Notes for a History of the Teaching of Algebra 1
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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. -
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). -
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. -
Taming the Unknown. a History of Algebra from Antiquity to the Early Twentieth Century, by Victor J
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 725–731 S 0273-0979(2015)01491-6 Article electronically published on March 23, 2015 Taming the unknown. A history of algebra from antiquity to the early twentieth century, by Victor J. Katz and Karen Hunger Parshall, Princeton University Press, Princeton and Oxford, 2014, xvi+485 pp., ISBN 978-0-691-14905-9, US $49.50 1. Algebra now Algebra has been a part of mathematics since ancient times, but only in the 20th century did it come to play a crucial role in other areas of mathematics. Algebraic number theory, algebraic geometry, and algebraic topology now cast a big shadow on their parent disciplines of number theory, geometry, and topology. And algebraic topology gave birth to category theory, the algebraic view of mathematics that is now a dominant way of thinking in almost all areas. Even analysis, in some ways the antithesis of algebra, has been affected. As long ago as 1882, algebra captured some important territory from analysis when Dedekind and Weber absorbed the Abel and Riemann–Roch theorems into their theory of algebraic function fields, now part of the foundations of algebraic geometry. Not all mathematicians are happy with these developments. In a speech in 2000, entitled Mathematics in the 20th Century, Michael Atiyah said: Algebra is the offer made by the devil to the mathematician. The devil says “I will give you this powerful machine, and it will answer any question you like. All you need to do is give me your soul; give up geometry and you will have this marvellous machine.” Atiyah (2001), p. -
Early History of Algebra: a Sketch
Math 160, Fall 2005 Professor Mariusz Wodzicki Early History of Algebra: a Sketch Mariusz Wodzicki Algebra has its roots in the theory of quadratic equations which obtained its original and quite full development in ancient Akkad (Mesopotamia) at least 3800 years ago. In Antiq- uity, this earliest Algebra greatly influenced Greeks1 and, later, Hindus. Its name, however, is of Arabic origin. It attests to the popularity in Europe of High Middle Ages of Liber al- gebre et almuchabole — the Latin translation of the short treatise on the subject of solving quadratic equations: Q Ì Al-kitabu¯ ’l-muhtas.aru f¯ı é ÊK.A®ÜÏ@ð .m. '@H . Ak ú¯ QåJjÜÏ@ H. AJº Ë@ – h. isabi¯ ’l-gabriˇ wa-’l-muqabalati¯ (A summary of the calculus of gebr and muqabala). The original was composed circa AD 830 in Arabic at the House of Wisdom—a kind of academy in Baghdad where in IX-th century a number of books were compiled or translated into Arabic chiefly from Greek and Syriac sources—by some Al-Khwarizmi2 whose name simply means that he was a native of the ancient city of Khorezm (modern Uzbekistan). Three Latin translations of his work are known: by Robert of Chester (executed in Segovia in 1140), by Gherardo da Cremona3 (Toledo, ca. 1170) and by Guglielmo de Lunis (ca. 1250). Al-Khwarizmi’s name lives today in the word algorithm as a monument to the popularity of his other work, on Indian Arithmetic, which circulated in Europe in several Latin versions dating from before 1143, and spawned a number of so called algorismus treatises in XIII-th and XIV-th Centuries. -
Fundamental Theorem of Arithmetic”
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector HISTORIA MATHEMATICA 19 (1992), 177-187 On a Seventeenth Century Version of the “Fundamental Theorem of Arithmetic” CATHERINE GOLDSTEIN URA D 0752 Bat 425, UniversitC de Paris XI, F-91405 Orsay, France The complete proof of the uniqueness of prime factorization is generally attributed to Gauss. We present here some related results published by Jean Prestet in 1689 which, we think, could be useful for any comparative study of this question. o 19~2 Academic PKSS, IX. La preuve de l’unicitt de la decomposition d’un entier en facteurs premiers est generale- ment attribuee a Gauss. Nous presentons ici les resultats correspondants publies par Jean Prestet en 1689, que nous pensons utiles a prendre en compte dans toute etude comparative de ces questions. 0 1992 Academic Press, Inc. Der vollstandige Beweis fur die Eindeutigkeit der Primfaktorzerlegung wird im allge- meinen Gauss zugeschrieben. Wir stellen hier einige verwandte Ergebnisse vor, die Jean Prestet 1689 veroffentlichte, die unserer Ansicht nach fur eine vergleichende Untersuchung dieser Frage ntltzlich sein konnte. o I!%% Academic PKSS. II-K. AMS 1991 subject classification: 01 A 45, 1 l-03. KEY WORDS: number theory, prime factorization, Gauss, Prestet. The first explicit proof of the so-called “fundamental theorem of arithmetic,” the uniqueness of the factorization of a natural integer into a product of primes, is generally attributed to Carl Friedrich Gauss [Gauss 1801, Second Section Sect. 16]-see for instance [Boyer 1968,551] or [Bourbaki 1960, 110-I 111. -
Islamic Mathematics
Islamic Mathematics Elizabeth Rogers MATH 390: Islamic Mathematics Supervised by Professor George Francis University of Illinois at Urbana-Champaign August 2008 1 0.1 Abstract This project will provide a summary of the transmission of Greek mathe- matics through the Islamic world, the resulting development of algebra by Muhammad ibn Musa al-Khwarizmi, and the applications of Islamic algebra in modern mathematics through the formulation of the Fundamental Theo- rem of Algebra. In addition to this, I will attempt to address several cultural issues surrounding the development of algebra by Persian mathematicians and the transmission of Greek mathematics through Islamic mathematicians. These cultural issues include the following questions: Why was the geometry of Euclid transmitted verbatim while algebra was created and innovated by Muslim mathematicians? Why didn't the Persian mathematicians expand or invent new theorems or proofs, though they preserved the definition-theorem-proof model for ge- ometry? In addition, why did the definition-theorem-proof model not carry over from Greek mathematics (such as geometry) to algebra? Why were most of the leading mathematicians, in this time period, Mus- lim? In addition, why were there no Jewish mathematicians until recently? Why were there no Orthodox or Arab Christian mathematicians? 0.2 Arabic Names and Transliteration Arabic names are probably unfamiliar to many readers, so a note on how to read Arabic names may be helpful. A child of a Muslim family usually receives a first name ('ism), followed by the phrase \son of ··· "(ibn ··· ). For example, Th¯abitibn Qurra is Th¯abit,son of Qurra. Genealogies can be combined; for example, Ibr¯ah¯ımibn Sin¯anibn Th¯abitibn Qurra means that Ibr¯ah¯ımis the son of Sin¯an,grandson of Th¯abit,and great-grandson of Qurra. -
Algebra and Geometry Throughout History: a Symbiotic Relationship
Early History of Algebra and Geometry Modern History The Final Stage: Abstract Algebra Algebo-Geometric Set Up Back to Algebra Algebra and Geometry Throughout History: A Symbiotic Relationship Marie A. Vitulli University of Oregon Eugene, OR 97403 January 25, 2019 M. A. Vitulli A Symbiotic Relationship Early History of Algebra and Geometry Modern History Babylonian and Egyptian Algebra and Geometry The Final Stage: Abstract Algebra Greek and Persian Contributions Algebo-Geometric Set Up Algebraic Notation Matures Back to Algebra Stages in the Development of Symbolic Algebra Rhetorical algebra: equations are written in full sentences like “the thing plus one equals two”; developed by ancient Babylonians Syncopated algebra: some symbolism was used, but not all characteristics of modern algebra; for example, Diophantus’ Arithmetica (3rd century A.D.) Symbolic algebra - full symbolism is used; developed by François Viète (16th century) M. A. Vitulli A Symbiotic Relationship Early History of Algebra and Geometry Modern History Babylonian and Egyptian Algebra and Geometry The Final Stage: Abstract Algebra Greek and Persian Contributions Algebo-Geometric Set Up Algebraic Notation Matures Back to Algebra Babylonian Algebra: Plimpton 322 ca. 1800 B.C.E. Babylonians used cuneiform cut into a clay tablet with a blunt reed to record numbers and figures. They had a base 60 true place-value number system. Plimpton 322 contains a table with 2 of 3 numbers of what are now called Pythagorean triples: integers a; b; and c satisfying a2 + b2 = c2. These are integer length sides of a right Figure: Plimpton 322 triangle. M. A. Vitulli A Symbiotic Relationship Early History of Algebra and Geometry Modern History Babylonian and Egyptian Algebra and Geometry The Final Stage: Abstract Algebra Greek and Persian Contributions Algebo-Geometric Set Up Algebraic Notation Matures Back to Algebra Babylonian Algebra: YBC 7289 ca. -
A Short History of Complex Numbers
A Short History of Complex Numbers Orlando Merino University of Rhode Island January, 2006 Abstract This is a compilation of historical information from various sources, about the number i = √ 1. The information has been put together for students of Complex Analysis who − are curious about the origins of the subject, since most books on Complex Variables have no historical information (one exception is Visual Complex Analysis, by T. Needham). A fact that is surprising to many (at least to me!) is that complex numbers arose from the need to solve cubic equations, and not (as it is commonly believed) quadratic equations. These notes track the development of complex numbers in history, and give evidence that supports the above statement. 1. Al-Khwarizmi (780-850) in his Algebra has solution to quadratic equations of various types. Solutions agree with is learned today at school, restricted to positive solutions [9] Proofs are geometric based. Sources seem to be greek and hindu mathematics. According to G. J. Toomer, quoted by Van der Waerden, Under the caliph al-Ma’mun (reigned 813-833) al-Khwarizmi became a member of the “House of Wisdom” (Dar al-Hikma), a kind of academy of scientists set up at Baghdad, probably by Caliph Harun al-Rashid, but owing its preeminence to the interest of al-Ma’mun, a great patron of learning and scientific investigation. It was for al-Ma’mun that Al-Khwarizmi composed his astronomical treatise, and his Algebra also is dedicated to that ruler 2. The methods of algebra known to the arabs were introduced in Italy by the Latin transla- tion of the algebra of al-Khwarizmi by Gerard of Cremona (1114-1187), and by the work of Leonardo da Pisa (Fibonacci)(1170-1250). -
Tabak J Algebra Sets Symbols
algebra Revised Edition Openmirrors.com THE HISTORY OF algebra sets, symbols, and the language of thought Revised Edition John Tabak, Ph.D. Openmirrors.com ALGEBRA: Sets, Symbols, and the Language of Thought, Revised Edition Copyright © 2011, 2004 by John Tabak, Ph.D. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval systems, without permission in writing from the publisher. For information contact: Facts On File, Inc. An imprint of Infobase Learning 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Tabak, John. Algebra : sets, symbols, and the language of thought / John Tabak—Rev. ed. p. cm.—(The history of mathematics) Includes bibliographical references and index. ISBN 978-0-8160-7944-5 (alk. paper) ISBN 978-1-4381-3585-4 (e-book) 1. Algebra—History. I. Title. QA151.T33 2011 512.009—dc22 2010021597 Facts On File books are available at special discounts when purchased in bulk quantities for businesses, associations, institutions, or sales promotions. Please call our Special Sales Department in New York at (212) 967-8800 or (800) 322-8755. You can find Facts On File on the World Wide Web at http://www.infobaselearning.com Excerpts included herewith have been reprinted by permission of the copyright hold- ers; the author has made every effort to contact copyright holders. The publisher will be glad to rectify, in future editions, any errors or omissions brought to its notice. Text design by David Strelecky Composition by Hermitage Publishing Services Illustrations by Dale Williams Photo research by Elizabeth H. -
Stages in the History of Algebra with Implications for Teaching∗
VICTOR J. KATZ STAGES IN THE HISTORY OF ALGEBRA WITH IMPLICATIONS FOR TEACHING∗ ABSTRACT. In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. But besides these three stages of expressing algebraic ideas, there are four more conceptual stages which have happened along side of these changes in expressions. These stages are the geometric stage, where most of the concepts of algebra are geometric ones; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea, and finally, the abstract stage, where mathematical structure plays the central role. The stages of algebra are, of course not entirely disjoint from one another; there is always some overlap. We discuss here high points of the development of these stages and reflect on the use of these historical stages in the teaching of algebra. KEY WORDS: algebra, abstract stage, equation-solving stage, function stage, geometric stage, rhetorical stage, symbolic stage, syncopated stage This article is about algebra. So the first question is, what do we mean by algebra? Few secondary textbooks these days give a definition of the subject, but that was not the case two centuries ago. For example, Colin Maclaurin wrote, in his 1748 algebra text, “Algebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose, and found convenient. -
The Book Al-Jabr Wa Al-Muqābalah
Madiha & Wan Suhaimi, “The Book Al-Jabr,” Afkar 15 (2014): 135-162 THE BOOK AL-JABR WA AL-MUQĀBALAH: A RESEARCH ON ITS CONTENT, WRITING METHODOLOGY AND ELEMENTARY ALGEBRA BY AL-KHWĀRIZMĪ Madiha Baharuddin PhD candidate Department of Aqidah and Islamic Thought Academy of Islamic Studies University of Malaya, Kuala Lumpur [email protected] Wan Suhaimi Wan Abdullah, PhD Associate Professor Centre for Advanced Studies on Islam, Science and Civilisation (CASIS) Universiti Teknologi Malaysia, Kuala Lumpur [email protected] Khulasah Islam suatu ketika dahulu pernah mencapai zaman kegemilangan tamadunnya. Banyak karya-karya yang ditulis hasil dari penguasaan pelbagai bidang ilmu oleh tokoh-tokoh kesarjanaan Islam. Kajian ini akan membincangkan sebuah hasil karya seorang tokoh tersohor, al-Khwarizmi melalui bukunya al-Jabr wa al-Muqabalah. Kertas kerja ini akan cuba memperlihatkan pembahagian bab-bab kandungan kitab ini, gaya penulisannya serta kedudukan kitab ini dari sudut sumbernya. Kajian ini juga difokuskan kepada asas algebra yang dikemukakan oleh beliau dalam bahagian pertama kitab ini. Ia merangkumi istilah-istilah matematik yang kerap digunakan di dalam kitab ini, contoh-contoh permasalahan yang terlibat dan juga cara penyelesaian yang ditunjukkan oleh al-Khwarizmi terhadap masalah-masalah tersebut. Ia diharap dapat memberi peluang kepada pembaca khususnya di Malaysia untuk mempelajari sesuatu hasil dari sejarah ketamadunan Islam, 135 Madiha & Wan Suhaimi, “The Book Al-Jabr,” Afkar 15 (2014): 135-162 menjadi nilai tambah kepada sumber dan bahan yang sedia ada, dan seterusnya memudahkan pembaca untuk memahami pokok perbincangan kitab ini untuk dibincangkan dalam konteks yang lebih meluas. Kata Kunci: al-Khwarizmi, al-Jabr wa al- Muqabalah, algebra, persamaan kuadratik.