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David Konstantinov
David Konstantinov

Learning Modern Algebra : From Early Attempts T... BEST


This book coversabstract algebra from a historical perspective by using mathematicsfrom attempts to prove Fermat's last theorem, as the titleindicates. The target audience is high school mathematicsteachers. However, typical undergraduate students will also derivegreat benefit by studying this text. The book is permeated withfascinating mathematical nuggets that are clearlyexplained




Learning modern algebra : from early attempts t...



This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily "end up on the blackboard."


A year ago, I did a one-semester long course on Abstract Algebra at my university. When we started, I was excited, because I knew the material presented in this course would be somewhat different from the things I had learned previously (including subjects like calculus, linear algebra, logic, graph theory, et cetera). I also knew Abstract Algebra is a very broad subject, and many modern subjects are rooted in its matter, including Algebraic Geometry, Algebraic Topology, Algebraic Number Theory, and many more subjects.


I had run into the exact same situation when I took a course in Abstract Algebra 1: called Group Theory. I also had a "Real" math course, it was a very rigorous proof oriented linear algebra class. But I ended up dropping the group theory course. The problem was the prof taught graduate level courses for 10 years, then after 10 years was asked to teach an intro Group Theory course. He went extremely fast, (I dropped it right before the midterm), so fast that he finished the course by 2/3 of the duration of the course. For this prof, he thought this course was a very easy course.The problem with Group theory is that there is a lot of material and it is very dense and inter-connects so much that it is overwhelming, its almost like learning a new language. ALSO I agree with orlandpm above when he says: "They like to think that some people are not "cut out" to be mathematicians, and they are happy to sideline these people"This the type of mentality I see a lot when one goes higher in mathematics! This amounts to a kind of elitism.IF University Math Departments really want students to progress further in the more Abstract-Pure mathematics, it should change the courses. In my opinion they should take the content of the course and reduce it in half and go deeper with this, than to cover so much. Also the proofs in Abstract algebra are more difficult to visualize than in Linear Algebra. They should make this one course and create 2 course called : Group Theory 1 and Group Theory 2. In this way, they can increase revenue as well.To me Group Theory is the course that most students encounter when trying to go further into Abstract math, and due to these kinds of experiences, the course in Group Theory is now kind of used to turn away many students from math.It has been my experience, that Group Theory has been used to block people going further in university abstract math.


Father of Modern Algebraic Notation Jennifer Orlansky The practice of using letters rather than numbers to represent bothknown (but unspecified) and unknown quantities marked the beginningsof modern algebra as we know it today. Frangois Viète (Latin:Vieta), a great French mathematician, is credited with the inventionof this system, and is therefore known as the "father of modernalgebraic notation" [3, p. 268]. In other words, Viète's maininnovation in algebra was the use of variable coefficients. Althoughhe is best known for his "pioneering work o n symbolism" [3, p. 268],Viète also contributed much to the field of mathematics; hisachievements have paved the way for the work of many mathematiciansand have sparked new ideas and research that influenced the evolutionof modern algebra. Viète was born in 1540 in Fontenay-le-Comte[10, p. 63]. Like many early mathematicians, Viète's primaryoccupation was not in the area of mathematics. He was a lawyer, likehis father, and received his legal degree from the University ofPoitiers in 15 60. He became a tutor for a young girl named Catherinede Parthenay, which prompted him to study astronomy, a subject inwhich she was interested [9, p,2]. He also served as a member of theroyal privy council under King Henry III and King Henry IV, deciphering codes during the war against Spain [10, p. 63]. Viète had a great fascination with mathematics, and used hisspare time to study this area of interest, as well as cryptography. He died in Paris in 1603. A chief goal of Viète's explorationwas "to join Euclidean geometry with a generalized numerical algebra"[3, p. 309]. To him, algebra was an art. He believed that his "new"algebra was really just an improvement of old ideas that had been"wrapped in g eometrical garb" [6, p. 94], as he explained in thededication of his Isagoge to Catherine: These things which are new arewont in the beginning to be set forth rudely and formlessly and mustthen be polished and perfected in succeeding centuries. Behold, theart which I present is new, but in truth so old, so spoiled anddefiled by the barbari ans, that I considered it necessary, in orderto introduce an entirely new form into it, to think out and publish anew vocabulary, having gotten rid of all its pseudo-technical terms... (Viète cited in [5, p. 131]. In an attempt to transform this"filth", as he called it, to "the great art," Viète wroteseveral works, hoping to enhance the "spoiled and defiled" system ofmathematics to his standards of superiority. Various mathematiciansin history influenced Viète, however, there were two primaryGreek sources that served as major contributors to Viète'swork: (1) Pappus' seventh book of the Mathematical Collections and (2)Diophantus' Arithmetic [7, p. 154]. In V iète's time, mostpeople were using Arabic terms when dealing with mathematics. SinceViète felt Arabic was a barbaric language, he sought books thatused Greek terms. His search resulted in Pappus' the MathematicalCollections. Pappus, "the last great mathematician of the Alexandrian school," was a Hellenisticmathematician, probably born about 340 A.D. [2, p. 49]. TheMathematical Collections is the only work of Pappus still inexistence. It originally consisted of eight books, the first, andporti ons of the second, of which are now missing [2, p. 49]. In theMathematical Collections, Pappus supplied "the geometers of his timewith a succinct analysis of the most difficult mathematical works andfacilitate[d] the study of them by explanatory lemma s" [2, p. 49]. Pappus distinguishes two kinds of analysis, zetetic and porisitic,terms which Viète borrowed and added to them a third stage,called rhetic or exegetic. These stages of analysis will be discussedat greater length later in the paper. The second important influenceon Viète was Diophantus' Arithmetic. Diophantus "was one ofthe last and most fertile mathematicians of the second Alexandrianschool. He flourished around 250 A.D." [2, p. 60]. Arithmetic, saidto have been written in 13 books, introduced the idea of an algebraic equation expressed inalgebraic symbols [2, p. 60]. Viète expanded this notion ofalgebraic symbols. According to Klein, some historians ofmathematics: see the Diophantine science as the primitive "preliminarystage" of modern algebra. From the point of view of modern algebraonly a single additional step seems necessary to perfect Diophantinelogistic: the thoroughgoing substitution of "general" numeri calexpressions for the "determinate numbers," of symbolic for numericalvalues - a step which was, subsequent to a great deal of progress inthe treatment of equations in general, finally taken by Vieta [7, p. 139]. Diophantus can be placed in the second of three categories ofalgebra, with respect to notation, described by Cajori. The firstclass is Rhetorical algebras, in which no symbols are used, everythingbeing written out in words. Rhetorical algebra was use d in someArabic works, the Greek works of Iamblichus and Thymaridas, and theearly Italian writers. The second class is Syncopated algebras, inwhich, as in the first class, everything is written out in words,except that abbreviations are used for cert ain frequently recurringoperations and ideas. Syncopated algebra can be found in the works oflater western Arabs, Diophantus and some of the later European writersdown to about the seventeenth century (with the exception ofViète and Oughtred). The t hird class is Symbolic algebras, inwhich all forms and operations are represented by a fully developedalgebraic symbolism, as for example, x2 + 10x + 7. Symbolic algebrais the form employed by Viète, Oughtred (1574-1660), theHindus, and the Europeans since the middle of the seventeenth century (Nesselmann, cited in [2,p. 111]. Viète combined the methods of analysis explained byPappus with the methods used by Diophantus to create one of his mostprominent works, In artem analyticem isagoge or Introduction to theAnalytic Art. This book was written during the second ofViète's "highly fruitful periods of leisure and research" [3,p. 267], 1584-1589. Another well-known production of Viètewas his Harmonicum Coeleste or Harmonic Construction of the Heavens,which was composed during his first period of great discoveries(1564-15 68). "In 1579 Vieta published his Canon mathematicus seu adtriangular cum appendicibus, which contains very remarkablecontributions to trigonometry. It gives the first systematicelaboration in the Occident of the methods of computing plane andspheri cal triangles by the aid of the six trigonometric functions"(Braunmhl cited in [2, p. 137]. Viète was a forerunner inmany aspects of modern day algebra. He was "the first to applyalgebraic transformation to trigonometry, particularly to themultisection of angles" [2, p. 138]. He established "the earliestexplicit expression for ( by an infi nite number of operations" [2, p. 143]. He "employed the [use of] the Maltese cross (+) as theshort-hand symbol for addition, and the (-) for subtraction" [2, p. 139]. Although the Germans introduced these characters, they were notcommonly used before Viète's time. They became more popular after Viètebegan to use them. Among the various concepts propagated byViète, his most well known "contribution to the development ofalgebra was his espousal and consistent utilization of the letters ofthe alphabet - he called these species - to represent both theconstant and the v ariable terms in all equations" [9, p. 5]. "Diophantus seems the most likely source for Vieta's use of the word'species'" [7, p. 321]. Viète referred to this new algebra,which used letters, as logistica speciosa, as opposed to the oldalgebra, logisti ca numerosa, which used numbers. "Letters had beenused [previously] to represent the quantities that entered into anequation, but there had been no way of distinguishing quantitiesassumed to be known from those unknown quantities that were to befound " [1, p. 335]. As a solution to this dilemma, Viètedesignated consonants for known quantities, and used vowels forunknown quantities. The creation of this system, also known asliteral notation (a notation in which letters stand in place ofnumbers), allowed mathematicians to solve equations in a more generalform, rather than having to work out each specific equation on anindividual basis. Whereas one once had to deal with each separateequation as a unique problem to be solved, one can now derive auniversal way of solving all problems alike. For instance, "literalnotation made it possible to build up a general theory of equations -to study not an equation like [6x2 - 5x - 1 = 0] but the quadraticequation ax2 + bx + c = 0" [1, pp.335-336]. Th is was a significantbuilding block in the foundation of algebra as we know it today. Eventhough this invention was quite important, "the vowel-consonantnotation of Vieta had a short existence" [1, p. 336]. AfterViète's death, Descartes began using the "letters at thebeginning of the alphabet...for given quantities, and those near the end (especially [z]) for the unknown. This rule was rapidlyassimilated into seventeenth century practice and has survived tomodern times" [1, p. 336]. On essential fact to add about hisnotation is that "he restricted the use of letters to positive quantities" [6, p. 94]. Unlike today's algebra, where we use lettersto denote both positive and negative numbers, Viète did notinclude negative quantities in his logistica speciosa. How werequantities expressed before Viète's introduction of logisticaspeciosa? "For unknown quantities [Diophantus] had only one symbol, ?"[2, p. 61]. He too did not recognize negative quantities. If asolution was not positive, he deemed the answer impossible and the problem to be faulty. Brahmagupta, on the otherhand, called the unknown quantity yavat-tavat. "When several unknownquantities occurred, he gave, unlike Diophantus, to each a distinctname and symbol. The first unknown was designat ed by the generalterm 'unknown quantity.' The rest were distinguished by names ofcolors, as the black, blue, yellow, red or green unknown" [2, p. 93]. "The Indians were the first to recognize the existence of absolutelynegative quantities" [2, p. 93]. In addition to Viète's conception of algebraic notation, healso offered some valuable insight on powers. "Prior to Viète,it was common practice to use different letters or symbols for thevarious powers of a quantity" [4, p. 131]. For example, althou ghDiophantus only had a single variable, he used three different symbolsto represent that one quantity: he signified our x, x2 and x3 withthree different symbols: ?, ?? and ??, respectively. The threedissimilar symbols make it difficult to comprehend that one is a square or cube of the other. Perhaps if we saw '?' and'??' or '? ?' we might more easily be able to decipher that '??' or '??' is the square of '?'. Viète, on the other hand, simplifiedthis by using A, A quadratum and A cubum, which wa s later shortenedto A, A q and A c. Even though this does not display powers asexponents as we use them today, his way makes it much easier tounderstand that A quadratum is the square of A and A cubum is the cubeof A. Our "present system of indices - x, x2, x3, and so on," wasintroduced by Descartes [4, p. 131], although there are examples ofvery large exponents (up to 45) where Viète resorts to numbersrather than his usual abbreviations, as is with the equation putforward by Adrianus Romanus. "In 1593 the Dutch mathematicianAdrianus Romanus proposed to all the mathematicians the problem ofsolving a certain equation of degree 45. The ambassador of theNetherlands at the court of the French king Henri IV claimed thatnobody in France would be able to solve this problem" [10, p. 65]. Viète proved theambassador wrong, and solved the equation the very next day. Branching off from powers is Viète's Law of Homogeneity. Waerden states: only magnitudes of "like genus" can be compared oradded. Thus, where we would write a quadratic equation as bx2 + dx =z, Viète writes "B in A Quadratum, plus D plano in A, aequari Zsolido". Namely, by Viète's Law of Homogeneity, if A (our x)and B (our b) are line segments, D must be a plane area and Z avolume. Hence he writes "D plano" and "Z solido" [10, pp.63-64]. Insimpler terms, the combined powers of each term must be the same inorder to add, subtract, multiply or divide them. Thus, in the examplenoted above, B (our b) is a line segment (or a length, side or root)with a power of 1; A Quadratum (our x2) is a plane (or a square) witha power of 2; together they comprise a power of 3 (a length and aplane). Furthermore, D plano in A (our dx) is also a length (A or x:power of 1) and a plane (D or d: power of 2) with a combined power of3; and Z solido (our z) is a solid (or a cube) with a power of 3. Thecombined power of bx2 is 3, (b's power is 1, x2's power is 2), of dxis 3 (d's power is 2, x's power is 1) and z is 3. Hence, you areadding three terms, all with the same power of 3. However, this lawis very confusing and it "implies a serious restriction of thealgebraic formalism...Omar Khayyam managed to circumvent thisrestriction by introducing a unit of length e" [10, p. 64]. 041b061a72


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