11111 I .6 MICROCOPYRESOLUTION TESTCHART NATIONAL BUREAUOF STANDARDS-1963-A DOCUMENT RESUME ED 054 941 SE 012 334 TITLE Historical Topics in Algebra. INSTITUTION National Council of Teachers ofMathematics, Inc., Washington, D.C. PUB DATE 71 NOTE 81p.; Reprint from Thirty-first Yearbookof the NCTM, p233-332 AVAILABLE FROMNational Council of Teachers of Mathematics,1201 Sixteenth Street, N.W., Washingtion, D.C.20036 ($1.00) EDRS PRICE MF-$0.65 HC Not Available from EDRS. DESCRIPTORS *Algebra; Enrichment; *History; *Instructional Materials; *Mathematical Concepts; Mathematics; *Secondary School Mathematics ABSTRACT This is a reprint of the historicalcapsules dealing with algebra from the 31st Yearbook ofNCTM,"Historical Topics for the Mathematics Classroom." Included aresuch themes as the change from a geometric to an algebraic solutionof problems, the development of algebraic symbolism, the algebraiccontributions of different countries, the origin and developmentof topics in algebra, and the search for generality and abstractstructures. (Author/JG) ; 4;,!;19A,.!,41P;',,i'vi'.,040-14Aki.,04,.;YIv.'iwii+,kii I ; 1614',1,,0 0411.4. 044:Ar..,14 LL' ! Ye"rt, ; ...10.",HM#"*"1"""1.11' f40 '14440444414,104 %IVA01.41014,10141**01.0604044041001#44,06.100.4.44,441001010444110)1100#0410.40P0h1.14.14.404000016160p44 - - - - - - Jfaiostreams iff the Now of Algebra- -1650 EGYPT Rhind papyrus GREECE I Pythagoreans Euclid Archirnedes Apollonius rik Mop) GREECE II Diophantus Pappus j 628 INDIA ARABIAN Brahmagupta 825 EMPIRE aI-Khowarizmi 1100 Omar Khayyam EUROPE Bhaskara 1202 Fibonoces Liber abaci 1450 Printing 1494 Pacioll's SUrna 1545 Ferrari, Torten lia, Cardano 1572 Bornbelli 1600 Viete 1700 Newton a I Khwari.zm BABYLONIA Bagridel INDIA Alexandria EGYPT ARABIA The special contents of this book are copyright (:) 1971 by The National Council of Teachers of Mathematics, Inc. All rights reserved. Printed in the United States of America. Yrry "PERMISSION TO REPRODUCE THIS COPY- of Congress Catalog Card Number: 79-171459 RIGHTED MATERIAL BY MICROFICHE ONLY :HAS BEEN GRANTED BY Al C:77 in TO ERIC AND ORGANIZATIONS OPERATING ,U,NDER AGREEMENTS WITH THE U.S. OFFICE OF EDUCATION. FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM REQUIRES PER- : MISSION OF THE COPYRIGHT OWNER." Preface This booklet is a separate presentation of the"capsules" dealing with algebra in the Thirty-first Yearbook of the NationalCouncil of Teachers of Mathematics, Historical Topics for theMathematics Classroom. Paperback publication makes this materialavailable in an economical and flexible form for use in algebra classrooms orby in- dividuals whose mathematical interest at the moment is primarilyin algebra. "What is new today becomes old tomorrow," even inmathematics. The "new math" of a few years ago is now commonplacein many elementary and S'econdary schools across the country.Of course such terms as "new math" and "traditional math" still carrymeaning for those professionally involved in the teaching ofmathematics, although these terms may not carry exactly the same meaningfor all persons. But such designations are related to chronologicalintervals and COD- ceptual patterns that encompaPR only a small Wirt. 01 thyovr"1 his- tory and development of iu The Thirty-first Yearbook of the NCTM is aconstant reminder to its readers that mathematics does indeed have ahistory and that there are values to be derived from using some ofthis history in the present-day classroom. As stated in its preface, the primaryobjective of that yearbook is "to make ,available to mathematicsclasses im- portant material from the history and development ofmathematies. with the hope that this will increase the interest of students inmaithe- matics and their appreciation for the cultural aspects of thesubliect." In the "overview" of the history of algebra givenin the yearbook, John K. Baumgart states: "Although originally 'algebra'referred to equations, the word today has a much broader meaning,and a,at,ios- factory definition requires a two-phase approach:(1) Early (ele- mentary) algebra is the study of equations and methods forsolving them. (2) Modern (abstract) algebra is the studye. mathematical structures such as groups, rings, and fieldsto mentiononly a few. Indeed, it is convenient to trace the development ofalgebra in terms PREFACE of these two phases, since the division is both chronological and con- ceptual." The accompanying algebra capsules give brief sketches of some of the individual topics that are part of each of these developments. A glance at the table of contents readily reveals that no attempt has been made to separate them into these two groups nor to give them in any sort of chronological order. Included are such themes asthe change from a geometric to an algebraic solution of problems, the de- velopment of algebraic symbolism, the algebraic contributions of dif- ferent countries at different times, the origin and development of cer- tain special topics in algebra, the search for generality and abstract structures. While the capsule treatment bas the advantage of permitting concen- tration on just one topic at a time, it also limits the discussion of the interrelationships with other algebraic topics and with other areas of mathematics as well. 'The overview of the history of algebra in the parent yearbook will be of help in giving the reader orientation and a general picture of some of these developments. The Thirty-first Yearbook includes, in addition to the material on algebra, overviews and capsules on the history of numbers amd numerals, computation. e'Pon- trigonometry, calculus, and modern mathem Logethey , essay inthe history of mathematics as a teaching tool. Those persons who contributed to the preparation of the entire project are acknowledged in the prefac of the yearbook. Now thanks are expressed not only to them but -a:,mtembers of the Publications Committee of1. the NCTM for their entum-agement and recommenda- tion that this portion be made av0,-abIt.- separately as one of the Council's supplementary publications. ARTHUILE. IIALLERBERG Chairman orf the Eclitoriail Panel Thirty-Tj.st Yearbook cl the NCTM iv Contents Preface lii Reference Symbols and Bibliography vi EXCERPT FROM YEARBOOK OVERVIEW Origin of the Word "Algebra," John. K.Baumgart CAPSULE DISCUSSIONS OF ALGER'? Equations and the Ways They Were:CA,.11, _erneth Cummins The Binomial Theorem, Wade Ellis 6 Continued Fractions, Elaine J. Tatham 9 Oughtred and the Slide Rule, Richard M. Park 13 Homer's Method, Richard M. Park 16 Solution of Polynomial Equations of Third and HigherDegrees, Rodney Hood 18 Vectors, Leonard E. Puller 21 DeterminaL ts and Matrices, Leonard E. Puller 23 Boolean Algebra, Anice Seybold 26 Congruence (Mod ,Sui. Perlis 30 Complex Numbers (The Story of -\47--1), Eugene W. Hellmich 32 Quaternions, Sam Perlis 37 Early Greek Algebra, Gertrude V . Pratt 4C Hindu Algebra, Perna E. Wrestler 43 Arabic Algebra, 820-1250, Cecil B. Read 47 Algebra in Europe, 1200-4850, Sister M. Stephanie Sloyen 51 Function, Joe Kennedy and Esther Ragan 54 Mathematical Induction 55 Fundamental Theorem of Algebra 58 Descartes's Rule of Signs, Donald W. Western 60 Symmetric Functions, Dorothy Wolfe 63 Discriminant, Dorothy Wolfe 65 Interest and Annuities, L. S. Shively 67 Exponential Notation, Donald J. Dessart 69 Rule of False Position, Waldeck E.Mainville, Jr. 74 REFERENCE SYMBOLS AND BIBLIOGRAPHY Cross-references witbin the text to related capsulesare indicated by giving the number of the capsule in boldfacetype within square brackets. For cE:psules with numbers outside the intervalfrom {66} through 00) the reader is referred to the Thirty-firstYearbook. Complete bibliographical information for coded referencesappear- ing in the text within slashesor indicated under "For Further Read- ing" is given in the extensive bibliography of the yearbook. The following selected :bibliography, though brief,may serve two purposes here: to identify the more commonly cited references and to provide school libraries with a listing of books of generalinterest in the history of mathematics. Bell, Eric Temple (d). Men of Mathematics. New York: Simon & Schuster,1937 ; paper, 1965. Boyer, Carl B. (g). A History of Mathematics. New York: John Wiley (Cr, Sons, 1968. Dantzig, Tobias (b). Number, the Language of Science. 4th ed.,rev. and enl. New York: Macmillan Co., 1954; Free Preis, paper, n.d. Eves, Howard (c). An Introduction io the History- of Mathematics. 3d ed., rev. New York: Holt., Rinehart'Sr. Winston, 1969. , Historical. Topics for the Mathematics classrooni. Washington, DC:: National Council of Teachers of Mathematics, 1969. Newman, JamcS Roy, ed. The World 'of Mathematics.' 4vols., New York: Simon & Schuster, 1956;paper, 1962. _ Sanford, Vera(d). A Short- History Of Mathematics. Boston: Houghton Mifflin Co.,1930., Smith, David 'Eugene '(a). History of Mathem.utics. 2vols. 1923, 1925. New York: Dover Publications, 1958. Van der Waerden, B. L. Science Awakening. Translated by Arnold Dresden. New York: Oxford University Press,1961 ; John Wiley & Sons, Science EditioLs, 1963. Origin of the Word"Algebra" from the yearbook overview by JOHN K. BAUMGART xotic and intriguing is the originof the word "algebra." It does not submit to a neat etymology asdoes, for example, the word"arith- metic," which is derived from theGreek arithmos ("number"). Algebra is a Latin variant of theArabic word al-jabr (sometimes transliterated al-jebr) as employed inthe title of a book, Ifisabal- jabr w'al-muqabalah, written in Baghdadabout A.D. 825 by the