DOCSLIB.ORG

HPM 2000 Conference History in Mathematics Education Challenges for a New Millennium a Satellite Metting of ICME-9

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
HPM 2000 Conference History in Mathematics Education Challenges for a New Millennium a Satellite Metting of ICME-9 Proceedings of the HPM 2000 Conference History in Mathematics Education Challenges for a new millennium A Satellite Metting of ICME-9 Vol. I Edited by Wann-Sheng Horng & Fou-Lai Lin August 9-14, 2000, Taipei, Taiwan Department of Mathematics National Taiwan Normal University ORGANIZERS International Study Group on the Relations Between History and Pedagogy of Mathematics Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, ROC International Programme Committee Chair: Jan van Maanen, the chair of HPM, University of Groningen, The Netherlands Glen Van Brummelen, Bennington College, USA John Fauvel (the former chair of HPM), Open University, UK Gail FitzSimons, Monash University, Australia Lucia Grugnetti, University of Parma, Italy Wann-Sheng Horng, National Taiwan Normal University, Taiwan Victor Katz, University of the District of Columbia, USA Eleanor Robson, Oxford University, UK Chikara Sasaki, University of Tokyo, Japan Man-Keung Siu, University of Hong Kong, SAR, China Conference Venue and Secretariat Department of Mathematics, National Taiwan Normal University 88, Sec. 4, Ting Chou Road, Taipei, Taiwan, ROC Tel: 886-2-29312611 Fax: 866-2-29332342 Proceedings of the HPM 2000 Conference History in Mathematics Education： Challenges for a new millennium Editors： Wann-Sheng Horng Department of Mathematics National Taiwan Normal University Fou-Lai Lin Department of Mathematics National Taiwan Normal University Assistant editor： Jer-Nan Huang, Yi-Wen Su, Feng-Chu Chen, Wen-Pei Wang Tsang-Yi Lin, Chin-Yee Teo, Ching-Ju Chiou Cover design by Jovi Cheng & Jer-Nan Huang Published by Department of Mathematics, National Taiwan Normal University Printed by Sponsored by National Science Council, Taiwan National Youth Commission, Taiwan National Taiwan Normal University Chang Chau-Ting Memorial Foundation Chiu Chang Math. Press Yuan T. Lee Foundation Peng Wan-Ru Foundation Science Monthly Yuan-Liou Publishing Co., LTD. Copyrights of the Digital Edition: © 2016 left to the authors. Pagination has been adjusted for this edition. Table of Contents Vol. I I. Plenary Lecture 8/10 Two pots and one lid: 1 The first arithmetic textbooks in the Netherlands before 1600 Marjolein Kool 8/11 Introduction of Western Sciences in China, Japan and Korea: 11 A Comparative Comment Seong-Rae Park 8/12 Becoming a mathematician in East and West: 18 some cross-cultural considerations. Christopher Cullen 8/13 The Interplay between Algorithm and Proof in Ancient China Karine Chemla (The text is not published) 8/14 The Use of Technology in Teaching Mathematics with History 27 - Teaching with modern technology inspired by the history of mathematics - Masami ISODA II. Symposia S1：History of Asian and Pacific mathematics S1-01 Theoretical Research Tendency of Mathematics in Pre-Qin Times 35 Viewed from Suan Shushu Shuchun Guo S1-02 An investigation on Chang Qiujian Suanjing 41 Yan-Chyuan Lin i S1-03 The Notion of Volume in the Jiu Zhang Suan Shu, 43 and Japanese Mathematics Shigeru Jochi S2：Mathematics education before S2-01 A History of Calculus Education in Japan 55 Osamu Kota S2-02 The Brief Sketch of Mathematics Education for 64 Immediately Follow the World War II in Japan Naomichi Makinae S2-03 Mathematics Textbooks and Terminology: 72 The Impact of Calvin Winson Mateer's Work on the Transformation of Traditional Chinese Mathematics Education into a Modern One Yibao Xu S2-04 Teaching Fortification as part of Practical Geometry : 75 A Jesuit case Frédéric Metin S3：The effectiveness of history in teaching mathematics: empirical studies S3-03 An Episode in the Development of Mathematics Teachers' 83 Knowledge? The Case of Quadratic Equations Greisy Winicki-Landman S3-04 Teachers’ Teaching Beliefs and Their 89 Knowledge about the History of Negative Numbers Feng-Jui Hsieh S3-05 99 What Are Teachers’ Views of Mathematics ii An Investigation of How They Evaluate Formulas in Mathematics Chia-Jui Hsieh, Feng-Jui Hsieh S3-07 Building teaching, learning and curriculum of calculus following 113 the path of its historical development as well as along lines based on introspective reports of homogeneity in pattern of processes of invention or reinvention Ada Sherer S3-08 The Royal Observatory in Greenwich; ethnomathematics 133 in teacher training Wendy S. Troy S3-10 History of mathematics: why, and is that enough? 141 Andy Begg S3-11 Lifelong Learning and the History of Mathematics: 148 An Australian Perspective Gail E. FitzSimons S3-12 The Quarrel between Descartes and Fermat to 156 introduce the notion of tangent in high school Michèle Grégoire S3-13 Using Mathematics History and PCDC Instruction Model to 163 Activate Underachievement Students’ Mathematics Learning Ya-Hui Hsiao, Ching-Kuch Chang S3-14 On the effectiveness of history in teaching 173 probability and statistics Ewa Lakoma S3-15 The Use of History of Mathematics To Increase 180 Students’ Understanding of the Nature of Mathematics Po-Hung Liu S3-16 Epistemological complexity of multiplication and division 191 iii from the view of dimensional analysis Yutaka OHARA S3-17 Method of Indivisibles in Calculus Instruction 198 Vrunda Prabhu, Bronislaw Czarnocha S3-18 Using History to Enliven the Study of Infinite Series 206 Bob Stein S3-20 217 Using mathematical text in classroom： the case of probability teaching Yi-Wen Su Vol. II II. Symposia S3：The effectiveness of history in teaching mathematics: empirical studies S3-23 The Practical Use of the History of Mathematics 1 and Its Usefulness in Teaching and Learning Mathematics at High School: The development of materials in teaching calculus (not in these Proceedings) Kumilo Tsukahara S3-26 The Development of Mathematics in Sub-Saharan Africa: 17 Challenges for the new millennium Luckson Muganyizi Kaino S3-27 The Introduction of History of Mathematics 27 in Norwegian schools Bjørn Smestad S3-28 Justification in mathematics and Procedures on Which 31 It Is based: A historical approach for didactical purposes Constantinos Tzanakis, Michael Kourkoulos iv S3-30 Several Characteristics of 52 Taiwan's high school mathematics curriculum Jya-Yi Wu Yu S3-31 F&B Mathematics Teaching in Vocational School: 61 A Team Work with HPM perspective Yu-Yi Lin S3-32 Using Mathematics Text in Classroom: 67 the case of Pythagorean Theorem Yen Fu Ming S3-33 Theories of ratio and theoretical music: an education approach 83 Oscar João Abdounur S3-35 The antiphairesis: a site of an educational dialogue 94 among Mathematics, History and Philosophy Ernesto Rottoli, Gianstefano Riva S3-36 103 ~~： A teaching report of “ Using ancient mathematical text in classroom”: Pascal’s triangle Hui-Yu Su S4：West and East, Contrast and transfer of mathematical S4-02 Multiculturalism in history: voices in 19th century 115 mathematics education east and west John Fauvel S4-04 On Computing the Volume of Sphere in the East and West: 121 A comparative study from the educational perspectives Yiu-Kwong Man, Yin-Kue Lo v S6：History of science and science education (especially for local participants) S6 From Atayal World to Science World: 130 A Pilot Study on a Science Class for Atayal Junior High Students Li-Yu Fu III. Workshop Antipodean Fibonacci Originals 142 Coralie Daniel IV. Round-Table RT-01 Why study values in mathematics teaching: 149 contextualising the VAMP project? A. J. Bishop, P. Clarkson, G. FitzSimons, W.T. Seah RT-01 Methodology Challenges and Constraints in the VAMP Project 157 Philip Clarkson, Alan Bishop, Gail FitzSimons, Wee Tiong Seah RT-01 Conceptions of Values and Mathematics Education held by 165 Australian Primary Teachers: Preliminary Findings from VAMP Gail E. FitzSimons, Wee Tiong Seah, Alan J. Bishop, Philip C. Clarkson RT-01 Score-ism As Their Pedagogical Value of Two Junior 174 High Mathematics Teachers Ching-Kuch Chang RT-01 An Elementary Teacher’s Pedagogical Values in Mathematics 180 Teaching: Clarification and Change Yuh-Chyn Leu, Chao-Jung Wu V. Panel P1-01 198 Ptolemy’s theorem and chord table vi Cheng-Mei Hsu P1-03 199 The Use of Dynamic Geometry Software and History of Mathematics in Teaching Conic Sections Jer-Nan Huang, Tai-Yih Tso P2-02 — 211 Ancient Mathematical Texts used in the Classroom—Quadratic Equations Hui-I Lauo P2-03 Using History of Mathematics in Teaching Radical Root Numbers 223 Ching-Ju Chiou vii viii Two pots and one lid： The first arithmetic textbooks in the Netherlands before 1600 Marjolein Kool Hogeschool Domstad, Utrecht A historical arithmetic problem Somebody has two silver pots and one silver lid. The price of the lid is 16 guilders. If he places the lid on the first pot, this combination will cost 4 times more than the second pot. If he places the lid on the second pot, this combination will cost three times more than the first pot. What will be the value of each pot? If you have tried to solve this problem, you probably wrote down two equations: x + 16 = 4y and y + 16 = 3x and had got the answer quite soon. I found this problem in a Dutch arithmetic manuscript of 1568. The author didn't use equations, letters and arithmetical symbols to solve this problem. These mathematical aspects, we are so familiar with, were hardly known in the sixteenth century Netherlands. Sixteenth century arithmeticians could solve problems like this one, just by using words, numbers and a few lines, as you can see in figure 1. They used the so called 'regula falsi' or 'rule of false position'. This rule is used to find the required unknown number with the help of two arbitrarily chosen numbers. Let me explain this. The value of the first pot with the lid is 4 times the value of the second pot. Imagine the first pot costs 4 guilders, than the second pot will cost 5 guil- ders. The second pot with the lid will cost three times the first pot. Three times 4 is 12, but here we have 21, so we have 9 guilders too much.
Load more

Recommended publications
  • Differential Calculus and by Era Integral Calculus, Which Are Related by in Early Cultures in Classical Antiquity the Fundamental Theorem of Calculus
    History of calculus - Wikipedia, the free encyclopedia 1/1/10 5:02 PM History of calculus From Wikipedia, the free encyclopedia History of science This is a sub-article to Calculus and History of mathematics. History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known Background historically as infinitesimal calculus, Theories/sociology constitutes a major part of modern Historiography mathematics education. It has two major Pseudoscience branches, differential calculus and By era integral calculus, which are related by In early cultures in Classical Antiquity the fundamental theorem of calculus. In the Middle Ages Calculus is the study of change, in the In the Renaissance same way that geometry is the study of Scientific Revolution shape and algebra is the study of By topic operations and their application to Natural sciences solving equations. A course in calculus Astronomy is a gateway to other, more advanced Biology courses in mathematics devoted to the Botany study of functions and limits, broadly Chemistry Ecology called mathematical analysis. Calculus Geography has widespread applications in science, Geology economics, and engineering and can Paleontology solve many problems for which algebra Physics alone is insufficient. Mathematics Algebra Calculus Combinatorics Contents Geometry Logic Statistics 1 Development of calculus Trigonometry 1.1 Integral calculus Social sciences 1.2 Differential calculus Anthropology 1.3 Mathematical analysis
    [Show full text]
  • Notices of the American Mathematical Society
    ISSN 0002-9920 of the American Mathematical Society February 2006 Volume 53, Number 2 Math Circles and Olympiads MSRI Asks: Is the U.S. Coming of Age? page 200 A System of Axioms of Set Theory for the Rationalists page 206 Durham Meeting page 299 San Francisco Meeting page 302 ICM Madrid 2006 (see page 213) > To mak• an antmat•d tub• plot Animated Tube Plot 1 Type an expression in one or :;)~~~G~~~t;~~i~~~~~~~~~~~~~:rtwo ' 2 Wrth the insertion point in the 3 Open the Plot Properties dialog the same variables Tl'le next animation shows • knot Plot 30 Animated + Tube Scientific Word ... version 5.5 Scientific Word"' offers the same features as Scientific WorkPlace, without the computer algebra system. Editors INTERNATIONAL Morris Weisfeld Editor-in-Chief Enrico Arbarello MATHEMATICS Joseph Bernstein Enrico Bombieri Richard E. Borcherds Alexei Borodin RESEARCH PAPERS Jean Bourgain Marc Burger James W. Cogdell http://www.hindawi.com/journals/imrp/ Tobias Colding Corrado De Concini IMRP provides very fast publication of lengthy research articles of high current interest in Percy Deift all areas of mathematics. All articles are fully refereed and are judged by their contribution Robbert Dijkgraaf to the advancement of the state of the science of mathematics. Issues are published as S. K. Donaldson frequently as necessary. Each issue will contain only one article. IMRP is expected to publish 400± pages in 2006. Yakov Eliashberg Edward Frenkel Articles of at least 50 pages are welcome and all articles are refereed and judged for Emmanuel Hebey correctness, interest, originality, depth, and applicability. Submissions are made by e-mail to Dennis Hejhal [email protected].
    [Show full text]
  • Occasion of Receiving the Seki-Takakazu Prize
    特集：日本数学会関孝和賞受賞 On the occasion of receiving the Seki-Takakazu Prize Jean-Pierre Bourguignon, the director of IHÉS A brief introduction to the Institut des Hautes Études Scientifiques The Institut des Hautes Études Scientifiques (IHÉS) was founded in 1958 by Léon MOTCHANE, an industrialist with a passion for mathematics, whose ambition was to create a research centre in Europe, counterpart to the renowned Institute for Advanced Study (IAS), Princeton, United States. IHÉS became a foundation acknowledged in the public interest in 1981. Like its model, IHÉS has a small number of Permanent Professors (5 presently), and hosts every year some 250 visitors coming from all around the world. A Scientific Council consisting of the Director, the Permanent Professors, the Léon Motchane professor and an equal number of external members is in charge of defining the scientific strategy of the Institute. The foundation is managed by an 18 member international Board of Directors selected for their expertise in science or in management. The French Minister of Research or their representative and the General Director of CNRS are members of the Board. IHÉS accounts are audited and certified by an international accountancy firm, Deloitte, Touche & Tomatsu. Its resources come from many different sources: half of its budget is provided by a contract with the French government, but institutions from some 10 countries, companies, foundations provide the other half, together with the income from the endowment of the Institute. Some 50 years after its creation, the high quality of its Permanent Professors and of its selected visiting researchers has established IHÉS as a research institute of world stature.
    [Show full text]
  • Transnational Mathematics and Movements: Shiing- Shen Chern, Hua Luogeng, and the Princeton Institute for Advanced Study from World War II to the Cold War1
    Chinese Annals of History of Science and Technology 3 (2), 118–165 (2019) doi: 10.3724/SP.J.1461.2019.02118 Transnational Mathematics and Movements: Shiing- shen Chern, Hua Luogeng, and the Princeton Institute for Advanced Study from World War II to the Cold War1 Zuoyue Wang 王作跃,2 Guo Jinhai 郭金海3 (California State Polytechnic University, Pomona 91768, US; Institute for the History of Natural Sciences, Chinese Academy of Sciences, Beijing 100190, China) Abstract: This paper reconstructs, based on American and Chinese primary sources, the visits of Chinese mathematicians Shiing-shen Chern 陈省身 (Chen Xingshen) and Hua Luogeng 华罗庚 (Loo-Keng Hua)4 to the Institute for Advanced Study in Princeton in the United States in the 1940s, especially their interactions with Oswald Veblen and Hermann Weyl, two leading mathematicians at the IAS. It argues that Chern’s and Hua’s motivations and choices in regard to their transnational movements between China and the US were more nuanced and multifaceted than what is presented in existing accounts, and that socio-political factors combined with professional-personal ones to shape their decisions. The paper further uses their experiences to demonstrate the importance of transnational scientific interactions for the development of science in China, the US, and elsewhere in the twentieth century. Keywords: Shiing-shen Chern, Chen Xingshen, Hua Luogeng, Loo-Keng Hua, Institute for 1 This article was copy-edited by Charlie Zaharoff. 2 Research interests: History of science and technology in the United States, China, and transnational contexts in the twentieth century. He is currently writing a book on the history of American-educated Chinese scientists and China-US scientific relations.
    [Show full text]
  • Proquest Dissertations
    University of Alberta Qin Jiushao and His Mathematical Treatise in Nine Sections in Thirteenth-Century China by Ke-Xin Au Yong A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Arts in History History and Classics ©Ke-Xin Au Yong Fall 2011 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de ('edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-81281-5 Our file Notre reference ISBN: 978-0-494-81281-5 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats.
    [Show full text]
  • Shaw' Preemia
    Shaw’ preemia 2002. a rajas Hongkongi ajakirjandusmagnaat ja tuntud filantroop sir Run Run Shaw (s 1907) omanimelise fondi, et anda v¨alja iga-aastast preemiat – Shaw’ preemiat. Preemiaga autasustatakse ”isikut, s˜oltumata rassist, rahvusest ja religioossest taustast, kes on saavutanud olulise l¨abimurde akadeemilises ja teaduslikus uuri- mist¨o¨os v˜oi rakendustes ning kelle t¨o¨o tulemuseks on positiivne ja sugav¨ m˜oju inimkonnale.” Preemiat antakse v¨alja kolmel alal – astronoomias, loodusteadustes ja meditsiinis ning matemaatikas. Preemia suurus 2009. a oli uks¨ miljon USA dollarit. Preemiaga kaas- neb ka medal (vt joonist). Esimesed preemiad omistati 2004. a. Ajakirjanikud on ristinud Shaw’ preemia Ida Nobeli preemiaks (the Nobel of the East). Shaw’ preemiaid aastail 2009–2011 matemaatika alal m¨a¨arab komisjon, kuhu kuuluvad: esimees: Sir Michael Atiyah (Edinburghi Ulikool,¨ UK) liikmed: David Kazhdan (Jeruusalemma Heebrea Ulikool,¨ Iisrael) Peter C. Sarnak (Princetoni Ulikool,¨ USA) Yum-Tong Siu (Harvardi Ulikool,¨ USA) Margaret H. Wright (New Yorgi Ulikool,¨ USA) 2009. a Shaw’ preemia matemaatika alal kuulutati v¨alja 16. juunil Hongkongis ja see anti v˜ordses osas kahele matemaatikule: 293 Eesti Matemaatika Selts Aastaraamat 2009 Autori~oigusEMS, 2010 294 Shaw’ preemia Simon K. Donaldsonile ja Clifford H. Taubesile s¨arava panuse eest kolme- ja neljam˜o˜otmeliste muutkondade geomeetria arengusse. Premeerimistseremoonia toimus 7. oktoobril 2009. Sellel osales ka sir Run Run Shaw. Siin pildil on sir Run Run Shaw 100-aastane. Simon K. Donaldson sundis¨ 1957. a Cambridge’is (UK) ja on Londonis Imperial College’i Puhta Matemaatika Instituudi direktor ja professor. Bakalaureusekraadi sai ta 1979. a Pembroke’i Kolled- ˇzistCambridge’s ja doktorikraadi 1983.
    [Show full text]
  • Historical Development of the Chinese Remainder Theorem
    Historical Development of the Chinese Remainder Theorem SHEN KANGSHENG Communicated by C. TRUESDELL 1. Source of the Problem Congruences of first degree were necessary to calculate calendars in ancient China as early as the 2 na century B.C. Subsequently, in making the Jingchu [a] calendar (237,A.D.), the astronomers defined shangyuan [b] 1 as the starting point of the calendar. If the Winter Solstice of a certain year occurred rl days after shangyuan and r2 days after the new moon, then that year was N years after shangyuan; hence arose the system of congruences aN ~ rl (mod 60) ~ r2 (mod b), . ~ ' where a is the number of days in a tropical year and b the number of days in a lunar month. 2. Sun Zi suanjing [c] (Master Sun's Mathematical Manual) Sun Zi suanjing (Problem 26' Volume 3) reads: "There are certain things whose number is unknown. A number is repeatedly divided by 3, the remainder is 2; divided by 5, the remainder is 3; and by 7, the remainder is 2. What will the num- ber be ?" The problem can be expressed as x --= 2 (mod 3) ~ 3 (mod 5) ~- 2 (rood 7). SUN ZI solved the problem as we do, giving x ~ 140 + 63 -k 30 ~=- 233 ~ 23 (rood 105). 1 Shangyuan is a supposed moment that occurred simultaneously with the midnight of jiazi [v] (the first day of the 60 day cycle), the Winter Solstice and the new moon. 286 SHEN KANGSHENG In speaking of the algorithm yielding these addends in the solution he continued: "In general, for G1 ~ 0(mod 5) ~ 0 (mod 7) = 1 (rood 3), take G1 = 70, G2 ~ 0(rood 3) ~ 0(mod 7) ~ 1 (mod 5), take G2 = 21, G3 ~ 0 (mod 3) ~ 0 (mod 5) ~ (1 rood 7), take G3 = 15.
    [Show full text]
  • ON MATHEMATICAL TERMINOLOGY: CULTURE CROSSING in NINETEENTH-CENTURY CHINA the History of Mathematical Terminologies in Nineteent
    ANDREA BRÉARD ON MATHEMATICAL TERMINOLOGY: CULTURE CROSSING IN NINETEENTH-CENTURY CHINA INTRODUCTORY REMARKS: ON MATHEMATICAL SYMBOLISM The history of mathematical terminologies in nineteenth-century China is a multi-layered issue. Their days of popularity and eventual decline are bound up with the complexities of the disputes between the Qian-Jia School who were concerned with textual criticism and the revival of ancient native Chinese mathematical texts, and those scholars that were interested in mathematical studies per se1 (and which we will discuss in a more detailed study elsewhere). The purpose of the following article is to portray the ‘introduction’ of Western mathematical notations into the Chinese consciousness of the late nineteenth century as a major signifying event, both (i) in its own right within the context of the translation of mathematical writ- ings from the West; and (ii) with respect to the way in which the tradi- tional mathematical symbolic system was interpreted by Qing commentators familiar with Western symbolical algebra. Although one might assume to find parallel movements in the commentatorial and translatory practices of one and the same person, nineteenth-cen- tury mathematical activities will be shown to be clearly compartmen- talized. In fact, for the period under consideration here, it would be more appropriate to speak of the ‘reintroduction’ of Western mathematical notations. Already in 1712 the French Jesuit Jean-François Foucquet S. J. (1665–1741) had tried to introduce algebraic symbols in his Aer- rebala xinfa ̅ (New method of algebra). This treatise, which was written for the Kangxi emperor, introduced symbolical 1 In particular, the ‘three friends discussing astronomy and mathematics’ (Tan tian san you dž ) Wang Lai (1768–1813), Li Rui (1763–1820) and Jiao Xun nj (1765–1814).
    [Show full text]
  • Elementary Algebra Aei from Wikipedia, the Free Encyclopedia Contents
    Elementary algebra aei From Wikipedia, the free encyclopedia Contents 1 Additive identity 1 1.1 Elementary examples ......................................... 1 1.2 Formal definition ........................................... 1 1.3 Further examples ........................................... 1 1.4 Proofs ................................................. 2 1.4.1 The additive identity is unique in a group ........................... 2 1.4.2 The additive identity annihilates ring elements ........................ 2 1.4.3 The additive and multiplicative identities are different in a non-trivial ring .......... 2 1.5 See also ................................................ 2 1.6 References ............................................... 2 1.7 External links ............................................. 3 2 Additive inverse 4 2.1 Common examples .......................................... 4 2.1.1 Relation to subtraction .................................... 4 2.1.2 Other properties ........................................ 4 2.2 Formal definition ........................................... 5 2.3 Other examples ............................................ 5 2.4 Non-examples ............................................. 6 2.5 See also ................................................ 6 2.6 Footnotes ............................................... 6 2.7 References ............................................... 6 3 Algebraic expression 7 3.1 Terminology .............................................. 7 3.2 In roots of polynomials .......................................
    [Show full text]
  • From China to Paris: 2000 Years Transmission of Mathematical Idea S
    FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEA S EDITED BY YVONNE DOLD-SAMPLONIUS JOSEPH W. DAUBEN MENSO FOLKERTS BENNO VAN DALEN FRANZ STEINER VERLAG STUTTGART FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEAS BOETHIUS TEXTE UND ABHANDLUNGEN ZUR GESCHICHTE DER MATHEMATIK UND DER NATURWISSENSCHAFTEN BEGRIJNDET VON JOSEPH EHRENFRIED HOFMANN FRIEDRICH KLEMM UND BERNHARD STICKER HERAUSGEGEBEN VON MENSO FOLKERTS BAND 46 FRANZ STEINER VERLAG STUTTGART 2002 FROM CHINA TO PARIS: 2000 YEARS TRANSMISSION OF MATHEMATICAL IDEAS EDITED BY YVONNE DOLD-SAMPLONIUS JOSEPH W. DAUBEN MENSO FOLKERTS BENNO VAN DALEN FRANZ STEINER VERLAG STUTTGART 2002 Bibliographische Information der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet these Publikation in der Deutschen National- bibliographic; detaillierte bibliographische Daten sind im Internet Uber <http:// dnb.ddb.de> abrufbar. ISBN 3-515-08223-9 ISO 9706 Jede Verwertung des Werkes auBerhalb der Grenzen des Urheberrechtsgesetzes ist unzulhssig and strafbar. Dies gilt insbesondere fair Ubersetzung, Nachdruck. Mikrover- filmung oder vergleichbare Verfahren sowie fi rdie Speicherung in Datenverarbeitungs- anlagen. ® 2002 by Franz Steiner Verlag Wiesbaden GmbH. Sitz Stuttgart. Gedruckt auf s3urefreiem. alterungsbesthndigem Papier. Druck: Druckerei Proff. Eurasburg. Printed in Germany Table of Contents VII Kurt Vogel: A Surveying Problem Travels from China to Paris .................... 1 Jens Hoyrup: Seleucid Innovations in the Babylonian "Algebraic" Tradition and their
    [Show full text]
  • CHINESE, INDIAN, and ARABIC MATHEMATICS 1. Chinese
    CHINESE, INDIAN, AND ARABIC MATHEMATICS FRANZ LEMMERMEYER 1. Chinese Mathematics One of the earliest mathematicians of China was Liu Hui (ca. 260 AD). His tools were similar to that of Greek mathematicians, and he even proved theorems; in particular he used similar triangles to solve problems in surveying (e.g. finding the distance between two islands): this is reminiscent of Thales’ use of similar triangles to measure the height of pyramids, or to the method Eratosthenes used for determining the circumference of the earth. He estimated π by approximating the circle with regular n-gons for n = 92 and 184 and knew the principle of exhaustion for circles; he also worked on the volume of the sphere (apparently, the works of Archimedes were not known in China). Zu Chongzhi (429-500) computed π to seven digits. Around 600 AD, Indian works on mathematics were translated into Chinese. Wang Xiaotong (ca. 625) showed how to compute roots of cubic equations nu- merically. Qin Jiushao (1202 – 1261) treated linear systems of congruences (Chinese remainder theorem) and discussed the Euclidean algorithm for computing greatest common divisors. Chinese mathematics had a ‘silver’ period from 300–700, and a ‘golden’ period during the 13th century (the Chinese version of Pascal’s triangle is from this period). Western mathematics was introduced in the 17th century. 2. Indian Mathematics Aryabatha (476–550) introduced and tabulated the sine function, and worked out the solution of linear diophantine equations like ax + by = c, where a, b, c are integers. Although Aryabatha used letter to denote numbers (like the Greeks), he might also have known the decimal system.
    [Show full text]
  • Muramatsu's Method Aryabhata (499 AD) Brahmagupta (640 AD
    π in Other Eastern Countries Japan India In the Edo Period of Japan (1603 to 1868), 3.16 was used for π. But Aryabhata (499 AD) as people recognized that this value was not accurate, different values for π were calculated. Wasan scholars such as Muramatsu Shigekiyo, Seki Takakazu, Kamata Toshikiyo, Takebe Katahiro, and Matsunaga Aryabhata usedp the perimeter of a 384-sided poly- Yoshisuke all made calculations of π, and accomplished results gon to find π ≈ 9:8684. Later Aryabhata would comparable to their European mathematician counterparts. publish an “approximation” for his square root value, 3.1416. Note that his approximation is Muramatsu’s Method more accurate than his official value! Shigekiyo Muramatsu published Sanso in 1663. Muramatsu served the Asano family and possibly had a math insti- Brahmagupta (640 AD) tute in Edo (present day Tokyo). In his Brahmagupta used a pattern of inscribed polygons that work, Sanso, Muramatsu arranged prob- increased in the number of sides to calculate the perime- lems published earlier in Japanese math- ter of a circle.p Using these data points, he came to believe ematics texts without answers. Mura- that π = 10: matsu classified these problems into dif- ferent levels according to how difficult he thought they’d be to learn. Asano Crest It is in this book that he also showed the calculation of π from the regular inscribed polygon of 32,768 sides. He correctly obtained the Ramanujan (1887-1920) value 3.1415926. Thus, this book was the first to contain a mathematical calculation of π in Japan. 1 Srinivasa Ramanujan found several formulas for π.
    [Show full text]