Reinvention of Early Algebra

Total Page:16

File Type:pdf, Size:1020Kb

Reinvention of Early Algebra REINVENTION OF EARLY ALGEBRA Developmental research on the transition from arithmetic to algebra The illustration on the cover is a mathematical riddle from a manuscript by Paolo Dagomari (ca.1281-1370). Source: Tropfke (1980), p. 577 CIP-data Koninklijke Bibliotheek, Den Haag Amerom, B.A. van Reinvention of early algebra. Developmental research on the transition from arith- metic to algebra. Utrecht: CD-ß Press, Center for Science and Mathematics Educa- tion. - (CD-ß wetenschappelijke bibliotheek ; 41) Thesis Utrecht University. - With ref. - With a summary in Dutch. ISBN 90-73346-48-7 Subject headings: mathematics education, early algebra, history of mathematics Cover: Zuidam Uithof Utrecht Press: Wilco, Amersfoort Copyright: Freudenthal Instituut, Utrecht 2002 REINVENTION OF EARLY ALGEBRA Developmental research on the transition from arithmetic to algebra Heruitvinden van aanvankelijke algebra Ontwikkelingsonderzoek rond de overgang van rekenen naar algebra (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof.dr. W.H. Gispen, ingevolge het besluit van het College van Promoties in het openbaar te verdedigen op donderdag 16 mei 2002 des middags te 16.15 uur door Barbara Ann van Amerom geboren op 13 maart 1969 te Dordrecht Promotoren: Prof. dr. J. de Lange Faculteit Wiskunde en Informatica, Universiteit Utrecht Prof. dr. K.G.E. Gravemeijer Faculteit Sociale Wetenschappen, Universiteit Utrecht Faculteit Wiskunde en Informatica, Universiteit Utrecht Co-promotor: Dr. J.A. van Maanen Faculteit Wiskunde en Natuurwetenschappen, Rijksuniversiteit Groningen Preface To be honest, my appointment to the project ‘Reinvention of algebra’ was a rather unexpected turn of events. I never had the ambition to pursue an academic career. During the final month of my teacher training I still had every intention to become a secondary school mathematics teacher. However, when I was given the opportuni- ty to combine my two main interests in mathematics – history and didactics – the choice was easily made. Looking back, there have been no regrets. Soon after the project started I took a part-time teaching job on the side, and just as I had hoped, the combination of theory and practice has been inspiring and fulfilling. At this time I wish to commemorate Leen Streefland, who helped me get started on the project and who supervised my early work. It has been an honor and a privilege to have known him and to have worked with him. He made me feel competent as if I were his equal and not a rookie, but even now there is so much more that I could have learned from him. Of course I could have never completed this book without the help of others. First of all I want to thank my supervisors Jan van Maanen, Jan de Lange and Koeno Gravemeijer, for their honest and constructive criticism, advice and support. I am es- pecially grateful to them for letting me decide my own course, and on occasions that I seemed to go astray they helped me just in time to find my way back. A special thanks goes to Jan van Maanen, who has guided me since my university years and who brought this project to my attention. He has been more than a supervisor to me, he has also been my friend. It was never too much trouble for him to travel from Groningen early in the morning in order to work together on my project the whole day. Even in rough times he continued to believe in me, sometimes more than I did myself, reminding me not to be too hard on myself. I also extend my thanks to my colleagues at the Freudenthal Institute, for their ex- pertise, their interest and support – in meetings, during coffee and lunch breaks or while waiting for the xerox machine to finish. In particular I want to thank Hanneke Beemer for designing and testing the student unit The Fancy Fair, and Mieke Abels for testing it again with her students and giving suggestions for improvement after- wards. I am very grateful to Marjolein Kool en Julie Menne for their advice and en- couragements when the going got rough. Moreover, Marjolein contributed some re- ally good ideas and historical materials for the student activities. Meeting the dead- line for this book would not have been possible without the assistence of Betty Heijman, Nathalie Kuipers, Ank van der Heiden-Bergsteijn, Marianne Moonen- Harmsen, Anneleen Post and Ellen Hanepen. v Uiteraard wil ik ook de docenten bedanken die hebben deelgenomen aan de verschil- lende onderwijsexperimenten. Dit zijn de leerkrachten van groep 7 en/of 8 van RK Basisschool De Kameleon te Utrecht, Koningin Julianaschool te Woudenberg, RK Basisschool ‘t Zonnewiel te Almere, Basisschool Rijnsweerd te Utrecht, CBS Het Anker te Brielle, OBS Lorentz te Hilversum en Basisschool Klaverblad te Amster- dam, en de brugklasdocenten verbonden aan RK Scholengemeenschap College De Klop te Utrecht, Almere College te Dronten en Mendelcollege te Haarlem. In het bij- zonder gaat mijn waardering uit naar Jaap Griffioen voor zijn op- en aanmerkingen op het lesmateriaal. Verder wil ik Monique Schipper en Natasja Capelle vermelden vanwege hun inzet bij de lesobservaties en -verslagen. Niet in de laatste plaats heb ik veel steun gehad van vrienden en familie. Mede dank zij hen is het me gelukt enkele tegenvallers te incasseren en sterker uit de strijd te komen. Met dit boek sluit ik een drukke, spannende en ook zeer dankbare periode in mijn leven af. De afgelopen maanden heb ik naar een dubbele bevalling toegeleefd – van een kind en van een boek – en nu is het tijd om daarvan te gaan genieten. vi Table of contents 1 Introduction 1.1 Reasons for this study 1 1.2 Aims of the study 1 1.3 Duration and partition of the study 2 1.4 Reading guide for this book 2 2 Learning and teaching of school algebra 2.1 Introduction 3 2.2 Traditional school algebra 3 2.3 Approaches to algebra 4 2.4 Typical learning difficulties 7 2.5 Symbolizing 10 2.5.1 Symbolizing and schematizing 11 2.5.2 Symbolizing equations 12 2.5.3 Models for equation solving 14 2.6 Arithmetic and algebra 17 2.6.1 A dual relationship 18 2.6.2 Accesses to algebra in the Dutch arithmetic curriculum 24 2.7 Pre-algebra: on the way to algebra 26 2.8 Algebra from a geometrical perspective 29 2.9 Early algebra in the Dutch curriculum 31 2.10 Conclusion 32 3 History in mathematics education 3.1 Introduction 35 3.2 Arguments for using history 35 3.3 History of algebra: a summary 37 3.4 Implementing historical elements of algebra 44 3.4.1 Reinvention of early algebra 45 3.4.2 Historical influences on the experimental learning strand 46 3.4.3 Expectations 47 4 Research plan and methodology 4.1 Introduction 49 4.2 Research questions 49 4.2.1 Main research questions and hypotheses 49 4.2.2 Sub-questions 50 4.3 Research method 51 4.3.1 Curriculum development and Realistic Mathematics Education 51 4.3.2 Developmental research 56 4.3.3 Explorative research 57 4.4 Research plan 57 4.4.1 Important changes of focus in the study 59 4.4.2 Audience 60 4.4.3 Educational design 60 vii 4.4.4 Teaching experiments 62 4.4.5 Analysis of student work 64 4.4.6 Peer review 67 4.5 Analysis of test tasks: a paradigmatic example 67 4.5.1 Test task Number Cards 67 4.5.2 Solution strategies for Number Cards 68 4.5.3 Progressive formalization 75 5 Design process 5.1 Introduction 83 5.2 The orientation phase 83 5.2.1 RME design heuristics 85 5.2.2 Mathematical content 88 5.2.3 Case studies 94 5.2.4 Evaluation and reflection 112 5.3 Additional try-outs for orientation 118 5.3.1 Mathematical starting level 118 5.3.2 Systems of equations 120 5.3.3 Pre-algebraic strategies from the past 123 5.4 Pilot experiment 124 5.4.1 Classroom experiences and results 126 5.4.2 A small group of students: experiences and results 145 5.5 Evaluation and reflection 158 5.6 Peer review 161 5.7 Conclusions 164 6 Final phase of the study 6.1 Introduction 171 6.2 State of affairs after the peer review 171 6.3 Research questions 173 6.4 Pre-algebra units revised 173 6.5 Field test 177 6.5.1 Experimental groups 177 6.5.2 Execution of the teaching experiment 179 6.6 Results of the Number Cards task 183 6.6.1 Purpose and expectations of Number Cards 183 6.6.2 The analysis: strategies observed 184 6.6.3 Notations 186 6.6.4 Summary of Number Cards results 188 6.6.5 Comparison with unit tasks 188 6.6.6 Theoretical reflections 194 6.6.7 Formulation of conjectures 196 6.7 Results of student work 199 6.7.1 Overview of results 200 6.7.2 Separate developments ofalgebraic reasoning and symbolizing 203 6.7.3 Symbolizing 238 viii Table of contents 6.7.4 Regression of strategy use 246 6.7.5 Understanding relations between quantities 253 6.7.6 Effect of history on the learning and teaching of early algebra 261 6.7.7 Role of the unknown 275 6.7.8 Recapitulation versus reinvention 277 6.8 Answers to the research questions 279 6.8.1 Conjectures 279 6.8.2 Sub-questions 280 6.8.3 Main research questions 283 6.9 Personal reflections 284 7 Discussion and recommendations 7.1 Reflections on the study’s turn-about 287 7.2 Discussion of the results 288 7.2.1 Symbolizing 288 7.2.2 Symbolizing versus reasoning 291 7.2.3 Regression of strategy use 292 7.2.4 Discontinuities between arithmetic and algebra 293 7.2.5 Indications of a suitable learning trajectory: crossing the gap 295 7.2.6 History as a didactical tool 297 7.2.7 Reinvention versus recapitulation 298 7.2.8 Further research 298 7.3 Recommendations for teachers and educational designers 299 References 303 Samenvatting 309 Appendix 323 Curriculum Vitae 339 ix x 1 Introduction 1.1 Reasons for this study educational argument In recent years, many research projects on mathematics education have focused on learning difficulties related to algebra, like the translation of word problems into simple algebraic equations.
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]
  • Lesson 2: the Multiplication of Polynomials
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M1 ALGEBRA II Lesson 2: The Multiplication of Polynomials Student Outcomes . Students develop the distributive property for application to polynomial multiplication. Students connect multiplication of polynomials with multiplication of multi-digit integers. Lesson Notes This lesson begins to address standards A-SSE.A.2 and A-APR.C.4 directly and provides opportunities for students to practice MP.7 and MP.8. The work is scaffolded to allow students to discern patterns in repeated calculations, leading to some general polynomial identities that are explored further in the remaining lessons of this module. As in the last lesson, if students struggle with this lesson, they may need to review concepts covered in previous grades, such as: The connection between area properties and the distributive property: Grade 7, Module 6, Lesson 21. Introduction to the table method of multiplying polynomials: Algebra I, Module 1, Lesson 9. Multiplying polynomials (in the context of quadratics): Algebra I, Module 4, Lessons 1 and 2. Since division is the inverse operation of multiplication, it is important to make sure that your students understand how to multiply polynomials before moving on to division of polynomials in Lesson 3 of this module. In Lesson 3, division is explored using the reverse tabular method, so it is important for students to work through the table diagrams in this lesson to prepare them for the upcoming work. There continues to be a sharp distinction in this curriculum between justification and proof, such as justifying the identity (푎 + 푏)2 = 푎2 + 2푎푏 + 푏 using area properties and proving the identity using the distributive property.
    [Show full text]
  • James Clerk Maxwell
    James Clerk Maxwell JAMES CLERK MAXWELL Perspectives on his Life and Work Edited by raymond flood mark mccartney and andrew whitaker 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Oxford University Press 2014 The moral rights of the authors have been asserted First Edition published in 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013942195 ISBN 978–0–19–966437–5 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only.
    [Show full text]
  • Act Mathematics
    ACT MATHEMATICS Improving College Admission Test Scores Contributing Writers Marie Haisan L. Ramadeen Matthew Miktus David Hoffman ACT is a registered trademark of ACT Inc. Copyright 2004 by Instructivision, Inc., revised 2006, 2009, 2011, 2014 ISBN 973-156749-774-8 Printed in Canada. All rights reserved. No part of the material protected by this copyright may be reproduced in any form or by any means, for commercial or educational use, without permission in writing from the copyright owner. Requests for permission to make copies of any part of the work should be mailed to Copyright Permissions, Instructivision, Inc., P.O. Box 2004, Pine Brook, NJ 07058. Instructivision, Inc., P.O. Box 2004, Pine Brook, NJ 07058 Telephone 973-575-9992 or 888-551-5144; fax 973-575-9134, website: www.instructivision.com ii TABLE OF CONTENTS Introduction iv Glossary of Terms vi Summary of Formulas, Properties, and Laws xvi Practice Test A 1 Practice Test B 16 Practice Test C 33 Pre Algebra Skill Builder One 51 Skill Builder Two 57 Skill Builder Three 65 Elementary Algebra Skill Builder Four 71 Skill Builder Five 77 Skill Builder Six 84 Intermediate Algebra Skill Builder Seven 88 Skill Builder Eight 97 Coordinate Geometry Skill Builder Nine 105 Skill Builder Ten 112 Plane Geometry Skill Builder Eleven 123 Skill Builder Twelve 133 Skill Builder Thirteen 145 Trigonometry Skill Builder Fourteen 158 Answer Forms 165 iii INTRODUCTION The American College Testing Program Glossary: The glossary defines commonly used (ACT) is a comprehensive system of data mathematical expressions and many special and collection, processing, and reporting designed to technical words.
    [Show full text]
  • A List of Arithmetical Structures Complete with Respect to the First
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 257 (2001) 115–151 www.elsevier.com/locate/tcs A list of arithmetical structures complete with respect to the ÿrst-order deÿnability Ivan Korec∗;X Slovak Academy of Sciences, Mathematical Institute, Stefanikovaà 49, 814 73 Bratislava, Slovak Republic Abstract A structure with base set N is complete with respect to the ÿrst-order deÿnability in the class of arithmetical structures if and only if the operations +; × are deÿnable in it. A list of such structures is presented. Although structures with Pascal’s triangles modulo n are preferred a little, an e,ort was made to collect as many simply formulated results as possible. c 2001 Elsevier Science B.V. All rights reserved. MSC: primary 03B10; 03C07; secondary 11B65; 11U07 Keywords: Elementary deÿnability; Pascal’s triangle modulo n; Arithmetical structures; Undecid- able theories 1. Introduction A list of (arithmetical) structures complete with respect of the ÿrst-order deÿnability power (shortly: def-complete structures) will be presented. (The term “def-strongest” was used in the previous versions.) Most of them have the base set N but also structures with some other universes are considered. (Formal deÿnitions are given below.) The class of arithmetical structures can be quasi-ordered by ÿrst-order deÿnability power. After the usual factorization we obtain a partially ordered set, and def-complete struc- tures will form its greatest element. Of course, there are stronger structures (with respect to the ÿrst-order deÿnability) outside of the class of arithmetical structures.
    [Show full text]
  • Randomized Linear Algebra for Model Order Reduction Oleg Balabanov
    Randomized linear algebra for model order reduction Oleg Balabanov To cite this version: Oleg Balabanov. Randomized linear algebra for model order reduction. Numerical Analysis [math.NA]. École centrale de Nantes; Universitat politécnica de Catalunya, 2019. English. NNT : 2019ECDN0034. tel-02444461v2 HAL Id: tel-02444461 https://tel.archives-ouvertes.fr/tel-02444461v2 Submitted on 14 Feb 2020 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. THÈSE DE DOCTORAT DE École Centrale de Nantes COMUE UNIVERSITÉ BRETAGNE LOIRE ÉCOLE DOCTORALE N° 601 Mathématiques et Sciences et Technologies de l’Information et de la Communication Spécialité : Mathématiques et leurs Interactions Par Oleg BALABANOV Randomized linear algebra for model order reduction Thèse présentée et soutenue à l’École Centrale de Nantes, le 11 Octobre, 2019 Unité de recherche : Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629 Thèse N° : Rapporteurs avant soutenance : Bernard HAASDONK Professeur, University of Stuttgart Tony LELIÈVRE Professeur, École des Ponts ParisTech Composition du Jury : Président : Albert COHEN Professeur, Sorbonne Université Examinateurs : Christophe PRUD’HOMME Professeur, Université de Strasbourg Laura GRIGORI Directeur de recherche, Inria Paris, Sorbonne Université Marie BILLAUD-FRIESS Maître de Conférences, École Centrale de Nantes Dir.
    [Show full text]
  • The Infinite and Contradiction: a History of Mathematical Physics By
    The infinite and contradiction: A history of mathematical physics by dialectical approach Ichiro Ueki January 18, 2021 Abstract The following hypothesis is proposed: \In mathematics, the contradiction involved in the de- velopment of human knowledge is included in the form of the infinite.” To prove this hypothesis, the author tries to find what sorts of the infinite in mathematics were used to represent the con- tradictions involved in some revolutions in mathematical physics, and concludes \the contradiction involved in mathematical description of motion was represented with the infinite within recursive (computable) set level by early Newtonian mechanics; and then the contradiction to describe discon- tinuous phenomena with continuous functions and contradictions about \ether" were represented with the infinite higher than the recursive set level, namely of arithmetical set level in second or- der arithmetic (ordinary mathematics), by mechanics of continuous bodies and field theory; and subsequently the contradiction appeared in macroscopic physics applied to microscopic phenomena were represented with the further higher infinite in third or higher order arithmetic (set-theoretic mathematics), by quantum mechanics". 1 Introduction Contradictions found in set theory from the end of the 19th century to the beginning of the 20th, gave a shock called \a crisis of mathematics" to the world of mathematicians. One of the contradictions was reported by B. Russel: \Let w be the class [set]1 of all classes which are not members of themselves. Then whatever class x may be, 'x is a w' is equivalent to 'x is not an x'. Hence, giving to x the value w, 'w is a w' is equivalent to 'w is not a w'."[52] Russel described the crisis in 1959: I was led to this contradiction by Cantor's proof that there is no greatest cardinal number.
    [Show full text]
  • History of Algebra and Its Implications for Teaching
    Maggio: History of Algebra and its Implications for Teaching History of Algebra and its Implications for Teaching Jaime Maggio Fall 2020 MA 398 Senior Seminar Mentor: Dr.Loth Published by DigitalCommons@SHU, 2021 1 Academic Festival, Event 31 [2021] Abstract Algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities (letters and other general sym- bols) defined by the equations that they satisfy. Algebraic problems have survived in mathematical writings of the Egyptians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts. In this paper, we will discuss historically famous mathematicians from all over the world along with their key mathematical contributions. Mathe- matical proofs of ancient and modern discoveries will be presented. We will then consider the impacts of incorporating history into the teaching of mathematics courses as an educational technique. 1 https://digitalcommons.sacredheart.edu/acadfest/2021/all/31 2 Maggio: History of Algebra and its Implications for Teaching 1 Introduction In order to understand the way algebra is the way it is today, it is important to understand how it came about starting with its ancient origins. In a mod- ern sense, algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities defined by the equations that they sat- isfy. Algebraic problems have survived in mathematical writings of the Egyp- tians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts, but these concepts had a heavier focus on geometry [1]. The combination of all of the discoveries of these great mathematicians shaped the way algebra is taught today.
    [Show full text]
  • Arithmetical Proofs in Arabic Algebra Jeffery A
    This article is published in: Ezzaim Laabid, ed., Actes du 12è Colloque Maghrébin sur l'Histoire des Mathématiques Arabes: Marrakech, 26-27-28 mai 2016. Marrakech: École Normale Supérieure 2018, pp 215-238. Arithmetical proofs in Arabic algebra Jeffery A. Oaks1 1. Introduction Much attention has been paid by historians of Arabic mathematics to the proofs by geometry of the rules for solving quadratic equations. The earliest Arabic books on algebra give geometric proofs, and many later algebraists introduced innovations and variations on them. The most cited authors in this story are al-Khwārizmī, Ibn Turk, Abū Kāmil, Thābit ibn Qurra, al-Karajī, al- Samawʾal, al-Khayyām, and Sharaf al-Dīn al-Ṭūsī.2 What we lack in the literature are discussions, or even an acknowledgement, of the shift in some authors beginning in the eleventh century to give these rules some kind of foundation in arithmetic. Al-Karajī is the earliest known algebraist to move away from geometric proof, and later we see arithmetical arguments justifying the rules for solving equations in Ibn al-Yāsamīn, Ibn al-Bannāʾ, Ibn al-Hāʾim, and al-Fārisī. In this article I review the arithmetical proofs of these five authors. There were certainly other algebraists who took a numerical approach to proving the rules of algebra, and hopefully this article will motivate others to add to the discussion. To remind readers, the powers of the unknown in Arabic algebra were given individual names. The first degree unknown, akin to our �, was called a shayʾ (thing) or jidhr (root), the second degree unknown (like our �") was called a māl (sum of money),3 and the third degree unknown (like our �#) was named a kaʿb (cube).
    [Show full text]
  • Chapter 8: Exponents and Polynomials
    Chapter 8 CHAPTER 8: EXPONENTS AND POLYNOMIALS Chapter Objectives By the end of this chapter, students should be able to: Simplify exponential expressions with positive and/or negative exponents Multiply or divide expressions in scientific notation Evaluate polynomials for specific values Apply arithmetic operations to polynomials Apply special-product formulas to multiply polynomials Divide a polynomial by a monomial or by applying long division CHAPTER 8: EXPONENTS AND POLYNOMIALS ........................................................................................ 211 SECTION 8.1: EXPONENTS RULES AND PROPERTIES ........................................................................... 212 A. PRODUCT RULE OF EXPONENTS .............................................................................................. 212 B. QUOTIENT RULE OF EXPONENTS ............................................................................................. 212 C. POWER RULE OF EXPONENTS .................................................................................................. 213 D. ZERO AS AN EXPONENT............................................................................................................ 214 E. NEGATIVE EXPONENTS ............................................................................................................. 214 F. PROPERTIES OF EXPONENTS .................................................................................................... 215 EXERCISE ..........................................................................................................................................
    [Show full text]
  • The Persian-Toledan Astronomical Connection and the European Renaissance
    Academia Europaea 19th Annual Conference in cooperation with: Sociedad Estatal de Conmemoraciones Culturales, Ministerio de Cultura (Spain) “The Dialogue of Three Cultures and our European Heritage” (Toledo Crucible of the Culture and the Dawn of the Renaissance) 2 - 5 September 2007, Toledo, Spain Chair, Organizing Committee: Prof. Manuel G. Velarde The Persian-Toledan Astronomical Connection and the European Renaissance M. Heydari-Malayeri Paris Observatory Summary This paper aims at presenting a brief overview of astronomical exchanges between the Eastern and Western parts of the Islamic world from the 8th to 14th century. These cultural interactions were in fact vaster involving Persian, Indian, Greek, and Chinese traditions. I will particularly focus on some interesting relations between the Persian astronomical heritage and the Andalusian (Spanish) achievements in that period. After a brief introduction dealing mainly with a couple of terminological remarks, I will present a glimpse of the historical context in which Muslim science developed. In Section 3, the origins of Muslim astronomy will be briefly examined. Section 4 will be concerned with Khwârizmi, the Persian astronomer/mathematician who wrote the first major astronomical work in the Muslim world. His influence on later Andalusian astronomy will be looked into in Section 5. Andalusian astronomy flourished in the 11th century, as will be studied in Section 6. Among its major achievements were the Toledan Tables and the Alfonsine Tables, which will be presented in Section 7. The Tables had a major position in European astronomy until the advent of Copernicus in the 16th century. Since Ptolemy’s models were not satisfactory, Muslim astronomers tried to improve them, as we will see in Section 8.
    [Show full text]
  • The Reverse Mathematics of Cousin's Lemma
    VICTORIAUNIVERSITYOFWELLINGTON Te Herenga Waka School of Mathematics and Statistics Te Kura Matai Tatauranga PO Box 600 Tel: +64 4 463 5341 Wellington 6140 Fax: +64 4 463 5045 New Zealand Email: sms-offi[email protected] The reverse mathematics of Cousin’s lemma Jordan Mitchell Barrett Supervisors: Rod Downey, Noam Greenberg Friday 30th October 2020 Submitted in partial fulfilment of the requirements for the Bachelor of Science with Honours in Mathematics. Abstract Cousin’s lemma is a compactness principle that naturally arises when study- ing the gauge integral, a generalisation of the Lebesgue integral. We study the ax- iomatic strength of Cousin’s lemma for various classes of functions, using Fried- arXiv:2011.13060v1 [math.LO] 25 Nov 2020 man and Simpson’s reverse mathematics in second-order arithmetic. We prove that, over RCA0: (i) Cousin’s lemma for continuous functions is equivalent to the system WKL0; (ii) Cousin’s lemma for Baire 1 functions is at least as strong as ACA0; (iii) Cousin’s lemma for Baire 2 functions is at least as strong as ATR0. Contents 1 Introduction 1 2 Integration and Cousin’s lemma 5 2.1 Riemann integration . .5 2.2 Gauge integration . .6 2.3 Cousin’s lemma . .7 3 Logical prerequisites 9 3.1 Computability . .9 3.2 Second-order arithmetic . 12 3.3 The arithmetical and analytical hierarchies . 13 4 Subsystems of second-order arithmetic 17 4.1 Formal systems . 17 4.2 RCA0 .......................................... 18 4.3 WKL0 .......................................... 19 4.4 ACA0 .......................................... 21 4.5 ATR0 .......................................... 22 1 4.6 P1-CA0 ......................................... 23 5 Analysis in second-order arithmetic 25 5.1 Number systems .
    [Show full text]