift from qualitative perception to . . . quantificational perception” (p. 49). That shift flattened the richly symbolic, “venerable” world picture of the Middle Ages and replaced it with a more instrumental modernism. Crosby wants to document and explain the acceleration. He divides his explanation of this acceleration into two parts, roughly, material causes and efficient causes. In other words, he will account first for the combustibles and the oxygen and then identify the match that set them ablaze. The “necessary but insufficient causes” of his story emerge as familiar elements in standard accounts of the European Renaissance of learning and science. Among them are the recovery of ancient texts and the growth of universities. Inexplicably, given his insistence on the temporal and logical priority of quantitative thinking over “science and technology as such,” these background causes of quantification include also the compass, the mechanical clock, printing, calendrical reforms, Hindu-Arabic numerals, Copernican astronomy, and advances in mathematics. Their stories reach beyond the temporal span of the book, 1250–1600. So also do his illustrations of the “venerable model,” which, even according to his own evidence, was still very much alive in 1600. For the efficient cause—the “match” that ignited this conflagration of Western knowledge and power—Crosby invokes new forms of visualization. The striking of the match seems to have taken several centuries. We have no unmoved mover to light it. And it has a threefold aspect: music, painting, and bookkeeping. For music, the...

Pandulf of Capua's De calculatione: An Illustrated Abacus Treatise and Some Evidence for the Hindu-Arabic Numerals in Eleventh-Century South Italy

Le codex Vaticanus Ottobonianus latinus 1354 contient l'unique copie d'un bref traite de mathematiques sur le calcul. L'auteur de ce traite est Pandulfe de Capoue. Le traite renseigne sur une variete de chiffres hindou-arabes. Dans une premiere partie, l'A. decrit le manuscrit, montre dans une seconde partie qu'il est relation avec le Mont Cassin puisque Pandulfe est un moine italien du XI e siecle. Il s'agit ensuite de degager les sources, le style et le contexte historique de l'oeuvre. En appendice, l'A. propose une edition et une traduction du traite qui se veut un enseignement de calcul pour les novices

Thus spake al- [formula omitted]wārizmī: A translation of the text of Cambridge University Library Ms. Ii.vi.5

The rules for manipulation of the Hindu-Arabic numerals 1, 2, 3, … and 0, otherwise known as algorism, became widely used in the West through Latin translation of Arabic from about the 12th century. Our principal Latin manuscript is Cambridge University Library Ms. Ii.vi.5. This manuscript has been published by Vogel and Yushkevich, but we present the first English translation. We have added a short introduction.

Ideas

The ancient Egyptian numerals used as far back as 3400 B.C. had groupings by ten but no place values. The use of these symbols will help students understand our base-ten system and the efficiency of our place-value notation. The basic rules for writing ancient Egyptian numerals (Egyptians now use Hindu-Arabic numerals, as we do) are relatively simple.

Abacists versus Algorists

For several hundred years after the introduction into Europe, possibly the beginning of the 8th century or earlierl, of the Hindu-Arabic numera. system, there existed a controversy over its use for accounting purposesl The proponents for the new system were called Algorists and those who defended the older Roman numeral system were called Abacists. Both factions had staunch supporters and even as late as the beginning of the sixteenth century a book on mathematics2 contained the woodcut (Fig. 1) showing Pythagoras calculating with the aid of the abacus and Boethius using Hindu-Arabic numerals in the modern method.

New numerals for base-five arithmetic

It may be of interest to teachers to discover that much can be learned (and perhaps taught) about our base-ten number system by considering a system having a base other than “ten”. Some of us have been teaching a long time and perhaps have forgotten how difficult the concepts of mathematics arc for the young child. To illustrate this difficulty, let the teachers be exposed to a set of number symbols they have never seen before and then try to work with them. The usual presentation of other number bases employs Hindu-Arabic numerals. This results in much difficulty because one must “unlearn” most of the number facts previously known. Children are usually uncomfortable in the presence of a statement like “4+4 = 10”, a true statement in baseeight arithmetic. The authors of this paper have developed a system of numerals which makes the transition from base-ten to another base less precipitous and perhaps more graciously understood. These numerals have had strong intuitive appeal to children. A child seems to come out of the experience with a better understanding of a number system.

Hindu-Arabic Numerals

The Numerals We Use seem SO simple and practical that we take them for granted. We tend to belittle them because they are familiar. On the other hand, we are painfully aware of the difficulties children have in learning to use them. So it is appropriate to survey the development of Hindu-Arabic numerals, not because we will at once relay the information to our pupils, but because we do our best teaching when the subject the children are investigating is one that commands our respect and admiration.

Numerals and the sexuality of scientific investigation

1. A simple method is described for obtaining a formal design of graphic symbols from the current Hindu-Arabic numerals. 2. This design has genital implications to West Europeans. 3. The psychosexual climate of post-feudal Europe has been such that general unawareness of this formal design might be expected. 4. Some observations by Freud are recalled in the light of which this design is of special interest.

The Abacus in the Teaching of Arithmetic

FoR over a thousand years the abacus was an indispensable part of the equipment of arithmeticians. However, with the adoption of the Hindu-Arabic numerals by Western people, the abacus was discarded as a means of calculating. Its only place in the schools was that of a plaything. Recent arithmetic books, for example, DailyLife Arithmetics,' have again included the abacus in arithmetical instruction, using it this time to show a historical phase of arithmetic. This use of the abacus is excellent although there is doubt that full advantage of its historical possibilities is taken. For example, nearly every modern person has a feeling that the Romans were much handicapped because it is so difficult to do our type of computation with Roman numerals. Such conclusions are, of course, invalid because few Romans made calculations in that way; they used the abacus. A little work with an abacus will demonstrate that in addition and subtraction the Roman method is, in some ways, superior to our pencil-and-paper methods. Multiplication and division are, of course, not so easily performed on the abacus as with our pencil-andpaper methods. The abacus can be used, then, not merely to show something the Romans used, but also to show the advantages and the disadvantages that these people had in calculating. In a similar way the abacus can be used to show how the Chinese and the Japanese calculate. These uses of the abacus, however, are relatively minor as compared with its other uses in teaching arithmetic. One of the outstanding features of our modern system of arithmetic is its use of the idea of place value; that is, that the value of any number depends on its position. Another important phase of modern arithmetic is its emphasis on, and use of, the collection idea

Introduction of Hindu-Arabic Numerals into Western Europe

EVIDENCE is sometimes adduced to indicate that the Hindu-Arabic numerals, or closely allied forms of them, were known in western Europe before this knowledge could have passed through Muslim Spain. For example, Alcuin of York (735–804) is said to show at least partial knowledge of the numerals. If this is so, whence did the information come?

The Hindu-Arabic Numerals. By D. E. Smith and L. C. Karpinski. (Ginn & Co.)

The Hindu-Arabic Numerals. By D. E. Smith and L. C. Karpinski. (Ginn & Co.)

BOOK REVIEWS

Book reviewed in this article:The Theory and Practice of Technical Writing, by Samuel C. EarleThe Hindu‐Arabic Numerals, by David Eugene Smith and Louis C. Karpinski.Essentials of Biology, presented in problems, by George William Hunter, A. M.A History of the Ancient World, by George Willis BotsfordGuide to the Insects of Connecticut, prepared under the direction of Wilton Everett Britton, Ph. D.Five Hundred Regents' Questions in Biology and Zoölogy, by S. C. KimmLaboratory Manual of First Year Science, by Joseph L. Thalman and Ada L. WeckelPrinciples of Rural Economics, by Thomas N. CarverThe School of To‐morrowMicrobiology for Agricultural and Domestic Science Students, edited by Charles E. Marshall.A Laboratory Manual for the Solution of Problems in Biology, by Richard W. Sharpe