Lecture 19. Hindu-Arabic Numeral System
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Early Etruscan Inscriptions
ja trvw (λ^ J uu^aunA oWVtC OnXv ^^a^JV /yu* A^-vr^ EARLY ETRUSCAN INSCRIPTIONS FABRETTI 2343-2346 GEORGE HEMPL THIS PAPER IS REPRINTED FROM THE MATZKE MEMORIAL VOLUME PUBLISHED BY THE UNIVERSITY XTbe TUntpereitp prese STANFORD UNIVERSITY CALIFORNIA 1911 IN MEMORY OF JOHN ERNST MATZKE EARLY ETRUSCAN INSCRIPTIONS FABRETTI 2342-2346 How I, a Germanic scholar, came to be interested in Venetic and Etrus- can, I have told in my report on the results of my Italic studies. This report has been delayed, chiefly by the difficulties inherent in such an under- taking, but it will now be published in a very short time. The present paper is an abstract from it. A few weeks before his death, Professor Matzke urged me to hasten the publication of my report. He said that my silence was being misinterpreted, and that I owed it not only to myself but also to my friends to publish something at once—if only a fragment. I was touched by what he said and the way in which he said it. It was almost exactly what another friend, Otto Jespersen, had written me from Copenhagen not long before, and what still others, as if by concert, now began to urge upon me. I saw the force of their arguments and decided to drop everything else and complete my report. And now that Fate has sud- denly cut short the life of one of them, I can find no more appropriate tribute to lay on his grave than the fragment he so recently urged me to publish. -
Nondecimal Positional Systems
M03_LONG0847_05_AIE_C03.qxd 9/17/07 10:38 AM Page 162 Not for Sale 162 CHAPTER 3 Numeration and Computation 3.2 Nondecimal Positional Systems In the preceding section, we discussed several numeration systems, including the decimal system in common use today. As already mentioned, the decimal system is a positional system based on 10. This is probably so because we have ten fingers and historically, as now, people often counted on their fingers. In the Did You Know box at the end of this section, we will give applications of positional systems that are based on whole numbers other than 10. One very useful application of bases other than ten is as a way to deepen our understanding of number systems, arithmetic in general, and our own decimal system. Although bases other than ten were in the K–5 curriculum in the past, they are not present currently. However, nondecimal positional systems are studied in the “math methods” course that is a part of an elementary education degree and may return to elementary school in the future. We will discuss addition and subtraction in base five in the next section and multiplication in base six in the following one. Base Five Notation For our purposes, perhaps the quickest route to understanding base five notation is to reconsider the abacus of Figure 3.1. This time, however, we allow only 5 beads to be moved forward on each wire. As before, we start with the wire to the students’ right (our left) and move 1 bead forward each time as we count 1, 2, 3, and so on. -
13054-Duodecimal.Pdf
Universal Multiple-Octet Coded Character Set International Organization for Standardization Organisation Internationale de Normalisation Международная организация по стандартизации Doc Type: Working Group Document Title: Proposal to encode Duodecimal Digit Forms in the UCS Author: Karl Pentzlin Status: Individual Contribution Action: For consideration by JTC1/SC2/WG2 and UTC Date: 2013-03-30 1. Introduction The duodecimal system (also called dozenal) is a positional numbering system using 12 as its base, similar to the well-known decimal (base 10) and hexadecimal (base 16) systems. Thus, it needs 12 digits, instead of ten digits like the decimal system. It is used by teachers to explain the decimal system by comparing it to an alternative, by hobbyists (see e.g. fig. 1), and by propagators who claim it being superior to the decimal system (mostly because thirds can be expressed by a finite number of digits in a "duodecimal point" presentation). • Besides mathematical and hobbyist publications, the duodecimal system has appeared as subject in the press (see e.g. [Bellos 2012] in the English newspaper "The Guardian" from 2012-12-12, where the lack of types to represent these digits correctly is explicitly stated). Such examples emphasize the need of the encoding of the digit forms proposed here. While it is common practice to represent the extra six digits needed for the hexadecimal system by the uppercase Latin capital letters A,B.C,D,E,F, there is no such established convention regarding the duodecimal system. Some proponents use the Latin letters T and E as the first letters of the English names of "ten" and "eleven" (which obviously is directly perceivable only for English speakers). -
Vigesimal Numerals on Ifẹ̀ (Togo) and Ifẹ̀ (Nigeria) Dialects of Yorùbá
Vigesimal Numerals on Ifẹ̀ (Togo) and Ifẹ̀ (Nigeria) Dialects of Yorùbá Felix Abídèmí Fábùnmi (Ilé-Ifẹ̀) Abstract This study intends to bring Ifẹ̀ (Togo) into a linguistic limelight using the numeral systems. Numerals are a very important aspect of the day to day socio-economic and linguistic life of Ifẹ̀ (Togo) people. The traditional Ifẹ̀ (Togo) number system is vigesimal. In this study, forty- two different number words are listed for Yorùbá Ifẹ̀ (Nigeria) and Yorùbá Ifẹ̀ (Togo) and compared with Standard Yorùbá. We compared the Ifẹ̀ (Togo) number words and counting patterns with that of the Standard Yorùbá and Ifẹ̀ (Nigeria) and discovered that, by the nature of the components of these numbers, majority of the basic number words are either bisyllabic or trisyllabic, each syllable having the form VCV for the cardinals, and CVCV for the ordinals. There are irregularities in tonality; there are also alternations in the sequences of the vowel (oral and nasalized) and consonant sounds. This work finds out that Ifẹ̀ (Togo) has two counting patterns. In the first pattern, it uses addition solely to derive the number words but with a counting pattern where 'ten', 'twenty' and the added number units are taken as a whole. In the second counting pattern, subtraction is used to derive number words but this is applicable only to three numbers i. e. seventeen – /mɛ́ɛtadínóɡú/, eighteen – /méèʤìdínóɡu/ and nineteen – /mɔ̀kɔ̃dínoɡ́ u/. The Ifẹ̀ (Togo) dialect of Yorùbá mostly uses additive number positions. The dialect favours additive number positions more than the subtractive and the multiplicative positions. In other words, higher numbers are frequently used as bases for addition not as bases for multiplication in Ifẹ̀ (Togo). -
0. Introduction L2/12-386
Doc Type: Working Group Document Title: Revised Proposal to Encode Additional Old Italic Characters Source: UC Berkeley Script Encoding Initiative (Universal Scripts Project) Author: Christopher C. Little ([email protected]) Status: Liaison Contribution Action: For consideration by JTC1/SC2/WG2 and UTC Replaces: N4046 (L2/11-146R) Date: 2012-11-06 0. Introduction The existing Old Italic character repertoire includes 31 letters and 4 numerals. The Unicode Standard, following the recommendations in the proposal L2/00-140, states that Old Italic is to be used for the encoding of Etruscan, Faliscan, Oscan, Umbrian, North Picene, and South Picene. It also specifically states that Old Italic characters are inappropriate for encoding the languages of ancient Italy north of Etruria (Venetic, Raetic, Lepontic, and Gallic). It is true that the inscriptions of languages north of Etruria exhibit a number of common features, but those features are often exhibited by the other scripts of Italy. Only one of these northern languages, Raetic, requires the addition of any additional characters in order to be fully supported by the Old Italic block. Accordingly, following the addition of this one character, the Unicode Standard should be amended to recommend the encoding of Venetic, Raetic, Lepontic, and Gallic using Old Italic characters. In addition, one additional character is necessary to encode South Picene inscriptions. This proposal is divided into five parts: The first part (§1) identifies the two unencoded characters (Raetic Ɯ and South Picene Ũ) and demonstrates their use in inscriptions. The second part (§2) examines the use of each Old Italic character, as it appears in Etruscan, Faliscan, Oscan, Umbrian, South Picene, Venetic, Raetic, Lepontic, Gallic, and archaic Latin, to demonstrate the unifiability of the northern Italic languages' scripts with Old Italic. -
Texts in Transit in the Medieval Mediterranean
Texts in Transit in the medieval mediterranean Edited by Y. Tzvi Langermann and Robert G. Morrison Texts in Transit in the Medieval Mediterranean Texts in Transit in the Medieval Mediterranean Edited by Y. Tzvi Langermann and Robert G. Morrison Th e Pennsylvania State University Press University Park, Pennsylvania This book has been funded in part by the Minerva Center for the Humanities at Tel Aviv University and the Bowdoin College Faculty Development Committee. Library of Congress Cataloging- in- Publication Data Names: Langermann, Y. Tzvi, editor. | Morrison, Robert G., 1969– , editor. Title: Texts in transit in the medieval Mediterranean / edited by Y. Tzvi Langermann and Robert G. Morrison. Description: University Park, Pennsylvania : The Pennsylvania State University Press, [2016] | Includes bibliographical references and index. Summary: “A collection of essays using historical and philological approaches to study the transit of texts in the Mediterranean basin in the medieval period. Examines the nature of texts themselves and how they travel, and reveals the details behind the transit of texts across cultures, languages, and epochs”— Provided by Publisher. Identifiers: lccn 2016012461 | isbn 9780271071091 (cloth : alk. paper) Subjects: lcsh: Transmission of texts— Mediterranean Region— History— To 1500. | Manuscripts, Medieval— Mediterranean Region. | Civilization, Medieval. Classification: lcc z106.5.m43 t49 2016 | ddc 091 / .0937—dc23 lc record available at https:// lccn .loc .gov /2016012461 Copyright © 2016 The Pennsylvania State University All rights reserved Printed in the United States of America Published by The Pennsylvania State University Press, University Park, PA 16802–1003 The Pennsylvania State University Press is a member of the Association of American University Presses. -
Positional Notation Or Trigonometry [2, 13]
The Greatest Mathematical Discovery? David H. Bailey∗ Jonathan M. Borweiny April 24, 2011 1 Introduction Question: What mathematical discovery more than 1500 years ago: • Is one of the greatest, if not the greatest, single discovery in the field of mathematics? • Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? • Was fiercely resisted in Europe for hundreds of years after its discovery? • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, to- gether with the basic arithmetic computational schemes, which were discov- ered in India prior to 500 CE. ∗Bailey: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Email: [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. yCentre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. 1 2 Why? As the 19th century mathematician Pierre-Simon Laplace explained: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very sim- plicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appre- ciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. -
Zero Displacement Ternary Number System: the Most Economical Way of Representing Numbers
Revista de Ciências da Computação, Volume III, Ano III, 2008, nº3 Zero Displacement Ternary Number System: the most economical way of representing numbers Fernando Guilherme Silvano Lobo Pimentel , Bank of Portugal, Email: [email protected] Abstract This paper concerns the efficiency of number systems. Following the identification of the most economical conventional integer number system, from a solid criteria, an improvement to such system’s representation economy is proposed which combines the representation efficiency of positional number systems without 0 with the possibility of representing the number 0. A modification to base 3 without 0 makes it possible to obtain a new number system which, according to the identified optimization criteria, becomes the most economic among all integer ones. Key Words: Positional Number Systems, Efficiency, Zero Resumo Este artigo aborda a questão da eficiência de sistemas de números. Partindo da identificação da mais económica base inteira de números de acordo com um critério preestabelecido, propõe-se um melhoramento à economia de representação nessa mesma base através da combinação da eficiência de representação de sistemas de números posicionais sem o zero com a possibilidade de representar o número zero. Uma modificação à base 3 sem zero permite a obtenção de um novo sistema de números que, de acordo com o critério de optimização identificado, é o sistema de representação mais económico entre os sistemas de números inteiros. Palavras-Chave: Sistemas de Números Posicionais, Eficiência, Zero 1 Introduction Counting systems are an indispensable tool in Computing Science. For reasons that are both technological and user friendliness, the performance of information processing depends heavily on the adopted numbering system. -
AL-KINDI 'Philosopher of the Arabs' Abū Yūsuf Yaʿqūb Ibn Isḥāq Al
AL-KINDI ‘Philosopher of the Arabs’ – from the keyboard of Ghurayb Abū Yūsuf Ya ʿqūb ibn Is ḥāq Al-Kindī , an Arab aristocrat from the tribe of Kindah, was born in Basrah ca. 800 CE and passed away in Baghdad around 870 (or ca. 196–256 AH ). This remarkable polymath promoted the collection of ancient scientific knowledge and its translation into Arabic. Al-Kindi worked most of his life in the capital Baghdad, where he benefitted from the patronage of the powerful ʿAbb āssid caliphs al- Ma’mūn (rg. 813–833), al-Muʿta ṣim (rg. 833–842), and al-Wāthiq (rg. 842–847) who were keenly interested in harmonizing the legacy of Hellenic sciences with Islamic revelation. Caliph al-Ma’mūn had expanded the palace library into the major intellectual institution BAYT al-ḤIKMAH (‘Wisdom House ’) where Arabic translations from Pahlavi, Syriac, Greek and Sanskrit were made by teams of scholars. Al-Kindi worked among them, and he became the tutor of Prince A ḥmad, son of the caliph al-Mu ʿtasim to whom al-Kindi dedicated his famous work On First Philosophy . Al-Kindi was a pioneer in chemistry, physics, psycho–somatic therapeutics, geometry, optics, music theory, as well as philosophy of science. His significant mathematical writings greatly facilitated the diffusion of the Indian numerals into S.W. Asia & N. Africa (today called ‘Arabic numerals’ ). A distinctive feature of his work was the conscious application of mathematics and quantification, and his invention of specific laboratory apparatus to implement experiments. Al-Kindi invented a mathematical scale to quantify the strength of a drug; as well as a system linked to phases of the Moon permitting a doctor to determine in advance the most critical days of a patient’s illness; while he provided the first scientific diagnosis and treatment for epilepsy, and developed psycho-cognitive techniques to combat depression. -
Two Editions of Ibn Al-Haytham's Completion of the Conics
Historia Mathematica 29 (2002), 247–265 doi:10.1006/hmat.2002.2352 Two Editions of Ibn al-Haytham’s Completion of the Conics View metadata, citation and similar papers at core.ac.uk brought to you by CORE Jan P. Hogendijk provided by Elsevier - Publisher Connector Mathematics Department, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, Netherlands E-mail: [email protected] The lost Book VIII of the Conics of Apollonius of Perga (ca. 200 B.C.) was reconstructed by the Islamic mathematician Ibn al-Haytham (ca. A.D. 965–1041) in his Completion of the Conics. The Arabic text of this reconstruction with English translation and commentary was published as J. P. Hogendijk, Ibn al-Haytham’s Completion of the Conics (New York: Springer-Verlag, 1985). In a new Arabic edition with French translation and commentary (R. Rashed, Les mathematiques´ infinitesimales´ du IXe au XIe siecle.´ Vol. 3., London: Al-Furqan Foundation, 2000), it was claimed that my edition is faulty. In this paper the similarities and differences between the two editions, translations, and commentaries are discussed, with due consideration for readers who do not know Arabic. The facts will speak for themselves. C 2002 Elsevier Science (USA) C 2002 Elsevier Science (USA) C 2002 Elsevier Science (USA) 247 0315-0860/02 $35.00 C 2002 Elsevier Science (USA) All rights reserved. 248 JAN P. HOGENDIJK HMAT 29 AMS subject classifications: 01A20, 01A30. Key Words: Ibn al-Haytham; conic sections; multiple editions; Rashed. 1. INTRODUCTION The Conics of Apollonius of Perga (ca. 200 B.C.) is one of the fundamental texts of ancient Greek geometry. -
History of Mathematics in Mathematics Education. Recent Developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang
History of mathematics in mathematics education. Recent developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang To cite this version: Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang. History of mathematics in mathematics education. Recent developments. History and Pedagogy of Mathematics, Jul 2016, Montpellier, France. hal-01349230 HAL Id: hal-01349230 https://hal.archives-ouvertes.fr/hal-01349230 Submitted on 27 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HISTORY OF MATHEMATICS IN MATHEMATICS EDUCATION Recent developments Kathleen CLARK, Tinne Hoff KJELDSEN, Sebastian SCHORCHT, Constantinos TZANAKIS, Xiaoqin WANG School of Teacher Education, Florida State University, Tallahassee, FL 32306-4459, USA [email protected] Department of Mathematical Sciences, University of Copenhagen, Denmark [email protected] Justus Liebig University Giessen, Germany [email protected] Department of Education, University of Crete, Rethymnon 74100, Greece [email protected] Department of Mathematics, East China Normal University, China [email protected] ABSTRACT This is a survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the HPM domain). Section 1 explains the rationale of the study and formulates the key issues. -
The "Greatest European Mathematician of the Middle Ages"
Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci). His names Fibonacci Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci.