California State University, Northridge
lVIATHEIV;ATICS SElVIINAR COURSE
A paper submitted in partial satisfaction of the requirements for the degree of Master of Science in
Mathematics
by
Harry James Pappas
May 1972 The paper of Harry James Pappas is approved:
California State University, Northridge May 1972 TABLE OF CONTENTS
Page I, INTRODUCTION 1 Bibliography 15 II. THE MATHEMATICS SEMINAR 17 Bibliography 25 III. HISTORY OF MATHEMATICS 26 Behavioral Objectives 26 Preface 26 Introduction to the Topic 27 Examples 53 Suggested Problems/Topics for Student Research 63 References for Student Research 64 Bibliography 68 IV. NON-EUCLIDEAN GEOMETRY 69 Behavioral Objectives 69 Preface 69 Introduction to the Topic 81 Examples 96 Suggested Problems/Topics for Student Research 103 References for Student Research 103 Bibliography 106 v. LINEAR PROGRAMMING 108 Behavioral Objectives 108 Preface 108 Introduction to the Topic 109 Examples 114 Suggested Problems/Topics for Student Research 134 References for Student Research 135 Bibliography 137 TABLE OF CONTENTS
Page VI, GAME THEORY 138 Behavioral Objectives 138 Preface 138 Introduction to the Topic 140 Examples 144 Suggested Problems/Topics for Student Research 153 References for Student Research 153 Bibliography 155 VII. THEORY OF NUMBERS 157 Behavioral Objectives 157 Preface 157 Introduction to the Topic 159 Examples 165 Suggested Problems/Topics for Student Research 176 References for Student Research 177 Bibliography 179 CHAPTER I INTRODUCTION
We should provide courses that reveal mathematics as a way to understanding "almost every corner of human thought."(3, P• 46) The role of mathematics in modern society is a very significant one. Mathematical methods have penetrated many other fields of knowledge, changing them in substance and power, and have actually brought new disciplines into being. Mathematics has become an indispensable tool in economics, sociology, biology, linguistics, etc. New mathematics- based fields have been created, such as computer science, information theory, and cybernetics. Today any domain that desires rational thinking is more and more in debt to mathematics. Within mathematics itself has come a great revolution since the turn of the century; its foundations have been reappraised and unifying mathematical disciplines have been created, making possible a better view of classical mathe matics and opening new areas and methods of research and application. Until 1930, mathematics applications were restricted to small areas accessible only to a very limited number of people. Now mathematics has become an important "social factor," making the planning of a good mathematics 2 curriculum an educational challenge and responsibility of the highest order.(l6 ) Although there has been much difference of opinion among secondary mathematics teachers as to just how curri culum should be changed, there seems to be general agreement that changes were necessary--thus giving rise to so-called accelerated programs, such as a full year of calculus; or enrichment, within existing courses, by introducing at a simplified or intuitive level topics that will be studied in depth at a later time.(S) A current problem in high school mathematics curricula is what mathematical topics ought to be taught to seniors and advanced juniors. Woodby (1965) in a survey of emerging twelfth•grade programs, concluded that no particular program seems to be the most appropriate one at the present; however, he noted that both acceleration and enrichment were included in what he labeled "strong mathematical programs." Many writers feel that acceleration is often overemphasized; for example, Grossman (1962) argues that too often enrichment is slighted in favor of acceleration. The twelfth-grade mathematics program offers an excel lent opportunity for enrichment by means of seminars, mathematics laboratories, or independent study. (5) Although at present some degree of agreement has been evolved regarding the mathematics curriculum through grade eleven, various proposals are still under discussion 3 involving the twelfth-grade program: some probability theory, matrix algebra and elementary functions, calculus, computer mathematics, and number theory. The program remains undecided throughout the country, Essentially, the updating of mathematics programs involves a change in content and approach. The change in content presents a desire to impart knowledge of topics now of greater importance than formerly, such as inequalities, number fields and probability, necessitating a drop from the curriculum of such items as extensive logarithmic calculations and solid-geometry proofs. However, no longer can a student learn in school all the mathematics he might be expected to use in later life. By concentrating on any topic, certain others are omitted, and students are unprepared for possible new uses of mathematics. This, then, ties to a change in approach. By emphasiz ing the structure of the subject, it is desired that students become sufficiently skillful in handling mathematical systems that they will be able to learn new mathematics more easily in the future. (7) As Wagner< 20 • p. 454) so aptly stated: Since no one can predict with certainty his future profession, much less foretell which math skills will be required in the future by a given profession, it is important that math be taught that students will be able in later life to learn new mathematical skills which the future will surely demand of them. 4
Basically, in the evolving mathematics curriculum (as only one part of our changing knowledge and information explosion), the teacher functions differently to promote a different type of learning, Emphasis is shifting away from the mastery of old knowledge to techniques for developing access to new knowledge (in this process existing knowledge remains important and cannot be discarded; it is simply reorganized). Thus, the teacher functions less autocrati cally and instead directs student inquiry, creating an environment where learning is maximized, The teacher functions as a resource and a coordinator, retaining the decision-making role in curriculum. Much of the classroom activity is open-ended, with the teacher the operational curriculum writer or developer, initiating changes and modifying existing format where necessary,(ll) In class, the mathematics teacher is aware of teaching only a small fragment of one of the great mathematics systems, The teacher should convey this to the students. Almost every course has little mathematics significance except as it relates to the mathematics that follows and to that which has preceded, To equalize this emphasis, teachers should feel more of a responsibility for giving new meaning, new interpraation to previous work and to developing better programs for the training of mathematics scholars. 5
In selecting subject matter for a course, a good teacher of mathematics should think through these three points:(l2) 1. Selecting material so that the resulting structure gives the impression of unity and completeness and provides an adequate founda tion for future extensions. 2. The utilitarian aspects of mathematics that have made possible the great technological and scientific achievements should be stressed- this can coexist with the pure aspects of mathematics, all too often stressed to the exclusion of all else. No teacher has a better opportunity to achieve a balance between the practical and the impractical. 3. Mathematics, along with all other subjects, should accept the responsibility for providing experiences to enhance cultural breadth and appreciation. A teacher of mathematics must believe and emphasize that there is a litera ture of mathematics as important to man's development as the great classics. It has been clearly shown as far back as 1950 that high IQ often accompanies low creativity because of adult and peer lack of attention or sympathy toward creativity. However, this originality, lack of conformity, and a tendency to see something different and new even in widely accepted usages and procedures can and should be fostered and directed. The gifted are reluctant to accept the obvious and reach the final solution too soon; they can combine ideas that are usually considered unrelated. Repetitions, reviews and 6
routine methods of presentation bore the gifted. Their pace of learning is high; they keep seeing new things and raising new questions. In mathematics, it becomes not just a ques- tion of solving harder and harder problems for these students, but imaginative thinking--as mathematics has areas of "simplicity, ingenuity, beauty and utility, like linear programming, finite mathematics, nonmetrical geometry, and theory of games."(lB, P• 396) Teachers must cope with the phenomenon of highly creative individuality mixed with a good deal of ambiguity and absence of rigor in speculation. The content of mathematics courses taught continues to be determined largely by the basic textbook. Although most schools have reorganized their mathematics curricula to some degree since 1960,<14) this has been effected mainly by the adoption of recently copyrighted textbooks and sub sequent changes in course content. Student boredom with textbooks among the gifted should lead in addition to such work as, for example, the more stimulating records of the triumphs, breakthroughs, and pitfalls of the masters. "The genius of Archimedes in working with a very inconvenient numeration system, the simple but profound contributions of Euler and Gauss in the theory of numbers, the absence of rigor in Newton, and the way Descartes expounded his coor dinate geometry without the use of negatives--all provide a psychological appeal for the gifted youngsters."(l8, P· 396) 7
Even among the gifted there is not a homogeneous class- there can be seen instead roughly three levelss(lB) (1) a flair for generalization and extension; (2) ingenuity used to devise new techniques, like Farey's series, and logarithms; and (3) breaking into a totally new line of thinking, like Newton's calculus and Galois' group theory. It seems little enough to expect that the educated person and certainly the gifted, whether he proposes to be a mathematics major or not, should be introduced to the following before he finishes his study of mathematics: 1. The historical growth of major mathematical concepts. 2. The growth of Greek mathematics as a model for systems of thought in every field. 3. The philosophical and religious controversies raised by mathematical and scientific dis coveries. 4. The influence in many fields of the great mathematics discoveries of Euclid, Kepler, Descartes, Newton, Einstein. 5. The mathematical basis of music. 6. The mathematical influence in art and archi tecture. 7. Mathematical forms in nature. B. Probability, statistics, and statistical inference in the social and biological sciences. 9. Computers and their social significance. 8
10. Physical and philosophic implications of the development of non-Euclidean geometry. 11. The process of generalizing a formula from a set of empirical data. (J, PP• 48-49) All of these topics can be presented in a manner com prehensible to high school students and in a form which will allow the interrelationships between various fields to be seen. Frequently, the historical beginnings provide an excellent basis to begin numerous topics. It is a little known fact that, to assist in learning, there has been a tremendous increase since the 1960's(4) in the number and variety of well-written reference books about mathematics published each year, representing a rich source of informa tion on the historic, cultural, social, biographic, philo sophic, and artistic aspects of mathematics, in addition to pure and applied mathematics knowledge. Such aids encourage the use of laboratory methods, problem sets, discussion topics and interdepartmental assistance from colleagues in physics, history, art, music and philosophy. The variation, of course, will be in amount and intensity of material covered. One such example of learning in mathematics for the well-educated was instituted with such a philosophy in mind. At the suggestion of the late Professor Max Beberman, the director of the University of Illinois Committee on School Mathematics (UICSM), and under the leadership and instruction 9 of Hyman Gabai, a sta·ff member of UISCM, an experimental mathematics course was taught to several 11 bright" twelfth grade University High students in 1965-66 (no judgment was made as to the feasibility of teaching such a class to "average" high school seniors). While not settling the question of what to teach in mathematics during the senior year, the course offers insights into possibilities for other such courses. The objectives of the course were to: 1. Help the students gain a deeper understanding and new insights into the foundations and some of the fundamental concepts of mathematics. 2. Broaden the students' backgrounds anQ provide them with a wider view of the nature of mathe matics and the vast range of problems to which it may be applied. 3. Provide the students with a preview of some topics that are usually discussed in first year college mathematics courses. Course content included these topics: 1, Introduction to the Real Number System 2. Absolute Value and Inequalities J, Mathematical Induction and Other Topics 4. Combinations, Permutations, and Propability 5. Introduction to the Theory of Games 6. The Complex Number System 7. The Fundamental Theorem of Algebra 8. Theory of Polynomial Equations 9. Quadratic Equations in Two Variables 10
10. The Completion Postulate 11. Topological Concepts 12. Limits and Continuity Plans, ideas and sources of reference constantly changed due to the newness of the c By its very nature, heuristic does not determine one particular approach in any aspect of classroom instruction, It does define a broad category of instructional methods, delineated by f~ur general characteristics:<6 • P• 494 ) 1. Approaches content through problems. 2. Reflects problem-solving techniques in the logical construction of instructional pro cedures. J, Demands the flexibility for uncertainty and alternate approaches. 4. Seeks to maximize student action and parti cipation in the teaching-learning process, Finally, much of the problem of any learning experience lies in (1) the background and knowledge of the teacherp and (2) the conditions prevalent for effective learning, With regard to mathematics instruction, it is the recom mendation of the National Council of Teachers of Mathematics that every teacher of secondary mathematics should have completed successfully a five-year program in mathematics, culminating in the master's degree, Teachers of mathematics in grades nine to twelve need, in addition to a strong program of general education, at least 24 semester hours, including a full-year program in aftculus, in courses selected from: 1. Analysis--trigonometry, plane and solid analytic geometry9 and calculus. 13 2. Foundations of mathematics--theory of sets, mathematical or symbolic logic, postulation systems, real and complex number systems. 3. Algebra--matrices and determinants, theory of numbersp theory of equations, and structure of algebra. 4. Geometry--Euclidean and non-Euclidean, metric and projective, synthetic and analytic. 5. Statistics--probability and statistical in ference. 6. Applications--mechanics, theory of games, linear programming, and operations research.(l5) And, with regard to conditions for effective learning, the following give some idea of the diversity necessary to assist the students 1. The student must have experiences that give him an opportunity to practice the kind of behavior implied by the objective. 2. The learning experiences must be such that the student obtains satisfactions from carrying on the kind of behavior implied by the objective. 3. The motivation of the learner; that is, the impelling force for his own active involvement, is an important condition. 4. The learner must find previous ways of reacting unsatisfactory so that he is stimulated to try new ways. 5. The learner must have some guidance in trying to carry on the new behavior he is to learn. 14 6. The learner should have ample and appropriate materials on which to work. 7. The learner should have time to carry on the behavior, to practice it until it has become part of his repertoire, B. The learner should have opportunity for a good deal of sequential practice. Mere repetition is inadequate and quickly becomes ineffective. 9. The learner should set standards for himself that require him to go beyond his performance, but standards that are attainable. 10. The learner must have means of judging his per formance to tell how well he is doing.(l9, PP• 27-28 ) 15 CHAPTER I BIBLIOGRAPHY 1. Betz, William. "Five Decades of Mathematical Reform- Evaluation and Challenge," The Mathematics Teacher (October, 1967), PP• 600-610. 2. Carnett, George s. "Is Our Mathematics Inferior?," The Mathematics Teacher (October, 1967), pp. 582-587. Eilber, Charles L. "College Preparatory Mathematics: Preparation for What," The Mathematics Teacher (January, 1968), pp. 46-49. 4. Gabai, Hyman. "An Experimental Twelfth-Grade Mathematics Course," The Mathematics Teacher (April, 1967), pp. 375-380. Goff, Gerald K,, and Johnson, Hiram D. "An Unexpected Field," The Mathematics Teacher (April, 1971), PP• 301-304. 6. Higgins, Jon L. "A New Look at Heuristic Teaching," The Mathematics Teacher (October, 1971), pp. 487-494. Hollingshead, Irving. "Number Theory--A Short Course for High School Seniors," The Mathematics Teacher (March, 1967), pp. 222-227. 8. Holmes, Joseph E. "Enrichment or Acceleration?," The Mathematics Teacher (October, 1970), pp. 471-4'?J:"""" 9. Johnson, Mauritz. "Who Discovered Discovery," Phi Delta Kappan (November, 1966), pp. 120-123. 10. Jones, Phillip s. "Discovery Teaching--From Socrates to Modernity," The Mathematics Teacher (October, 1970), pp. 501-509. 11. Laux, Dean M. "A New Role for Teachers?," Phi Delta Kappan (February, 1965), pp. 265-268. 12. Newsom, Carroll V. "A Philosophy for the Mathematics Teacher," The Mathematics Teacher (January, 1969), pp. 19-23. 16 13. Phenix, Philip. .B§alms of Meaning. New York: McGraw- Hill Book Company, 1964. 14. Schaefer, Sister Mary G, "Revision of Secondary Mathe matics in a Selected Number of Schools," The Mathe matics Teacher (February, 1968), pp. 1.57-161. 1.5. Secondary School Curriculum Committee, National Council of Teachers of Mathematics. "The Secondary Mathematics Curriculum," The Mathematics Teacher (May, 19.59), pp. 389-417. 16. Steiner; Hans-Georg. "Some Aspects of a Modern Pedagogy of Mathematics," The Mathematics Teacher (May, 1970), pp. 441-44.5. Strutmatter, Kenneth, and Rockwood, Gerald J. "Twelfth Grade High School Mathematics: Calculus or ?," The Mathematics Teacher (January, 1968), pp. 39-41.--- 18. Sunwasan, P. K. "Detection and Care of the Gifted in Mathematics," The Mathematics Teacher (April, 1968), pp. 396-399. 19. Tyler, Ralph w. "New Dimensions in Curriculum Develop ment," Phi Delta Kappan, (September, 1966), pp. 2.5-28. 20. Wagner, John. "Objectives and Activities of SMSG," The Mathematics Teacher (October, 1960), pp. 4.54-4.56.--- 21. Whitehead, Alfred N. "Mathematics and Liberal Education," The Mathematics Teacher (May, 1968) (original publi cation, 1912), pp • .509-.516. 22. Young, J. W. A. "The Teaching of Mathematics," The Mathematics Teacher (March, 1968), pp. 287-29~ 17 CHAPTER II THE MATHEMATICS SEMINAR A great discovery solves a great problem, but there is a grain of discovery iri the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play you own inventive facilities, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.(), P• v) The traditional mathematics curriculum of our secondary schools has been undergoing change for many years. Many of the suggestions for improvements have been published--these suggestions have shown greatest diversity with regard to the twelfth-grade program. However, at present, there is in general, agreement on the mathematics curriculum through grade eleven. The curriculum will continue to change at all levels in content and extent to meet the demands of our ever- changing environment. Mathematics that were formerly offered in the freshman and sophomore years of college during the mid nineteenth century have been either pushed down into the high school or removed from the curriculum altogether. This has happened because of pressures from the scientific world, pressures from students who want to learn more and pressures 18 from the next higher educational level--pressures which have become even more intense in recent years and which show no sign of abating in the future. Both commercial and experimental modern mathematics programs usually have the broad objective of updating the subject to meet the changing needs of the modern society. Therefore, essentially, the updating of mathematics programs consists of two parts--a change in content and a change in approach.( 2) The purpose of this paper is to attempt to enhance the current mathematics curriculum of the twelfth-grade (and pos sibly the eleventh-grade) program by proposing: The establishment of a Mathematics Seminar course for advanced students of mathematics to be offered as a one-semester elective and/or as a summer school enrichment course to those students who have success- fully completed at least trigonometry, which implies completion of at least three years of college preparatory mathematics, including one year of algebra, one year of geometry 0 and one year of a combination of advanced algebra and trigonometry or an equivalent mathematics background under different mathematics course titles. Special application will be made to Santa Monica High School where the writer has been a member of the Mathematics Department since 1964o 19 The writer intends to present this paper to Santa Monica High School in order to establish such a Seminar Course as soon as it is feasible to include it in the existing curriculum, At this time, the Director of Curriculum for the high school and the Chairman of the Mathematics Department believe that the Mathematics Seminar Course appears to have merit for in clusion in the curriculum as an enrichment elective and quite possibly as an acceptable college-preparatory mathematics course. It is envisioned that this Mathematics Seminar will enrich the background of those interested students who meet these requirements: 1. Successful completion of at least trigonometry 2. Recommendation of two mathematics teachers 3. Completion of grade eleven (with possible exceptions) It is perhaps desirable at this point to indicate the current (1972-73) mathematics program at Santa Monica High School in order to give a complete picture of the place of the proposed Mathematics Seminar: Grade 10: Two semesters: Modern Geometry or One semester: Intermediate Mathematics (Algebra 3) One semester: Trigonometry Grade 11: One semester: Intermediate Mathematics (Algebra 3) One semester: Trigonometry or 20 Grade 11: One semester: Algebra 4 (includes mathematical (Cont.) induction; theory of equations; matrix algebra; systems of equations; including quadratics, equations of degree 3 and greater; analytic geometry; vectors and matrices; and basic calculus) One semester: Probability and Statistics Grade 12: One semester: Algebra 4 One semester: Probability and Statistics or One semester: Calculus (at Santa Monica High School; can take Advanced Placement Test after completion of course for college credit) or One semester/ One year: Calculus (at UCLA or Santa Monica College for college credit. UCLA administers a general ability and mathematics test before allow ing admittance to the course. One semester: Modern Algebra (by teacher recom mendation only; class consists of 10-15 students, which is about 1/3 of 1% of the school enroll ment of 2800-3300. Text by Birkhoff and MacLane. Although the Seminar Course could be easily adapted to modular or other innovative scheduling, it is designed at I'resent to adapt to and enrich the Santa Monica High School scheduling. The writer foresees no undue problems in adjusting to a different pattern, should the need arise: One semester: 75 class hours (five 50-minute class meetings per week for 18 weeks) Summer School: 65 class hours (five 130-minute class meetings per week for 6 weeks) 21 The word seminar can denote the following: A small group of students engaged in advanced study and original research under a member of the faculty and meeting regularly to exchange informa tion and hold discussions.<4> Study, branch of learning; course, course of study; three R's, ABC's; reading, writing and arithmetic; liberal arts; humanities, humanism.(5, P• 560 •7) A small group of advanced students in a college or graduate school engaged in original research under the guidance of a professor who meets regu larly with them for reports and discussions. A course of study so pursued, A scheduled meeting of such a group. A meeting for an exchange of xeas in some areas; conference,(l, P• ll78) Specifically, it is the writer's proposal to develop the Mathematics Seminar Course in the following manner: 1. To introduce each topic (selected topics for the first Seminar Course are listed below and developed in detail in subsequent chapters) to the Seminar group. 2. To list the behavioral objectives for each topic. 3. To lead the group (by lecture, demonstration, discussion, research where applicable) appro priately into each topic (this will vary from topic to topic and from teacher to teacher). 4. To supply a thorough bibliography for each topic and to maintain as complete a classroom library as possible. 22 5. To disseminate the problems or topics for seminar members to investigate (with variation as warranted; students work alone, in pairs, or in small groups). 6. To have student presentation of research shared by all seminar members by always writing work and solutions on ditto masters and by use of the various following methods of presentation: a. Chalkboard b, Overhead projector c. Film, film strip, or slides d. Discussion group, led by person(s) who have researched the problem(s) or topic(s). e. Other methods originated by seminar members. 7. To have pertinent guest speakers. 8, To take field trips that will supplement the group's interest in mathematics. 9. To have evaluation of student performance by the student himself, his peers, and the instructor. (Examinations are optional,) The ideal size for the seminar probably is 8-12 students, but due to high school district policies regarding class size, a more realistic number is undoubtedly 15-25 students. The seminar classroom should contain or have easy access to a duplicating machine and a copying machine. The classroom library, mentioned above, should include, minimally, the following: 1. Many books on each topic included in the seminar course. 23 2. Mathematics journals (from current issues going back at least four or five years): a. The Mathematics Monthly b. The Mathematics Teacher c. School Science and Mathematics d. Scientific American e. Others deemed appropriate by students and/or teacher 3. Mathematics Dictionary. 4. Four-volume set of the World of Mathematics by Newman. A seminar is an open-ended and ever-changing course, vary- ing in structure, content and approach as topics, students, and teachers change. To initiate the course and give it a fundamer1tal structure from which to proceed in future years, the writer has selected the following topics to comprise the first Mathematics Seminar Course (these topics have been selected based upon enhancing the current Santa Monica High School mathematics curriculum and are thought to represent imperative topics of enrichment in mathematics at the high school level by mathematicians ar1d mathematics curriculum specialists): 1. History of Mathematics 2. Non-Euclidean Geometry 3. Linear Programming 4. Game Theory 5. Number Theory 24 In the chapters to follow, each of these five topics will be presented separately, giving: 1. Approximate length of time to be devoted to topic 2. Behavioral objectives 3. Introduction to topic 4. Topic presentation, including: a, Definition of terms, where applicable b. Explanation of concepts c, Examples, where applicable d. Suggested problems/topics for seminar members to pursue e. Bibliography of teacher's presentation 5. Suggested student bibliography 25 CHAPTER II BIBLIOGRAPHY 1. The American Herita e Dictionar of the En lish Lan ua e. Boston= Houghton-Mifflin Company, 19 9. 2. Hollingshead, Irving. "Number Theory--A Short Course for High School Seniors," The Mathematics Teacher, (March, 1967), pp. 222-227. 3. Polya, G. How to Solve It. Garden City, New York; Doubleday and Company, Anchor Books, 1957. 4. The Random House. The Random House Dictionary of the English Language. New York: 1966. 5. Roget's International Thesaurus, Third Edition. New York: Thomas Crowell Company, 1962. 26 CHAPTER III HISTORY OF MATHEN~TICS Approximate Length of Time for Unit Three to four weeks. Behavioral Objectives 1. The student will develop an awareness of the vast ness of the history of mathematics. 2. The student will gain more mathematical knowledge by solving ancient problems, using both ancient and modern methods, J, The student will demonstrate an historical knowl edge of mathematics by researching some problem studies from the past and sharing this knowledge with his peers. 4. The student will research famous mathematicians; eras· of mathematics; schools of mathematical thought; interrelationships of ancient mathe matics, mathematicians, and their countries; gaining an appreciation of the development of mathematics over the centuries. 5. The student will be continually exploring the historical aspects of any of his future mathe matical endeavors. Preface The writer has often felt that the mathematics curricula of most schools is seriously lacking in almost complete 27 neglect of the historical aspects of the various studies of mathematics. Not only will the study of the history of mathe matics greatly enhance the appreciation and knowledge of mathematics for all involved, but it should add a great deal of interest to a subject considered abstract and technically remote to many students. Introduction to the Topic The history of mathematics, even that of elementary mathematics, is so vast that only an introduction to the subject is possible in a one-semester course.<2• P• 3) Obviously, the purpose of offering a three- or four-week unit on the history of mathematics as part of this Mathematics Seminar can, at best, only be to expose and motivate the students to the history of mathematics. A most appropriate introduction to this topic would be to first expose the students to a chronological table of the history of mathe- matics: A Chronological Table ( 2 • PP • 40 3-LHl) -4700 Possible beginning of Babylonian calendar. -4228 Probably introduction of Egyptian calendar. -4000 Discovery of metal. -3500 Writing in use. -3100 Approximate date of a royal Egyptian mace in a museum at Oxford. -2900 Great pyramid of Gizeh erected. -~2400 Babylonian tablets of Ur. 28 -2200 Date of many mathematical tablets found at Nippur; mythical date of the lo-shu, the oldest known example of a magic square. -1950 Hammurabi, King of Babylonia. -1850 Moscow papyrus (25 numerical problems, "greatest Egyptian pyramid"); oldest extant astronomical instru ment, -1750 Plimpton 322 dates somewhere from -1900 to -1600. -1650 Rhind, or Ahmes, papyrus (85 numerical problems)~ -1600 Approximate date of many of the Babylonian tablets in the Yale collection. -1500 Largest existing obelisk; oldest extant Egyptian sundial, -1350 Date of later mathematical tablets found at Nippur; Rollin papyrus (elaborate bread problems). -1167 Harris papyrus (list of temple wealth). -1105 Possible date of the Chou-pei, oldest Chinese mathemati cal work. -753 Rome founded. -650 Papyrus had been introduced into Greece by this date. -600 Thales; beginning of demonstrative geometry. -540 Pythagoras (geometry, arithmetic, music). -500 Possible date of the Sulvasutras (religious writings showing acquaintance with Pythagorean numbers and with geometric constructions). -4·80 Battle of Thermopylae. -460 Parmenid.es (sphericity of the earth). -450 Zeno (paradoxes of motion). -440 Hippocrates of CHbs (reduction of the duplication prob lem, lunes, arrangement of the propositions of geometry in a scientific fashion); Anaxagoras (geometry), -430 Antiphon (method of exhaustion). -425 Hippias of Elis (trisection with quadratix); Theodorus of Cyrene (irrational numbers); Socrates. -410 Democritus (atomic theory) -404 Athens finally defeated by Sparta -400 Archytas (leader of Pythagorean school at Tarentum, application of mathematics to mechanics). -380 Plato (mathematics in the training of the mind, Plato's Academy). 29 -375 Theaetetus (incommensurables, regular solids). -370 Eudoxus (incommensurables, method of exhaustion, astronomy). -350 Menaechmus (conics); Dinostratus (quadrature with quad ratrix, brother of Menaechmus); Xenocrates (history of geometry); Thymaridas (solution of systems of simple equations). -340 Aristotle (systematizer of deductive logic), -336 Alexander the Great began his reign. -335 Eudemus (history of mathematics). -332 Alexandria founded, -323 Alexander the Great died. -320 Aristaeus (conics, regular solids). -300 Euclid (Elements, perfect numbers, optics, data). -280 Aristarchus (Copernician system), -260 Conon (astronomy, spiral of Archimedes); Dositheus (recipient of several papers by Archimedes). -250 Stone columns erected by King Asoka and containing arliest preserved examples of our present number symbols. -240 Nicomedes (trisection with conchoid), -230 Eratosthenes {prime number sieve, size of the earth). -225 Apollonius (conic sections, plane loci, tangencies, circle of Apollonius); Archimedes (greatest mathemati cian of antiquity, circle and sphere, computation of , area of parabolic segment, spiral of Archimedes, infinite series, method of equilibrium, mechanics, hydrostatics). -180 Hypsicles (astronomy, number theory); Diocles (dupli cation with cissoid). -140 Hipparchus (trigonometry, astronomy, star catalogue), -100 Probable date of carvings on the walls of a cave near Poona. -75 Cicero discovered the tomb of Archimedes. 75 Possible date of Heron (machines, plane and solid men suration, root extraction, surveying). 100 Nicomachus (number theory); Menelaus (spherical trig onometry); Plutarch. 150 Ptolemy (trigonometry, table of chords, planetary theory, star catalogue, geodesy, Almagest). 200 Probable date of inscriptions carved in caves at Nasik. 30 250 Probable date of Diophantus (number theory, syncopation of Algebra). 300 Pappus (Mathematical Collection, commentaries, isoperi metry, projective invariance of cross ratio, Castillon Gramer problem, arbelos theorem, generalization of Pythagorean theorem, centroid theorems, Pappus' theorem). 320 Iamblichus (number theory). 390 Theon of Alexandria (commentator, edited Euclid's Elements). 410 Hypatia of Alexandria (commentator, first women men tioned in the history of mathematics, daughter of Theon of Alexandria). 460 Proclus (commentator). 480 Chinese value of 7T as 355/113. 500 Metrodorus and the Greek Anthology. 505 Varahamihira (Hindu astronomy). 510 Boethius (writings on geometry and arithmetic became standard texts in the monastic schools); Aryabhata the Elder (astronomy and arithmetic). 530 Simplicius (commentator). 560 Eutocius (commentator). 622 Flight of Mohammed from Mecca. 628 Brahmagupta (algebra, eyelid quadrilaterals). 641 Last library at Alexandria burned. 710 Bede (calendar, finger reckoning). 711 Saracens invade Spain. 766 Brahmagupta's works brought to Bagdad. 775 Alculin called to the court of Charlemagne. 790 Harun al-Rashid (caliph patron of learning). 820 Mohammed ibn Musa al-Khowarizmi (wrote influential treatise on algebra and a book on the Hindu numerals, astronomy, "algebra," "algorithm"); al-Mamun (caliph patron of learning). 850 Mahavira (arithmetic, algebra), 870 Tabit ibn Qorra (translator of Greek works, conics, algebra, magic squares, amicable numbers). 871 Alfred the Great began his reign, 900 Abu Kamil (algebra). 920 Al-Battani, or Albategnius (astronomy). 31 980 Abul-Wefa (geometric constructions with compasses of fixed opening, trigonometric tables), 1000 Alhazen (optics, geometric algebra); Gerbert, or Pope Sylvester II (arithmetic, globes). 1020 Al-Karkhi (algebra). 1042 Edward the Confessor became king. 1066 Norman Conquest. 1095 First Crusade. 1100 Omar Kha¥yam (geometric solution of cubic equations, calendar). 1120 Plato of Tivoli (translator from the Arabic); Adelard of Bath (translator from the Arabic). 1130 Jabir ibn Aflah, or Geber (trigonometry). 1140 Johannes Hispalensis (translator from the Arabic); Robert of Chester (translator from the Arabic). 1146 Second Crusade, 1150 Gherardo of Cremona (translator from the Arabic); Bhaskara (algebra, indeterminate equations). 1202 Fibonacci (arithmetic, algebra, geometry, Fibonacci sequence, Liber abaci). 1225 Jordanus Nemorarius (algebra). 1250 Sacrobosco (Hindu-Arabic numerals, sphere); Nasir eddin (trigonometry); Roger Bacon (eulogized mathematics); rise of European universities. 1260 Campanus (translation of Euclid's Elements, geometry). 1271 Marco Polo began his travels, 1303 Chu Shi-kie (algebra, numerical solution of equations). 1325 Thomas Bradwardine (arithmetic, geometry, star polygons). 1349 Black Death destroyed a large part of the European population. 1360 Nicole Oresme (coordinates, fractional exponents). 1431 Joan of Arc burned, 1435 Ulugh Beg (trigonometric tables). 1450 Nicholas Cusa (geometry, calendar refo~m) ; printing from movable type. 1453 Fall of Constantinople 1460 Georg von Peurbach (arithmetic, astronomy, table of sines), 32 1470 Regiomontanus, or Johann Muller (trigonometry). 1478 First printed arithmetic, in Treviso, Italy. 1482 First printed edition of Euclid's Elements. 1484 Nicholas Chuquet (arithmetic, algebra); Borghi's arith metic. 1489 Johann Widman (arithmetic, algebra, +and- signs), 1491 Calandri's arithmetic, 1492 Columbus discovered America. 1494 Pacioli (Suma, arithmetic, algebra, double entry book keeping). 1500 Leonardo da Vinci (optics, geometry) 1506 Scipione del Ferro (cubic equation); Antonio Maria Fior (cubic equation). 1510 Albrecht Durer (curves, perspective, approximate tri section, patterns for folding the regular polyhedra). 1514 Jakob Kobel (arithmetic). 1518 Adam Riese (arithmetic). 1521 Luther excommunicated, 1522 Tonstall's arithmetic. 1525 Rudoff (algebra, decimals); Stifel (algebra, number mysticism); Buteo (arithmetic). 1530 Da Coi (cubic equation); Copernicus (trigonometry, planetary theory). 1545 Ferrari (quartic equation); Tartaglia (cubic equation, arithmetic, science of artillery); Cardano (algebra). 1550 Rhaeticus (tables of trigonometric functions); Scheubel (algebra); Commandino (translator, geometry). 1556 First work on mathematics printed in the New World. 1557 Robert Recorde (arithmetic, algebra, geometry, =sign). 1558 Elizabeth became Queen of England. 1570 Billingsley and Dee (first English translation of the Elements). 1572 Bombelli (algebra, irreducible case of cubic equations). 1573 Valentin Otho found early Chinese value of ff , namely 355/113. v-- 1575 Xylander, or Wilhelm Ho+tzman (translator). .3.3 1580 Francois Viete, or Vieta (algebra, geometry, trigono metry, notation, numerical solution of equations, theory of equations, infinite product converging to 2/rr)• 158.3 Clavius (arithmetic, algebra, geometry, calendar). 1590 Cataldi {continued fractions); Stevin (decimal fractions, compound interest table, statics, hydrostatics). 159.3 Adrianus Romanus {value of r,r, problem of Apollonius). 1595 Pitiscus {trigonometry). 1600 Thomas Harriet (algebra, symbolism); Jobst Burgi (log arithms); Galilee {falling bodies, pendulum, projectiles, astronomy, telescopes, cycloid); Shakespeare. 160.3 Accademia dei Lincei founded (Rome). 1608 Telescope invented. 1610 Kepler (laws of planetary motion, volumes, star poly hedra, principle of continuity); Ludolf van Ceulen {computation of 7T). 1612 Bachet de Meziriac {mathematical recreations, edited Diophantus' Arithmetica). 1614 Napier (logarithms, rule of circular parts, computing rods). 1615 Henry Briggs (common logarithms, tables). 1619 Savilian professorships at Oxford established, 1620 Gunter (logarithmic scale, Gunter's chain in surveying); Paul Guldin {centroid theorems of Pappus); Snell (geo metry, trigonometry, refinement of classical method of computing 77, loxodromes). Mersenne (number theory, Mersenne numbers, clearing house for mathematical ideas); Oughtred (algebra, sym bolism, slide rule, first table of natural logarithms); Mydorge (optics, geometry); Albert Girard (algebra, spherical geometry). 16.35 Fermat (number theory, maxima and m1n1ma, probability, analytic geometry, Fermat's last "theorem"); Cavalieri (method of indivisibles). Descartes (analytic geometry, f + v = e + 2, folium, ovals, rule of signs). 1640 Desargues (projective geometry); de Beaune (Cartesian geometry); Torricelli (physics, geometry, isogonic center); Frenicle de Bessy (geometry); Roberval (geo metry, tangents, indivisible~; de la Loubere (curves, magic squares). 16:4.3 Louis XIV crowned. 1649 Charles I executed, 1650 Blaise Pascal (conics, cycloid, probability, Pascal triangle, computing machines); John Wallis (algebra, imaginary numbers, arc length, exponents, symbol for infinity, infinite product converging to nrz, early integration); Frans van Schooten (edited Descartes and Viete); Gregoire de Saint-Vincent (circle squarer, other quadratures); Wingate (arithmetic); Nicolaus Mercator (trigonometry, astronomy, series computation of logarithms); John Fell (algebra, incorrectly credited with the so-called Fell equation). 1660 Sluze (spirals, points of inflection); Viviani (geo metry); Brouncker (first president of Roy~l Society, rectification of parabola and cycloid, infinite series, continued fractions). 1662 Royal Society founded (London). 1663 Lucasian professorships at Cambridge established. 1666 French Academy founded (Paris). 1670 Barrow (tangents, fundamental theorem of the calculus); James Gregory (optics, binomial theorem, expansion of functions into series, astronomy); Huygens (circle quadrature, probability, evolutes, pendulum clocks, optics); Sir Christopher Wren (architecture, astronomy, physics, rulings on hyperboloid of one sheet, arc length of cycloid). 1671 Giovanni Domenico Cassini (astronomy, Cassinian curves). 1675 Greenwich observatory founded. 1680 Sir Isaac Newton (fluxions, dynamics, hydrostatics, hydrodynamics, gravitation, cubic curves, series, numerical solution of equationsp challenge problems); Johann Hudde (theory of equations); Robert Hooke (physics, spring-balance watches); Seki Kowa (deter minants, multinomial theorem, calculus). 1682 Leibniz (calculus, determinants, multinomial theorem, symbolic logic, notation, computing machines). 1685 Kochanski (approximate rectification of circle). 1690 Marquis de !'Hospital (applied calculus, indeterminate forms); Halley (astronomy, mortality tables and life insurance, translator); Jakob (James Jacques) Bernoulli (isochronous curves, clothoid, logarithmic spiral, probability); de la Hire (curves, magic squares, maps); Tschirnhausen (optics, curves, theory of equations). 1700 Johann (John, Jean) Bernoulli (applied calculus); Giovanni Ceva (geometry); David Gregory (optics, geo metry); Parent {solid analytic geometry). 35 1706 William Jones (first use of 1T for circle ratio). 1715 Taylor (expansion in series, geometry). 1720 De Moivre (actuarial mathematics, probability, complex numbers, Stirling's formula). 1731 Alexis Clairaut (solid analytic geometry). 1733 Saccheri (forerunner of non-Euclidean geometry). 1740 Maclaurin (higher plane curves, physics); Frederick the Great became King of Prussia. 1750 Euler (notation, eirr = -1, Euler line, quartic equation, ~-function, beta and gamma functions, applied mathe matic~. 1760 Comte de Buffon (calculation of 7T by probability). 1770 Lambert (non-Euclidean geometry, hyperbolic functions, map projection, irrationality of 7!). 1776 United States independence. 1780 Lagrange (calculus of variations, differential equa tions, mechanics, numerical solution of equations, attempted rigorization of calculus in 1797, theory of numbers). · 1789 French Revolution. 1794 Ecole Normale Superieure and Ecole Polytechnique founded; Monge (descriptive geometry, differential geometry of surfaces). 1797 Mascheroni (geometry of compasses). 1804 Napoleon made emperor. 1805 Laplace (celestial mechanics, probability, differential equations); Legendre (Elements de geometrie, theory of numbers, elliptic functions, method of least squares, integrals). 1810 Gergonne (geometry; editor of Annales). 1815 "The Analytical Society" at Cambridge. 1820 Gauss {polygon construction, number theory, differential geometry, non-Euclidean geometry, fundamental theorem of algebra, astronomy, geodesy); Poinsot (geometry). 1822 Feuerbach {geometry of the triangle). 1824 Thomas Carlyle (English translation of Legendre's Geometrie). 1825 Bolyai and Lobatchevsky (non-Euclidean geometry); Abel (elliptic functions). 36 1826 Crelle's Journal; principle of duality. 1830 Cauchy (rigorization of analysis, functions of a com plex variable); Poncelet (projective geometry, ruler constructions); Galois (groups, theory of equations); Babbage (computing machines); Peacock (algebra). 1836 Liouville's Journal. 1837 Trisection of an angle and duplication of cube proved impossible. 1839 Cambridge Mathematical Journal, which in 1855 became Quarterly Journal of Pure and Applied Mathematics. 1840 Steiner (geometry). 1841 Archiv der Mathematik und Physik. 1842 Nouvelles annales de mathematigues. 1843 Hamilton (quaternions), 1847 Staudt (freed projective geometry of metrical basis). 1850 Chasles (higher geometry, history of geometry); Cayley unvariants, matrices and determinants, hyperspace); H. G. Grassmann (calculus of extension); Plucker (higher analytic geometry); Mannheim (standa~zed the modern slide rule), 1854 Riemann (Riemann surfaces, Riemann integral, Riemannian geometry), 1855 Zacharias Dase (lightning calculator); Dirichlet (theory of numbers). 1865 London Mathematical Society founded; Proceedings of London Mathematical Society. 1872 Societe Mathematique de France founded; Klein's Erlanger Prograrom; Dedekind (irrational numbers). 1873 Brocard (geometry of the triangle). 1878 American Journal of Mathematics. 1880 George Cantor (irrational numbers, transcendental num bers, transfinite numbers). 1881 Gibbs (vector analysis). 1882 Lindemann (transcendence of zr, squaring of circle proved impossible), 1884 Circolo Matematico di Palermo founded, 1887 Rendiconti. 1888 Lemoine (geometry of the triangle, geometrography); American Mathematical Society founded (at first under a different name); Bulletin of the American Mathemati cal Society. 37 1889 Peano (axioms for the natural numbers), 1890 Weierstrass (arithmetization of mathematics); Deutsche Mathematiker-Vereinigung organized, 1892 Jahresbericht. 1895 Poincare (Analysis situs). 1896 Prime number theorem proved by Hadamard and de la Vallee Poussin. 1899 Hilbert (Grundlagen der Geometrie, formalism). 1900 Transactions of American Mathematical Societ ; Russell and Whitehead Principia mathematic~, logicism), 1906 Frechet (abstract spaces). 1907 Brouwer (intuitionism). 1916 Einstein (general theory of relativity), 1917 Hardy and Ramanujan (analytical number theory), 1931 Godel's theorem, 1934 Gelfand's theorem, 1963 P, J, Cohen on the continuum hypothesis, The vastness of the preceding chronological table cer tainly allows for an almost infinite number of arrangements of topics in the history of mathematics. Given such a short amount of time, this particular segment of the Seminar must indeed be extremely flexible and left to the discretion of the instructor and the developing interests expressed by students. To aid the instructor in narrowing the selection, the writer has included a variety of materials to develop as topics within this unit, There will be much additional hrrstorical information presented in this paper within the Introduction of the other topics that follow, i.e., history of non-Euclidean geometry, history of linear programming and game theory; history of 38 number theory; all of which should further enhance the students' historical perspective, The mathematical importance of the history of mathematics can be shared with students by exposing them to various numeral systems and by showing solutions to some mathematical problems according to the methods of the times and also by modern methods. These thoughts may give the students motiva tion to undertake some of the Suggested Problems/Topics for Student Research in this section. The material that follows includes both a variety of historical topics and problems. California Indian Arithmetic(4) The earlies record of California Indian arithmetic is from the first Spanish Entry into San Francisco Bay, 1775, by Father Vicente Maria, S.F. The second oldest record is at the end of a confessional manual written for Franciscan mis sionaries at Ventura, California, about 1812, by Father Jose Senan, O.F.M. The distribution of Indian number systems throughout the state of California is shown on the map which follows. Addition, subtraction and multiplication were found in the formation of Indian number words; however, there exists no direct evidence of division or sophisticated properties of numbers and their operations or change of base, Refer to the following table, Indian Number Words. 39 Base 4 Base 5 '* Base 10 ~ San Francisco. CALIFORNIA INDIAN NUl\llERAL SYSTEMS INDIAN NUMBER WORDS* San San Luis Obispo Santa Barbara Santa Cruz Is. Buenaventura I 1 • tsxumu paka p~;------,~:;:~"-~·"·----~~aqueet ...... _"~ .. ~,=- i 2 ecin ickom ickomo Jisxum ashcom 3 mica masox masex jmasex maseg 1 . 4 1 paks i ckumu ckumu ckumu lscumu 5 tiyewi yiti-pakas yiti-paka 4+1 lsit-isma litipaques I1 6 , ksua-sya · yiti-ckom yiti-ckomo 4+2 jSit-isxum yetishcom I 7! kcua-mice yiti-masox yiti-masex 4+3 jsit-masex ~itimaseg ~ 8 [ ckomo malawa • malawa 1malawa lmalahua •· 9 l cumo-tcimaxe · ts 'pa tspa jspa ·etspa l1o ! tuyimili · tciya · kel-ckomo ika-ckum cashcom Ill.! tiwapa na-paka 10+1 tulu jtelu Ftelu 1 h2 t takotia · na-ickom 10+2 masex-eakumu !masekapa-ckumu :. aseg scumu I l t ' , 3x4 1 ~· f ,13 I wak-cumu 12+1 !I na-masox 10+3 r kel-paka 12+1 j ~aseg scumu canpa ,. I ! : ! ! ~ queet 3x4+1 14 ! wak1-esin 12+2 · na-ckumu 10+4 l kel-icko 12+2 ! ~eshcom laliet ~ I ! I· i ~ 2 less than 16 ~ 15 1 wakl-mice 12+3 I na-yitipakas . kel-masex 12i:3 i ·waqueet cihue ~.: 1 I I 10+4+1 l 1 ~ 1 less than 16 g 116 ~- psusi t na-yi tickom . pet a i ~chigipsh I' f ! 10+4+2 ! f ~ ' 17 1 , na-yi timesxo ~ 1 !chigipsh canpa- I t t •· l queet 16+1 · 18 ' na-malawa f •·eshcom cihue scu- I rt 10+8 ~ 1 muhuy 20-2 j :19 ' : na-ts 'pa 10+9 J jPaqueet cihue scu- i ~ muhuy 20-1 B 20 ,! . ickom-a-tciua ickom-c-kelckom~ isxum-pas- Iscumuhuy -~ LL______--~·--·-~··••'""·"-'" ..;,..,.~·----~--.. "~-·-··- 2~-~ 2... J.. _, __ ,. ______~-~.~-,,_.t~~~"'--~~~~~... """"~~~·""'--~·--··--·--·- f *All of these Indians spoke the same language. Numbers developed slower than language and various tribes favored certain bases. 41 The Arabic Mentality(6 ) An understanding of the "Arabic Mentality" is necessary for an understanding of the Arabic contributions to mathe matics. Their attitude toward mathematics, their concept of proof, even toward knowledge in general, becomes clear as they are understood. To the Muslim, history is a series of accidents that in no way affect the nontemporal principles of Islam. He is more interested in knowing and "realiz ing" these principles than in cultivating originality and change as intrinsic virtues. The symbol of Islamic civilization is not a flowing river, but the cube of Kaaba, the stability of which symbolizes the permanent and immutable character of Islam. One might say that the aim of all the Islamic science- and, more generally speaking, of all the medieval and ancient cosmological sciences--is to show the unity and interrelatedness of all that exists, so that in contemplating the unity of the cosmos, man can be led to the unity of the Divine Principle, of which the unity of Nature is the image. The ortho doxy based on this creed is intangible, and therefore not so closely bound to specific form~lations of ) dogmatic theology as in Christianity.\6, PP• 21-22 The spirit of Isl&a emphasizes ••• the unity of Nature, that unity that is the aim of the cosmo logical sciences, and that is adumbrated and pre figured in the continuous interlacing of arabesques uniting the profusion of plant life with the geo metric crystals of the verse of the Quran. • • • Herein one can already see why mathematics was to make such a strong appeal to the Muslim: its abstract nature furnished the bridge that Muslims were seeking between multiplicity and unity. It provided a fitting texture of symbols for the uni verse--symbols that were like keys to open the cosmic text. We should distinguish at once between the two types of mathematics practiced by Muslims: one was the science of algebra, which was always related to geometry and trigonometry, the other was the science of numbers, as understood in the Pythagorean sense. The Pythagorean number has a symbolic as well as a 42 quantitative aspect; it is a project of Unity, which, however, never leaves its source, , •• To study numbers thus means to contemplate them as symbols and to be led thereby to the intelligible world. So also with the other branches of mathematics. Even where the symbolic aspect is not explicityly stated, the connection with geometric forms has the effect upon the mind of freeing it from dependence upon mere physical appearance, and in that way preparing it for its journey intg the i~tel6igible world and, ultimately, to Unity.~ • PP· 5-2) The Arabic mathematicians in Bagdad had a foundation in algebra based on long tradition, Indigenous to the land was Babylonian mathematics, enhanced by Greek mathematics. Both of these sources were enriched with the importation of Hindu mathematics. Soon after its founding in 762, Bagdad hosted the scholar of Ujjain, Kanka. He was invited by the caliph Abbasid Al-Mansur to lecture on the arithmetic, algebra and astronomy of Brahmagupta of Ujjain, So impressed was the caliph that he commissioned the Arabic scholar al-Fazari to complete a translation of Brahmagupta's Brahma Sphuta Siddhanta in 770. The algebraic content of Chapter 18 of the Brahma increased mathematical knowledge and expressed some old ideas in new terms: 1, Rules for all operations on positive and negative numbers, including the notice that there are two square roots (negative root also by implication) of every number. 2. Classification of equations in one and more un knowns, both linear and quadratic, together with their general (rhetorical) solutions: ax= c x + y = a and x - y = b ax2 : bx = c (two roots) X - y = a and xy = b ax - by = + c and ax + by = + c ax +by + z = c ax2 + c = y2, 43 The contents of the other chapters of the Brahma consisted of astronomy, rejections and corrections of erroneous mathemati cal ideas, arithmetic and mensuration, spherics, and two chapters which have been lost. In 815 the caliph al-Ma'mun founded the Bait al-hikmah or House of Wisdom, a center of higher learn1ng that included an observatory and a library. It was a gathering place for scholars and translators, and almost the entirety of Greek learning was translated into Arabic to prepare for the assimi lating of that knowledge by Islam. This translation was not merely to increase knowledge but because Islamic society was being challenged by theologians and philosophers of religious minorities, especially by Christians and Jews who lived within its borders. The Muslims found themselves on the losing side in religious and philosophic debates. They could not defend their principles of faith through logical arguments as could the "infideils," nor could they bring logical proofs to demon strate the truth of the tenets of Islam. The authority of the caliphs was based on religious law in Islamic society. To strengthen the law and thereby enhance its own authority, which in turn safeguarded the interests of the Muslim com munity, the early Abbasid caliphs engaged the Arabic scholars in the study of Greek science and philosophy. Of these scholars, the first outstanding mathematician in Bagdad was Muhammad ibn Musa al-Khwarazmi (863). Not only did he synthesize the previous work of the Babylonians, Greeks and Indians, but he made his own contributions. His Algebra (al-Jabr wa'l-muqabalah), the first Muslim work on algebra, gave its name to this science in both East and West. He introduced the Indian numerals to the Muslim world and through his work on arithmetic, the West learned of the numerals called "Arabic." He wrote the first extensive Muslim work on geography, revising much of Ptolemy and drawing new 44 geographical and celestial maps. His astronomical tables are among the best in Islamic astronomy. His influence is attested to by the fact that "Algorism," the Latinization of his name, al-Khwarazmi, for a long time meant arithmetic in most European languages and is used today for any recurring method of calculation which has become an established rule. It has even entered into the technical vocabulary of modern computation techniques.<6 • P• 45) He became bitterly disappointed at the appointment of Thabit ibn Qurrah (826?-901) instead of himself to translate the Greek masters for the caliphs. In turn, he ignored the Greek method of logical proof and developed an intuitive demonstration of his own. His Algebra became well known throughout the Muslim world and was a model for all subsequent algebra texts. It was brought into the Western world in Spain through the translation of Robert of Chester (about 1144) (who also was the first to translate the Quran), and was translated into Latin by Gerard of Cremona (about 1188). John of Seville used three equations from al-Khwarazmi in his Liber Algorismi de Practica Arismetrice (1148); Abu Kamil borrowed a large number of illustrative examples from al-Khwarazmi for his Algebra in 900; and Leonardo da Pisa borrowed from both Abu Kamil and al-Khwarazmi for his monumental Liber Abaci in 1202. The emphasis in al-Khwarazmi's Algebra is on the treatment of quadratic equations. His Babylonian predecessors had reduced all quadratic equations to six types: x + y = a and xy = b 2 2 x ! y = a and x ! y = b. al-Khwarazmi's contribution was to reduce these six to three, two of which he derived from one of the above: + ax = b, the equation for y X - y =a and xy = b e2x2 _ ax = b, the equation for x. 45 The third equation is x2 + b = ax. The importance of these is that all problems involving quadratics can be expressed by one of these three equations. Foundations of Pythagorean Philosophy(4) The so-called Pythagoreans were engaged in the study of mathematical objects. They were the first to advance this study. Having been brought up in it, they regarded the principles of mathematical objects as '_the principles of all things. Since of mathematical objects numbers are by nature first, and (a) they seemed to observe in numbers, rather than in fire or earth or water, many likenesses to things, both existing and in generation (and so they regarded such and such an attribute of numbers as justice, such other as soul or inellect, another as opportunity, and similarly with almost all of the others), and (b) they also observed numerical attributes and ratios in the objects of harmonics; since, then, all other things appeared in their nature to be likenesses of numbers, and numbers to be first in the whole of nature, they came to the belief that the elements of numbers are the elements of all things and that the whole heaven is a har mony and a number. And whatever facts in numbers and harmonies could be shown to be consistent with the attributes, the parts, and the whole arrangement of the heavens, these they collected and fitted into a system; and if there was a gap somewhere, they readily made additions in order to make their whole system connected. I mean, for example, that since ten is considered to be complete and to include every nature in numbers, they said that the bodies which travel in the heavens are also ten; and since the visible bodies are nine, they added the so-called "Counter-Earth" as the tenth body. • • • The elements of a number are the EVEN and the ODD, the ODD being finite and the EVEN being infinite; the ONE is com posed of both of these (for it is both even and odd); a number comes from the ONE; and, as we said, the whole heaven is numbers. (Aristotle, Metaph. 985b- 986a, passim.)- 46 The monad is the first principle of all things, From its forms and from numbers the elements arose. And he declared that the number and form and mea- sure of each of these is somehow as follows: FIRE is composed of twenty-four right-angled triangles, surrounded by four equilaterals. And each equi lateral consists of six right-angled triangles, whence they compare it to a pyramid. AIR is com posed of forty-eight triangles, surrounded by eight equilaterals. And it is compared to the octahedron, which is surrounded by e~t equilateral triangles, each of which is separated into six right-angled triangles so as to become forty-eight in all, And WATER is composed of one hundred and twenty triangles, surrounded by twenty equilaterals, and it is com pared to the icosahedron, which is composed of one hundred and twenty equilateral triangles. And AETHER is composed of twelve equilateral pentagons and is like a dodecahedron. And EARTH is composed of forty-eight triangles, and is surrounded by six equilateral pentagons, and it is like a cube, For the cube is surrounded by six tetragons, each of which is separated into eight triangles, so that they become in all forty-eight, (Plato, Phaedrum, 16,) A Brief History of the Five Polyhedrons A regular polygon is a plane figure which is bounded by straight lines and has equal sides and equal interior angles. There is of course an infinite number of such figures. In three dimensions the analogue of the regular polygon is the regular polyhedron: a solid which is bounded by regular polygons and has congruent faces and congruent interior corner angles. These forms are not infinite, as might be supposed, There are only five: the regular tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron. (See figures that follow,) The first systematic study of the five regular solids appears to have been made by the ancient Pythagoreans. They believed that the tetrahedron, hexahedron, octahedron and icosahedron respectively underlay the structure of the 47 THE FIVE PLATONIC SOLIDS* (Regular Polyhedrons) TETRAHEDRON OCTAHEDRON ICOSAHEDRON HEXAHEDRON DODECAHEDRON *For each of these solids, the Number of Vertices - Number of Edges + Number of Faces = 2. 48 traditional four elements: fire, earth, air and water. The dodecahedron was obscurely identified with the entire universe. Because these notions were elaborated in Plato's Timaeus, the five regular polyhedrons came to be known as the Platonic solids, The beauty and fascinating mathematical properties of these five forms haunted scholars from the time of Plato through the Renaissance. The analysis of the Platonic solids provides the climactic final book of Euclid's Elements, Johannes Kepler believed for a time that the orbits of the six planets known in his day could be obtained by nesting the five solids in a certain order within the orbit of Saturn. Today the mathematician no longer views the Platonic solids with mystical reverence, but their rotations are studied in connection with group theory, and they continue to play a colorful role in recreational mathematics.<4) Mathematics of Africa There has been little written about the history of the mathematics of Africa. Within this vast continent approxi mately 750 different languages (not simply dialects) are spoken. Following are presented a few of the numeration systems in use today. Research into the mathematics of Africa remains an open topic, of current interest to many students with the rising aspirations and economic development of the emerging countries within Africa, 49 AFRICAN NUN~RATION(B) English Swahili Lugandu • ...... c.~ --=~:!ol~~V< Magic Sguares(J) One of the oldest of the Chinese mathematical classics is the I-king, or Book of Permutations. In this appears the configuration which follows, known as the le-shu. (J, P• 5) ¢ • • • • • • • o • r '-1 9 :l t + 3 5 7 0 8 I b 50 The le-shu is the oldest known example of a magic square, and mythology claims that it was first seen by the Emperor Yu. in about 2200 B.C., decorating the back of a divine tortoise along the bank of the Yellow River. It is a square array of numerals indicated by knots in strings. An nth order magic square is a square array of n 2 distinct integers so arranged that the n numbers along any row, column, or main diagonal have the same sum, called the magic con- stant of the square. The magic square is said to be normal is the n 2 numbers are the first n 2 positive integers. Show that the magic constant of an nth order normal magic square is n(n2 + 1)/2. Solution, Let the figure below be a magic square of nth order, i.e., it has n x n = n 2 blocks and it contains the integers 1, 2, ••• , n 2 in its various blocks. Since the sum of each of its columns equals the magic constant and since there are n columns, we can find the magic constant by adding all of the blocks and dividing by n or 1 + 2 + ••• + n 2 = k where k is the magic constant n and since because i=l 2 2 2 1 (1 + 1) + (1 + 2) + ••• + (n - 2) + (n - 1) + s-.n = + n~ + S 2 = n2 + (n2 - 1) + (n2 - 2) + ••• + (2 + 1) + (1 + 1) + 1 n Therefore 1 2 3 ·- --~··~·· ' . . ' . I t o t , :r--:---r-:~"--r-71 ·. : ~ . nL·---·~ -A~~.-~1· '' "'_· 51 Indian Mathematics(4 ) While the origins of Indian mathematics remain hidden in antiquity, there are some indications of early interest in mathematics. The decimal system itself must be quite ancient since Valmiki, father of Sanskrit poetry and author of the Ramayana (about 100 B.C.) has a fantastic number for the size of the army inimical to the demon king Ravana: 1o12 + (36)1o4 +lo5 = l,ooo,ooo,46o,ooo. As would be expected, the form of the Hindu numerals has undergone considerable change over the centuries. There is noticeable transition from ancient tally, or stroke, numera tion to the numerals found in the Baskhshali manuscript about 1150 A.D. : 0 1 8 0 7) The modern Nagari script for the same numerals is, 0 With regard to various operations, the addition and subtraction processes are obvious. The Bhaskara II Manu script of about 1170 A.D. states: "Add the figures in the same places in the direct or inverse order." This means to add from either the right or left. The same is possible in subtraction: "Subtract the numbers according to their places in the direct or inverse order." Multiplication could be done in four ways, one of which is similar to our modern method and another is known as the "gelosia" method. The gelosia method was transmitted to Europe and takes its name from Venice where the arrangement of the boxes resembled Venetian blinds. See figure below. 52 Division was accomplished much as today but from the left (inverse order). This method was quite popular in Europe from the fifteenth through seventeenth centuries, where it came to be known as the "galley" method, A broad outline of a galley could be sketched about the completed division problem. The method originated in India about the fourth century A.D. Squaring numbers and finding their square roots pro ceeded by unusual algorithms. Both of these processes were known by the Hindu mathematician. The Brahmagupta manuscript of about 628 gives this rule for squaring: "The product of the sum and the difference of the number (to be squared) and an assumed number plus the square of the assumed number give the square: m2 = (n + x) (m - x) + x2 53 Examples This section includes several historically original mathe matics problems from the past, with some of the problems both translated and solved using the "tools" of the time and modern tools; and others simply translated and solved using modern methods. Original Algebra Problem of Newton(?) Given the sides of any parallelogram, AB, BC, DC, and AC, and one of the diagonals, BC; find the other diagonal AD. See the figure belews Let E be the concourse of the diagonals, and to the diagonal BC let fall the perpendicular AF, and I. ACq - ~d + BCg = CF and also ACg ... AEg + ECg = CF II. 2EC Modern Solution. I, Prove: AC 2 AB 2 + BC 2 = CF c 2BC CF2 + AF 2 = AC 2 2 AF + FB 2 = AB 2, AF 2 = AB 2 - FB 2 CF 2 + AB 2 FB 2 = AC 2 CF 2 + AB 2 - (BC - CF) 2 = AC 2 2 2 2 CF + AB BC + 2CB • CF - CF2 = AC 2 2CB • CF = AC 2 - AB 2 + BC 2 CF = AC:~ AB 2 + BC 2 2BC 54 2 2 II, Prove: AC 2 - AE + EC = CF 2EC CF2 + AF 2 = AC 2 AF2 + FE2 = AE2, AF2 = AE2 - FE2 CF2 + AE 2 FE2 = AC 2 2 2 2 2 CF + AE (EC CF) = AC CF 2 + AE 2 EC 2 + 2EC • CF - CF 2 = AC 2 2 2EC • CF = AC 2 - AE 2 + EC 2 2 2 CF = AC - AE + EC 2EC Wherefore, since EC = tBC and AE = tAD, it will be ACQ ABO + BCg AGo - !ADq + ~BCQ4 - 2BC ~ ~BC- and having reduced, AD ='\/2ACq + 2ABq - BCq. Since I =II, we have: 2 2 2 2 AC 2 - AB 2 + BC AC AE + EC EC = iBC 2BC = 2EC AE = iAD (.!.AD) 2 + (iBC) 2 2~;Bc) 2 AC 2 - AB 2 + BC 2 _ AC 2 - iAD 2 + tBC 2 2BC - BC BC(AC 2 - AB 2 + BC 2) = 2BG{AC 2 - ~AD 2 + iBC 2) 2 2 2 AC 2 - AB + BC = 2AC - iAD 2 + tBC 2 2 2 2 2 tAD = AC + AB - iBC AD = \f2Ac 2 + 2AB 2 - BC 2 • Q.E.D. (Only+ sign is used in measure of lengths.) Babylonian Algebra<4) Find the length and width of an excavation given the semiperimeter as 6;30 and the area as 7;30. Halve the length and width which I have already added together, and you will get 3;15. Square 3;15 and you will get 10;33,45. Subtract 7;30 from 10;33,45 and you will get 3;3,45. Take the square root of this and you have 1;45. 55 Add it to the one, and subtract it from the other, Thus, you get the length, the width; 5 (GAR) is the length, 1;30 GAR is the width. Modern Solution, (Interpretation Base 60) 1 + w = 6;30; lw = 7;30 or 1 + w = 6!; lw = 7! 1 ; w = 3;15 or 1 ; w = 3i 2 2 (3;15) = 10;33,45 or (3i) = 10 9/16 - 7;30 1 2 3; 3,45 =( ~ w) - lw =3 1/16 12 + 2lw + w2 2 2 .=;...... ;....:::::.:;4::.;.;.__..;.;. - 1 w = 1 - 2lw + w = 4(3;3,45) = 12i (1 4(3;3,45) or 1 - w 2 = 1;45 = 1 3/4 1 + w 1 - w 2 + 2 = 5 or 1 = 5(60°) = 5 + w 1 1 - w = 1;30 or w = 1(6o0) 2 2 +Jo(6o-l) = lt. Numerical Algebra of Dio:Qhantus(4 ) To find two numbers such that their sum and product are given. The square of half the sum must exceed the product by a square number, Given sum 20, given product 96; 2z the differente of the required numbers. Therefore, the required numbers are 10 + z and 10 - z. Hence, 100 - zz = 96. There fore, z = 2. The numbers are 12, 8. Modern Solution. x + y; xy x + y = xy + z2 2 (x + y = 20; xy = 96) x - y = 2z X= 10 + z; X= 10- Z 2 2 10 - z = 96 2 100-96 = (x ; Y) • xy 4 = (x 2 Y)2 2 = X - -Y or x - y = 4 2 Therefore z = 2 and x = 12; x = 8, Babylonian Problem(4 ) Given that the width of the rectangle is * less by con struction than the length and 40 is the measure of the diagonal. How long are the length and width? You set 1 for the length, put 1 for the extension. 15, the *• you subtract from 1; you get 45. Set 1 as the length, set 45° as the width, square 1, the length, you get 1. Square 45, the width, you get 33'45". Of 1 and 33'45" you make the sum: 1.33.45 you get. What is the root of this square? 1.15 is the root of the square. Consider that 40, the diagonal that was given, provides the inverse of the 1.15 diagonal. 48 you find. Multiply 48 by 40 the diagonal that was given. 32 you get. 32 is the length. Multiply 32 by 45 the width you have assumed, 24 you get, 24 is the width. Modern Solution, w = - i, d = 40 1 = 1 i Of 60 = 15 1 - 15 X 60 = 45 X 60-l 12 = 12 = 1 w2 = 452 = 33'45" = 33 x 6o-1 + 45 x 6o-2 2 2 _x. + w = 1 + 33'45" = 1 x 6o'0 + 33 x 6o-1 + 45 x 6o-2 2 2 ,(e ·+ w = Aj1,33,45 = 1,15 = 1 X 60° + 15 X 60-l 1 - 1 - 4 1.15 - 75/60 - 5 57 ~ of 60 = ;48; (48 X 60-l) X 40 = 32 (32) • 1 = 32 == 1 32 X (45 X 60-l) = 24 = w. (Note: The Babylonians did not use the Pythagorean Theorem directly because they did not substitute w = ~~.) Arabic Algebra(4 ) This problem is taken from the Book of Increase and Decrease (Divine Numeration), which Abraham compiled and arranged from the same book of the Indians. The missing word ( ) is "lanx" from the Latin and is assumed to mean scale or plate. What remains from a square after taking its third and fourth is eight. What is the square? Assume the to be 12. Its third and fourth arise and so you subtract 7, and 5 remains. This you compare with the 8 (what was left of the square) and it will be dvious that you erred by 3. Keep this. Then take a second which is divisible by the first, such as 24. Its third and fourth make 14, and what remains is 10. Comparing this with 8 what remained of 8, it is obvious that you erred by 2. Now multiply the error of the second _____ namely 2, by the first , which is 12, and you get 24. Then multiply the error of the first , or 3, by the second , which is 24, and you get 72. Now combine 24 and 72, whereby one error is decreased and the other increased. For if both were increased or decreased, you would subtract the less from the greater. Therefore after you have added the 24 and the 72, the sum is 96. Then add the two errors which are 3 and 2, and 5 arises. Finally divide 96 by 5. What arrives for you is 19 dragmas and a fifth of a dragma, And so this is the rule: 12 for the unknown, subtract its third and fourth, and 5 remains. Now tell what you 58 multiply by 5 to get 12. Now that is the unknown. This is 22 . Multiply therefore 22 by 8, and there is 195.1 5 5 Modern Solution. x2 x2 - x2 = 8 3 r,: 12 - 4 - 3 = 5 1 8 - 5 = 3 error 24 - 8 - 6 = 10 10 - 8 = 2 error2 Therefore 24e1 + 12e2 _ 2£ = ~ el + e2 - 5 1 5 5 ~~) = 12 8 (1~) = ~ Problem of False Position(4 ) Divide 100 loaves among 5 men in such a way that the shares received shall be in arithmetical progression and that 117 of the sum of the largest three shares shall be equal to the sum of the smallest two. What is the difference of the shares? Do it thus: Make the difference of the shares 5~. Then the amounts that the 5 men receive will be 23, 17~, 12, 6~, 1 (assumed and called False Position), total 60. As many times as is necessary to multiply 60 to make 100, so many times must these terms be multiplied to make the true series. 1 I 6o 213 I 40 The total, 1 213, times 60 makes 100. Multiply by 1 213: 23 it becemes 38 113 17i it becomes 29 116 12 it becomes 20 6i it becemes 10 213 116 1 it becomes 1 213 Total 60 it becomes 100. 59 Modern Solution. l+x+l+ 2x + 1 + 3x + 1 + 4x Therefore 1 (3 + 9x) = 2 +X 7 3 + 9x = 14 + 7x 2x = 11 X = 5~ Heron's Formula Presented below is a proof of the formula given by Abu 'Abdallah Muhammad bin Ahmad al-Shanni. Little is known about this geometer except that he lived in the Middle East at about the tenth century, that he wrote on the construction of the regular heptagon, and that he solved a particular cubic equation. Area of Prove Triangle use of the following: Lemma. In the figure above, v • is a segment of the hypo- tenuse of any right triangle. One endpoint of v ' is the mid- point of the hypotenuse, the other end is anywhere on it. Segments x' and y' are parallel to the legs x and y respec tively as shown. Then (4 6_) 2 = xx 'yy', where D. is the area of the oblique triangle drawn in heavier lines than the rest of the figure. Or to state it in the manner of al-Shanni, 60 four times the area of the triangle is the mean proportional between rectangles bounded by x and x', andy andy' respec- tively. To prove the lemma, al-Shanni constructs the figure above, an isoceles right triangle with legs of length z so that the latter is the mean proportional between the legs of the first figure, i.e., z2 = xy. In like manner, segment v has one extremity on the midpoint of the hypotenuse, and the other so placed that z• 2 = x'y'. By similar triangles, 2~ = ~, , and zz • = 2vw = 4 6•, where L::s is the area of the triangle drawn heavily on the figure above. v' Also by similar triangles, ~ w• , since .2£' = :Y..' • X y Therefore L• = since each is the same fraction, v v• w= w• , of the equivalent isosceles triangles having as sides z, w, w; andy, w', w• respectively. Hence 2 2 2 (4 6> = {4 6•) = (zz •) = xx 'yy', which is the lemma. An A _ ~.lxx' Ei.' equivalent statement is '--.::1 - IV """1i • 4 I Now, consider the general triangle with sides a, b, and c as displayed in the following figure, 61 From one extremity of side a lay off two segments equal to b, one in extension of a and the other interior to a, so that a rectangle of diagonals 2b and sides x• and y' emerges as shown. From the vertex at which a and c intersect, draw x and y parallel to x' and y' respectively, The configuration which results is that of ~he lemma. Now make two applications of the theorem of Ptolemy which states that in any cyclic quadrilateral the sum of the pro duct of pairs of opposite sides equals the product of the diagonals. One trapezoid in the figure has a + b in diagonal; another has c. Both are isoceles, hence cyclic. The theorem yields c 2 + xx' = (a + b) 2 , and (a - b) 2 + yy' = c 2 , or xx' = (a + b) 2 - c 2 , and yy' = c 2 - (a - b) 2 • Now apply the lemm~ to t~e trirt~le orz:ide~ _a.,~, ,d . J2 C to obtain 6= Aj~ I~ =via+ b4_- c 1 c -¢a- bJ which completes the proof of the theorem. Al-Shanni makes no mention either of Heron or of Archimedes, although both were well known to the Muslim mathematicians of the Middle Ages. Neither does he allege that he is the first to prove the relation. Modern Solution. To prove the area of a triangle whose sides are a, b, and cis equal to 1/s(s- a)(s- b)(s- c) where s = l/2(a+b+c). 62 We shall first find the altitude h in terms of a, b, and c. 1, Let h be the _L from C to AB and x be the projection of b on AB, 2 2 2 2. In Figure 1, h + (c - x) = a ; In Figure 2, h 2 + (x- c) 2 = a 2• 3. In either case, h 2 + c 2 2xc + x2 4. -(h2 + x2 5. 6, Solving, s. 9. - a2)(2bc- b2 - c 2 + a2), 10. 4c2 = ({ 2 2 2 2 11. Grouping, h 2 b + c ) - a ] G - ( b - c ) ] 4c2 12. Factoring, h 2 = (b + c + a)(b + c- a)(a- b + c)(a + b- c). 4c2 for convenience, let a + b + c = 2s. Then b + c - a == 2s - 2a = 2(s -a), a + c - b = 2s 2b = 2(s b), and a+ b- c = 2s 2c = 2(s c). Substituting in (12), we have 13. h 2 = 2s • 2 ( s - a) • 2 ( s - b) • 2 ( s - c) • 4c2 14. h 2 = ~ 2 s(s -a)(s- b)(s- c), . -!'f- k1'!!al ? -.t!-~IW'l"'f"..!T 2 a)(s b) (s c), 1.5. h = c J./s(s - - - 16. Area of /1ABC = i base • altitude. ... 1c 17. Area of ~ABC = 2 • ~ 1\js(s - a)(s - b) (s c). e,.. ~ .. 1' • ~..... -~"''lL_,. 18. Area of 6ABC = 1\j s(s - a)(s - b) (s - c). Q.E.D. Suggested Problems/Topics for Student Research 1, Do historical research of some numeral system(s). 2. Research California mathematics, 3. Research the history of mathematics in the United States. 4. Prepare a geneology of the history of mathematics of certain countries during a selected number of years, .5. Research mathematical discoveries and inventions since 1930. 6. Write on the impact of "Sputnik" of 19.57 on the reemphasis of science and mathematics in our educational system. 7. Write about some famous mathematician(s). 8, Precise articles pertaining to the history of mathematics from those listed in the References for Student Research, 9. Write about the history of mathematics as it relates to the history of mankind. 10. Write a paper relating the history of mathematics to that of other disciplines or areas of study, 11. Solve ancient mathematical problems, using both ancient and modern methods: a. Magic squares, b, Select some geometric constructions that are trivial using straight edge and modern com pass; then attempt the constructions using the straight edge with Euclidean compass and some ingenuity. 64 c. Work on one or more of these famous "classics": (1) Squaring the circle (2) Duplication of the cube (3) Trisection of the angle. d. Prove the extension of the Pythagorean theorem given by Pappus in Book IV of the Mathematical Collection. e. Show that a quadratic surd cannot be equal to the sum of a nonzero rational number and a quadratic surd. f, Compute by "regula duorum falsorum" (often called the rule of double false position) to three decimal places, the root of x3 - 36x + 72 = 0; and give some historical background to this method. g, Any of the suggested problem studies from: Eves, Howard, An Introduction to the History of Mathematics, Third Edition, San Francisco: Holt, Rinehart and Winston, 1969. 12, Other items pertinent to the history of mathematics suggested by students and approved by instructor. References for Student Research Aaboe, Asger. Episodes from the Early History of Mathematics. New York: Random House, Inc., 1964. Anonymous. "Japanese Mathematics," The Mathematical Gazette, 3:268-270. 1 2 6 Archibald, R, c. "Historical Notes on the Relation e / 7T- i ," The American Mathematical Monthly, 2~cll6-121, Archibald, R. C. Outline of the History of Mathematics, Sixth Edition. Menasha, Wisconsin: Mathematical Association of America, 1949. Ball, Walter w. A Short Account of the History of Mathematics. New York: Dover Publications, Inc., 1960. Beeler, Madison S. (Editor). "Ventureno Numerals," u. c. Pub lications in Linguistics, 34:13-18. Beeler, Madison s. (Editor). "The Ventureno Confesionario of Jose Senan, O.F.lVI.," u. C. Publications in Linguistics, 47:76-79. Boyer, Carl B. A History of Mathematics. New York: John Wiley and Sons, 1968. 65 Caballeria y Collell, Juan, History of the City of Santa Barbara. Santa Barbara, California, 1892, Careccio, John. "Mathematical Heritage of Zambia," The Arithmetic Teacher, May, 1970. --- Cheng9 D. c. "On the Mathematical Significance ef the Chinese Ho Tu and Lo Shu," The American Mathematical Monthly, 32:499-504. Clagett, Marshall, Greek Science in Antiquity. New York: MacMillan and Company, 1969. Coolidge, J. L, "The Number c," The American Mathematical Monthly, 57:591-602. Dantzig, Tobias. The Bequest of the Greeks. New York: Charles Scribner's Sons, 1955. Datta, B., and Singh, A. N. History of Hindu Mathematics. Bombey: Asia Publishing House, 1962. Dixon, R. B., and Kroeber, A. L. "Numeral Systems of the Languages of California," American Anthropologist, 9:663-690. Eels, w. C. "Number Systems of North American Indians," The American Mathematical Monthly, 20:263-272 9 293-299. Eves 9 H. "A History of Mathematics Time Strip," The Mathe matics Teacher, 54:452-454. Gardnero Martin, "The Five Platonic Solids," Scientific American, December, 1958, pp. 126-128, Bureau of Ghana Languages. A Luganda Grammar. London, 1961. Hofmann, Joseph E. Classical Mathematics: A Concise History of the Classical Era in Mathematics. New York: Philo- sophical Library, 1959, Jelitai, J, "The History of Mathematics in Hungary before 1830," National Mathematics Magazine, 12:125-130. Jones, P. s. "American Mathematics~" The Mathematics Teacher 0 49•30-33. Jones, P. s. "From Ancient China 'til Today," The Mathematics Teacher, 49:607-610. Karpinski, L. c. "Mathematics in Latin America," Scripta Mathematica, 13:59-63. Kokomoor, F. w. "The Status of Mathematics in India and Arabia during the 'Dark Ages' of Europe," The Mathematics Teacher, 29:224-231. Kroeber, A. L. Handbook of the Indians of California. Berkeley: California Book Company, Ltd, 1953. 66 Larney, B. M. "How the Middle Ages Counted," School Science and Mathematics, 31:919-930. Larivee, J. A. "A History of Computers," The Mathematics Teacher, 51:469-473. Lasserre, Francois. The Birth of Mathematics in the A§e of Plato. New York: American Research Council, 196 • Locke, L. L. "The History of Modern Calculating Machines; An American Contribution," The American Mathematical Monthly, 31:422-429. Miller, G. A. "History of Mathematics in America," School Science and Mathematics, 35:292-296. Miller, G. A. "Primary Facts of the History of Mathematics," The Mathematics Teacher, 32:209-211. More, Louis T. Isaac Newton, A Biography. New York: Dover Publications, Inc., 1962. Ore, Oystein. Number Theory and Its History. New York: McGraw-Hill Book Company, 1948. Osborn, J. "The Closing Years of' Greek Mathematics," ~ Mathematics Teacher, 56:540-545. Read, c. B. "The Second Half of the Century in the History of Mathematics," The Mathematics Teacher, 53:463-466. Sarton, George. The Study of the History of Mathematics. New York: Dover Publications, Inc., 1957. Scott, Joseph F. A History of Mathematics from Antiquity to the Beginning of the Nineteenth Century. London: Taylor and Francis, 1969. Shelton, J. B, "A History of Mathematics Chart," The Mathe matics Teacher, 42:563-567. Simons, L. G. "Eighteenth Century Algebra in America," Scripta Mathematica, 3:355-356. Smith, c. L. "On the Origin of'>' and'<'," The Mathematics Teacher, 57:479-481. Smith, David. History of Mathematics. New York: Dover Publications, Inc., 1953. Srinivasiengar, c. N. The History of Ancient Indian Mathe matics. Calcutta: World Press Private Company, 1967. Tully, Jean, and Keniston, Rachel. High School Geometry. New York: Ginn and Company, 1960. Turnbull, Herbert w. The Great Mathematicians. New York: New York University Press, 1961. 67 von Baravalle' H. "The Number rr, II The Mathematics Teacher, 45&340-348. Wolfers, Edward P, "The Original Counting Systems of Papua and New Guinea," The Mathematics Teacher, 64:77-83. Wood, w. D. "An Historical Outline of the Development of Mathematics in the United States during the Last Fifty Years," The Pentagon, 7:52-68, Wren, F, L. "The 'New Mathematics' in Historical Perspective," The Mathematics Teacher, 62:579-585. Wren, F. L., and Rossman, R. "Mathematics used by American Indians North of Mexico," School Science and Mathematics, 3 3 : 36 3-37 2 • Zaslavsky, Claudia. "Black African Traditional Mathematics," The Mathematics Teacher, April, 1970. 68 CHAPTER III BIBLIOGRAPHY 1. Demeryad, A. P. Mathematical Games and Pastimes. New York: MacMillan Company, 1964. 2. Eves, Howard. An Introduction to the History of Mathe matics. San Francisco: Holt, Rinehart and Winston, 1969. Freitag, Herta T. and Arthur H. "The Magic of a Square," The Mathematics Teacher, January, 1970, pp. 5-14. 4. Hughes, Barnabas B, O.F.M. History of mathematics descriptive material from course, The History of Mathematics, Valley State College, Northridge, Calif ornia, 1972. Krueger, Marcella B, "A Brief History of Mathematics in Rhyme--The Early Greeks to the Seventeenth Century," The Mathematics Teacher, May, 1961, pp. 366-368. 6. Nasr, Seyyed H. Science and Civilization in Islam. Cambridge: Harvard University Press, 1968. Whiteside, Derek T, The Mathematical Works of Isaac Newton, Volume II. New York: Johnson Reprint Corporation, 1967. 8. Zaslavsky, Claudia. "Black African Traditional Mathe matics;" The Mathematics Teacher, April, 1970; PP• 345-356, CHAPTER IV NON-EUCLIDEAN GEOtlliTRY Approximate Length of Time for Unit Three to four weeks. Behavioral Objectives 1. The student will gain an historical perspective of both Euclidean and non-Euclidean geometry. 2. The student will be able to distinguish between Euclidean and non-Euclidean geometry. 3. The student will gain an appreciation of the genius of Euclid and some of the non-Euclidean geometers, such as: Proclus, Playfair, Lobatchevsky, Riemann, Gauss, Bolyai. 4. The student will become knowledgeable and some what sophisticated about the parallel postulate substitutes. 5. The student will be able to enhance his peers' knowledge of non-Euclidean geometry by research ing some aspect and sharing it. Preface "Which geometry is true?" The world we live in is probably more of a non-Euclidean one than a Euclidean one, according to modern physics. However, as far as mathematics is concerned, the question is totally irrelevant and meaning less. The real question is, "Which geometry is the most 70 convenient?" To this there is a simple answer& Euclidean geometry is the simplest and most convenient to use and will probably always be used in elementary applications.(S, P• 215) All of the members of this Seminar have been well exposed to Euclidean geometry during a full-year course in geometry and therefore have a fairly sound knowledge of its definitions, postulates, and theorems. Non-Euclidean geometry, however, is almost completely unknown to most students at this time, Thus, the purpose of this section is to explore some of the important aspects of non-Euclidean geometry, The discovery of non-Euclidean geometry in the 19th century is generally considered to mark the beginning of modern mathematics, just as the impressionist painting of the same period is con sidered to mark the beginning of modern art, There was revolution in the air--political revolution, artistic revolution, scientific revolution. The time was ripe for challenging tradition in all fields and mathematics was no exception,(l5, P• 3) Because most students in this Seminar have been away from the regular geometry course for li to 2i years, the writer is including at this pointe 1. A list of Euclid's Definitions, Postulates, and the First Thirty Propositions of Book I. 2. Hilbert's Axioms for Euclidean Plane Geometry, 3. Birkhoffvs Postulates for Euclidean Plane Geometry, 4. The SMSG Postulates for Euclidean Geometry. 71 Euclid's Definitions 9 Postulates, and tae First Thirty Propositions of Book I Definitions• 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. B. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9. And when the lines containing the angle are straight, the angle is called rectilineal. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle. 13. A boundary is that which is an extremity of anything. 14. A figure is that which is contained by any boundary or boundaries. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. 16. And the point is called the centre of the circle. 17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 18. A semicircle is the figure contained by the diameter and the circumference cut off by it, And the centre of the semicircle is the same as that of the circle. 72 19. Rectilineal figures are those which are contained by straight lines; trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. 20. Of trilateral figures; an equilateral triangle is that which has three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal, 21, Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled tri angle that which has an obtuse angle, and an acute angled triangle that which has its three angles acute. 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled, And let quadrilaterals other than these be called trapezia~ 23. Parallel straight lines are straight lines which, ~eing in the same plane and being produced indefinitely 1n both directions, do not meet one another in either direction. The Postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another, 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced in definitely, meet on that side on which are the angles less than the two right angles. The Common Notions: 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 73 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. The First Thirty Propositions of Book I: 1. On a given finite straight line, to construct an equi lateral triangle. 2. To place at a given point (as an extremity) a straight line equal to a given straight line. 3. Given two unequal straight lines, to cut off from the greater a straight line equal to the less. 4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely, those which the equal sides subtend, In isosceles triangles, the angles at the base are equal to one another, and, if the equal straight lines be pro duced further, the angles under the base will be equal to one another. 6. If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another. Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely, each to that which has the extremity with it. 8. If two triangles have the two sides equal to two sides respectively; and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines. 9. To bisect a given rectilineal angle. 10. To bisect a given finite straight line. 11. To draw a straight line at right angles to a given straight line from a given point on it. 74 12. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. 13. If a straight line set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles. 14. If with any straight line; and at a point on it, two straight lines not lying on the same side make the ad jacent angles equal to two right angles, the two straight lines will be in a straight line with one another. 15. If two straight lines cut one another, they make the vertical angles equal to one another, 16. In any triangle if one of the sides be prod~ced, the exterior angle is greater than either of the interior and opposite angles. 17. In any triangle two angles taken together in any manner are less than two right angles. 18. In any triangle the greater side subtends the greater angle. 19. In any triangle the greater angle is subtended by the greater side, 20. In any triangle two sides taken together in any manner are greater than the remaining one. 21. If on one of the sides of a triangle, from its extremi ties, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle, 22. Out of three straight lines, which are equal to three given straight lines, to construct a triangles thus it is necessary that two of the straight lines taken to gether in any manner _should be greater than the remaining one. 23. On a given straight line and at a point on it, to con struct a rectilineal angle equal to a given rectilineal angle. 24. If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base. 25. If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other. 75 26. I£ two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that sub tending one o£ the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle, 27. I£ a straight line falling on two straight lines makes the alternate angles equal to one another, the straight lines will be parallel to one another. 28. I£ a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another. 29. A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. 30. Straight lines parallel to the same straight line are also parallel to one another, Hilbert's Axioms for Euclidean Plane Geometry Group I Axioms o£ Connection: I-1, Through any two distinct points A, B, there is always a line m, I-2, Through any two distinct points A, B, there is not more than one line m, I-3. On every line there exist at least two distinct points. There exist at least three points which are not on the same line. I-4. Through any three points, not on the same line, there is one and only one plane. Group II Axioms o£ Order: II-1, If point B is between points A and C, then A, B, C are distinct points on the same line, and B is between C and A. II-2, For any two distinct points A and C, there is at least one point B on the line AC such that C is between A and B. 76 II-3. If A, B, C are three distinct points on the same line, then only one of the points is between the other two. DEFINITION. By the segment AB is meant the set of all points which are between A and B. Points A and B are called the end points of the segment. The segment AB is the same as segment BA. II-4. (Pasch's Axiom) Let A, B, C be three points not on the same line and let m be a line in the plane A, B, C, which does not pass through any of the points A, B, c. Then if m passes through a point of the segment AB, it will also pass through a point of segment AC or a point of segment BC. NOTE II-4': This postulate may be replaced by the separation axiom: A line m separates the points of the plane which are not on m, into two sets such that if two points X and Y are in the same set, the seg ment XY does not intersect m, and if X and Y are in different sets, the segment XY does intersect m. In the first case X and Y are said to be on the same side of m; in the second case, X and Y are said to be on opposite sides of m, DEFINITION. By the ray AB is meant the set of points consisting of those which are between A and B, the point B itself, and all points C such that B is between A and c. The ray AB is said to emanate from point A. A point A, on a given line m, divides m into two rays such that two points are on the same ray if and only if A is not between them, DEFINITION. If A, B, C are three points not on the same line, them the system of three segments AB, BC, CA and their endpoints is called the triangle ABC. The three segments are called the sides of the triangle, and the three points are called the vertices. Group III Axioms of Congruence: III-1. If A and B are distinct points on line m, and if A' is a point on line m' (not necessarily distinct from m), then there is one and only one point B' on each ray of m' emanating from A' such that the segment A'B' is congruent to the segment AB. III-2. If two segments are each congruent to a third, then they are congruent to each other, (From this it can be shown that congruence of segments is an equiva lence relation: i.e., AB ~ AB; if AB ~ A'B', then A' B ' ~ AB ; and if AB e CD and CD ~ EF, then AB !! EF • ) 77 III-3. If point C is between A and B, and point C' is between A' and B', and if the segment AC ~segment A'C', and the segment CBs segment C'B', then segment AB ~segment A'B'. DEFINITION. By an angle is meant a point (called the vertex of the angle) and two rays (called the sides of the angle) emanating from the point. If the vertex of the angle is point A and if B and C are any two points other than A on the two sides of the angle, we speak of the angle BAC or CAB or simply of angle A. III-4. If BAC is an angle whose sides do not lie on the same line and if in a given plame, A'B' is a ray emanating from A', then there is one and only one ray A'C' on a given side of line A'B', such that~ B'A'C' ~+:'BAG. In short, a given angle in a given plane can be laid off Qn a given side of a given ray in one and only one way. Every angle is congruent to itself. DEFINITION. If ABC is a triangle then the three angles BAC, CBA, And ACB are called the angles of the triangle. Angle BAC is said to be included by the sides AB and AC of the triangle. III-5. If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then each of the remaining angles of the first triangle is congruent to the corresponding angle of the second triangle. Group IV Axiom of Parallels (for a plane): IV-1. (Playfair's Postulate) Through a given point A not on a given line m there passes at most one line, which does not intersect m. Group V Axioms of Continuity: V-1. (Axiom of Measure or the Archimedean Axiom) If AB and CD are arbitrary segments, then there exists a number n such that if segment CD is laid off n times on the ray AB starting from A, then a point E is reached, where n • CD = AE, and where B is between A and E. V-2. (Axiom of Linear Completeness) The system of points on a line with its order and congruence relations cannot be extended in such a way that the relations existing among its elements as well as the basic properties of linear order and congruence resulting from the Axioms I-III and V-1 remain valid. 78 NOTE V': These axioms may be replaced by Dedekind's axiom of continuity: For every partition of the points on a line into two non-empty sets such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set. Birkhoff's Postulates for Euclidean Plane Geometry UNDEFINED ELEn~NTS AND RELATIONS: (a) points A, B, ••• ; (b) sets of points called lines, m, n, ••• ; (c) distance between any two points: d(A, B) a real non-negative number with d(A, B)~ d(B; A); (d) angle formed by three ordered points A, 0, B, (A 'I 0, Bi 0)-j:AOB, a real number (mod 277). The point 0 is called the vertex of the angle. POSTULATE I Postulate of line measure: The points A, B, ••• of any line m can be put into 1:1 correspondence with the real numbers X so that rxB - XA I= d(A, B) for all points A, B. DEFINITIONS. A point B is between A and C (A i C) if d(A, B) + d(B, C) = d(A, C). The points A and C together with all points B between A and C form seg ment AC. The half-line m' with endpoint 0 is defined by two points o:, A in line m (A 'I 0) as the set of all points A' of m such that 0 is not between A and A'. If A, B, Care three distinct points the three segments AB, BC, CA are said to form a triangle ABC with sides AB, BC, CA and vertices A, B, c. If A, B, Care in the same line,~ABC is said to be degenerate. POSTULATE II Point-line postulate: One and only one line m contains two given points P, Q (PI Q). If two distinct lines have no points in common they are parallel, A line is always regarded as parallel to itself. POSTULATE III Postulate of angle measure: The half-lines m, n, ••• through any point 0 can be put into 1:1 correspondence with the real numbers a (mod 2~) so that if AIO and B~O are points of m and n, respectively, the difference a - a (mod 2n) is -f. AOB. n m 79 DEFINITIONS. Two half-lines m, n through 0 are said to form a straight * if .~ mOn = w. Two half-lines m, n, through 0 are said ts form a right ~ if ~mOn +w/2, in which case we also say that m is perpen- dicular to n. POSTULATE IV Postulate of similarity: If in tw0 triangles ABC and A'B'C' and fer some constant k>O, d(A', B') = kd(A, B), d(A', C') = kd(A, C) and also.;:B'A'C' = + ~BAC, then also d(B'. c') = kd(BC), -9:' c 'B I At = + 4.CBA, ana 4 A' C 'B' = :!: ~ACE. - DEFINITIONS. Any two geometric figures are similar if there exists a 1:1 correspondence between the points of the two figures such that all corresponding distances are in proportion and corresponding angles are either equal or all negatives of each other. Any two geometric figures are congruent if they are similar with k = 1, The SMSG Postulates for Euclidean Geometry UNDEFINED TERMS. Point, line, plane, POSTULATE 1 Given any two different points, there is exactly one line which contains both of them, POSTULATE 2 The Distance Postulate• To every pair of dif ferent points there corresponds a unique positive number. POSTULATE 3 The Ruler Postulate: The points of a line can be placed in correspondence with the real numbers in such a way that a. To every point of the line there corresponds exactly one real number. b, To every real number there corresponds exactly one point of the line. c. The distance between two points is the absolute value of the difference of the corresponding numbers. POSTULATE 4 The Ruler Placement Postulate: Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive. POSTULATE 5 Every plane contains at least three non-collinear pointsp and space contains at least four non-coplanar points. POSTULATE 6 If two points lie in a plane, then the line containing these points lies in the same plane. 80 POSTUnATE 7 Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane. More briefly, any three points are ceplanar, and any three non-collinear points determine a plane. POSTULATE 8 If two different planes intersect, then their intersection is a line. POSTULATE 9 The Plane Separation Postulate: Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that a. Each of the sets is convex and b. If P is in one set and Q is in the other, then the segment PQ intersects the line. POSTULATE 10 The Space Separation Postulate: The points of space that do not lie in a given plane form two sets such ~~ a. Each of the sets is convex and b. If P is in one set and Q is in the other, then the segment PQ intersects the plane. POSTULATE 11 The Angle Measurement Postulate: To every angle LBAC there corresponds a real number between 0 and 180, POSTULATE 12 The Angle Construction Postulate: Let ~be a ray on the edge of the half-plane H. For every number r between 0 and 180 there is exactly one ray ~. with P in H, such that m LPAB = r. POSTULATE 13 The Angle Addition Postulate: If Dis a point in the interior of LBAC, then m ZBAC = m LBAD + m LDAC. POSTULATE 14 The Supplement Postulate: If two angles form a linear pair, then they are supplementary. POSTULATE 15 The S.A.S. Postulate: Given a correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence. POSTULATE 16 The Parallel Postulate: Through a given external point there is at most one line parallel to a given line. POSTULATE 17 To every polygonal region there corresponds a unique positive number. POSTULATE 18 If two triangles are congruent, then the triangular regions have the same area. POSTULATE 19 Suppose that the region R is the union of two regions R1 and R2• Suppose that R1 and R2 intersect at most in a finite number of segments and points. Then the area of R is the sum of the areas of R and R • 1 2 81 POSTULATE 20 The area of a rectangle is the product of the length of its base and the length of its altitude. POSTULATE 21 The volume of a rectangular parallelpiped is the product of the altitude and the area of the base. POSTULATE 22 Cavalieri's Principle: Given two solids and a plane. If for every plane which intersects the solids and is parallel to the given plane the two intersections have equal areas, then the two solids have the same volume. Introduction to the Topic The discoverers of non-Euclidean geometry f'ared somewhat like the biblical king Saul. Saul was looking for some donkeys and found a kingdom. The mathematicians wanted merely to piek a hole in old Euclid and show that one of his postulates which he thought was not deducible from the others is, in fact, so deducible. In this they failed. But they found a new world, a geometry in which there are infinitely many lines parallel to a given line and passing through a given point; in which the sum of the angles in a triangle is less than two right anglesr and which is nevertheless free of contra dictions. (lO, P• l) Non-Euclidean geometry dates back to about 1830 when the first publication by Lobatchevsky appeared. The name non-Euclidean was used by Gauss to de·scribe a system of geometry which differs from Euclid's in its properties of parallelism.(5, P• vii) Originally geometry began as a practical science of' measurement and was used about 2000 B.C. in Egypt. Thales (640-546 B.C.) brought it to Greece and began the process of abstraction by which positions and straight edges are idealized into points and lines. 82 Pythagoras and his disciples added much progress; and, among others, Hippocrates attempted a logical presentation in the form of a chain of propositions based on a few definitions and assmaptions. Then, of course, Euclid (about 300 B.C.) greatly improved on this by writing his Elements. The geometry that is taught now is essentially a part of the Elements. The primary point in establishing non-Euclidean geometry is to drop Euclid's parallel axiom and to substitute for it another suitable hypothesis. If the other Euclidean axioms are retained, then it is possible, by the customary rules of deduction, to derive from this new set of axioms a new geometry--non-Euclidean. After lengthy exposure to Euclidean geometry and its parallel axiom, it seems at first impossible to accept the non-Euclidean concept that parallel lines do meet at a point. But, in fact, non-Euclidean geometry is indeed possible and plausible and governed by laws similar in structure to a Euclidean geometry, Imagine, for instance, the spatial situation on the surface of a sphere of a very large diameter with creatures whose entire existence is limited to this two-dimensional surface. As long as they know only a limited portion of their world, i.e., only a piece of the spherical surface small compared to the diameter, they would have a geometry different in nothing from our Euclidean Plane Geometry. 83 Their straight lines would be, in fact, arcs of great circles, these being the lines of shortest distance between two points (and the shapes of stretched strings) on the surface of the sphere. As soon, however, as the surface inhabitants extend their experience beyond the immediate neighborhood, they will be forced to change their hypotheses. There are no "parallel" lines on the spherical surface; all great circles intersect each other in two points. Now it is indeed possible--by means which cannot be explained here in any detail--to describe the situation on the surface of a sphere in such a way that all statements become ·equivalent to those of a plane geometry in which the parallel axiom does not hold, But the same relation as that between a plane and a very large spherical surface exists also between the three-dimensional space described by Euclidean geometry and a "curved" space not satisfying the parallel postulate,<12• P• 1729) Gauss (1777-1855) was the first scholar who recognized the existence of non-Euclidean geometry, and he also dis covered many of its theorems. Unfortunately, he printed none of his findings; however, his work now known was taken from his notebooks and published much later. Gauss did not communicate his findings to the world of science because of his use of differential geometry (based on differential calculus) instead of synthetic methods (similar to those used in elem~ntary geometry) which are more appr~priate. Meanwhile, the Hungarian mathematician, Janos Bolyai (1802-1860), wrote a paper entitled "Appendix Scientiam 84 Spatii Absolute Veram Exhibens" ("Appendix Giving an Absolutely True Science About Space"). Nikolai Lobatchevsky (1793-1856), a Russian mathematician and a professor at Kazan University, also published a paper, "On the Principles of Geometry." Although Bolyai's paper appeared a few years later than Lobatchevsky's, their work was completely independent. Amazingly, however, the thoughts of these two scholars were closely related and both based on the properties of the horospherea A horosphere is a surface consisting of all points symmetrical to a given point 0 about the lines of a bundle of parallels. Refer to Figure 1, s Figure 1 Clarification of terms: Bundle• A set of all straight lines parallel in the same direction. Point S: The centre of the horosphere and the half lines connecting the points of the horosphere with S as the radii of the horosphere. S is also an infinitely distant point of the bundle. Because of the publication of Gauss' correspondence, in which he gave great praise to Lobatchevsky and Bolyai, the 85 concepts of non-Euclidean geometry spread in the 1860's and 1870's. Much interest in geometry was caused by the natural evolution of scientific discovery; and many scholars turned to non•Euclidean geometry, supplementing and refining its methods. Riemann (1826-1866) created a very general science called Riemannian geometry, whose special cases were Euclidean and non-Euclidean geometries. It is essentially due to Riemann ~rl full recognition was given to spherical geometry as a t~e of non-Euclidean geometry. It is known that two lines intersect in at.. most one point on a plane and in two diametrically opposite points on a sphere. If diametrically opposite points on a sphere are considered as a single point, then an elliptic geometry model is obtained. Therefore, in elliptic geometry, points corres pond to diameters of the sphere and lines correspond to planes through the center of the sphere. Elliptic (or Riemannian) geometry is very important as a consistent geometry, where most of Euclid's postulates hold, but in which the parallel postulate fails because any two lines on a plane intersect. Elliptic plane geometry also plays an important role since the geometry of ideal points and id~ lines (i.e., when parallel lines are assumed to intersect at a point on the horizon line, or the ideal line or the line at infinite of the plane, the ideal point is 86 its point at the horizon) associated with Euclidean space is an elliptic geometry. Hyperbolic (or Lobatchevskian) geometry can be distinguished from Euclidean geometry by the following property: Given any line m and a point P that is not a point of m9 there are at least two lines through P which do not intercept m. As illustrated by Figure 2, there are then infinitely many lines t through P which do not intersect m. This pro• perty is based upon the existence of two ideal points He and Rw and the lines PHe and PHw are each parallel to m in hyper bolic geometry.(ll, P• 33l) Figure 2 If a fixed circle is intersected at right angles by other circles, then a hyperbolic plane geometry model is obtained, The "ideal" points are on the fixed circle; the ordinary points are those inside the fixed circle. Refer to Figure 3. Notice that two lines r and s intersect if they have an interior point P in common. Lines r and m and also s and m are parallel if they have a point, Q or T, of the fixed circle 87 in common. In all other cases lines m and n are non- intersecting b~t not parallel. The comparison of properties of spherical and Euclidean plane geometry may now be extended to include elliptic and hyperbolic geometries. The abbreviations (P), (S), (E), and (H) are used, respectively, for Euclidean plane geometry, spherical geometry, elliptic geometry, and hyperbolic geometry.(ll, PP• 332-333) Any two lines: (P) : intersect in one and only one point or are parallel. (S) I intersect at a pair of diametrically opposite points. (E) 1 intersect in one and only one point. (H) 1 intersect in one and only one point, are parallel, or are non-intersecting. Given any line m and a point P that is not on m, there exist through P and parallel to m1 88 (P): exactly one line, (S)a no lines. (E) 1 no lines. (H)I two lines. The sum of the interior angles of any triangle is: (P): 180° (S): greater than 180° but less than 540°, (E): greater than 180°, (H): less than 180°. Similar triangles have the same shape and: (P): may be of different sizes, (S)c have the same size. (E): have the same size, (H) I have the same size, Lines perpendicular to the same line: (P): are parallel. (S) I all pass through two points (poles of the given line). (E): all pass through a single point (pole of the given line), (H): are non-intersecting (also not parallel) lines, Another way of comparing Euclidean and non-Euclidean plane geometry is illustrated by the following table: 89 Comparison Table for Euclidean and Non-Euclidean Plane Geometrx Beginning End of Sentence Euclidear: Lobachevskian Riemannian pf Sentence Two distinct at most at most one one (single point lines inter- one elliptic) sect in two (double elliptic) Given line L one and · at least two no lines through P .and point P only one lines parallel not on L, line to L lthere exist I lA line ·is I is is not ~ separated I 1 into two 1 g parts by a I f 1 point Parallel are equi~ are never do not ! lines ·distant l equidistant exist 1 i I If a line must i may or may --- i intersect fintersects not ~ the other ~one of two I ~parallel ~lines, it I t I I f The valid {right r acute angle obtuse I hypothesis I t :saccheri angle ~ angle l hypothesis I is the I m 1 I !Two distinct are I are parallel intersect ~ I lines perpen-. parallel i ~ I dicular to I 1 ~ lthe same linei I I ~ The angle sum equal to . less than greater ~ 180° of a triangle . than I is I~ ~ I~ h¥. Th: area of' a! indepen- I proportional p~ollt'- ( of' its tr1angle is dent I to the defect t1onal to f sum f .the excess Two triangles similar congruent congruent with equal corresponding angles are 90 Usually elliptic and hyperbolic geometries are called the non-Euclidean geometries. It is important to realize that there are other geometries such as finite and spherical geo metry which are not Euclidean; that is, not Euclidean and non-Euclidean have different meanings, The only plane geometries in which lines are continuous and in which t~o types of transformations such as translations and rotations are possible are Euclidean, sphericalv elliptic, and hyperbolic. Many of the most important geometries fit into a hier archy that is somewhat similar to a family tree. (ll' P• 335) Refer to Figure 4. /ROJECTIVE li~ SIMILARITIES EQUIAREAL ,-__ ~ID~= -.- - ELLIPTIC HYPERBOLIC SPHERICAL Figure 4 The structure in Figure 4 can be used to show the signi ficance of the various postulates of Euclidean geometry to the other geometries. Since two lines intersect in two points 91 on a sphere, spherical geometry is not a special case of projective geometry. Topology includes the study of the properties of Euclidean geometry that depend upon continuity and may be considered as a very general geometry, having many of the other geometries as special cases. These other geometries can be obtained by placing additional restrictions upon the transformations and figures considered in topology. For example, in projective geometry, lines are considered; in affine geometry, parallel lines. Euclidean geometry can be considered as a special case of affine geometry in which corresponding figures must be of the same shape (~) and ~ size (~);by assuming the Pythagorean distance formula, this same restriction can be placed on affine geometry. The study of non-Euclidean geometry through the history of the parallel axioms seems to be the easiest and most appropriate approach. And Euclid's fifth postulate presents a good understanding: If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles.(lO, P• 21 ) Refer to Figure 5. 92 1 m Figure 5 Euclid defined parallel lines as straight lines which lie in the same plane and are produced indefinitely in both directions and do not meet one another in either direction. Many mathematicians thought that the fifth postulate was deducible from the others. But if this were so, then, strictly speaking, it should not be included among the pos tulates. Naturally, there were many arguments to support this point of view. Hence, many mathematicians knew that parallel postulates converse was a provable proposition. The converse is formulated as follows: Proposition A: Consider a line falling on two intersecting lines. Then the sum of exactly one pair of interior angles on the same side is less than two right angles. This converse is equivalent to a theorem familiar to most students of geometry, i.e.: Theorem 1: If two lines are cut by a trans versal and the alternate interior angles are con gruent, then the lines do not intersect. 93 Figure 6 If Theorem 1 were false, ct1 = J3 in Figure 6 or ~+_J3 = 180° and g1 and g2 could intersect. The above would therefore contradict Proposition A, which asserts that if g1 and g2 intersect, then ~ + J3 < 180°. So, Proposition A implies Theorem l,(lO, p. 22 ) (The proof that Theorem 1 implies Proposition A remains unsolved here as an exercise.) Euclid organized plane geometry as a deductive system and listed ten axioms as the foundation of the system. They were the following:( 2) Postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any center and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. 94 Common notions: 1, Things which are equal to the same thing are also equal to one another, 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remaind ers are equal. 4. Things which coincide with one another are equal to one another, 5. The whole is greater than the part. As stated earlier, the fifth postulate became a subject of controversy, and many mathematicians thought it erroneous to include it among the axioms. Since they felt it was not self-evident as an axiom should be, and since the converse is a theorem that can be proved, it seemed reasonable that the so-called fifth postulate should also be a theorem, So, for 2,000 years, from the first century B.C. to the ninteenth century A.D., mathematicians tried to prove that the fifth postulate was a consequence of Euclid's nine other axioms. Amazing as it may seem, all attempts to prove the fifth pos tulate failed! Many mathematicians thought they were sue- cessful, but in every case the proofs were fallacious, Usually the proofs assumed some equivalent proposition to the fifth postulate so the mathematicians really assumed what they were trying to prove. 95 Following are some of the best known alternatives for the fifth postulate:()) Proclus: Parallels remain at a finite distance from one another, Playfair: Through a given point, not on a given line, only one parallel can be drawn to the given line. Legendre: There exists a triangle in which the sum of the three angles is two right angles. Laplace, There exist two non-congruent triangles with Saccheri: the angles of one equal, respectively, to the angles of the other. Legendre, Through any point within an angle less than Lorenz: two-thirds of a right angle, there is a line that meets both sides of the angle. Gauss: If K denotes any integer, there exists a triangle whose area exceeds K. Bolyai: Given any three points not on a straight line, there is a circle that passes through them, The failure of all attempts to prove the fifth postulate led to a new conviction in the minds of Carl Friedrich Gauss (1777-1855), Nicolai Ivanovitsch Lobatchevsky (1793-1856) and Johann Bolyai (1802-1860). They decided that no one had succeeded in proving the fifth postulate as a consequence of the other axioms because it was really independent of the other axioms. This conviction had two important implications. First, it implied that Euclid was correct in taking the fifth postulate as an axiom. Secondly, it implied that some 96 contrary postulate could be substituted for it. If this is done, another kind of geometry is obtained, one different from Euclid's and hence called a non-Euclidean geometry. It has since been shown that this new kind of geometry is as free of contradiction as Euclid's geometry, and therefore equally valid as a mathematical system. Thus, the final out come of 2000 years of investigation of the fifth postulate was that Euclid was both "vindicated and dethroned.'' He was vindicated when it was shown that he was correct in consider- ing the fifth postulate to be an axiom. He was dethroned when it was shown that a contrary axiom could take its place. Euclidean geometry is no longer dominant. It is only one of several possible geometries.(2) Examples The Sum of the Angles of a Triangle To prove an important and insufficiently known theorem concerning the angle sum of a triangle, we introduce the following lemma. Lemma. Given LABC and LA. Then there exists a triangle, ,6A B , L\A B 1 1 c1 such that 1 1 c1 has the same angle sum as 6ABc , and LA1 ~ !LA. F Figure 7 97 Proof. Let E be the midpoint of BC, and let F be chosen on .AE such that .AE = EF and E is between A and F. Then ~ BEA ;:; ~ CEF and their corresponding angles are equal. We show .6AFC is the L.}A1B1c1 we are seeking. By labeling the angles as in the diagram we have L2 = L2', L3 = L3' and LA + LB + LC = Ll + L2 + L3 + L4 = Ll + L2'+ LJ'+ L4 = LOAF +LAFC + LFCA. To complete the proof, observe that LA = Ll + L2 which implies LA = Ll + L2'. In this equation, one of the terms on the right-hand side, Ll or L2', must be less than or equal to one-half the term on the left. A. If Ll ~~LA, relabel A as A1 ; if not, relabel F as A1 • Then relabel the other two vertices of 4AFC as B1 and cl and the lemma is proved. In intuitive terms the lemma says that we can replace a triangle by a "slenderer" one without altering its angle sum. In effect this is proved by cutting off L1 ABE from .6.ABC and pasting it back as ~FOE. The lemma is less trivial than may be seen at first sight, for in neutral geometry we cannot assume that the angle sum is constant for all triangles--this is a Euclidean theorem whose proof depends on the parallel postulate. Hence the lemma is important because it indicates that given a triangle we can construct a noncongruent triangle having the same angle sum. The lemma implies, by an easy argument, the existence of an infinite sequence of noncongruent triangles, all having the same angle sum as a given triangle. Assume Euclid's Fifth Postulate to Prove Playfair Assume that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced 98 indefinitely, meet on that side on which are the angles less than the two right angles. To prove: Through a given point, not on a given line, only one parallel can be drawn to the given line. Proof. Figure 8 Assume 41 < 90°. Therefore 4 1 + 4 3 < 180° (by Euclid • s Fifth Postulate) and we are finished. Assume Playfair to Prove Euclid's Fifth Postulate Figure 9 Assume 41 + 4 2 < 180° and 4 1 + 4 3 :;: 180° (Supplementary) Therefore 42 < 43. RP ff) (Assuming Playfair only ~ parallel) m meets f at S) and RP IJ ~ insures that PS inside. 99 Assume Euclid to Prove Legendre Assume (i) the sum of the angles of any triangle is two right angles to prove (ii) there exists a triangle in which the sum of the angles is two right angles. Proof. Given line a and point P not on a. Construct PQ_l a and S •psj_ PQ. Therefore PS does not intersect line a (by Euclid). s• Figure 10 Assume (i) and that b is a line through point P (not PS). Ray PT is either inside angle SPQ or angle S 0 PQ. Suppose PT is inside angle SPQ. Let R be a point on a line a such that angle PRQ < angle SPT. Do this by finding a point R on a such that angle PRQ is less than any given angle. Figure 11 = • 3) R0 R1 PR 0 4 1 + 4 2 = 2 Right 4 • 2 • ( 4 + + 2 is not more than two right 4- (by Legendre). 100 Therefore 2 <4 3) i 4 1 or 4PR1 Q 'S i { 4 1). Construct R1R2 = PR1 • WLOG 4 PR2Q ~ i {i 4 1). Thus, the points R1 , R2 , ••• R50 , ••• are such that Rn- l Rn = PR n-l and 4 PRn Q $ i ( 4 1) If n is big, 4 PRnQ < any given angle. Therefore, point R exists on line a such that 4PRQ < 4.SPT, and if the sum of the angles of a triangle is two right angles, 4 PRQ + 4 RPQ = 1 right angle = 4 SPR + 4 RPQ, There fore 2j SPR == 4 PRQ and 4 SPR < 4 SPT. Or ~ PT between PQ and PR and it ~ intersect line a between R and Q. Sacderi-Legendre Theorem The angle sum of any triangle is less than or equal to 180° •. Proof. Suppose the contrary. Then there exists a 0 triangle, .6. ABC, with angle sum 180° + p , where p is a positive number. Now, applying the preceding lemma {refer to Proof of The Sum of the Angles of a Triangle), there exists a more "slender" triangle, ,e1 A1 B1 c , with the same 0 1 angle sum as Ll ABC, 180° + p , such that LAl ~ i LA. By applying the lemma to ~ A1B1c1 , we see that there exists 0 a triangle, ~A 2 B 2c 2 , with the same angle sum, 180° + p , such that LA2 ~ i LA1 ~ i LA. By continuing in this fashion we construct a sequence of triangles 101 0 each with angle sum 180° + p , such that, for every positive integer n, LAn ~ ~ LA. Clearly, we can select n sufficiently large so that L~ is as small as we please, in particular so that 0 LAn ~ P • 0 Since L~ + LBn +LOn= 180° + p , it follows that 1 LBn + L.:n-0 ~ 180° This contradicts Euclid's Proposition 17 of Book I, which states: "In any triangle, two angles taken together in any manner are less than two right angles." The supposition is therefore false, and the theorem follows. As a specific example of this theorem, suppose p = 1 and LA = 25°. Thus in our original triangle, .4 ABC , we have LA+ LB + LC = 181° and LA= 25°. By the lemma, there exists + + = a triangle, 6 A1B1c1 , such that LA1 LB1 Lc1 181° and 0 LA1 ~ ~· In like fashion there exists a triangle, ~ A2B2c2, 0 such that LA2 + LB 2 + Lc 2 = 181° and LA2 ~ ~ • To see the contradiction, apply the lemma three times more to obtain 0 IJ,.A5B5c5 in which LA5 + LB5 + Lc 5 = 181° and LA5 ~ ~ < 1°. Consequently LB5 + c 5 > 180° which is impossible. Colollary. The angle sum of any quadrilateral is less than or equal to 360°. The corollary implies Saccheri's con clusion that the hypothesis of the obtuse angle is false. Similarly, the theorem denies that the angle sum of a triangle 0 can exceed 1Bo • But the possibility that the angle sum of a triangle may be less than 180°, which corresponds to Saccheri's hypothesis of the acute angle, forces itself on our attention. 102 An Evolution of Non-Euclidean Geometry Saccheri (166Z-1?33l "The great clown"; developed many important theorems of r------~----~f non-Euclidean geometry F. Btl ai (1775-1856) Ga~s (1777-1855)~Bartels (1?69-18)6] father of J, Bolyai; first to consid- teacher of Gauss; fellow student of er a system deny- later teacher of Gauss ing Postulate Lobatchevsky Five; not pub- 1 lished I ? ..., J, Bolyai (1802-1860)~----~'----)Lobatchevsky (1793-1856) Hungarian army officer; Professor at University of developed geometry of acute Kazan; pupil of Bartels; angle, published in 1832; developed geometry of acute praised by Gauss; generally angle; first published in ignored otherwise; never 1829-30; published much published again because of math; never recognized for disappointment; claimed work in geometry in his time; Lobatchevsky copied him defended his theories A Chronology of Non-Euclidean Geometry Gauss F, B.OlyaL. J. Bolyai Lobatchevsky About 1795-1800 1796-1799 student 182.3 refers Before 1823 developed ideas with Gauss at to discovery student of Bartels of non-Euclidean Gottingen; often . of non at Kazan geometry discussed problem Euclidean of parallels with· geometry 1823 began inves 1824 refers to Gauss tigations of theories in a 1825 commits parallels letter to 1804 sent "Theori~· theory to Schumacher Parallelarum" to writing 1826 first pub Gauss lished work on 1832 theory his theory About 1820 warned published as· son J. Bolyai not appendix to 1829-30 theory to get involved ~ work of :published in full with problem of father :in Russian parallels , ~.; ~1840 first pub- 'lished in Germany; ·criticism from 1 · J, Bolyai 10.3 Suggested Problems/Topics for Student Research 1, Research one of the other possible geometries, 2. Assume Euclid's Fifth Postulate to prove Playfair, J, Assume Laplace, Saccheri to prove Playfair. 4. Research the Legendre, Lorenz alternative to the Fifth Postulate as to why the two-thirds of a right angle is specified, Research the geometry of Bolyai-Lobatchevsky (hyperbolic geometry), 6. Research the mathematics of one or more of the following men and relate the findings to non Euclidean geometry: Proclus, Saccheri, Lambert, Riemann, Klein, Hilbert, Bolyai, Gauss, Lobatchev sky. Prove theorem(s) concerned with non-Euclidean geometries, 8. Write a paper on the different types of geometries, 9. Develop material for a class symposium on "What non-Euclidean concepts should be taught in a regular high school geometry course?" 10. Prove that Theorem 1 implies Proposition A (refer to Introduction to the Topic in this chapter) 11. Develop ideas on the probable/possible inter relationship of some non-Euclidean geometers. 12, Devise a thorough chronology of non-Euclidean geometry, including several additional non Euclidean geometers to those listed in this chapter, 1.3. Other topics suggested by students and approved by instructor. Suggested References for Student Research Adler, Claire F. Modern Geomet~. San Francisco: McGraw Hill Book Company, 1967. Adler, Irving. A New Look at Geometry. New York: The New American Library, 1966. Blumenthal, Leonard M, A Modern View of Geometry. San Francisco: w. H. Freeman and Company, 1961. 104 Bonola, Roberto. Non-Euclidean Geometry: A Critical and Historical Study of Its Developments, New York: Dover Publications, Inc., 1955, · Coolidge, Julian L. A History of Geometrical Methods, New York: Dover Publications, Inc,, 1940, Coxeter, H. S. M, and R, R. S. Non-Euclidean Geometry, Toronto: University of Toronto Press, 1965, Daux, P, H. "The Founding of Non-Euclidean Geometry," Mathe matics News Letter (April-May, 1933), pp, 12-16. Eves, Howard. An Introduction to the History of Mathematics, San Francisco: Holt, Rinehart and Winston, 1969, Fitzpatrick, Sister M, M, "Saccheri, Forerunner of Non Euclidean Geometry," The Mathematics Teacher, 57:323-332. Forder, H. G. Geometry. London: Hutchinson and Company, ' 1960. Gillispie, c. c. Dictionary of Scientific Biography, Volume II. New York: Charles Scribners and Sons, 1970. Halsted, G. B, "A Class-Book of Non-Euclidean Geometry," The American Mathematical Monthly, 8:84-87. --- Halsted, G. B, "Facsimile Editions of John Bolyai's 'Science Absolute of Space,'" The American Mathematical Monthly, 17:31-33. Halsted, G. B, "Gauss and the Non-Euclidean Geometry," The American Mathematical Monthly, 7:247-252. Halsted, G, B. "A Non-Euclidean Geometry," The American Mathematical Monthly, 9:153-159. Halsted, G. B, "Non-Euclidean Geometry," The American Mathe matical Monthly, 7:123-133. Halsted, G. B. "Non-Euclidean Geometry, Historical and Expository," The American Mathematical Monthly, 1-5. Halsted, G. B. "The Popularization of Non-Euclidean Geometry," The American Mathematical Monthly, 8:31-35. Halsted, G, B. "Simon's Claim for Gauss in Non-Euclidean Geometry," The American Mathematical Monthly, 11:85-86. Halsted, G. B. "Supplementary Report on Non-Euclidean Geometry," The American Mathematical Monthly, 8:216-230. Khayyam, o. (Translated by Amir-Moez, A. R,) "Discussion of the Difficulties in Euclid," Scripta IVIathematica, 24: 275-303. Kulczycki, Stefan. Non-Euclidean Geometry. New York: Academic Press, 1964. 105 Leborbeiller, P, "The Curvature of Space," Scientific American, 191:80-86. Lewis, F. P. "History of Parallel Postulate," The American Mathematical Monthly, 27cl6-23. Meschkowski, Herbert. Non-Euclidean Geometry. New York: Academic Press, 1964. Meserve, Bruce E., and Sobel, Max A. Mathematics for Secondary School Teachers. Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1962. Moise, Edwin E, Elementary Geometry from an Advanced Stand point. Reading, Massachusetts: Addison-Wesley, 1963. Newman, James R. The World of Mathematics, Volume III, New York: Simon and Schuster, 1956. Redei, L, Foundations of Euclidean and Non-Euclidean Geometries According to F, Klein. New York: Pergamon Press, 1968. Rolwing, R, H., and Levine, M, "The Parallel Postulate," ~ Mathematics Teacher, 62:665-669. Smith, David E. History of Mathematics, Volume II, Boston: Ginn and Company, 1953. Sommerville, D, M, Y, The Elements of Non-Euclidean Geometry. New York: Dover Publications, Inc., 1958. Struik, D. J. A Concise History of Mathematics. New York: Dover Publications, Inc., 1967, Toth, I. "Non-Euclidean Geometry Before Euclid," Scientific American, 221:87-98. Tuller, Annita, A Modern Introduction to Geometries. New York: D. Van Nostrand Company, Inc., 1967. Vucinich, Alexander, Science in Russian Culture. Stanford: Stanford University Press, 1963. Wolfe, Harold E, Introduction to Non-Euclidean Geometry. San Francisco: Holt, Rinehart and Winston, 1945. 106 CHAPTER IV BIBLIOGRAPHY 1. Adler, Claire F. Modern Geometry. San Francisco: McGraw~ Hill Book Company, 1967. 2. Adler, Irving. A New Look at Geometry. New York: The New American Library, 1966. Blumenthal, Leonard M. A Modern View of Geometry. San Francisco: w. H. Freeman and Company, 1961. 4. Bonola, Roberto. Non-Euclidean Geometry: A Critical and Historical Study of Its Developments. New York: Dover Publications, Inc., 1955. Coxeter, H. s. M. and F. R. s. Non•Euclidean Geometry. Toronto: University of Toronto Press, 1965. 6. Eves, Howard. An Introductian to the History of Mathe matics. San Francisco: Holt, Rinehart and Winston, 1969. 7. Forder, H. G. Geometry. London: Hutchinson and Company, 1960. 8. Golos, Ellery B. Foundations of Euclidean and Non Euclidean Geometry. New York: Holt, Rinehart and Winston, 1968. Kulczycki, Stefan. Non-Euclidean Geometry. New York: Pergamon Press, 1961. 10. Meschkowski, Herbert. Non-Euclidean Geometry. New York: Academic Press, 1964. 11. Meserve, Bruce E., and Sobel, Max A. Mathematics for Secondary School Teachers. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1962. 12. Newman, James R. The World of Mathematics, Volume III. New York: Simon and Schuster, 1956. 13. Redei, L. Foundations of Euclidean and Non-Euclidean Geometries According to F. Klein. New York: Pergamon Press , 196 8. 107 14. Sommerville, D. M. Y. The Elements of Non*Euclidean Geometry. New York: Dover Publications, Inc., 1958. 1). Tuller, Annita. A Modern Introduction to Geometries. New York: D. Van Nostrand Company, Inc., 1967. 16. Wolfe, Harold E. Introduction to Non-Euclidean Geometry. San Francisco: Holt, Rinehart and Winston, 1945. 108 CHAPTER V LINEAR PROGRAMMING Approximate Length of Time for Unit Three to four weeks, Behavioral Objectives 1, The student will gain some insight into the recent historical development of linear programming. 2. The student will become familiar with the termin ology of linear programming. 3. The student will gain an appreciation of some applications of linear programming to economic and mathematical models. 4, The student will develop the mathematical skills necessary to solve linear programming problems, using: (a) geometric methods, (b) algebraic methods, and (c) the simplex method, 5. The student will research some area of linear pro gramming or solve various problems with linear programming techniques and share the results with the Seminar class. Preface World War II actually initiated the development of linear programming. The use of scientific methods for various tac tical problems during the war introduced a new field of endeavor known as "operations research," These military problems were so successfully solved that there was a 109 natural evolution of the new techniques towards the solution of industrial problems, business operations, and government research. Problems such as minimizing production costs, maximizing profits, determining the best product allocation, or the best use of limited resources were undertaken by operations research methods,(lJ, P• 235) The mathematical technique known as "linear programming" is the tool used in coping with these kinds of problems. This technique of linear programming is to find the best solution from among all solutions of a system of linear in- equalities. When using linear programming, basic resources, such as labor, time, and raw materials (known as constraints or restrictions) are taken into consideration in maximizing profits or minimizing costs. Programming in the mathematical sense is finding a solution to certain kinds of optimization problems and is not the same as the word "programming" used in connection with writing instructions for a computer.<19 P• lOl) Introduction to the Topic Linear programming is concerned with special classes of maximum and minimum problems which arise frequently in economic applications.<2 • P• l) The variables are usually processing or scheduling variables in some physical situation; the inequalities are obtained from the physical constraints of these variables; and the criterion for "best solution" is 110 the value of some given linear function of all the variables. As the term is used today, linear programming includes the formulation of the problem in linear programming terms, algorithms for finding the best solution, and the analysis of the effect of changes in the values of problem parameters. When a solution fails to exist, the system is said to be infeasible or to have no feasible solution. When the best solution is infinite in one or more variables, the system is said to be unbounded.(5, P• 5l) Optimization problems are concerned with finding the minimum or maximum of some combination of the unknown quan---- tities (variables). When the quantities to be minimized or maximized and the requirements on the variables are all linear, such problems are called linear programming prob lems.(l, P• lOl) The methods of linear programming (simplex and combinatorial) are applied to the solution of diverse problems dealing with optimal situations, as stated above. The solution of a linear programming problem is a set of values of the variables which (1) satisfies the constraints, and (2) maximizes or minimazes the objective. It is not to be confused with a "solution of the constraints" which needs to satisfy only (1) above.(l, P• 103) Some typical linear programming problems include those such as oil refinery scheduling (maximizing the profit per barrel or maximizing the profits for the operation), paper 111 trimming (filling orders with as little trim loss as possible), and production allocation. For all linear programming problems, four possibilities exist with regard to solutions:(l, PP• 105-l06) 1. No solution exists; the constraints are inconsistent. 2. A unique, bounded solution exists and occurs at an extreme point. 3. An infinite number of bounded solutions exists, and at least one of them occurs at an extreme point. 4. The solution exists but is unbounded. The solution of a simple linear programming problem, where the function ax + by + c defined over a convex polygon takes on its maximum (minimum) value at a vertex point of the con vex polygon, can be obtained by finding the vertices of the convex polygon or by substituting the coordinates of each vertex into the function. The largest value of the function is, of course, the maximum; the smallest value, the mini mum. (11, P• 332) TERIVIS Algorithm: A mathematical procedure whose routine application (iteratively or recursively) yields the solution to a particular class of problem. Basic variables (primal): Those m primal variables, in an m-constraint linear programming problem, which are permitted to be nonzero and nonbounded at an iterative stage in the solution. The remaining variables (non~ basic variables) are fixed at zero. In a bounded variabl~ code, ~orne v)ariables may be fixed at their bounds.t5, PP• 41-59 Constraints: The inequalities which must be satisfied.(l, P• 101) 112 Convex set: Geometrically, a set of points (region) that contains all the points on the line segment joining any two points of the set. A point which does not lie on any line segment joining two other points of the set is called an extreme point or vertex of the set. Convex programmin9: Optimization of a convex function over a convex reg1on (set). Linear programming and quadra tic p~ogramming ar~ special cases of convex program ming.l5, PP• 4I-59J Duality: To every linear-programming problem there corres ponds another linear-programming problem which is called the dual. The optimal solution of the primal and the optimal solution of the dual are intimately connected. If the optimal solution to one is known, then the optimal solution to the other is readily available. Situations can a~ise where)the dual is easier to solve than the primal.ll, P• 24~ The unsymmetric dual problems:(2, P• 7l) The primal problems Find a column vector X=(x ,x , ••• xn) 1 2 which minimizes the linear functional, f=cX, subject to the conditions AX=b and X~ o. This statement of the general linear-programming problem assumes that the number of rows of A is less than the number of columns of A. The dual problem: Find a row vector W=(w1 ,w2 , ••• wm) which maximizes the linear functional, g=Wb, subject to the conditions WA~c. Notes: In the dual problem, the variables wi are not restricted to be nonnegative. For both problems, c=(crc2, ••• cn) is a row vector, b=(b1 ,b2, ••• bm) is a column vector, and A=(aij) is the matrix of coefficients. By transposing the rows and columns, including the right hand side and the objective function, reversing the inequalities, and(maximizing instead of minimizing, the dual is obtained. 1, PP• 248-249) Further, duality serves as a bridge betwee~ linear programming and the theory of games.l2, P· 7lJ · Duality theorems for linear programming:(5, P• 46) Main theorem: If both the primal and dual problems have a finite optimum, then the optimum values are equal. Corollary: If either problem has a feasible finite optimum, then so does the other, and the optimum values are equal. 113 Corollary: A feasible but unbounded solution to one problem implies no feasible solution for the other. Corollary: No feasible solution to one problem implies that the other is either unbounded or infeasible. Weak theorem of the alternative: A variable and its complementary slack are not both nonzero. Strong theorem of the alternative: Among all alternate optima, at least one solution exists in which a vari able and its complementary slack are not both zero, and the one of the pair that is zero in this solution is zero in all alternate solutions. Theorem: The dual of the dual is the primal. Equilibrium theorem (canonical): Let vector x = (Ci) be a nonnegative solution of xA = b and let vector;¢ y = (J'tj) be a solution of Ay ~ c. Then x maximizes xc, and y ( ) minimizes yb if and only if .1?. = 0 whenever a. y > v.. 2 ,p. 82 s 1 1 01 Extreme point: A point of a convex set which does not lie on a line segment joining any other two points of the set; also called a "vertex.'' Feasible solution: A solution to the constraint equations in which all variables satisfy their sign restrictions. A feasible solution which corresponds to a feasible basis is a basic feasible solution, Geometric solution: A graphic method of solving a linear programming problem, by plotting the half-planes deter mined by the constraints and the lines of constant value for the functional. Its use is restricted to problems with, at most, two structural variables. Iteration: A single cycle of operations in a solution algorithm made up of a number of such cycles. Nonbasic variable: A variable in a linear programming itera tive stage whose value is fixed at zero, or is bound, and whose column v~ctor tnB~efore does not appear in the current basis.\5, P• J Objective or objec1ive function: The quantity to be maximized or minimized. I, P• 101) Simplex method: A computational routine for obtaining the optimal solution to a linear programming problem. It is an iterative elimination procedure at each stage yield ing a basic solution, and it rests primarily on the following two principles: (a) elementary row operations on the constraint matrix leave the set of feasible solutions unchanged, and (b) the number of nonzero values in an optimal solution is never more than the number of constraint equations. 114 The following steps comprise one iterative stage in the simplex method: 1. A test of whether the current solution is optimal and/or feasible. 2. If not both optimal and iasible, a choice of an entering variable and a departing variable. A pivot step so a$ to read off)the new solution; then back to (1),\ 5 • PP• 5 8-59 Examples Example of Feed Blending(5, PP• 7-8) A poultry farmer needs to supplement the vitamins in the feed he buys. He is considering two products, each of which contains the four vitamins required, but in differing amounts. Naturally, he wants to meet (or exceed) the minimum vitamin requirements at least cost. Should he buy one product or the other, or should he mix the two? The facts are summarized below: Cost Per Ounce Product 1 (3¢) Product 2 (4¢) Vitamin 1 5 units 25 units Vitamin 2 25 units 10 units Vitamin 3 10 units 10 units Vitamin 4 35 units 20 units The farmer must provide 0 per hundred potinds of feed, at least 50 units of vitamin 1, 100 units of vitamin 2p 60 units of vitamin 3, and 180 units of vitamin Let P represent 4. 1 the number of ounces of product 1 purchased, and P2 the number of ounces of product 2. Therefore, the objective is to minimize 3P1 + 4P2• The constraints this time are the minimum requirements. Each of the four vitamins can be obtained in varying amounts from either product. Whatever combination of products is bought, the sum of the units of a given vitamin in the two must equal or exceed the minimum requirement for that vitamin. Thus P the following four •,~greater than or equal to" constraints are: 115 1, 5Pl + 25P2 ~50; 2. 25P1 + lOP2 ~ 100 3. lOP1 + lOP2 ~ 60; 4. 35P1 + 20P2 ~ 180 (The nonnegativity requirement on the variables, i.e., the number of ounces of P1 and P2.) 10 ·•. :·----·.·-· ...· :.~--- ~- .~:- FEASIBLE REGION 6 __._,-. 2 --- :- -. ..:: .· .. -----~- :< 10 12 Number of ounces of Product 1 bought Note: There is no restriction on the amount by which the minimum can be exceeded. Solution. The figure below shows only that part of each constraint line that is on the border of the feasible region, and the dotted lines are total cost lines, Shown are the feed additive constraints and the total cost lines for 12¢, 19¢, 30¢, and 40¢. " ·FEASIBLE REGION "~1>~· ~:o:.·-.-~·,. 2 12 116 Point P is the intersection of 50 and + lOP2 = 60. So, lOP1 + lOP2 = 60) lOP1 + lOP2 = 60 5P1 + 25P2 = 50 -lOP1 - 50P2 =-100 ----=------= ) coordinates - 4oP2 = -40 or P2 1 of the low- Therefore P = 5 est point, P, 1 are {5, 1) Since 3P1 + 4P2 minimizes, the lowest cost line (LC) has a slope of - 3/4 and, of course, passes through the point (5, 1), Therefore, to obtain the equation of LC, slope and point form are used, i.e., p 2 - 1 P =~or- 3P + 15 = 4P - 4 or 1 - 5 1 2 3P1 + 4P2 = 19¢ The conclusions are that: 1. 12¢ line is not feasible anywhere and 40¢ line is feasible everywhere. 2. Since the 19¢ line is the lowest cost line that touches the feasible region, the required vitamin content cannot be obtained,for less than 19¢. 3. 3P1 + 4P2 = 19¢ passes through (P1 , P2) or (5, 1) and this indicates: 5 ounces of P1) {satisfy the four constraints 1 ounce of P2 ;==>\Minimizes the cost (19¢) One of the most valuable features of the methods used to solve general linear programming problems is that the methods guarantee that the optimum solution is the best. 117 Problem and Its Dual: Maximize x + y subject to (; ~-5~ ! ~ · · 1 · h {Y = Jx + 2) {2y = 4 or Th~s w~l be max~mum w en Y =-Jx + 2 or{Y = 2, x = o. Therefore Max. = 0 + 2 = 2. ) Dual: Minimize 2x + 2y subject to (-.3~! ~Y ~ i . This will be minimum when (-3~ ! ~y : i) or 6y ~ 4 or y ~ 2/3, x ~ 1/3. Therefore Min, = 2(1/.3) + 2(2/.3) = 2. Therefore, by the Equilibrium Theorem~ 2 is the maximum at point (0, 2). Geometric interpretation: It is desired to determine where the family of lines, x + y = c is maximal in the area y :;; .3x + 2 . bounded by y ~-.3x + • So, look at the fam~ly of lines, ( 2 y = -x + c (i.e., slope of -1). y Maximum Point (0, 2) x + y = 2 Therefore c = 2 lines represent the Y = Jx + of lines with m = -1) -.3x + 2 Canonical form'a Maximize x + canonical form is: Maximize x + + y + 1W1 = 2 + y + 1W2 = 2. 118 (8) The Simplex Method: 1 y 2 3 1 0 1 Therefore x = 0; y = 2; w1 = 0; w2 = 0 and Maximum at (o, 2). Since Maximum = x + y + ow1 + ow2 , the result is = 0 + 2 + 0 + o = 2. Example of Three Inequalities and Two Unknowns To Show the Effect of Doubling and Tripling on the Cost Function Daily Supply of Vitamins in Any Food Bread Milk x of A y of B Iron 1/8 3/8 Calcium 1/6 1/6 B2 3/10 1/10 (Cost Function: 1) C = X + y; X ;a- 0, y ;a- 0 l/8x + 3/8y ;a- 1; l/6x + l/6y ~ 1; 3/lOx + 1/lOy ~ 1 4 Extreme Points: (8, 0), (5, 1), (2, 4), (0, 10). Cost at extreme points: 8, 6, 6, 10. Therefore x + y = 6 is the minimum cost line from ((2, 4) to (5' 1)). 119 y 2-xL..t-U>f. jiJ T /01 ·- Suppose that the price of milk (B) is doubled. Then 0 = x + 2y or 0(8, 0) = 8 0(2, 4) = 10. Therefore, the 0(5, 1) = 7 0(0;10) = 20 minimum cost is at (5, 1). Let the price of bread triple. Then o = Jx + y or 0(8;0) = 24 0(2, 4) = 10. Therefore, the 0(.$,1) = 16 0(0,10) = 10 minimum cost line is from ((0, 10) to (2, 4)). E Problem and Its Dual and Equilibrium Theorem: Maximize Jx + 2y subject to ?x + 6y ~ 84; 4x + 2y ~ 32. The above will be maximum when [4~ ! ~~ : ~~J or when ~ : ~~~~. Therefore Max. = 3(12/5) + 2(56/5) = 148/5 = 29 3/5. Dual: Minimize 84x + 32y subject to(?x + 4y ~ 3. This (6x + 2y ;? 2 will be minimized when {7x + 4y = 3 or when x = 1/5. \6x + 2y = 2 y = 2/5 Therefore Min. = 84(1/5) + 32(2/5) = 148/5 = 29 3/5. Therefore, by the Equilibrium Theorem, the maximum is 29 3/5 and the optimal solution is (12/5, 56/5). 120 Analysis (Graph) by Dual: Geometrically, find where the family of lines of the form 3x + 2y = C or y = -~x + ~ have a maximum value (C) in the area bounded by ?x + 6y ~ 84 and 4x + 2y ~ 32. (Note: the dual might be more easily pictured.) Dually, it is desired to find where the family of lines, 84x + 32y = C or y = -~x + 3~ have a minimal value (C) in the area bounded by ?x + 4y ~ 3 and 6x + 2y ~ 2. \ \\UJ. (ll ' \ ' \ ~\ \ \ \ \ '\ \ (1) ?x + 4y -·3 \ \ \ \ ( 2) 6x + 2y = 2·. . . ",_ \ Minimum point \ (1/5, 2/5) '·. \ • Therefore \ 84x + 32y = 29 3/5 (Min.) \ -\ \~ \ \ \ (Pencil lines repr~een~ \ the family ~f lines,_ \ with m = -21/8) \ ' \, \~. \ \ \ \ '·' \ Simplex Method (In Canonical Form): In canonical form: Max. Jx + 2y + ow1 + ow 2 , such that (1) ?x + 6y + w1 = 84; (2) 4x + 2y + w2 = 32. 121 3 2 0 0 X y 0 w 84 6 1 0 1 ( ~~~ ~ 32 __:__.. :~3 0 -3 -2 0 0 ~~= :7·· :===~~~--- (o w1 -~8-~ ~ J X 8 l ~172--~ 24 0 ~1/2 0 3/4 epeat 0 w1 56/5 0 1 2/5 If x = 12/5, w2 = O, it is found in (4x + 2y + w =32) 3 X 12/5 1 0 -1/5 2 that y = 56/5. Therefore, 29 3/5 0 0 1/5 Max. = 3(12/5) + 2(56/5) = 29 3/5. Example With Slack Variables Maximize x1 + x2 where (1) 2x1 + x2 $ (2) x1 + 3x2 Constraints xl . x 2 ~ 0 The slack variables are x + x such that 3 4 (3) 2x + x + x = 2 1 2 3 (4) x1 + 3x2 + x4 = 3 x2 x1 ? 0; x2 ~ 0 x ~ 0; x 4 ~ 0 3 c 122 The preceding graph shows convex set OABC with extreme points o, A, B, and C such that: Extreme Point 0 A B c xl = 0 1 3/5 c x2 = 0 0 4/5 1 x3 = 2 0 0 1 x4 = 3 2 0 0 The variables which must be zero at the Extreme Point are nonbasic. Example: At point o, x1 and x2 are nonbasic At point Bp x and x are nonbasic. 3 4 Variables which may be zero or nonzero are basic, Solution. At point 0, x x 0 and x 2, x 3. Therefore, 1 = 2 = 3 = 4 = (5) x = 2 - 2x - x and (6) x = 3 - x 3x • By the Gauss 3 1 2 4 1 2 method, point C: fx 3 = 1 - 5/3x1 + l/3x4 x2 = 1 - l/3x1 - l/3x4 Look at point 0 and Equations (5) and (6) where x = 0, x = 0 1 2 These cannot be decreased because (x1 ~ 0, x2 ~ 0); but they can be increased (x1 and x 2). Since the objective is x1 + x 2 , it would make the objective larger. The rest of the solution is left as a problem for the students in the Seminar. Example of a Dual Problem(5, PP• l0?-108) G. lV.' • • 11 + 4 b . t t ( ?x + 6 Y ::; 84 1ven: 1ax1m1ze x y, su Jec o: 4x + 2y ~ 32 • The above system appears in tabular form as follows: 11 4 84 ~ ~ K 32 :?' 4 2 It can be seen that the above formulation was prepared hori zontally. If the new variables (z1 and z2) are assigned to the first and second rows, then it is possible to prepare a formulation vertically. The matrix now appears as: 123 11 4 z1 84 ~ ~ ~ z2 32 ~ 4 2 If the old variables x and y are crossed out, then the formu lation of the~ problem may result as follows: Minimize 84z1 + 32z2 , subject to 7z1 + 4z2 ~ 11; 6z1 + 2z 2 ~ 4, or -84 -32 zl z2 11 $ 7 4 4 $ 6 2 Note that the objective (maximization or minimization) of the dual problem is the reverse of the primal and that the constant column of the primal becomes the objective function of the dual. Similarly, the objective function of the primal is the constant column of the dual problem. It can be seen also that the inequality signs are reversed in the dual prob lem. These changes and comparisons are completely general and apply equally well to any problem formulation. Some examples of primals and their duals: (1) Primal• Dual: Min. x - 3x + 2x , subject to + 12w + 2 3 5 Max~7w1 2 lowj, x + 3x - x + 2x 7 su jec ~ 1 2 3 5 = - 2x + 4x + x = 12 · w < 0 2 3 4 ~ - 4x 2 + Jx3 + ax5 + x6 ~ 10 Jw~ -2w2 -~~:J I $ 1 and x ;:- 0 -w + 4w +3w 1 < -3 2 1 1 2 3 - b = (7, t w2 i <- 0 c = (0, 1, -3, 0 ' 2' I 2w 1 + 8w 3 J ~ 2 . w ! 0 2 I- _.r·"'!3 A = (~ -~ -t 1 0 ••-., •.._~,~ ...... ,,....~...,.,·,r->- _,,_,_ ...>-.;r<' , .....-~-- 0 -4 3 0 8 1 0 0. 3 -2 -4 -1 4 3 0 1 0 2 0 8 0 0 1 124 (2) Primal: Max 28x + 28y such that[35x + 40y ,; 180~ or 4y $ X ' (7x + 8y $ 360) -x + 4y $ 0 Dual: Min. 360x + Oy such that y The Simplex Method Minimize x1 + 2~ 2 = z such that 2x1 + x2 $ 10 (Objective Function, i.e., OF) xl + x2 <- 6 -xl + x2 ~ 2 .;.:2x 1 + x2 $ 1 A 1 2 3 125 With slack variables, the constraints are: (1) 2x + x + x 10 1 2 3 = (2) x1 + x2 + x4 = 6 (3) -x + x + x 2 1 2 5 = (4)-2x1 + x2 + x6 = 1 With four equations, a basis will consist of four variables that are permitted to be nonzero; the other two variables, therefore, zero. Table 1: List of 15 possible ways to assign 2 var~~es = o and solve for other 4 variables. Note: l~)~ 15 • Solution # x x x x x x Vertex 1 2 3 4 5 6 1 0 0 10 6 2 1 I A I 2 0 10 0 -4 -8 -9 ! 3 o I 6 4 0 -2 -3 f 4 o I 2 8 0 -1 f 5 o 1 9 1 0 i B ' I ~I l 6 5 i 0 0 1 j 7 11 F f 6 ~ . 7 0 -2 0 ~ 8 13 I~ • ~ l 8 0 14 8 ~ 0 -3 -21 ~ J 9 -1/2' 0 11 13/2 f 3/2 0 J f 10 41 2 0 o,~ 4 7 ! E ' ~.' i ~ 11 8/3 :14/3 0 -4/3 (~ 0 5/3 ~ ~ 12 9/4 '11/2 0 -7/4- ~-5/4 0 ~ - ~ 13 2 0 f 0 1 ! D 2' 4 t ~ I il 14 5/3 ~13/3 7/J 0~-5/3 I 0 . 1· J tl 15 3 5 2~ 0 0 ~ c " 1 g In Table 1 above, 9 of the 15 solutions contain negative values; therefore, these 9 are not feasible. The other 6 are feasible and correspond to vertices A, B, C, D, E, F. The 6 lettered lines are basic solutions and were developed by setting 2 variables = 0 and solving equations for the other 4 variables. (The basic feasible solutions correspond to the vertices.) 126 Table 2: The 6 basic solutions of this problem, Vertex A 0 0 r 10 6 2 1 ~ I f B 0 1 ! 9 5 1 0 c 1 5 2 I 0 0 f I D 2 2 0 ~ 0 1 E 4 0 0 4 'i F 5 ~ I 0 1 7 11 In Table 2 above, each pair of adjacent lines represents two adjacent vertices. The two adjacent bases have all but one variable in common, e.g., basis for A is composed of x , x , x , and x ; the basis forB also contains x , x , 3 4 5 6 3 4 and x , with x in point A replaced by x in point B. 5 6 2 In broad outline, the simplex method is as follows:(5, PP• 38-39) 1. Select an initial feasible basis. If all the constraints are "less than or equal" types with positive right-hand sides, the slack variables immediately constitute such a basis; we shall sometimes encounter actual equations, so there will naturally be no slack variable for that constraint. - 2. If necessary, perform some algebraic operations that will express the objective function en tirely in terms of nonbasic variables. The variables that remain in the objective function after these operations will then all have the value zero, and the value of the objective function will therefore be constant. 3. Inspect the objective function to find the term with the largest positive coefficient. Giving this variable some positive value will increase the value of the objective function; this vari able is to be introduced into the basis. 4. By a simple test, then determine which variable is to go out of the basis. Effecting the re moval is a matter of some algebraic manipulations that constitute the bulk of the computational effort. These operations will modify the 127 coefficients in the objective function, so as once again to express it in terms of nonbasic variables only. 5. Inspect the objective function. If there are no terms with positive coefficients, there is no possible way to increase the value of the objective function by giving any nonbasic variable a positive value. The optimum has therefore been reached. If there are positive terms; repeat steps 3, 4, and 5. This process is the algebraic equivalent of moving from vertex to vertex in such a way as to guarantee improvement at every step. Since there is a finite number of vertices, the process must terminate after a finite number of repeti tions. Since improvement is guaranteed at every step, when finally there are no more positive terms in the objective function, the value of the objective function must be the largest it can possibly be. There can never be any question whether the result found by this method really is the best that could be determined, Preceding Example of Simplex Method Using Above Outline The slack variables from the previous problem, x , x , 3 4 x , and x , provide the initial feasible basis. If the 5 6 slack variables are nonzero, then x1 = x2 = 0 satisfies the constraints. Therefore, the basic solution is 2 zero and 4 nonzero variables, and it is feasible (the constraints are satisfied). The objective function is therefore expressed in terms of nonbasic variables; its value is zero. In the objective function then, x2 has the largest + coefficient; hence, it is the variable introduced into the basis to start improving it, Since x2 is a basis variable, the system is transformed so that x2 appears in only one equation, Subtract equation (4) from equations (1), (2), and (3) because (4) has the smallest 128 right-hand side. Do not make the right-hand side of any equation negative. Now these equations are developed: (Objective (OF -2(4)) Function) + 2x6 = z - 2 (5) 4x1 x6 = 9 ((1)-(4)) (6) = ((2)-(4)) 3x1 x6 5 (7) xl + x5 x6 = 1 ((3)-(4)) (8) -2x1 + x2 + x6 = 1 ( ( 4)) Since everything on the left-hand side of the objective function is 0 (nonbasic variables only), 0 = z- 2 or z = 2. Hence, the value of the objective function has increased from 0 to 2 by a change of basis. The basis is now values of B, i.e., x 1, x 9, x = 5, x 1. 2 = 3 = 4 5 = Since there is only one term of the objective function with a positive coefficient, x1 , look for further improvement' and notice that in the objective function there is only one choice, since only one term has a positive coefficient. This time the decision of which variable to remove from the basis is more complicated. The goal is to eliminate x1 from all equations except one, Equation (8) is no good becau~~x1 • Divide each equa tion ((5), (6), (7), (8)) by its coefficient of x1 , e,g., (Objective Function) 5x1 + 2x6 = z - 2 (5) xl x6 = 7+ t (6) xl x6 =j 3 3 (7) xl x6 = 1 (8) xl - xz = 1 2 2 Ask which equation's right-hand member is smallest (positive) or what ratio bi is smallest, The answer is equation (7). a .. lJ 129 Therefore, the variable that is now basic in equation (7), x , will be removed. In terms of equations (5) to (8), this 5 means that: (Objective Function) 1 5x5 + 7x6 = z - 7 ((OF)-5•(7)) (9) - 4x + 3x ( ( 5 ) -4. ( 7 ) ) x3 5 6 = 5 (10) 3x + 2x 2 ((6) -3(7)) x4 - 5 6 = (11) x1 + x5 x6 = 1 (same as (7)) (12) x + 2x = ((8) +2(7)) 2 5 - x6 3 Now the basis is x 5, x 2, x 1, x 3 and the 3 = 4 = 1 = 2 = value of the objective function has been increased to 7. OR the coordinates (x1 , x2) are (1, 3) which correspond to vertex C. Therefore the optimum is being approached. The next improvement is to add x6 to the basis and remove x4 • (Note that x6 was previously removed from the basis and is now being returned.) Therefore, (Objective, Function) = z -10 (13) = 2 (14) (15) = 2 (16) = 4 Therefore (x1 = Max. x1 + 2x2 = 2 + 2(4) = 10. 130 Example of Simplex Method Maximize 4ox1 + 6ox2 The standard form is: (1) 2x + x + x = 70 1 2 3 x. ~ 0 or (2) x + x + x = 40 J. 1 2 4 Min. - 4ox1 - 6ox = M. (3) x + 3x + x = 90 2 1 2 5 The slack variables, x , x , x , constitute a set of basic 3 4 5 variables (bi). The first extreme point is x = 70, x 40, 3 4 = x 90, when x = x o. The above is already in canonical 5 = 1 2 = form. A unit cha~ge in x2 of - 4ox1 - 6ox2 produces a greater decrease in M than a unit change in x1 • Therefore, x2 is to be the new basic variable (aiJ.), b '70 40 oo Now, examine the ratios, i • They are: Lf• -r• ~ a .. l.J > ri = 3 yields the smallest ~ Use equation 3 above to eliminate x2 in equations (1) and (2). This corresponds to examining: x = 70 - x x 70, 40, or 30 will make one 3 2 2 = x4 = 40 - x2 of the basic variables o. Only x 90 - 3x x 30 will preserve the basic 5 = 2 2 = solution. The canonical form for the basis variables x , x , x is: 2 3 4 (4) 5/3x + x l/3x 40 ((1) - 1/3(3)) 1 3 5 = (5) 2/3xi + x4 l/3x = 10 ((2) - 1/3(3)) 5 (6) l/3x + x + l/3x 30 (1/3(3)) 1 2 5 = -2ox +2ox + M + 1800 1 5 The new extreme point is x = o, x = O, x = 30, x = 40, 1 5 2 3 x 10, x 0, and therefore M 1800 at this point. 4 = 5 = = - Next, further decrease M from the present position of M = - 1800 with x = 0, x = O, and note that x must be 1 5 1 increased, keeping x = 0, 5 131 b. 40 10 JQ Examine the ratios, , , , or 24, 15, 90. a~~~ 573 273 173 10 lJ Therefore, since or 15 is the smallest,-~ use equa- 273 tion (5) for the elimination of x1 in equations (4) and (6). Corresponding to the basic feasible solution; x1 = 15, x 25, x = 15, x = o, x = O, the following is the canoni 2 = 3 4 5 cal form: x - 2.5x + .5x = 15 ((4) 5/2(5)) 3 4 5 + 1.5x4 .5x5 = 15 (J/2 • (5)) .5x4 + .5x5 = 25 ((6) 1/2(5)) 30x + 30x = M + 2100 4 5 Since x and x are to be non-negative (they are now= 0), 4 5 M = - 2100 is the optimal solution when x1 = 15, x2 = 25, and x3 = 15. For hand computation, it is convenient to interpret all of this in matrix form: a •. a'11 a'12 ••• 1n b' 1 a'21 a'22 ••• a' 2n b' 2 au a' ml m2 • • • a' mn b' m cu 1 c' 2 • • • c' m. -M' Ste]2 1. xl x2 x3 x4 x5 b 2 1 1 0 0 70 1 1 0 1 0 40 1 3-i~ 0 0 1 90 -40 -60 0 0 0 0 Ste:Q 2. Examine the last row of constraints and notice that x2 is to be the new basic variable introduced since the ratios 4 are ~' ~, ~ - :::::;>~ivot on * Row 3, Column 2, i.e., (J, 2) element to obtain: 5/3 0 1 0 - 1/3 40 (1) - 1/3(3) 2/3 0 0 1 - 1/3 10 (2) - 1/3(3) 1/3 1 0 0 • 1/3 30 1/3 • (3) -20 0 0 0 + 20 1800 (3) + 20 • (3) 1.32 Step 3. Examine the last row and find the column with the smallest negative number, i.e., the first. Step 4. Divide the last column entries by the positive first column entries and choose the smallest. In this case, examine 40 - 10 57.3 - 24; 27.3 = 15; !7.3 = 90 ====}:;>_,., Choose 15~ Pivot on (2, 1) -==>~ eliminate x1 , i.e.: 0 0 1 - 2.5 .5 1 0 0 1.5 - .5 0 1. 0 .5 .5 0 0 0 + .30 + 10 Example of Simplex Method* Maximize x1 + 2x2 = z 2x1 + x2 $ 10 xl + x2 $ 6 where + x2 < 2 xl ~ 0 ~2x1 + x2 $ 1 x2 ~ 0 With the slack variables, these are the constraints: (1) 2x + x + 1x + ox + ox + ox 10 1 2 3 4 5 6 = (2) xl + x + ox + 1x + ox + ox = 6 2 3 4 5 6 Min. lVl = -x1 -2x2 + x + ox + ox + 1x + ox (.3)- xl 2 3 4 5 6 = 2 (4)-2x + x + ox + ox + ox + 1x 1 1 2 3 4 5 6 = Step 1. xl x2 x.3 x4 x5 x6 b 12 1 1 0 0 0 10 1 1 0 1 0 0 6 . - 1 1 0 0 1 0 2 Matrix Form - 2 1* 0 0 0 1 1 - 1 - 2 0 0 0 0 0 * To show the need for iteration technique. 133 Step 2. Examine the last row of constraints and note that x2 is to be the new basic variable introduced since the ratios are (1 ~, f, f• i) ~Pivot on * Row 4, Colunn2, i.e. (4, 2) element to obtain: (Object is to eliminate x2) 0 1 0 0 - 1 9 (1) - (4) 0 0 1 0 - 1 5 (2) - (4) 0 0 0 1 - 1 1 (3) - (4) 1 0 0 0 1 1 (4) 0 0 0 0 2 2 (5) + 2 • (4) Step 3. Examine the last row and find the column with the smallest negative number, i.e., the first. Step 4. Divide the last column entries by the positive first column entries and choose the smallest. In this case, examine 1~ (4, (*; 1; i; ~) ~ Choose 2 Pivot on 1) eliminate x1 , i.e.: ro 2 1 0 0 1 11 ~ l 0 3/2 0 1 0 1/2 13/2 1/2 0 0 1 -1/2 3/2 -1/2 0 0 0 -1/2 -1/2 u-5/2 0 0 0 -l/2 -1/2 Since there remain negative elements in the last row the optimal solution has not yet been reached. Therefore, more iteration is needed. (The rest of the problem remains for the Seminar students to complete.) Some Disadvantages of the Simplex Method(3) The difficulty with the simplex method is that it is too general; it is applicable to any system, and this generality brings complexity. Although it is a tremendously powerful 134 tool, it is more powerful than is usually necessary. "The use of the simplex method to solve a problem having a great deal of structure and a nonzero-element population of only five percent is equivalent to shooting a rabbit with an elephant gun. Our only excuse for this rather unsporting behavior is the fact that the elephant gun is all that we have at the moment. We are forced to use it until special guns are designed which can be used not only for the special rabbits of the transportation type but for general rabbits as well."(J, P• 82 ) Additionally, the simplex method c·an be very time consuming, even when considered on the time scale of electronic digital computers. Many numbers are computed and recorded which either are never needed at all in the process or are needed only in an indirect way. Also, it does not yield the inverse and the simplex multipliers. Suggested Problems/Topics for Student Research 1. Complete the solutions to the problems presented in this chapter. 2. Write a paper on the development of linear pro gramming. 3. Select problem(s) and solve with: a. A geometric model b. Algebraic methods, using matrices c. The simplex method 4. Write a paper that relates linear programming to real economic models. 5. Write a flow chart for the simplex method. 6. Write a Fortran (or other type) program to solve a linear programming problem. 7. Write a paper on the applications of linear pro gramming. 8. Other topics suggested by students and approved by instructor. 135 References for Student Research Charnes, F., and Cooper; T. Mana ement Models and Industrial Application of Linear Programming two volumes • New York: John Wiley and Sons, 1960. Dantzig, G. B., Orden, A., and Wolfe, P. "Generalized Simplex Method for Minimizing a Linear Form under Linear Inequal ity Restraints," Pacific Journal of Mathematics, 5:183- 195. Darn, W. s.; and Greenberg, H. J. Mathematics and Computing: With Fortran Programming. New York: John Wiley and Sons, 1967. Gale, David. The Theory of Linear Economic Models. New York: McGraw-Hill Book Company~ 1960. Garvin, Walter w. Introduction to Linear Programming. New York: McGraw-Hill Book Company; 1958. Gass, Saul I. Linear Programming Methods and Applications. New York: McGraw-Hill Book Company, 1958. Glicksman, Abraham. An Introduction to Linear Programming and the Theory of Games. New York: John Wiley and Sons, 1963. Hadley, s. Linear Programming. New York: Addison-Wesley, 1962. International Business Machines Corporation. General Infor mation Manual: An Introduction to Linear Programming. New York: 1964. Isaacs; Rufus. Differential Games. New York: John Wiley and Sons, 1965. Kemeny, J. G., Mirkil, H., Snell, J. L., and Thompson, G. L. Finite Mathematical Structures. New Jersey: Prentice Hall, Inc., 1959. Kemeny, J. G.; Schleifer, A., Snell, J. L,, and Thompson, G. L. Finite Mathematics with Business Applications. New Jersey: Prentice-Hall, Inc., 1962. Metzger, Robert W, Elementary Mathematical Programming. New York: John Wiley and Sons; 1958. National Council of Teachers of Mathematics. Enrichment Mathematics for High School. Washington, D. c., 1963. Newman, James R. The World of Mathematics, Volume II. New York: Simon and Schuster, Inc., 1956. Nielsen, Kaj L. Methods in Numerical Analysis. New York: The MacMillan Company, 1964. 136 Price, Wilson T,, and Miller, Merlin. Elements of Data Pro cessing Mathematics. San Francisco: Holt, Rinehart and Winston, Inc~, 1967. von Neumann, John, and Morgenstern, Oskar. Theory of Games and Economic Behavior. Princeton: Princeton University Press, 1955. 137 CHAPTER V BIBLIOGRAPHY 1. Dorn, w. s., and Greenberg, H. J. Mathematics and Com puting: With Fortran Programming. New York: John Wiley and Sons, Inc., 1967. 2. Gale, David. The Theory of Linear Economic Models. New Yorka McGraw-Hill Book Company, 1960. 3. Garvin, Walter w. Introduction to Linear Programming. New York: McGraw-Hill Book Company, 1960. 4. Gass, Saul I. Linear Programming Methods and Applications. New York: McGraw-Hill Book Company, 1958. 5. International Business Machines Corporation. General Information Manual: An Introduction to Linear Pro gramming. New York, 1964. 6. Isaacs, Rufus. Differential Games. New York: John Wiley and Sons, Inc., 1965. Kemeny, J. G., Mirkil, H., Snell, J. L.g and Thompson, G. L. Finite Mathematical Structures. New Jersey: Prentice Hall, Inc., 1959. 8. Metzger, Robert w. Elementary Mathematical Programming. New York: John Wiley and Sons, Inc., 1958. 9. National Council of Teachers of Mathematics. Enrichment Mathematics for High School. Washington, D. c., 1963. Newman, James R. The World of Mathematics, Volume II. New York: Simon and Schusterp Inc., 1956. 11. Nielsen, Kaj L. Methods in Numerical Analysis. New York: The MacMillan Company, 1964. 12. Notes from McGhee, John, on the Simplex Method, San Fer nando Valley State College, 1972. 13. Price, Wilson T., and Miller, Merlin. Elements of Data Processing Mathematics. San Francisco: Holt, Rinehart and Winston, Inc., 1967. 138 CHAPTER VI GAIVJE THEORY Approximate Length of Time for Unit Two to three weeks. Behavioral Objectives 1. The student will become familiar with various games and their strategies. 2. The student will gain some knowledge of game theory terminology. 3. The student will develop an understanding of the essential difference between games of strategy and games of pure chance. 4. The student will become acquainted with the begin nings of game theory and its application to war games. 5. The student will develop some skill in dealing with some strategies necessary for specific problem solving. Preface The theory of games was devised by John von Neumann in 1928 and has developed rapidly during the pastiwenty-eight years. Several years ago, Dr. von Neumann and Dr. Oskar Morgenstern developed a theory of games and economic behavior that has had a tremendous impact on society. It not only influences the marketing strategies of great corporations 139 but also the development of defense decisions. Von Neumann and Morgenstern worked through systems that incorporated conflicting interests, incomplete information, and the inter- play of free rational decision and choice. Their initial work was with zero sum two-person games and dual games. The stock market, for example, is similar to the two-person game, except that it is considered as an infinite, n-person game. Although the stock market is probably tempor arily too complex even for the game theoreticians, some day, it too, will become a serious candidate for quantification and equations.(lS, P• lO) When von Neumann and Morgenstern published their treatise, Theory of Games and Economic Behavior, during World War II, many scientists concerned with operational research on both sides of the Atlantic investigated and developed various theoretical aspects of tactics and strategy. Howeverp this theory really does not aid a person in becoming more pro ficient in any specific games; for example, it does not advise on poker bidding nor does it tabulate chess openings. Game theory is concerned with general aspects applicable to all games and with processes which obtain a special significance when a long succession of plays is being considered. Since it is mathematical theory, it uses several areas of mathe- matics, such as algebra and measure theory, and some entirely new concepts that had to be created as well. (l3) 140 Game theory is a mathematical discipline and as such has its own scientific interest, independent of any applications. Its character has been explored rather deeply during the last two decades, gradually progressing from the study of the zero-sum two-person game to noncooperative n-person games, and finally to the cooperative case of non-zero-sum. It is especially the latter development that brings the question of the applicability of game theory to economic transactions so that all will gain from its execution, which makes the game non-zero-sum.<16 • P• l) Actually, through its side effects, game theory influences economics and economic science. The well-known duality between the theory of the zero-sum two-person game and linear programming, proved by Gale, Kuhn, and Tucker,(6) established a relation between game theory and current economic theory that is both impor tant and interesting. Linear programming and any of its variants is only possible if there exists full control over all variables, directly or statistically, on which the out- come hinges. When even one single variable is not under the control of the planner, a logically and mathematically different situation is presented with which current economic theory cannot cope satisfactorily. Introduction to the Topic The theory of games concerns games of strategy, and in contrast to pure games of chance, their outcome does not 141 depend on chance alone, but also on various decisions which the players must make during the course of play, It is a branch of mathematics that aims to analyze various problems of conflict by abstracting common strategic features for study in theoretical "models"--termed "games" because they are patterned on actual games such as bridge and poker. By s.tressing strategic aspects; that is, aspects controlled by the participants, it goes beyond the classical theory of probability, in which the treatment of games is limited to aspects of pure chance.(9, P• 48) In layman's terms, game theory c·ould be described as an attempt to quantify and work through the actions of players in a game; to measure their options continuously. (l8 , P• lO) Typical examples of these games are parlor games in which the participating players must make decisions according to certain rules. The decisions and perhaps also certain random events (such as spinning a roulette wheel, rolling dice, and dealing cards) determine the course of play and therefore the winnings and losses of the players. Game theory is concerned not with any particular game but with all of them; not with technical matters but with theoretical matters. For example, "What is the best way to play chess?" is not a game-theoretical question, but, "Is there a best way to play chess?" is a game-theoretical question.(l?, p. 3) 142 The essential features of game theory situations are: (17, PP• 18-20) 1. There must be at least two players. 2. The game begins by at least one of the players choosing among a number of specified alternatives. J, "Move" refers to the situation in which the choice is made, i,e,, the specification of who is to make the choice and what alternativ~ are available to him. 4. After the choice associated with the first move is made, a certain situation results and this leads to or implicates the next choice and its alterna tives, etc. 5. There are two general types of games: perfect information, always a best way to play; and not perfect information. 6, If the game has successive choices (moves), then there is a termination rule. 7. The end of the game varies as do the payoffs. TERMS Constant-sum game: The game ( .z:-1 , , •• ; ~; A1 , • o,, An) where there exists a number, c, such that .~. Ai ( .s-1 , o • o 1 .a~) = 1=J c for all Goofspiel game: Rules: Each player is provided with a hand of n cards numbered 1 ton. On the first "move," each player chooses a card from his hand, and the player holding the higher card wins a score of a ; or if the cards are equal, then each player wins a 1 • On the next move, each player chooses one of 112 the cards remaining in his hand, and this time high man receives a score of a ; and in case of a tie, each gets a ; • Play continues2 in this way until all n cards have2 2 been played, after which the player with the higher score wins an amount equal to the difference in scores from his opponent. (The numoers a , ,.. . a )are l{nown to both players in advance,){), PP•1 lo-4 185n High number game: Rules: Each player chooses a positive integer, the player choosing the higher winning,{), P• 201) Low number game: Rules: Each player chooses a positive integer. The player choosing the lower integer wins 1, unless his choice was exaQtly 1 les$ than his opponent's, in which case he loses 2,\3, P• 202) · Matrix: Almost all elementary expositions of game theory begin with the game matrix and its attendant concepts, This is an excellent device for formulating and proving the basic theorems without which the subject could not exist, but aside from certain very simple instances, the matrix is not a satisfactory tool for solving games, It is a formulation, not an end, and for such pu~Doses a$ obtaining answers, a very unsatisfactory one,t8, P• 2) Max, Min.-Min. Max. theorem: If ~is a payoff function for which both minimax and maximin exist, then maximin~~ minimax ~ and(equality bolds if and only if? has a saddle point. J, P• 203) Mixed strategy: A probability dtst~ibution defined over the strategy set :£..of a player. \19 J n-person game: When the rules of the game are such that the players fall into n mutually exclusive sets in such a way that the people within each set have identical inter ests. These n sets of people with identical interests are referred to as "persons." Pure strategy: Ordinary strategies which may be looked upon as special kinds of mixed strategies. 144 Saddle points: Equilibrium points in zero-sum two-person games. The words "saddle point" refers to the familiar shape of a riding saddle. If one moves bacl{ward or forward from the center of the saddle, the surface rises; whil~ if one mQves sideways, the saddle surface drops off.l~, P• 202) Another way to express the above is to state that when the minimum of the max. is equal to the maximum of the min., a saddle point exists. Strategy: For games generally can be described as the set of instructions which tells a player what to do in all possible situations that may arise at each stage of the game. Symmetric games: A game that has strategies S and T and payoff tp is called symmetric if S: T and cp_ ( s, t): - c:p ( t, s) for all ( s, t) , where s in S and t in T. Theory of games: Attempts to study economic behavior by iso lating the aspect of conflicting interests• which ocQurs in its simplest form in games of strategy.\3, P• 181) Zero-sum two-person games: Whe(n the gains )and losses of the two persons are balanced. 13, P• 1286 Examples Example of a Two-Person Game Theory Problem Two players, A and B, play this game. Each player chooses a head (H) or a tail (T) of a coin without knowledge of the other 8 S choice. When choices do not match, A loses $1 to B. When choices match (HH), A wins $2. When choices match (TT)p A wins $1. How should A play? How should B play? ~ H T This matrix indicates A's win- H 2 -1 nings in all cases. CB·, loses the same amount, T -1 1 or wins the negative amount.) 145 Pr~theoretical analysis presents these thoughts: A should play H because he wins more. But then B, figuring this out, should play T to take advantage of A's greed. Then A, sensing this, should play T to doublecross B. B, figuring this out, should play H. But then A should • , • Obviously, this sort of infinite outmaneuvering leads no where; so how should A play? Suppose A plays conservatively, i.e., A assumes that B knows his move. Then regardless of A's move (H or T), A will lose $l(or win-$~. If A chooses H and T with certain proba bilities, then even A will not know what his move will be ~til he play~. Suppose A chooses H and T (each has P = i) and suppose A tells this to B (A plays iH +iT). What does B do? B will presumably maximize his expected winnings (or, equivalently; he will minimize A's expected winnings). The following figures show what happens whether B decides to play T or H. A will win (on the average) $0 or $i according as B plays T or H. So, from a position where A must lose $1, B has arrived at a position where A will expect to win $0 or $i. Refer to Figure 1. Therefore, B will play T, of course, in this situation, and A will break even. A can now consider other probabilities for Hand T, i.e., A can choose H with probability p and T with probability 1 - p (A plays p(H) + (1- p) (T)). Refer to Figure 2, 1 --1: - 2 1 .!. 1 ·--1: + 2 2 B plays T; E(W) ·= 0 B plays H; E(W) = i Figure i: A's winnings if A plays iH +iT 146 - p 2: 2p 1~-11 1<" T T l: 1 - p 1: ... -,.; 1 + p w B plays T; E(W) = 1 2p B plays H; E(W) = 1 + 3P Figure 2: A's winnings if A plays p(H) + 1- p)(T) Even if B knows A's strategy, all he can do is play T or H to minimize 1 - 2p and - 1 + 3p. Refer to Figure 3. By graphing the two lines: y = 1 - 2p and~:-1 + 3p and, for each p, choosing the least y value, the smaller of 1 - 2p and - 1 + 3P is found. Refer to the broken line in Figure 3. y 1 p Figure 3: A plays p(H) + (1- p) (T); A's guaranteed winnings From Figure 3, it is seen that if A plays conservatively, he will choose p = 2/5, thereby guaranteeing himself an expected winning of l/5 ($.20), regardless of how B plays, For any other choice of p, A can expect to win less if B plays properly, Therefore, A plays 2/5H + 3/5T and will win (on the average) $.20 regardless of B's move. (Note: It is felt that A will play T most of the time but will play H occasionally "to keep B honest." However, the above analysis does not ascribe any such motivations to A.) A merely wants to maximize his guaranteed expected winnings, 147 To analyze the problem from B's point of view, it is seen that he can reason similarly to A. Suppose B played most conservatively and told A that he intends to play q(H) + (1- q)(T). A's winnings can be computed on the assumption that A plays H or T. If B announces his move, B will assume that A will maximize A's winnings and will play T or H accord ing as 1 ~ 2q or - 1 + Jq is larger. y Figure 5: B plays q(H) + (1- q){T); A's maximal winnings 148 Summary: Both A and B can play H and T with any proba bilities. If A plays 2/5H + 3/5T, nothing B does will affect A0 S winnings. A will win $.20 on the average. If B plays 2/5H + 3/5T, nothing A does will affect A's winnings. A will win $,20 on the average, By the introduction of probability, both A and B have been given strategies, Baseball Game Problem Find the payoff matrix for the following game, called baseball. Rules: P2 , the pitcher, has a choice of two stra , tegies called fast ball and curve. P1 the batter, may either swing or take. If P1 swings on a curve or takes a fast ball, he is "out" and loses 1. If P1 takes a curve, the game is played a second time and if he again takes a curve, he "walks" and wins 1. If P1 swings on a fast ball, he wins an amount p > o. First Second If 2nd, If 2nd, J...- .• Pitch Pitch swing/swing take/swing take/take Max, --·---~ :o.':'ir,.'9.F'.A~·~J!'"""""=-~-~~~ fast fast p -1 -1 p ~~~~~k~~~~ curve fast -1 swing~ P take ---7> -1 p ·-~·+·-·---~~~-'t\<'~>'!M".~~~ curve curve -1 swing---? -1 take~ 1 1 - . [~~~~--""""""""""F""""~·-...J -J. Min. Max. is p if p < 1, Max. Min. is - 1. Since Min. Max.¥ Max. Min., there is no saddle point according to the theorem (List of Terms). Therefore, no strategy appears to be best for each of the players. This problem must be a mixed strategy, and to solve it will require the use of linear programming, (Perhaps a Seminar member will pursue this,) The above matrix could also be considered as a 4 x 4 if fast/curve and swing/take are included, Example of a Goofspiel Problem Consider 3-card goofspiel in which a 1, a 2, a 3. Show that the strategy 1 = 2 = 3 = "play card i on the ith move" is optimal. Is this strategy also optimal if a = 1, 1 a = 2, but a = 4? For a = 2, a = 3, a = 4? 2 3 1 2 3 The three cards give 3 • 2 • 1 or 6 arrangements, i.e., 1, 2, 3; 1, 3, 2; 2, 1, 3; 2, 3, 1; 3, 1, 2; 3, 2, 1. Therefore, the matrix is as follows; the elements in this matrix are written with respect to P : 1 p2 ¥--1~ a~~~ -a:-!2~3 ·~~~· 11"-3~--r-~a:;+a;-~ . "---~~--~~:;_~:;;;:;; , -~;;:;;;-·· -~~;;s-r:a +a +a 2 3 1 2 3 I . . I 2 2 ;-2- 1. "3 I ~a2 +~ al-a-;+;3 I ~ ...... ar~2 +a3 -al+a~ +a3 1-al-a2 +a3 pl ~. __ ..,,_,"~"''"'"'',...,'"''•"'+."':'~~~~~,..i!'~ .. I'C!,~~.-.~~..,..-...o.:o:r,""'>:'~,.:...... ,...-.,;:,~f~"C\o:::A~n..,~ll"'~...,.,.,.~,._.-,.,.,.._,.._...... ,~ -~"""·~.,· ····•~. · ~ r-2 3 1 lal+a2-a3 lal+a2-a3 I al+a2-a3 I -al+a2-a3 ~-al+a2~ I .·. . . . 1~-·~-- : 2 l 2__ .. --- , 2 ~ ~ 2 f al -a2-a3 .. al -a2 +~ ~ al +a2-a3 . al -a2 +a3 ,,.,.. I al-a2 +a3 ~ _5·~ 150 For a 1, a = 2; a = 3, the above matrix looks like 1 = 2 3 the following: p2 Row Min. 0 -2 Max, of Row Min.=O 0 Min, of Col.Max.=3 pl -2 Therefore Max.~ S -4 Min. q:> . -4 No saddle point Max. Max. of Row Min.=l Min, of Col. Max.=3! Therefore Max.~ S Min.p No saddle point Max. of Row Min.=-1 Min. of Col.Max.=4! Therefore Max. q; S Min. cp No saddle point 151 Simplified Poker Problem(J, p, 212 ) Find the payoff matrix for the following simplified poker game. Rulesa P1 draws either the card High or Low with equal probability (~). If P1 bets, P2 may "fold" and lose a or "call" and win or lose b according as P1 holds High or Low. If P passes, then may also pass, giving payoff 1 Pz zero; or P may bet, in which case he wins a if P holds Low 2 1 and loses b if P1 holds High. F c F C < -• ·"pl bets p B p..,.;~--~~.. P passes Hi ;Lo B 1 B !I l! + l! j b + b 'a2~ + a2 r ~ b + b ! B ,2 2· 2 2 12 2 >-----~------_! ---- ... ------·------· ---.- ·j Pia+o b a l! a, b+o B '2 2-2,2-21 2 pl r·----·· ...... ------···----- ·--f· ---· b P B a+~ ~-~p+~ 0 - 2 p p - ~- ~--~--;-----~ - ~ --,- ~: - ~ 0 + 0 ~~-----~:..;_~_,____ =a.:--~j ·o ~r a·-~ -__ -o-- ~-~---t~" ?___ __ a I a : a i b-a i l b b-a b ' a -2 1 -2-' 0 ! 2 :-2 2 I 2 ' l i I ° I : rb~a: -:~:1 ~:-/ ~·--~: :~:-r ~; ~~~::::~~s;~~~- ·p1 •s strategy is determined by seeing his High or Low card. Therefore, it is a 4 x 4 matrix. If P1 does not see his own card, then the matrix would be 2 x 4 (this problem could be pursued by a member of the Seminar). If P1 does not see his own card and P2 does not see the bet or pass, then the matrix would be 2 x 2 (problem for member of the Seminar). 152 Payoffs for Infinite Games of High Number and Low Numb~r Find the payoff matrix for infinite games of high number and low number, and show that the matrix is skew symmetric. Check for saddle point. Row Min. -1 -1 -1 -1 -1 -1 • • Max. /> ( Max, of Row Min. = -1 Max. of Row Min. = -1) \Min. of Col.Max. = 1 Min. of Col.Max. = 1:, "'> j Therefore Max. :s; Min. Therefore Max. $ Min. ) ""'"- ~ No saddle point No saddle point 1 \, ./' \.. \ Low Number Payoff P1 (Max.) Row Min,l 1 2 3 4 56 ••• -1 J II / ~ pJ ~ 1~-~fEtf~-f~Ff :_~;__ l -2 e 3 1L -~~- -~--=~--'I --.--~~-L----~----t_·, __ :~_J~ ___:~.-J · ~-- n----~-:.:.- · 4 : _1 __! -·- ~-- ! -2 i 0 j 2 j -1 I ... I -2 -2 l -1 I 1 ! ::~:- r : ~ J_ 2_ r - .. =-1 1 1 1, 1, 2 o, ••• 1 -2 """""·-~---·" -< tc t< ~' • ··-~~f-'·~·~--""~~ • 1 • r • • • • ... { • j • • • • • • •• ! • • • • • ... ~ L • • • - -·- J 1 2 2 2 2 2 Col. Max. 153 Suggested Problems/Topics for Student Research 1. Write a paper about the development of game theory. 2. Write a paper about specific games and their strategies. 3. Write a paper that compares games of strategy to games of pure chance. 4. Develop an "original" game and its strategies. 5. Solve some game theory problems from problem sets in any of the Suggested References for Student Research. 6. Write a paper relating game theory to the decision making processes involved in our national defense. 7. Write a paper regarding game theory applications that have been used in some of our great corpora tions. 8. Other game theory topics suggested by students and approved by instructor. Suggested References for Student Research Burger, Ewald, Introduction to the Theory of Games. New Jersey: Prentice-Hall, Inc., 1963. Dorn, Williams., and Greenberg, Herbert J. Mathematics and Computing: With Fortran Programming. New York: John Wiley and Sons, Inc., 1967. Garvin, Walter W. Introduction to Linear Programming. New York: McGraw-Hill Book Company, l9b0, Hausner, Melvin. Elementary Probability Theory. San Fran cisco: Harper and Row, 1971. International Business Machines Corporation. General Infor mation Manual: An Introduction to Linear Programming. New York, 1964. Kuhn, H. w., and Tucker, A. w. Contributions to the Theory of Games. New Jersey: Princeton University Press, 1959. McKinsey, J. c. c. Introduction to the Theory of Games. New York: McGraw-Hill Book Company, 1952. Newman, James R. The World of Mathematics, Volume II. New York: Simon and Schuster, Inc., 1956. 154 Parthasarathy, T., and Raghavan, T. E. s. Some Topics in Two Person Games. New York: American Elsevier Publishing Company, Inc., 1971. Price, Wilson T,, and Miller, Merlin. Elements of Data Pro cessing Mathematics. San Francisco: Holt, Rinehart, and Winston, Inc., 1967. Princeton University Conference. Recent Advances in Game Theory. New Jersey: Princeton University Press, 1962. Rapoport; Anatol, Two-Person Game Theory. Ann Arbor: The University of Michigan Press, 1966. Smith, Adam. The Money Game. New York: Random House, 1968. von Neumann, John, and Morgenstern, Oskar. Theory of Games and Economic Behavior. Princeton: Princeton University Press, 1955. 155 CHAPTER VI BIBLIOGRAPHY 1. Danskin, John M. The Theory of Max-Min. New York: Springer-Verlag, Inc.o 1967. 2. Dorn, Williams., and Greenberg, Herbert J. Mathematics and Computing: With Fortran Programming. New York: John Wiley and Sons, Inc., 1967. Gale, David. The Theory of Linear Economic Models. New York: McGraw-Hill Book Company, 1960. 4. Garvin, Walter w. Introduction to Linear Programming. New York: McGraw-Hill Book Company, 1960. Goldberg, Samuel, Introduction to Difference Equations. New York: John Wiley and Sons, Inc,, 1958. 6. Hausner, Melvin. Elementary Probability Theory. San Francisco: Harper and Row, 1971. 7. International Business Machines Corporation. General Information Manual: An Introduction to Linear Pro gramming. New York, 1964. 8. Isaacs, Rufus. Differential Games. New York: John Wiley and Sons, Inc., 1965. Kemeny, J, G., Mirkil, H., Snell, J. L., and Thompson, G. L. Finite Mathematical Structures. New Jersey: Prentice Hall, Inc., 1959. 10. Leitmann, George. Optimization Techniques with Applica tions to Aerospace Systems. New York: Academic Press, 1962. 11. McKinsey, J. c. c. Introduction to the Theory of Games. New York: McGraw-Hill Book Company, 1952. 12. National Council of Teachers of Mathematics. Enrichment Mathematics for High School. Washington, D. c., 1963. 13. Newman, James R. The World of Mathematics, Volume II. New York: Simon and Schuster, Inc., 1956. 156 14. Parthasarathy, T., and Raghavan, T. E. s. Some Topics in Two-Person Games. New York: American Elsevier Pub lishing Company, Inc., 1971. 15. Price, Wilson T., and Miller, Merlin. Elements of Data Processing Mathematics, San Francisco: Holt, Rinehart, and Winston, Inc., 1967. 16. Princeton University Caference, Recent Advances in Game Theory. New Jersey: Princeton Uni~rsity Press, 1962. 17. Rapoport, Anatol. Two-Person Game Theo~. Ann Arbors The University of Michigan Press, 19 • 18. Smith, Adam. The Money Gam~. New York: Random House, 1968. 19. von Neumann, John, and Morgenstern, Oskar. Theory of Games and Economic Behavior. Princeton: Princeton University Press, 1955. 157 CHAP~1 ER VII THEORY OF NUIVIBERS Approximate Length of Time for Unit Four to six weeks, ~havioral Objectives 1. The student will become familiar with some of the basic topics of number theory. 2. The student will improve his skill in the applica tion of deductive reasoning. 3. The student will gain knowledge and some proficiency in mathematical induction. 4. The student will gain a valuable review of elemen tary number material from a different point of view and will thus establish more important mathe matical ideas. 5. The student will gain renewed interest in mathe matics through the study of number theory as it includes number material already familiar to most students. 6. The student will develop a good foundation for the study of more advanced topics, including abstract algebra. Preface It is generally conceded that Pythagoras and his followers, in conjunction with the fraternity 0 s philosophy, took the first steps in the development 158 of number theory, and at the same time laid much of the basis of future number mysticism.<4, P• 53) Man has shown much curiosity about numbers since ear- liest times. The ancient Chinese and Greeks showed parti cular interest in the study of relationships among numbers. However, it was not until the seventeenth century that the first serious study of number theory was made by the famous French mathematician, Pierre Fermat (1601-1665), who is usually considered the founder of the theory of numbers.(l, P• l) Fermat was actually a jurist and a parliamentarian by pro fession, but his great diversion was mathematics, which he developed as a "master of masters." Although he is best remembered today for his work in the theory of numbers, Fermat was also one of the founders of analytic geometry and the calculus.(?, P• 500) Number theory has very few applications to the other sciences and is generally regarded as the purest branch of pure mathematics. It is interesting to note that many of its general results were discovered and suggested by special cases which were observed not only by mathematicians but also by amateurs.(l) One of the greatest mathematicians, physicists, and astronomers of all time, Carl Friedrich Gauss (1777-1855), indicated his partiality for mathematics in general and for the theory of numbers in particular by remarking that "mathematics is the queen of the sciences, but number theory 159 is the queen of mathematics."(6 , P• l) Gauss not only made many original contributions to number theory, but he also systematized all of the available materials and organized them in the form used today. It is in this respect that his work in the theory of numbers often is compared with the work of Euclid (about 300 B.C.) in geometry. Introduction to the Topic Much of elementary number theory arose out of the inves tigation of essentially three problems: perfect numbers, periodic decimals, and Pythagorean numbers.(lO, P• vii) Since number theory represents a very broad area of mathe matics, the writer has selected only a few topics that should be appropriate for the students of the Seminar course. Much of the material in this chapter gives an informal account of some of the more elementary results of number theory without going into any detailed proofs, since the subject lends itself nicely to such treatment. Some of the general theorems will be verified and may certainly be pursued by members of the Seminar. Before presenting some of the basic topics and theorems useful for this short course in number' theory, the writer will now include important terms and their definitions. TERW~ Amicable numbers: Two numbers are amicable if the sum of the divisors of the first number is equal to the second 160 number itself and if the sum of the divisors of the second number is equal to the first number (include all of the divisors of a number except itself).( The small- ) est pair of amicaie numbers is 220 and 284. 1, PP• 16-17 Archimedean property: If a< b and a > 0, the~ there ~xists a positive integer c such that ax c > b,\9, P• 4 J Composite numbers: Positive integers that are greater than 1 and not prime. Congruence: The idea of congruence was introduced by Gauss, and he expressed it as a= b(mod m). This means that a - b is exactly divisible by the number m, and it is read, "a is congruent to b(modulo m)."(l, P• JJ) Division algorithm: Let a and b be integers, b ~ o. There exist unique integers q and r, 0 ~ r < lbl , such that a = bq + r. The numbers q and r are 9alled th~ quotient and remainder when a is divided by bo\11, P• 8) Equivalence relation: On a set A is a relation P of ordered pairs of elements of A, having the properties: reflexive, symmetric, and transitive. Some examples of equivalence relations are: congruence relation modulo m, equality of numbers, and similarity of triangles. Fibonacci Sequence: Is a sequence of natural numbers, i.e., 1, 1, 2, J, 5, 8, 13, 21, J4, 55, ••• ,where each new term is(formed by adding the last term to its prede cessor. 1, P• 2) Greatest common divisor: If d is the largest common divisor of a and b, then it is called the greatest cqmmon divisor of a and, band is denoted by (a, b).t6, P• 22) Integral domain: Is a system (D,(0,G0) in which the basic properties of addition and mul ti~icat.ion of integers are val:id for t:Q.e binary operations \!.)and@ , respective- ly. t 9' p. 3) Mathematical induction: If a set S of positive integers is such that (i) the positive integer 1 is an element of S, and (ii) the positive integer k + 1 is an element of S whenever the positive integer k is an e(lement of S, then S is the set of all positive integers. 9, P• 6) Or, more simply, a proposition about positive integers that has the following two properties is true for all positive 161 integers: Property I--The proposition is true for the integer 1; Property II--If the proposition is true for the integer k, it mvst also be true for the integer (k + 1),\11, P• 2-3J Perfect numbe~: One that is equal to the sum of(all its ) positive divisors other than itself (Euclid).lO, P• 1 Some examples of perfect numbers are: 6, 28, 496. Periodic decimals: The decimal expansion a /10 +a /102 + ••• is said to be periodic if there exist 1positive2integers k and t such that ak = ak+t = ak+2t = ••• ak+l = ak+t+l = ak+2t+l = ••• • • • The smallest such t· is called the period of the expansion. If k = 1, the expansion is said to be purely periodic. Example: 1/3 = .311 ••• is purely periodic with period 1,(11, pp. 86-87) Prime numbers: Numbers whose only exact divisors are them selves and one. Pythagorean numbers: Numbers that satisfy the equation, a2 + b2 = c2. Relatively prime: Two numbers that have only the factor 1 in common. Residue class: A set of integers containing exactly those integers which are congruent, modulo m, to a fixed inte ger. Example: There are seven residue classes, modulo 7, namely the sets : ( ••• , -14, -7, 0, 7, 14, 21, ••• ) ( ••• , -13, -6, -1, 8, '15, 22, ••• ) ( ••• , -12, -5, 2, 9, 16, 23, ••• ) ( ... , -11, -4, 3, 10, 17, 24, ••• ) ( ••• , -10, -3, 4, 11, 18, 25, ••• ) ( ••• ,- 9, -2, 5, 12, 19, 26, ••• )(11, p.4J) ( ••• , - 8, -1, 6, 13' 20' 27 , ••• ) Unique factorization theorem (fundamental theorem of arith metic): An integer greater than 1 is either a prime number or is a product of prime numbers. Furthermore, the factorization into primes is unique except for the order of the factors.(ll, P• 17) 162 Well-ordering £rinciple: Every nonempty set of positive integers contains a least element (this is tn~ equiv~lent to the principle of mathematical induction).t6, P• 6 J Some of the basic topics that may seem appropriate for a four- to six-week exposure to number theory are:(5) 1. Mathematical induction 2. The divisors of a number, including the Euler phi function or totient, perfect numbers and amicable numbers 3. Some proven facts of number theory: a. Solution of congruences: a: b(mod m), ax :: b(mod m) b. Wilson's theorem: For any prime p, (p- 1)! :: - l(mod p) c. Pythagorean triples d. Diophantine equations: ax + bJ= c. The following list of theorems presented here is intended to correlate with and complement the above list of topics.(5) 1. For all a, n, b. E N (N is the set of natural n l n numbers), a 2. bi = ~ abi. i=l i=l 2. For all b, c, d E I (I is the set of integers), if b < c, then (b +d) < (c +d). 3. For all b, c €: I andn € N, if b > c then bn > en. 4. For all b, c E w (W is the set of whole numbers), there exists a unique pair of integers m E W and r E W such that c = mb + r and 0 :;; r < b • 5. For all b, n EN and b > 1, n may be expressed m i uniquely as :;2: ai b with all ai E W and ai <: b. i=O 163 6. For all b, c, dE I, if b/c and b/(c +d), then b/d, (b/c means b divides c). 7. For all a, bEN, if 0 ~a< p and 0 ~ b < p where p is a prime, then p is not a factor of ab. 8, For all a, b t N, if p is a prime and p/ab, then p/a or p/b. 9, For all n £ N and n > 1, n may be factored uniquely into a product of primes. 10. If (a, m) = 1 and (b, m) = 1, then ( ab, m) = 1. 11. If a/cd and (a, c) = 1, then a/d, (Universe is the set of integers.) 12. For any beN and c £ N, g = (b, c) is the least positive value of bx + cy, where x€I and y G_ I • lJ, For a, b, c, de I, m £ N, if a ~ b(mod m) and c; d(mod m), then a+ c: b + d(mod m):. 14. For all a, b, c, d € I and mEN, if a : b(mod m) and c =d(mod m), then ac; bd(mod m), 15. For a, b, c ~I, if ab = ac(mod m) and (a, m) = 1, then b =c(mod m). 16. If ai:; bi(mod m) for all i E:. (1, 2, J, ••• , r;), n n then TT ai = TT bi (mod m). i=l i=l 17. If ai ~ bi(mod m~ for all i € [1, 2, J, ••• , n) , then ~ ai = ~ bi (mod m). i=l i=l n . 18. If P(x) = ~ aix1 is a polynomial and b _ c(mod m), i=O then P(b) : P(c)(mod m). 164 19. For each b £I, m £ N, there is an r E: I such that b : r(mod m) and 0 ~ r < m. Furthermore, r is unique. 20, If (a1 , a 2 , ••• , a~(m)} is a reduced residue 1, , , system modulo m and (b, m) = then { ba1 ba2 ••• , baf(m)J is a reduced residue system modulo m. 21. If (b, m) = 1, then b~(m) • l(mod m) Euler's theorem is followed by its corollary, Fermat's theorem. 22. ax : b(mod m) always has a unique solution if (a, m) = 1. 23. ax : b(mod m) has no solution if (a, m) = d > 1 and d is not a factor of b. 24. If ax : b(mod m) and d/a, d/b, and d/m, then ~:~(mod~) has the same solution as ax: b(mod m). 25. The set of congruences x- ai(mod mi) where i E. [ 1, 2, ••• , r) has a solution if (mi, mj) = 1 for all pairs i, j, with if j, and this solution r is given by x = :2:_ m1m2 ••• mi-lmi+l'"'mrxi i=l (The formula is not valid for i = 1 since m0 is not defined), where each xi is found by solving the congruence m1m2 ••• mi-lmi+l"''mrxi _ ai(mod mi). 26. All of the primitive solutions of the Diophantine equation a2 + b2 = c 2 , with b even, are given by s 2 - r 2 = a, 2rs = b, and r 2 + s 2 = c, where s > r > 0, (r, s) = 1 and r and s are of opposite parity. 27. The set of residue classes modulo a prime form a field. 165 28. If a, bE I and ab s O(mod p), where pis a prime, then either a= O(mod p) orb: O(mod p). 29. For any prime p, (p- 1)! =- l(mod p), JO. If (n- 1)! :- l(mod n), then n is prime. 31. If (a, bi) = 1 fori= 1, 2, ••• , n, then (a, b1b2 ••• bn) = 1. Examples Four Examples of Mathematical Induction = n{n + 1} 1, Prove: For all integers ==- 1, 1 + 2 + ••• + n 2 = 1(1 + 1~ (i) For n = 1, 1 2 is true. k{k + 1} (ii) If for n = k, k ?: 1, 1 + 2 + • • • + k = 2 is true (assumption), then for n = k + 1, = k{k + ll 1 + 2 + • • • + k + k + 1 2 + k + 1 = = k2 + k + 2k + 2 = 2 k2 + 2k + 2 {k + 1} (k + 2l = 2 = 2 , which is of the form, n~n + 1~ 2 Q.E.D. 2. Prove that the sum of the squares of the first n positive odd integers is equal to 1n(4n2 - 1); i.e., prove that: 1 2 + 32 + 52+ ••• + (2n _ l)2 = n(4n~- 1), (i) For n = 1, 12 = l( 4 (l~) - l) or 1 = 1 is true. (ii) If for n = k, k ~ 1, 1 2 + 32 +52 + ••• + (2k- 1) 2 = 2 k(4k .~( 1) is ,true,: :th.en for n = k + 1, 2 12 + 32 +52 + ••• + (2k- 1) 2 + (2k + 1) = 2 k(4k - 1) + (2k + 1)2= 3 166 2 - k(4k2 - 1) + 3(2k + 1) - - 3 - k(2k- 1)(2k + 1) + 3(2k + 1) 2 = - 3 = <2k + 1 >ud 2k - 1 > + 3 <2k + 1 il = 3 = (2k + 1)(2k2 - k + 6k + 3) = 3 2 = (2k + 1)(2k + 5k + 3) = 3 = (2k + 1)~2k + 2}~k + 1} .= 3 {k + 1}(4k2 + 8k + = 2~ = 3 2 = {k + 1~ ~{k + 1~ - :iJ 3 2 which is of the form, n{4n - 1} Q.E.D • 3 • 3. Prove that (an - bn) is divisible by (a - b) for all positive integral values of n, where a I b. (i) For n = 1, (a1 - b1 ) is obviously divisible by (a- b). (ii) If for n = k, k ~ 1, (ak - bk) is divisible by (a - b) is true, then for n = k + 1, (ak+l - bk+l) is divisible by (a- b); i.e., ak.~l - abk + abk - bk+l = a(ak - bk) + bk(a - b) ~~~~~·\\;;w.....,...,.~· assumption (part (i)) Q.E.D. 4. Proof by mathematical induction:(9, P• 96 ) Theorem: If a =b(mod m) and n is a positive integer, then an; bn(mod m). Assume ak =bk(mod m), k positive integer. Then, by theorem 14, aka = bkb(mod m) or ak+l; bk+l(mod m). Hence, by the first principle of mathematical· induction, an ; bn(mod m) for any positive integer n, and the theorem is proved. 167 The Divisors of a Number The number of divisors of a number: When natural numbers are decomposed into their prime factors, each prime factor may occur any number of times, 4 For example, 80 = (2 )(5) and 225 = (52)(32). When all of the prime and composite divisors, including 1 and the number itself, are considered, it is important to note that these divisors play a key role in the development of number theory. The number 35 has only two distinct prime factors, 5 and 7, but it has four divisors, 1, 5, 7 and 35. These divisors can be counted systematically as shown in the prime decomposition 2 of 60, i.e., 60 = (2 ) (3) (5) and 60 has 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Note the following: The first two factors 2 give rise to three divisors, 1, 2 and 22 ; the factor 3 gives two divisors 1 and 3; the factor 5 gives two divisors 1 and 5; each of the divisors 1, 2 and 4 may be combined with each of the two divisors 1 and 3; now there are six divisors: 1, 2, 4; 3, 6, 12. Each of these six divisors may be combined with each of the divisors 1 and 5; now there are 12 divisors: 1, 2, 4; 3, 6, 12; 5, 10, 20; 15 f 30 f 60 I To summarize the above process, there is a rule in number theory which states: Let the natural number N have the factorization N = paqbrc ••• ,where p, q, r, ••• are the prime factors raised to the powers a, b, c, ••• , respectively. Therefore, the number of divisors of N, d(N), is found by the formula: d(N) .= (a + 1) (b + 1) (c + 1) ••• , where the dots indicate continuing until all of the exponents are exhausted.(l, PP• ll-l2) An example, using the formula, is: 45 = (32)(51 ), where p = 3; q = 5; a = 2; b = 1. Therefore, d(45) = (2 + 1)(1 + 1) = 6 divisors, i.e., 1, 3, 5, 9, 15, 45. 168 Euler £hi-function or totient: The number of positive integers less than or equal to a positive integer n and relatively prime to n is denoted by qp(n)(9, P• 72) It is also called the indicator of nor the totient of n and was first introduced about 1760 by the Swiss mathematician, Leonard Euler. Examples of the Euler phi-function are: (/) (1) = 1 Therefore, each element of R' is congruent to a different element of R; that is, ar1 , ar2, ••• , and ar~(m) are con gruent to r 1 , r 2, , •• , and r~(m)' but not necessarily in the order of appearance. Then ar _ r• (mod m) 1 1 ar2 :; r• 2(mod m) ~r : r• (mod m) I 3 3 art(m) = r't(m)(mod m), where r• 1 , r• 2, ••• , and r'p(m) are r 1 , r 2, ••• , and r~(m) in some rearranged order. By repeated applications of theorem 14, ar1ar2 ••• arf(m) E r•1r• 2 ••• r'f(m)(mod m) - r 1r 2 ••• r~(m)(mod m), and \P(m) ( a r 1r 2 ••• rf(m) _ r 1r 2 ••• r~(m) mod m). Since (ri' m) = 1 fori= 1, 2, 3, ••• , ~(m), then by theorem 31 (r1r 2 ••• rf(m)' m) = 1. Hence, by theorem 15, af(m) : l(mod m). Perfect numbers: One of the oldest areas of study in the theory of numbers is that of perfect numbers. From time immemorial, numer ologists have attributed special significance to the first two perfect numbers, 6 and 28, because God created the world in 6 days and it takes 28 days for the moon to circle the earth. (l, P• 14) In modern notation, a positive integer a is called perfect in case Cia)= 2a.<6 , P· l05) For example: eF(6) = 1 + 2 + 3 + 6 = 12 and ~(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56. All known perfect numbers are even and are as characterized in the following theorem:(6 , P• l05) 170 Theorem: If 2n - 1 is a prime, then a = 2n-l(2n - 1) is perfect and every even perfect number is of this form. Proof:(9, p. 67) Let a= 2n-1 (2n-l) where 2n- 1 is a prime. Then cr(a) = (1 + 2 + 22 + ... + 2n-l) (1 + (2ft - lil = = 2n 1 n 2 - 1 • 2 = 2n(2n - 1) = 2a, and it follows that a is an even perfect number. That is, an even integer is a perfect number if it is of the form 2n-l(2n-l) where 2n - 1 is a prime. Example: Prove that 33,550,336 is a perfect number. 33,550,336 = 212 C213 - 1) and 21 3 - 1 = 8191, which is a prime. Therefore, 33,550,336 is a perfect number, Amicable numbers: Amicable numbers were known to the Greeks about 320 A.D., about 500 years after the discovery of perfect numbers. The ancients believed that if two people were wearing talismans bearing these kinds of numbers, they would be friendly and in harmony, hence amicable.(l, P• l7) The numbers 1184 and 1210 are amicable because the sum of the aliquot divisors of either number equals the other number, i.e., 1210 = 1~2+4+8+16+32+37+74+148+296+592 and 118L~ = 1 +2+5+10+11 +22+55+110+121 +242+605. Pythagoras once said that a friend is "one who is the other I such as are 220 and 284."( 2, P• 26 ) A symbolic definition of amicable numbers is: If two positive integers m and n are amicable, then O"'"(m) = m + n = c:r(n). Some Proven Facts of Number Theory Solution of congruences: 1. a= b(mod m), a, b integers and m is a natural number. So, if a = b or a - b = 0, then the difference is 171 divisible by every modulus and writing a ; a(mod m) is point less. If m = 1, a: b(mod 1) is always true since (a - b) is always divisible by 1. Some examples of a : b(mod m) con gruences are: 9; 4(mod 5) since 9- 4 is divisible by 5; and 9 ; 3(mod 6) since 9 - 3 is divisible by 6. However, 9 is not congruent to 3(mod L~) since 9 - 3 is not divisible by 4. A practical way of defining congruence is to say that a : b(mod m) means that when a is divided by m, the same remainder is obtained as when b is divided by m. Amorig the many applications of congruences are: magic squares, calendar problems, card tricks, a large assortment of games, and arithmetic and algebra. 2. ax; b(mod m), the linear congruence. The solution of this type of congruence is an integer_ x such that ax ; b(mod m). While equations of the form ax= b always have a solution when a f o, congruences may not have solutions. For example, Sx ; 2(mod 3) has solutions because (Sx - 2); 3 is an integer when x = 1, 4, 7, 10, •••• The most important results of congruences of the first degree, ax: b(mod m) are the following:(l, P• 39) a. Let d be the greatest common factor of a and m. Then the congruence has no solu tion unless d is a factor of b. b. If d is a factor of b, then the congruence has d solutions. c. In particular, if d = 1, that is, if a and m are relatively prime, the congruence has exactly one solution. This theorem is very useful in one approach to obtaining the incongruent solutions of a linear congruence.(9, PP• 114-ll5) Theorem: If ac : bc(mod m)- and (c, m) = g, then a : b(mod m/g). Proof: If ac =bc(mod m), then ml(ac- be); that is, mjc(a- b). If (c, m) = g, then c = gc' and m = gm'. Now gm'jgc'(a- b), m'jc•(a- b), where (c', m') = 1. 172 Therefore, m'f(a -b). Hence, a: b(mod m'); that is, a : b(mod m/g). Example: If 114: 66(mod 16) or 6 • 19: 6 • ll(mod 16) then 19 E ll(mod 8), Wilson's theorem: Edward Waring published the theorem (If p is a prime, then the product of all the natural numbers up to and includ ing p- 1, that is (1)(2)(3) ••• (p- 1), increased by 1, is divisible by p) in 1770 and ascribed it to his student, John Wilson (1741-1793).(l, P• l9) The interpretation of Wilson's theorem is that the expression (p- 1)! + 1 is divisible by p, or that (p- 1)! =- l(mod p). Examples: If p = 5, then 4! + 1 = 25 is divisible by 5. If p = 11, then 10! + 1 = ],628,800 + 1 = 3,628,801, which is divisible by 11. The first proof of this theorem was given in 1770 by J, L. Lagrange (1736-1813). The proof of Wilson's theorem using Fermat's theorem is left as an exercise for interested members of the Seminar. Pythagorean triples: The Pythagorean theorem states that for any right triangle with legs a and band hypotenuse c, a 2 + b2 = c 2• Most stu dents are familiar with Pythagorean triples (also called triplets) such as: 3, 4, 5; 5, 12, 13; 8, 15, 17; and their multiples. The problem in number theory is how to find all of the solutions of a 2 + b2 = c 2• Obviously, there is an infinite number of Pythagorean triples because if a, b, c is a Pythagorean triple and k is a positive integer, then ka, kb, kc form an infinite set of them, Suppose that a, b, c is a Pythagorean triple and d =(a, b). Then d2/ c 2 and dfc. Also d =(a, b) =(a, c)= (b, c). If the factors of a, b, and care: a= a'd, 173 2 2 2 + b2 2 b = b'd, c = c'd, then a' + b' = a = £__ = d2 d2 this means that a', b', c' is a Pythagorean triple and the numbers are pairwise relatively prime. (ll, P• l63) Definition: A set of pairwise relatively prime positive integers, a, b, c, is called a primitive Pythagorean triple if a2 + b2 = c2.(11, p. 164) Theorem: If x, y, z is a primitive Pythagorean triple, then one of the numbers x, y is even and one is odd, Proof: Primitive indicates that x and y are obviously not both even. So, suppose that both are odd. Then x2 : y2 : l(mod 4) and z 2 E x2 + y2 : 2(mod 4), which is impossible since the square of an even integer is congruent to O(mod 4). Therefore, one of the numbers x, y is even and one is odd (this is known as opposite parity).(ll, p. 164) All of the primitive solutions of x2 + y2 = z2 are given by x = 2mn, y = m2 - n 2 , z = m2 + n2 where m and n are inte gers of the opposite parity. Some of the primitive solutions of x 2 + y2 = z2 are presented below: m n X y z 2 1 4 3 ~ 2£ 4 3 24 7 ·~ 8£ 7 6 84 13 ·~ 8 3 48 55 7; 9 4 72 65 9? 10 7 140 51 145 44 12t 11 2 117 ·~ 12 5 120 119 165 8 208 23~ 13 105 ·~ 14 9 252 115 277 Diophantine equations: Any equation whose solution is restricted to integral values of the unknowns is called a Diophantine equation, named and first considered by the Greek mathematician Diophantus, who lived in the third century A.D. Diophantine equations, like equations in algebra, may be of first degree or higher, may be in one or more unknowns, or may be simul taneous systems of equations, A Diophantine equation may have no solution, a finite number of solutions, or even an infinite number of solutions. They are often referred to as indeterminate equations because they do not determine x and y (in ax + by = c) completely, It is typical of Dio phantine equations to often have an infinite number of solutions and to "find them all," which is to obtain some· procedure that turns them out in a systematic fashion,(S, P• 65) It may be shown generally that if x = r and y = s is a particular solution of ax + by = c, then all integral solu tions may be found from the formulas x = r + bt and y = s - at, where t may be a positive or negative integer. If t = o, x =randy= s.(l, P• 45) First degree Diophantine equations are congruences of the first degree. For example, 6x + 5y = 35 may be written either as 6x =35(mod 5) or 5y: 35(mod 6). 6x: 35(mod 5) means that the expression 35 - 6x when divided by 5 must be an integer, So it is obvious that a Diophantine equation may have no solution just as a congruence may have no solution. The fundamental result for a Diophantine equation con cerning lattice points (of a linear equation) is: If the linear equation (1) ax + by = c, a, b, c integers, has any lattice point (r, s), then all lattice points are given by the formulas (2) x = r + bt; y = s - at. So, ax +by= c becomes: a(r + bt) + b(s - at) = ar + abt + bs - bat or ar - bs = c, because (r, s) is assumed to be a lattice point. 175 Proof that (2) gives all lattice points:(9, PP• 46-50) Prove that if (x', y') is any lattice point of (1), then there is an integer t' for which (2) will yield this lattice point. Given (x'y') a lattice point, then ax• +by' = c (3). Also, (r, s) is a lattice point by hypothesis, so ar + bs = c ( 4) • Therefore, a(x' - r) = b(s - y')(5). (Subtract (4) from (3), or (5) is a(x'b- r) = (s- y'), which shows that the left member of the equation is an integer because the right member is an integer. However, a and b have no factor in common, so (x' - r) is divisible by b, Lett' = x'b- r an integer and substitute t' fort in part o f f ormu1 a ( 2) , 1,e,,. x = r + bt ' = r + b (x' b - r) = X I. t' also equals s; y' from (5), soy in (2) becomes: y = s- a(s- y•) =yo, a Because x' and y' are the coordinates of any lattice point, the above proves that all lattice points are obtained from formula (2)(using a proper choice oft), Note: the above proof uses the fundamental theorem of arithmetic. Example using the above proof: 14 X + 22 Y := 50 (x 2 and y 1) is a solution 0 = 0 = g = (14, 22) = 2, a = 14, b = 22 So x = 2 + llt and y ~ 1 - 7t, t an integer. The topics chosen for this section of the Seminar course were carefully selected and based essentially on the ideas from the Ed,D. dissertation of Irving Hollingshead, "Number Theory in the Twelfth Grade Mathematics Program." There are, of course, many topics of number theory that are inappropriate 176 for the high school level and not advocated for addition to the present mathematics curriculum. Therefore, the writer is currently of the opinion that a course of approximately seven to nine weeks (a mini-course) would be as appropriate as the suggested four- to six-week exposure of this Seminar course. An exposure of more than approximately nine weeks of number theory would, of necessity, appear to become too complicated and sophisticated even for advanced mathematics students at the high school level. Suggested Problems/Topics for Student Research 1. Solve any of the problems from this chapter that were left for the reader. 2. Use mathematical induction to solve some of the popular problems of number theory (these can be found within the first few chapters of many books on number theory). 3. Research the topic: "Divisors of a Number." 4. Research some of the work that has been done attempting to prove a formula for all prime numbers. Work on the solutions of higher-order congruences that satisfy f(x) : O(mod m), where f(x) is an integral polynomial. 6. Study the proofs of Fermat's theorem and Wilson's theorem for understanding. Examine several other properties of Pythagorean triples, including any pertinent proofs. 8. Research Diophantine equations of higher degree. 9. Examine some conjectures and unsolved problems of number theory such as: twin prime problem, a conjec ture on a quadratic progression, primes between consecutive squares, the Goldbach conjecture, Fermat's last theorem, the method of infinite descent. 177 10. Examine other areas such as: complex numbers, Gaussian primes, Fermat's little theorem, Fell's equation, Chinese remainder theorem, prime divisors of n2 + a, quadratic residues. 11, Other topics that are of interest to the students and approved by the instructor. References for Student Research Barnett, I. A. "Introducing Number Theory in High School Algebra and Geometry, Part 1: Algebra," The Mathematics Teacher, 58:14-23. Barnett, I. A, "Introducing Number Theory in High School Algebra and Geometry, Part 2: Geometry," The Mathema tics Teacher, 58:89-101. Barnett, I. A. Some Ideas About Number Theory. Washington, D. C,: National Council of Teachers of Mathematics, 1963. Beiler, Albert H. Recreations in the Theory of Numbers, New York: Dover Publications, Inc., 1966. Bell, E, T, "Gauss and the Early Development of Algebraic Numbers," National Mathematics Magazine, 18:188-204, 219-233. Bell, E. T, "Successive Generalizations in the Theory of Numbers," The American Mathematical Monthly, 34:55-73. Buckeye, Donald A., and Ginther, John L, Creative Mathematics. San Francisco: Canfield Press, 1971. Carmichael, R. D. "A Lesson from the History of Numbers," School Science and Mathematics, 13:392-399. Carmichael, R. D. "Recent Researches in the Theory of Numbers," The American Mathematical Monthly, 39:139-160. Eves, Howard. An Introduction to the History of Mathematics, Third Edition. San Francisco: Holt, Rinehart and Winston, 1969. Herwitz, P. w. "The Theory of Numbers," Scientific American, 185:52-55. Hollingshead, Irving. "Number Theory--A Short Course for High School Seniors," The Mathematics Teacher, 60:222-227. Jones, Phillips. "From China 'til Today," The Mathematics Teacher, 49:607-610. Lehmer, D, N. "Hunting Big Game in the Theory of Numbers," Scripta Mathematica, 1:229-235. 178 Long, Calvin T. Elementary Introduction to Number Theory. Boston: D. c. Heath and Company, 1965. Newman, James R. The World of Mathematics, Volume I. New York: Simon and Schuster, 1956. Ogilvy, c. Stanley, and Anderson, John T. Excursions in Number Theory. New York: Oxford University Press, 1966. Pettofrezzo, Anthony J., and Byrkit, Donald R. Elements of Number Theory. New Jersey: Prentice-Hall, Inc., 1970. Pettofrezzo, Anthony J., and Hight, Donald w. Number Systems Structure and Properties. Illinois: Scott, Foresman and Company, 1969. Reid, c. "Perfect Numbers," Scientific American, 188:84-86, Rolf, H. L. "Friendly Numbers," The Mathematics Teacher, 60:157-160. Ross, Arnold. "Notre Dame's 1960 Summer Program for Gifted High School Children," The Mathematics Teacher, 54: 440-443. School Mathematics Study Group. Essays on Number Theory (Two Volumes). Connecticut: Yale University Press, 1960. Shanks, Daniel. Solved and Unsolved Problems in Number Theory. Washington, D. c.: Spartan Books, 1962. Shockley, James E. Introduction to Number Theory. New York: Holt, Rinehart and Winston, 1967. · Weil, Andre. Basic Number Theory. New York: Springer Verlag, Inc., 1967. 179 CHAPTER VII BIBLIOGRAPHY 1. Barnett, I, A. Some Ideas About Number Theory. Wash ington, D, C,: National Council of Teachers of Mathematics, 196J. 2. Beiler, Albert H. Recreations in the Theor~ of Numbers. New York: Dover Publications, Inc., 196 • Buckeye, Donald A., and Ginther, John L. Creative Mathematics. San Francisco: Canfield Press, 1971. 4. Eves, Howard. An Introduction to the History of Mathe matics, Third Edition. San Francisco: Holt, Rinehart and Winston, 1969. Hollingshead, Irving. "Number Theory--A Short Course for High School Seniors," The Mathematics Teacher, 60: 222-227. 6. Long, Calvin T. Elementary Introduction to Number Theory. Boston: D. C. Heath and Company, 1965. Newman, James R. The World of Mathematics, Volume I. New York: Simon and Schuster, 1956. 8. Ogilvy, c. Stanley, and Anderson, John T. Excursions in Number Theory. New York: Oxford University Press, 1966. Pettofrezzo, Anthony J,, and Byrkit, Donald R. Elements of Number Theory. New Jersey: Prentice-Hall, Inc., 1970. 10, Shanks, Daniel. Solved and Unsolved Problems in Number Theory. Washington, D, C.: Spartan Books, 1962. 11. Shockley, James E. Introduction to Number Theory. New York: Holt, Rinehart and Winston, 1967. 12. Uspensky, J, V., and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill Book Company, 1939, lJ. Weil, Andre. Basic Number Theory, New York: Springer- Verlag, Inc •. , 1967. -