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0387969187 9780387969183 Differential and Difference Equations Through Computer Experiments:Phaser is a sophisticated program for IBM personal com- puters, developed atBrown University by the author and some of his students, which enables usersto experiment with differential and difference equations and dynamical systems in an interactive environment using graphics. This book begins with a brief discussion of the geometric inter- pretation of differential equations and numerical methods, and proceeds to guide the student through the use of the program. To run Phaser, you need an IBM PC, XT, AT, or PS/2 with an IBM Color GRaphics Board (CGB), Enhanced Graphics Adapter (VGA). A math coprocessor is supported; however, one is not required for Phaser to run on the above hardware.
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Modelling With Circular Motion The School Mathematics Project 9780521408899 ISBN: 052140889X Pub Date: 1993 Publisher: Cambridge University Press Summary: The aim of 16-19 Mathematics has been to produce a course which, while challenging, is accessible and enjoyable to all students. The course develops ability and confidence in mathematics and its applications, together with an appreciation of how mathematical ideas help in the understanding of the world and society in which we live. This unit: - develops an understanding of work and energy through the modelling of rea...l situations involving circular motion; - provides insight into the potential of mathematics for modelling physical phenomena; - helps foster an appreciation of the links between mathematics and the real world; - develops a basis for further study in engineering and science; - fosters an ability to model both in familiar and unfamiliar contexts within the field of mechanics. School Mathematics Project Staff is the author of Modelling With Circular Motion The School Mathematics Project, published 1993 under ISBN 9780521408899 and 052140889X. Forty five Modelling With Circular Motion The School Mathematics Project textbooks are available for sale on ValoreBooks.com, twenty used from the cheapest price of $1.24, or buy new starting at $10
A handy, fast, reliable, precise tool if you need to perform complex mathematical calculations Scientific CalculatorPrecision 54 54 digits. Trigonometric, hyperbolic and inverse functions. A handy, fast, reliable, precise tool if you need to perform complex mathematical calculations Scientific CalculatorPrecision 63 Prebult Common Costants list with fundamental constants. Unlimited User Constants list. A handy, fast, reliable, precise tool if you need to perform complex mathematical calculations Scientific CalculatorPrecision combinatorial functions. A handy, fast, reliable, precise tool if you need to perform complex mathematical calculations Scientific CalculatorPrecision 36 36 digits. Trigonometric, hyperbolic, inverse and combinatorial functions. A handy, fast, reliable, precise tool if you need to perform complex mathematical calculations Scientific CalculatorPrecision combinatorial functions. Karen's Calculator is a high-precisioncalculator. Unlike ordinary calculators, it returns completely accurate results, even with operands containing thousands of Digits. If you're patient, operands and results containing hundreds of thousands of digits are possble too! Most calculators are only accurate when dealing with numbers containing a few digits -- usually 16 to 32. When asked to calculate using larger numbers, they either fail (displaying an error) or return results that are only approximately accurate. SCX is a general-purpose high-precisioncalculator drop-in replacement for the built-in Palm calculator. A programmable graphics calculator which lets you visualize expressions and formulas as graphs in a chart and can also create video clips from graphs. GraphiCal has over 50 built-in mathematical functions, can integrate and find roots and extrema. It is programmable, offers user defined variables and functions, programming interfaces for VB and VC and a very user-friendly graphics system.. Farsight Calculator,such as "Finance box","date calculations" and "Unit convertor" etc.16X is high resolution programmable timer software. It controls 16 Features: Support Network and Com Port connection between PC and relay board. Detect connection status between PC and relay board continuously. Remind after disconnection and recover after reconnection automatically. HiDigit is a new calculating software with extended capabilities. This is an essential application for math, algebra, calculus, geometry, physics and engineering students. The main advantage of the software is a simple input format even for the most complicated formulas. For example, you can enter 10pi instead of "10*pi". For complex numbers, you can use the following format - "1+2i". For percentages - "number + %". The other important feature of HiDigit is its highprecision - up to 15 decimals. AresCalc AresCalc contains a set of powerful tools,such as "Finance box","date calculations" and "Unit convertor" etc. Main Features 1ird
Linear Algebra WithRenowned for its thoroughness, clarity, and accessibility, this best-selling book by one of today's leading figures in linear algebra reform offers users a challenging yet enjoyable treatment of linear algebra that is infused with an abundance of applications and worked examples. Balancing coverage of mathematical theory and applied topics, the book stresses the important role geometry and visualization play in understanding the subject, and now comes with the new ancillary ATLAS computer exercise guide.Provides modern and comprehensive coverage of the subject, spanning all topics in the core syllabus recommended by the NSF sponsored Linear Algebra Curriculum Study Group. Offers new applications in astronomy and statistics, emphasizes the use of geometry to visualize linear algebra and aid in understanding all of the major topics, and previews some of the more difficult vector space concepts early on. MATLAB computing exercises provide users with experience performing matrix computations.For mathematicians.
... More About This Book pretty sure that involved ratios. The problem is, you can't quite remember. Here you get an adult refresher and real-life context—with examples ranging from how to figure out how many shingles it takes to re-roof the garage to the formula for resizing Mom's tomato sauce recipe for your entire family. Forget higher calculus—you just need an open mind. And with this practical guide, math can stop being scary and start being useful. Product Details ISBN-13: 9781435145238 Publisher: Adams Media Corporation Publication date: 6/30/2013 Pages: 256 Product dimensions: 5.60 (w) x 8.50 (h) x 0.90 (d) Meet the Author Laura Laing graduated from James Madison University with a BS in Mathematics. After teaching high school math for four years, she became a staff writer for Inside Business. Her articles have appeared in Parade, The City Paper, Baltimore Sun, and The Advocate
Mathematics Mathematics began with roots in the basic concepts of space and number and has flowered into many wonderful forms. The creation and discovery of new mathematics have never been more active or vital than they are today. Mathematics is sometimes called the science of pattern and order. It relies on logic as a standard of truth, but uses observation and even experimentation as means of discovering truth. Mathematicians think of their work as a blend of science and art, sometimes elegant and beautiful, describing deep and useful creations. In addition to theorems and theories, mathematics offers distinct modes of thought which are both versatile and powerful for understanding the world. Courses serve those who wish to make mathematics a part of a liberal arts education, those who desire a mathematics background for other disciplines, such as Computer Science, Economics or the natural sciences, those who wish to minor in Mathematics, and those who wish to major in Mathematics. Mathematics majors choose careers in education, industry, business, banking and insurance serving as teachers, statisticians, industrial mathematicians, computer programmers or analysts, actuaries and research workers in the biological, management or social sciences. Their training can also serve as a stepping stone to professional training or graduate work in a variety of fields. Course Listings MATH 102X Problem-Solving (.25) The course will offer students the opportunity to solve challenging mathematical problems unlike standard homework problems in any course. Class time will be spent studying problems, discovering solutions, writing up solutions formally, and discussing the important ideas of each solution. Most problems will be of the kind appearing on the Putnam Exam, an annual international mathematics competition. This course may be repeated for credit. Offering: Fall Instructor: Staff MATH 130 (QA) Contemporary Mathematics (1) A survey of contemporary topics in mathematics such as: voting systems and power, apportionment, fair division of divisible and indivisible assets, efficient distribution, scheduling and routing, growth and decay in nature and economics, symmetry and fractal geometry, probability and statistics. MATH 130 may not be taken for credit after any Mathematics course numbered above 140 has been completed. MATH 141 (QA) Calculus I (1) A first course in calculus-differential and integral calculus of algebraic and exponential functions, with applications. (MATH 141 counts for only .5 credit if the student has completed MATH 139 Brief Calculus.) MATH 142 (QA*) Calculus II (1) A second course in calculus: review of differential and integral calculus via trigonometric and logarithmic functions, techniques and applications of integration, polar coordinates and parametric equations, infinite series. MATH 220 (QA) Mathematics for Elementary Teachers (1) The objective of the course is to present mathematics in a format that prepares teachers to teach elementary school mathematics. Teachers need a firm foundation in the theory of mathematics as it pertains to the elementary school curriculum. They also need ideas and methods for teaching that will generate interest and enthusiasm among the students. Topics to be covered will include problem solving, mathematics as a method of communication, mathematics as a method of reasoning, and specifics of elementary school mathematics such as whole number operations, geometry and spatial sense, measurement and estimation, fractions and decimals, and patterns and relationships. MATH 251 (W) Foundations of Advanced Mathematics (1) This course is intended as the first course after calculus for those students intending to major or minor in mathematics. It provides an introduction to logic and the methods of proof commonly used in mathematics. Applications covered in the course are the foundations of set theory, the real number system, elementary number theory and other basic areas of mathematics. MATH 325 (QA) Mathematics for Teachers (1) The objective of this course is to present mathematics in a format that prepares teachers to teach mathematics in the public schools. Teachers need a firm foundation in the theory of mathematics as it pertains to their particular curricula. They also need ideas and methods for teaching that will generate interest and enthusiasm among the students. The course will emphasize mathematics as a method of communication and reasoning. Topics selected to be relevant to elementary, middle, and/or high school curricula will depend on the interests of the students, but will have a strong problem-solving emphasis. The course will require an extensive early field experience in the public school classroom. MATH 366 (QA) Applied Mathematics: Optimization (1) Formulation of problems in mathematical terms, solutions of the problems, interpretation and evaluation of the solutions. Topics will be chosen from inventory problems, growth and survival models, linear programming, scheduling, Markov chains, game theory and queuing problems. MATH 486 Topics in Mathematics (1) This course offers timely exposure to topics in mathematics which are not part of the regular curriculum. Examples of topics which might be offered: Cryptology, Differential Geometry, Vector Analysis, Topology. Offering: On demand Instructor: Staff MATH 490 Independent Research (.5) Directed research to investigate topics of special interest under the guidance of a faculty member. Topics chosen on the basis of the background and interests of the individual student. Prerequisite: Consent of instructor Offering: On demand Instructor: Staff MATH 491 Advanced Independent Study (.5) A course of directed research designed to enable the exceptional student to continue the investigation of topics of special interest under the guidance of a faculty member. Prerequisite: Consent of instructor Offering: On demand Instructor: Staff MATH 499 Seminar in Mathematics (1) Study selected in consultation with the mathematics faculty and presented to the class. The seminar serves as the Senior Year Experience and involves oral and written presentation of research and reading topics. Required for Mathematics majors.
Features detailed explanations and solutions to every sixth exercise in the text. Includes helpful hints and strategies for studying in this course. In addition, Visual Calculus, the popular, easy-to-use software for IBM compatible computers is included.
More About This Textbook Overview Numerical ability is an essential skill for everyone studying the biological sciences; but many students are frightened by the 'perceived' difficulty of mathematics, and are nervous about applying mathematical skills in their chosen field of study. Having taught introductory maths and statistics for many years, Alan Cann understands these challenges, and just how invaluable an accessible, confidence building textbook could be to the fearful student. Unable to find a book pitched at the right level that concentrated on why numerical skills are useful to biologists, he wrote his own. The result is Maths from Scratch for Biologists, a highly instructive, informal text that explains step by step how and why you need to tackle maths within the biological
What's the reliability of cancer tests, diabetes tests, and pregnancy tests? This brief discussion shows some functions to be used on your graphing calculator to visualize a graph of the accuracy of a... More: lessons, discussions, ratings, reviews,... With this one-variable function grapher applet and function evaluator, users can rotate axis/axes, change scale, and translate by using mouse or by entering data. The web site also contains informatio game explores functions in a different way: a and b = f(a) are drawn in a unique numerical line. When the user changes a, b = f(a) changes following a rule. The objective of the game is to discov... More: lessons, discussions, ratings, reviews,... This physics-exploration applet allows the user to experiment with different roller coaster track designs, then test. Friction and mass are modeled. Includes hot links to explanations of various phys... More: lessons, discussions, ratings, reviews,... Using this virtual manipulative you may: graph a function; trace a point along the graph; dynamically vary function parameters; change the range of values displayed in the graph; graph multiple functi... More: lessons, discussions, ratings, reviews,... Commercial site with one free access a day. Students use mapping diagram to create a relation, then they can check if it is a function from the mapping diagram, ordered pairs and graph. After studen... More: lessons, discussions, ratings, reviews,... The three applets on this page are: One-to one or Not?, Graphs of Inverse Functions, and Inverse Functions. The first one allows students to decide if a function is one-to-one, and if not, it shows aAn application problem that asks students to help an excavation company prepare a bid for one of their clients by calculating what they should charge to dig a hole. To solve the problem students must ... More: lessons, discussions, ratings, reviews,... Enter a set of data points and a function or multiple functions, then manipulate those functions to fit those points. Manipulate the function on a coordinate plane using slider bars. Learn how each co... More: lessons, discussions, ratings, reviews,... Windows software which allows the display of 2D and 3D diagrams both on one, and on different screens. Display 2D diagrams in the Cartesian and polar systems of coordinates. Display 3D diagrams in
Intermediate Algebra for College Students (5th Edition) 9780136007623 ISBN: 0136007627 Edition: 5 Pub Date: 2008 Publisher: Prentice Hall Summary: The goal of this book is to provide readers with a strong foundation in algebra by developing the problem-solving and critical thinking abilities. Topics are presented in an interesting format, incorporating real world sourced data and encouraging modeling and problem-solving. Robert F. Blitzer is the author of Intermediate Algebra for College Students (5th Edition), published 2008 under ISBN 9780136007623 a...nd 0136007627. Forty three Intermediate Algebra for College Students (5th Edition) textbooks are available for sale on ValoreBooks.com, thirty used from the cheapest price of $5.72, or buy new starting at $55 teacher 5th HC ed,just like the student version cover to cover but will have answers and or marginal notes,any extra labeling will be covered with black [more] ALTERNATE EDITION: A brand new never used teacher 5th HC ed,just like the student version cover to cover but will have answers and or marginal notes,any extra labeling will be covered with black book tape,includes a testprep cd,smoke free env[less]
This book is a survey of abstract algebra with emphasis on linear algebra. It is intended for students in mathematics, computer science, and the physical sciences. The first three or four chapters can stand alone as a one semester course in abstract algebra. However they are structured to provide the background for the chapter on linear algebra. Finishing the chapter on linear algebra gives a basic one year undergraduate course in abstract algebra. Contents: 1) Background and Fundamentals of Mathematics. 2) Groups. 3) Rings. 4) Matrices and Matrix Rings. 5) Linear Algebra. 6) Appendix. iii In tro duction In 1965 I rst taugh t an undergraduate course in abstract algebra. It w as fun to teac h b ecause the material w as in teresting and the class w as outstanding. Fiv e of those studen ts later earned a Ph.D. in mathematics. Since then I ha v e taugh t the course ab out a dozen times from v arious texts. Ov er the y ears I dev elop ed a set of lecture notes and in 1985 I had them t yp ed so they could b e used as a text. They no w app ear (in mo died form) as the rst v e c hapters of this b o ok. Here w ere some of m y motiv es at the time. 1) T o ha v e something as short and inexp ensiv e as p ossible. In m y exp erience, studen ts lik e short b o oks. 2) T o a v oid all inno v ation. T o organize the material in the most simple-minded straigh tforw ard manner. 3) T o order the material linearly T o the exten t p ossible, eac h section should use the previous sections and b e used in the follo wing sections. 4) T o omit as man y topics as p ossible. This is a foundational course, not a topics course. If a topic is not used later, it should not b e included. There are three go o d reasons for this. First, linear algebra has top priorit y It is b etter to go forw ard and do more linear algebra than to stop and do more group and ring theory Second, it is more imp ortan t that studen ts learn to organize and write pro ofs themselv es than to co v er more sub ject matter. Algebra is a p erfect place to get started b ecause there are man y \easy" theorems to pro v e. There are man y routine theorems stated here without pro ofs, and they ma y b e considered as exercises for the studen ts. Third, the material should b e so fundamen tal that it b e appropriate for studen ts in the ph ysical sciences and in computer science. Zillions of studen ts tak e calculus and co okb o ok linear algebra, but few tak e abstract algebra courses. Something is wrong here, and one thing wrong is that the courses try to do to o m uc h group and ring theory and not enough matrix theory and linear algebra. 5) T o oer an alternativ e for computer science ma jors to the standard discrete mathematics courses. Most of the material in the rst four c hapters of this text is co v ered in v arious discrete mathematics courses. Computer science ma jors migh t b enet b y seeing this material organized from a purely mathematical viewp oin t. PAGE 4 iv Ov er the y ears I used the v e c hapters that w ere t yp ed as a base for m y algebra courses, supplemen ting them as I sa w t. In 1996 I wrote a sixth c hapter, giving enough material for a full rst y ear graduate course. This c hapter w as written in the same \st yle" as the previous c hapters, i.e., ev erything w as righ t do wn to the n ub. It h ung together prett y w ell except for the last t w o sections on determinan ts and dual spaces. These w ere indep enden t topics stuc k on at the end. In the academic y ear 1997-98 I revised all six c hapters and had them t yp ed in LaT eX. This is the p ersonal bac kground of ho w this b o ok came ab out. It is dicult to do an ything in life without help from friends, and man y of m y friends ha v e con tributed to this text. My sincere gratitude go es esp ecially to Marilyn Gonzalez, Lourdes Robles, Marta Alpar, John Zw eib el, Dmitry Gokhman, Brian Co omes, Huseyin Ko cak, and Sh ulim Kaliman. T o these and all who con tributed, this b o ok is fondly dedicated. This b o ok is a surv ey of abstract algebra with emphasis on linear algebra. It is in tended for studen ts in mathematics, computer science, and the ph ysical sciences. The rst three or four c hapters can stand alone as a one semester course in abstract algebra. Ho w ev er they are structured to pro vide the bac kground for the c hapter on linear algebra. Chapter 2 is the most dicult part of the b o ok b ecause groups are written in additiv e and m ultiplicativ e notation, and the concept of coset is confusing at rst. After Chapter 2 the b o ok gets easier as y ou go along. Indeed, after the rst four c hapters, the linear algebra follo ws easily Finishing the c hapter on linear algebra giv es a basic one y ear undergraduate course in abstract algebra. Chapter 6 con tin ues the material to complete a rst y ear graduate course. Classes with little bac kground can do the rst three c hapters in the rst semester, and c hapters 4 and 5 in the second semester. More adv anced classes can do four c hapters the rst semester and c hapters 5 and 6 the second semester. As bare as the rst four c hapters are, y ou still ha v e to truc k righ t along to nish them in one semester. The presen tation is compact and tigh tly organized, but still somewhat informal. The pro ofs of man y of the elemen tary theorems are omitted. These pro ofs are to b e pro vided b y the professor in class or assigned as homew ork exercises. There is a non-trivial theorem stated without pro of in Chapter 4, namely the determinan t of the pro duct is the pro duct of the determinan ts. F or the prop er ro w of the course, this theorem should b e assumed there without pro of. The pro of is con tained in Chapter 6. The Jordan form should not b e considered part of Chapter 5. It is stated there only as a reference for undergraduate courses. Finally Chapter 6 is not written primarily for reference, but as an additional c hapter for more adv anced courses. PAGE 5 v This text is written with the con viction that it is more eectiv e to teac h abstract and linear algebra as one coheren t discipline rather than as t w o separate ones. T eac hing abstract algebra and linear algebra as distinct courses results in a loss of synergy and a loss of momen tum. Also with this text the professor do es not extract the course from the text, but rather builds the course up on it. I am con vinced it is easier to build a course from a base than to extract it from a big b o ok. Because after y ou extract it, y ou still ha v e to build it. The bare b ones nature of this b o ok adds to its rexibilit y b ecause y ou can build whatev er course y ou w an t around it. Basic algebra is a sub ject of incredible elegance and utilit y but it requires a lot of organization. This b o ok is m y attempt at that organization. Ev ery eort has b een extended to mak e the sub ject mo v e rapidly and to mak e the ro w from one topic to the next as seamless as p ossible. The studen t has limited time during the semester for serious study and this time should b e allo cated with care. The professor pic ks whic h topics to assign for serious study and whic h ones to \w a v e arms at". The goal is to sta y fo cused and go forw ard, b ecause mathematics is learned in hindsigh t. I w ould ha v e made the b o ok shorter, but I did not ha v e an y more time. When using this text, the studen t already has the outline of the next lecture, and eac h assignmen t should include the study of the next few pages. Study forw ard, not just bac k. A few min utes of preparation do es w onders to lev erage classro om learning, and this b o ok is in tended to b e used in that manner. The purp ose of class is to learn, not to do transcription w ork. When studen ts come to class cold and sp end the p erio d taking notes, they participate little and learn little. This leads to a dead class and also to the bad psyc hology of \O K ; I am here, so teac h me the sub ject." Mathematics is not taugh t, it is learned, and man y studen ts nev er learn ho w to learn. Professors should giv e more direction in that regard. Unfortunately mathematics is a dicult and hea vy sub ject. The st yle and approac h of this b o ok is to mak e it a little ligh ter. This b o ok w orks b est when view ed ligh tly and read as a story I hop e the studen ts and professors who try it, enjo y it. E. H. Connell Departmen t of Mathematics Univ ersit y of Miami Coral Gables, FL 33124 ec@math.miami.edu viii 1 2 3 4 5 6 7 8 9 11 10 Abstract algebra is not only a ma jor sub ject of science, but it is also magic and fun. Abstract algebra is not all w ork and no pla y and it is certainly not a dull b o y See, for example, the neat card tric k on page 18. This tric k is based, not on sleigh t of hand, but rather on a theorem in abstract algebra. An y one can do it, but to understand it y ou need some group theory And b efore b eginning the course, y ou migh t rst try y our skills on the famous (some w ould sa y infamous) tile puzzle. In this puzzle, a frame has 12 spaces, the rst 11 with n um b ered tiles and the last v acan t. The last t w o tiles are out of order. Is it p ossible to slide the tiles around to get them all in order, and end again with the last space v acan t? After giving up on this, y ou can study p erm utation groups and learn the answ er! 2 Bac kground Chapter 1 of A and B A [ B = f x : x 2 A or x 2 B g = the set of all x whic h are elemen ts of A or B An y set called an index set is assumed to b e non-v oid. Supp ose T is an index set and for eac h t 2 T A t is a set. [ t 2 T A t = f x : 9 t 2 T with x 2 A t g \ t 2 T A t = f x : if t 2 T ; x 2 A t g = f x : 8 t 2 T ; x 2 A t g Let ; b e the n ull set. If A \ B = ; then A and B are said to b e disjoint Denition Supp ose eac h of A and B is a set. The statemen t that A is a subset of B ( A B ) means that if a is an elemen t of A then a is an elemen t of B That is, a 2 A ) a 2 B : If A B w e ma y sa y A is con tained in B or B con tains A Exercise Supp ose eac h of A and B is a set. The statemen t that A is not a subset of B means Theorem (De Morgan's la ws) Supp ose S is a set. If C S (i.e., if C is a subset of S ), let C 0 the complemen t of C in S b e dened b y C 0 = S C = f x 2 S : x 62 C g Then for an y A; B S ( A \ B ) 0 = A 0 [ B 0 and ( A [ B ) 0 = A 0 \ B 0 Cartesian Pro ducts If X and Y are sets, X Y = f ( x; y ) : x 2 X and y 2 Y g In other w ords, the Cartesian pro duct of X and Y is dened to b e the set of all ordered pairs whose rst term is in X and whose second term is in Y Example R R = R 2 = the plane. PAGE 11 Chapter 1 Bac kground 3 Denition If eac h of X 1 ; :::; X n is a set, X 1 X n = f ( x 1 ; :::; x n ) : x i 2 X i for 1 i n g = the set of all ordered n -tuples whose i -th term is in X i Example R R = R n = real n -space. Question Is ( R R 2 ) = ( R 2 R ) = R 3 ? Relations If A is a non-v oid set, a non-v oid subset R A A is called a r elation on A If ( a; b ) 2 R w e sa y that a is related to b and w e write this fact b y the expression a b Here are sev eral prop erties whic h a relation ma y p ossess. 1) If a 2 A then a a (rerexiv e) 2) If a b; then b a (symmetric) 2 0 ) If a b and b a then a = b (an ti-symmetric) 3) If a b and b c; then a c (transitiv e) Denition A relation whic h satises 1), 2 0 ), and 3) is called a p artial or dering In this case w e write a b as a b Then 1) If a 2 A then a a 2 0 ) If a b and b a then a = b 3) If a b and b c; then a c Denition A line ar or dering is a partial ordering with the additional prop ert y that, if a; b 2 A then a b or b a Example A = R with the ordinary ordering, is a linear ordering. Example A = all subsets of R 2 with a b dened b y a b is a partial ordering. Hausdor Maximalit y Principle (HMP) Supp ose S is a non-v oid subset of A and is a relation on A This denes a relation on S If the relation satises an y of the prop erties 1), 2), 2 0 ), or 3) on A; the relation also satises these prop erties when restricted to S In particular, a partial ordering on A denes a partial ordering PAGE 12 4 Bac kground Chapter 1 on S Ho w ev er the ordering ma y b e linear on S but not linear on A The HMP is that an y linearly ordered subset of a partially ordered set is con tained in a maximal linearly ordered subset. Exercise Dene a relation on A = R 2 b y ( a; b ) ( c; d ) pro vided a c and b d: Sho w this is a partial ordering whic h is linear on S = f ( a; a ) : a < 0 g : Find at least t w o maximal linearly ordered subsets of R 2 whic h con tain S One of the most useful applications of the HMP is to obtain maximal monotonic collections of subsets. Denition A collection of sets is said to b e monotonic if, giv en an y t w o sets of the collection, one is con tained in the other. Corollary to HMP Supp ose X is a non-v oid set and A is some non-v oid collection of subsets of X and S is a sub collection of A whic h is monotonic. Then 9 a maximal monotonic sub collection of A whic h con tains S Pro of Dene a partial ordering on A b y V W i V W ; and apply HMP The HMP is used t wice in this b o ok. First, to sho w that innitely generated v ector spaces ha v e free bases, and second, in the App endix, to sho w that rings ha v e maximal ideals (see pages 87 and 109). In eac h of these applications, the maximal monotonic sub collection will ha v e a maximal elemen t. In elemen tary courses, these results ma y b e assumed, and th us the HMP ma y b e ignored. Equiv alence Relations A relation satisfying prop erties 1), 2), and 3) is called an e quivalenc e r elation Exercise Dene a relation on A = Z b y n m i n m is a m ultiple of 3. Sho w this is an equiv alence relation. Denition If is an equiv alence relation on A and a 2 A w e dene the e quivalenc e class con taining a b y cl ( a ) = f x 2 A : a x g PAGE 13 Chapter 1 Bac kground 5 Theorem 1) If b 2 cl ( a ) then cl ( b ) = cl ( a ) : Th us w e ma y sp eak of a subset of A b eing an equiv alence class with no men tion of an y elemen t con tained in it. 2) If eac h of U; V A is an equiv alence class and U \ V 6 = ; then U = V 3) Eac h elemen t of A is an elemen t of one and only one equiv alence class. Denition A p artition of A is a collection of disjoin t non-v oid subsets whose union is A In other w ords, a collection of non-v oid subsets of A is a partition of A pro vided an y a 2 A is an elemen t of one and only one subset of the collection. Note that if A has an equiv alence relation, the equiv alence classes form a partition of A Theorem Supp ose A is a non-v oid set with a partition. Dene a relation on A b y a b i a and b b elong to the same subset of the partition. Then is an equiv alence relation, and the equiv alence classes are just the subsets of the partition. Summary There are t w o w a ys of viewing an equiv alence relation \| one is as a relation on A satisfying 1), 2), and 3), and the other is as a partition of A in to disjoin t subsets. Exercise Dene an equiv alence relation on Z b y n m i n m is a m ultiple of 3. What are the equiv alence classes? Exercise Is there a relation on R satisfying 1), 2), 2 0 ) and 3) ? That is, is there an equiv alence relation on R whic h is also a partial ordering? Exercise Let H R 2 b e the line H = f ( a; 2 a ) : a 2 R g Consider the collection of all translates of H ; i.e., all lines in the plane with slop e 2. Find the equiv alence relation on R 2 dened b y this partition of R 2 F unctions Just as there are t w o w a ys of viewing an equiv alence relation, there are t w o w a ys of dening a function. One is the \in tuitiv e" denition, and the other is the \graph" or \ordered pairs" denition. In either case, domain and r ange are inheren t parts of the denition. W e use the \in tuitiv e" denition b ecause ev ery one thinks that w a y Chapter 1 Bac kground 9 Theorem Supp ose S X Y The subset S is the graph of a function with domain X and range Y i eac h v ertical strip in tersects S in exactly one p oin t. This is just a restatemen t of the prop ert y of a graph of a function. The purp ose of the next theorem is to restate prop erties of functions in terms of horizon tal strips. Theorem Supp ose f : X Y has graph Then 1) Eac h horizon tal strip in tersects in at least one p oin t i f is 2) Eac h horizon tal strip in tersects in at most one p oin t i f is 3) Eac h horizon tal strip in tersects in exactly one p oin t i f is Solutions of Equations No w w e restate these prop erties in terms of solutions of equations. Supp ose f : X Y and y 0 2 Y Consider the equation f ( x ) = y 0 Here y 0 is giv en and x is considered to b e a \v ariable". A solution to this equation is an y x 0 2 X with f ( x 0 ) = y 0 Note that the set of all solutions to f ( x ) = y 0 is f 1 ( y 0 ). Also f ( x ) = y 0 has a solution i y 0 2 image( f ) i f 1 ( y 0 ) is non-v oid. Theorem Supp ose f : X Y 1) The equation f ( x ) = y 0 has at least one solution for eac h y 0 2 Y i f is 2) The equation f ( x ) = y 0 has at most one solution for eac h y 0 2 Y i f is 3) The equation f ( x ) = y 0 has a unique solution for eac h y 0 2 Y i f is Righ t and Left In v erses One w a y to understand functions is to study righ t and left in v erses, whic h are dened after the next theorem. Theorem Supp ose X f Y g W are functions. 1) If g f is injectiv e, then f is injectiv e. PAGE 18 10 Bac kground Chapter 1 2) If g f is surjectiv e, then g is surjectiv e. 3) If g f is bijectiv e, then f is injectiv e and g is surjectiv e. Example X = W = f p g Y = f p; q g f ( p ) = p and g ( p ) = g ( q ) = p Here g f is the iden tit y but f is not surjectiv e and g is not injectiv e. Denition Supp ose f : X Y is a function. A left in v erse of f is a function g : Y X suc h that g f = I X : X X A righ t in v erse of f is a function h : Y X suc h that f h = I Y : Y Y Theorem Supp ose f : X Y is a function. 1) f has a righ t in v erse i f is surjectiv e. An y suc h righ t in v erse m ust b e injectiv e. 2) f has a left in v erse i f is injectiv e. An y suc h left in v erse m ust b e surjectiv e. Corollary Supp ose eac h of X and Y is a non-v oid set. Then 9 an injectiv e f : X Y i 9 a surjectiv e g : Y X : Also a function from X to Y is bijectiv e i it has a left in v erse and a righ t in v erse i it has a left and righ t in v erse. Note The Axiom of Choice is not discussed in this b o ok. Ho w ev er, if y ou w ork ed 1) of the theorem ab o v e, y ou unkno wingly used one v ersion of it. F or completeness, w e state this part of 1) again. The Axiom of Choice If f : X Y is surjectiv e, then f has a righ t in v erse h That is, for eac h y 2 Y it is p ossible to c ho ose an x 2 f 1 ( y ) and th us to dene h ( y ) = x Note It is a classical theorem in set theory that the Axiom of Choice and the Hausdor Maximalit y Principle are equiv alen t. Ho w ev er in this text w e do not go that deeply in to set theory F or our purp oses it is assumed that the Axiom of Choice and the HMP are true. Exercise Supp ose f : X Y is a function. Dene a relation on X b y a b if f ( a ) = f ( b ) : Sho w this is an equiv alence relation. If y b elongs to the image of f then f 1 ( y ) is an equiv alence class and ev ery equiv alence class is of this form. In the next c hapter where f is a group homomorphism, these equiv alence classes will b e called cosets. Chapter 1 Bac kground 13 in tegers N to Y : Sho w Q is coun table but the follo wing three collections are not. i) P ( N ), the collection of all subsets of N ii) f 0 ; 1 g N the collection of all functions f : N f 0 ; 1 g iii) The collection of all sequences ( y 1 ; y 2 ; : : : ) where eac h y i is 0 or 1. W e kno w that ii) and iii) are equal and there is a natural bijection b et w een i) and ii). W e also kno w there is no surjectiv e map from N to f 0 ; 1 g N i.e., f 0 ; 1 g N is uncoun table. Finally sho w there is a bijection from f 0 ; 1 g N to the real n um b ers R (This is not so easy T o start with, y ou ha v e to decide what the real n um b ers are.) Notation for the Logic of Mathematics Eac h of the w ords \Lemma", \Theorem", and \Corollary" means \true statemen t". Supp ose A and B are statemen ts. A theorem ma y b e stated in an y of the follo wing w a ys: Theorem Hyp othesis Statemen t A Conclusion Statemen t B Theorem Supp ose A is true. Then B is true. Theorem If A is true, then B is true. Theorem A ) B ( A implies B ). There are t w o w a ys to pro v e the theorem \| to supp ose A is true and sho w B is true, or to supp ose B is false and sho w A is false. The expressions \ A B ", \ A is equiv alen t to B ", and \ A is true i B is true ha v e the same meaning (namely that A ) B and B ) A ). The imp ortan t thing to remem b er is that though ts and expressions ro w through the language. Mathematical sym b ols are shorthand for phrases and sen tences in the English language. F or example, \ x 2 B means \ x is an elemen t of the set B ." If A is the statemen t \ x 2 Z + and B is the statemen t \ x 2 2 Z + ", then \ A ) B "means \If x is a p ositiv e in teger, then x 2 is a p ositiv e in teger". Mathematical Induction is based up on the fact that if S Z + is a non-v oid subset, then S con tains a smallest elemen t. 16 Bac kground Chapter 1 2) ( a; b ) = 1 ; i.e., the subgroup generated b y a and b is all of Z 3) 9 m; n 2 Z with ma + nb = 1. Denition If an y one of these three conditions is satised, w e sa y that a and b are r elatively prime This next theorem is the basis for unique factorization. Theorem If a and b are relativ ely prime with a not zero, then a j bc ) a j c Pro of Supp ose a and b are relativ ely prime, c 2 Z and a j bc Then there exist m; n with ma + nb = 1, and th us mac + nbc = c No w a j mac and a j nbc Th us a j ( mac + nbc ) and so a j c Denition A prime is an in teger p > 1 whic h do es not factor, i.e., if p = ab then a = 1 or a = p The rst few primes are 2, 3, 5, 7, 11, 13, 17,... Theorem Supp ose p is a prime. 1) If a is an in teger whic h is not a m ultiple of p then ( p; a ) = 1 : In other w ords, if a is an y in teger, ( p; a ) = p or ( p; a ) = 1. 2) If p j ab then p j a or p j b 3) If p j a 1 a 2 a n then p divides some a i : Th us if eac h a i is a prime, then p is equal to some a i Pro of P art 1) follo ws immediately from the denition of prime. No w supp ose p j ab If p do es not divide a then b y 1), ( p; a ) = 1 and b y the previous theorem, p m ust divide b Th us 2) is true. P art 3) follo ws from 2) and induction on n The Unique F actorization Theorem Supp ose a is an in teger whic h is not 0,1, or -1. Then a ma y b e factored in to the pro duct of primes and, except for order, this factorization is unique. That is, 9 a unique collection of distinct primes p 1 ; p 2 ; :::; p k and p ositiv e in tegers s 1 ; s 2 ; :::; s k suc h that a = p s 1 1 p s 2 2 p s k k Pro of F actorization in to primes is ob vious, and uniqueness follo ws from 3) in the theorem ab o v e. The p o w er of this theorem is uniqueness, not existence. PAGE 25 Chapter 1 Bac kground 17 No w that w e ha v e unique factorization and part 3) ab o v e, the picture b ecomes transparen t. Here are some of the basic prop erties of the in tegers in this ligh t. Theorem (Summary) 1) Supp ose j a j > 1 has prime factorization a = p s 1 1 p s k k Then the only divisors of a are of the form p t 1 1 p t k k where 0 t i s i for i = 1 ; :::; k 2) If j a j > 1 and j b j > 1, then ( a; b ) = 1 i there is no common prime in their factorizations. Th us if there is no common prime in their factorizations, 9 m; n with ma + nb = 1 ; and also ( a 2 ; b 2 ) = 1. 3) Supp ose j a j > 1 and j b j > 1. Let f p 1 ; : : : ; p k g b e the union of the distinct primes of their factorizations. Th us a = p s 1 1 p s k k where 0 s i and b = p t 1 1 p t k k where 0 t i Let u i b e the minim um of s i and t i Then ( a; b ) = p u 1 1 p u k k : F or example (2 3 5 11 ; 2 2 5 4 7) = 2 2 5. 3 0 ) Let v i b e the maxim um of s i and t i Then c = p v 1 1 p v k k is the le ast (p ositiv e) c ommon multiple of a and b: Note that c is a m ultiple of a and b; and if n is a m ultiple of a and b; then n is a m ultiple of c Finally if a and b are p ositiv e, their least common m ultiple is c = ab= ( a; b ) ; and if in addition a and b are relativ ely prime, then their least common m ultiple is just their pro duct. 4) There is an innite n um b er of primes. (Pro of: Supp ose there w ere only a nite n um b er of primes p 1 ; p 2 ; :::; p k : Then no prime w ould divide ( p 1 p 2 p k + 1).) 5) Supp ose c is an in teger greater than 1. Then p c is rational i p c is an in teger. In particular, p 2 and p 3 are irrational. (Pro of: If p c is rational, 9 p ositiv e in tegers a and b with p c = a=b and ( a; b ) = 1. If b > 1, then it is divisible b y some prime, and since cb 2 = a 2 this prime will also app ear in the prime factorization of a This is a con tradiction and th us b = 1 and p c is an in teger.) (See the fth exercise b elo w.) Exercise Find (180,28), i.e., nd the greatest common divisor of 180 and 28, i.e., nd the p ositiv e generator of the subgroup generated b y f 180,28 g Find in tegers m and n suc h that 180 m + 28 n = (180 ; 28). Find the least common m ultiple of 180 and 28, and sho w that it is equal to (180 28) = (180 ; 28). PAGE 26 18 Bac kground Chapter 1 Exercise W e ha v e dened the greatest common divisor (gcd) and the least common m ultiple (lcm) of a pair of in tegers. No w supp ose n 2 and S = f a 1 ; a 2 ; ::; a n g is a nite collection of in tegers with j a i j > 1 for 1 i n Dene the gcd and the lcm of the elemen ts of S and dev elop their prop erties. Express the gcd and the lcm in terms of the prime factorizations of the a i When is the lcm of S equal to the pro duct a 1 a 2 a n ? Sho w that the set of all linear com binations of the elemen ts of S is a subgroup of Z and its p ositiv e generator is the gcd of the elemen ts of S Exercise Sho w that the gcd of S = f 90 ; 70 ; 42 g is 2, and nd in tegers n 1 ; n 2 ; n 3 suc h that 90 n 1 + 70 n 2 + 42 n 3 = 2 : Also nd the lcm of the elemen ts of S Exercise Sho w that if eac h of G 1 ; G 2 ; :::; G m is a subgroup of Z then G 1 \ G 2 \ \ G m is also a subgroup of Z No w let G = (90 Z ) \ (70 Z ) \ (42 Z ) and nd the p ositiv e in teger n with G = n Z Exercise Sho w that if the n th ro ot of an in teger is a rational n um b er, then it itself is an in teger. That is, supp ose c and n are in tegers greater than 1. There is a unique p ositiv e real n um b er x with x n = c Sho w that if x is rational, then it is an in teger. Th us if p is a prime, its n th ro ot is an irrational n um b er. Exercise Sho w that a p ositiv e in teger is divisible b y 3 i the sum of its digits is divisible b y 3. More generally let a = a n a n 1 : : : a 0 = a n 10 n + a n 1 10 n 1 + + a 0 where 0 a i 9. No w let b = a n + a n 1 + + a 0 and sho w that 3 divides a and b with the same remainder. Although this is a straigh tforw ard exercise in long division, it will b e more transparen t later on. In the language of the next c hapter, it sa ys that [ a ] = [ b ] in Z 3 Card T ric k Ask friends to pic k out sev en cards from a dec k and then to select one to lo ok at without sho wing it to y ou. T ak e the six cards face do wn in y our left hand and the selected card in y our righ t hand, and announce y ou will place the selected card in with the other six, but they are not to kno w where. Put y our hands b ehind y our bac k and place the selected card on top, and bring the sev en cards in fron t in y our left hand. Ask y our friends to giv e y ou a n um b er b et w een one and sev en (not allo wing one). Supp ose they sa y three. Y ou mo v e the top card to the b ottom, then the second card to the b ottom, and then y ou turn o v er the third card, lea ving it face up on top. Then rep eat the pro cess, mo ving the top t w o cards to the b ottom and turning the third card face up on top. Con tin ue un til there is only one card face do wn, and this will b e the selected card. Magic? Sta y tuned for Chapter 2, where it is sho wn that an y non-zero elemen t of Z 7 has order 7. PAGE 27 Chapter 2 GroupsGroups are the cen tral ob jects of algebra. In later c hapters w e will dene rings and mo dules and see that they are sp ecial cases of groups. Also ring homomorphisms and mo dule homomorphisms are sp ecial cases of group homomorphisms. Ev en though the denition of group is simple, it leads to a ric h and amazing theory Ev erything presen ted here is standard, except that the pro duct of groups is giv en in the additiv e notation. This is the notation used in later c hapters for the pro ducts of rings and mo dules. This c hapter and the next t w o c hapters are restricted to the most basic topics. The approac h is to do quic kly the fundamen tals of groups, rings, and matrices, and to push forw ard to the c hapter on linear algebra. This c hapter is, b y far and ab o v e, the most dicult c hapter in the b o ok, b ecause group op erations ma y b e written as addition or m ultiplication, and also the concept of coset is confusing at rst. Denition Supp ose G is a non-v oid set and : G G G is a function. is called a binary op er ation and w e will write ( a; b ) = a b or ( a; b ) = a + b Consider the follo wing prop erties. 1) If a; b; c 2 G then a ( b c ) = ( a b ) c If a; b; c 2 G then a + ( b + c ) = ( a + b ) + c: 2) 9 e = e G 2 G suc h that if a 2 G 9 0 =0 G 2 G suc h that if a 2 G e a = a e = a: 0 + a = a +0 = a 3) If a 2 G 9 b 2 G with a b = b a = e If a 2 G; 9 b 2 G with a + b = b + a = 0 ( b is written as b = a 1 ). ( b is written as b = a ). 4) If a; b 2 G then a b = b a: If a; b 2 G then a + b = b + a Denition If prop erties 1), 2), and 3) hold, ( G; ) is said to b e a gr oup If w e write ( a; b ) = a b w e sa y it is a multiplic ative group. If w e write ( a; b ) = a + b 19 PAGE 28 20 Groups Chapter 2 w e sa y it is an additive group. If in addition, prop ert y 4) holds, w e sa y the group is ab elian or c ommutative Theorem Let ( G; ) b e a m ultiplicativ e group. (i) Supp ose a; c; c 2 G Then a c = a c ) c = c: Also c a = c a ) c = c: In other w ords, if f : G G is dened b y f ( c ) = a c then f is injectiv e. Also f is bijectiv e with f 1 giv en b y f 1 ( c ) = a 1 c (ii) e is unique, i.e., if e 2 G satises 2), then e = e: In fact, if a; b 2 G then ( a b = a ) ) ( b = e ) and ( a b = b ) ) ( a = e ) : Recall that b is an iden tit y in G pro vided it is a righ t and left iden tit y for an y a in G: Ho w ev er, group structure is so rigid that if 9 a 2 G suc h that b is a righ t iden tit y for a then b = e Of course, this is just a sp ecial case of the cancellation la w in (i). (iii) Ev ery righ t in v erse is an in v erse, i.e., if a b = e then b = a 1 Also if b a = e then b = a 1 : Th us in v erses are unique. (iv) If a 2 G; then ( a 1 ) 1 = a (v) The m ultiplication a 1 a 2 a 3 = a 1 ( a 2 a 3 ) = ( a 1 a 2 ) a 3 is w ell-dened. In general, a 1 a 2 a n is w ell dened. (vi) If a; b 2 G; ( a b ) 1 = b 1 a 1 : Also ( a 1 a 2 a n ) 1 = a 1 n a 1 n 1 a 1 1 (vii) Supp ose a 2 G: Let a 0 = e and if n > 0 ; a n = a a ( n times) and a n = a 1 a 1 ( n times). If n 1 ; n 2 ; :::; n t 2 Z then a n 1 a n 2 a n t = a n 1 + + n t : Also ( a n ) m = a nm Finally if G is ab elian and a; b 2 G then ( a b ) n = a n b n Exercise W rite out the ab o v e theorem where G is an additiv e group. Note that part (vii) states that G has a scalar m ultiplication o v er Z This means that if a is in G and n is an in teger, there is dened an elemen t an in G This is so basic, that w e state it explicitly Theorem Supp ose G is an additiv e group. If a 2 G let a 0 =0 and if n > 0, let an = ( a + + a ) where the sum is n times, and a ( n ) = ( a ) + ( a ) + ( a ), PAGE 29 Chapter 2 Groups 21 whic h w e write as ( a a a ) : Then the follo wing prop erties hold in general, except the rst requires that G b e ab elian. ( a + b ) n = an + bn a ( n + m ) = an + am a ( nm ) = ( an ) m a 1 = a Note that the plus sign is used am biguously \| sometimes for addition in G and sometimes for addition in Z In the language used in Chapter 5, this theorem states that an y additiv e ab elian group is a Z -mo dule. (See page 71.) Exercise Supp ose G is a non-v oid set with a binary op eration ( a; b ) = a b whic h satises 1), 2) and [ 3 0 ) If a 2 G 9 b 2 G with a b = e ]. Sho w ( G; ) is a group, i.e., sho w b a = e: In other w ords, the group axioms are stronger than necessary If ev ery elemen t has a righ t in v erse, then ev ery elemen t has a t w o sided in v erse. Exercise Supp ose G is the set of all functions from Z to Z with m ultiplication dened b y comp osition, i.e., f g = f g Note that G satises 1) and 2) but not 3), and th us G is not a group. Sho w that f has a righ t in v erse in G i f is surjectiv e, and f has a left in v erse in G i f is injectiv e (see page 10). Also sho w that the set of all bijections from Z to Z is a group under comp osition. Examples G = R G = Q or G = Z with ( a; b ) = a + b is an additiv e ab elian group. Examples G = R 0 or G = Q 0 with ( a; b ) = ab is a m ultiplicativ e ab elian group. G = Z 0 with ( a; b ) = ab is not a group. G = R + = f r 2 R : r > 0 g with ( a; b ) = ab is a m ultiplicativ e ab elian group. Subgroups Theorem Supp ose G is a m ultiplicativ e group and H G is a non-v oid subset satisfying 1) if a; b 2 H then a b 2 H and 2) if a 2 H then a 1 2 H PAGE 30 22 Groups Chapter 2 Then e 2 H and H is a group under m ultiplication. H is called a sub gr oup of G Pro of Since H is non-v oid, 9 a 2 H By 2), a 1 2 H and so b y 1), e 2 H The asso ciativ e la w is immediate and so H is a group. Example G is a subgroup of G and e is a subgroup of G These are called the impr op er subgroups of G Example If G = Z under addition, and n 2 Z then H = n Z is a subgroup of Z By a theorem in the section on the in tegers in Chapter 1, ev ery subgroup of Z is of this form (see page 15). This is a k ey prop ert y of the in tegers. Exercises Supp ose G is a m ultiplicativ e group. 1) Let H b e the c enter of G i.e., H = f h 2 G : g h = h g for all g 2 G g Sho w H is a subgroup of G 2) Supp ose H 1 and H 2 are subgroups of G: Sho w H 1 \ H 2 is a subgroup of G 3) Supp ose H 1 and H 2 are subgroups of G with neither H 1 nor H 2 con tained in the other. Sho w H 1 [ H 2 is not a subgroup of G 4) Supp ose T is an index set and for eac h t 2 T H t is a subgroup of G Sho w \ t 2 T H t is a subgroup of G 5) F urthermore, if f H t g is a monotonic collection, then [ t 2 T H t is a subgroup of G 6) Supp ose G = f all functions f : [0 ; 1] R g Dene an addition on G b y ( f + g )( t ) = f ( t ) + g ( t ) for all t 2 [0 ; 1]. This mak es G in to an ab elian group. Let K b e the subset of G comp osed of all dieren tiable functions. Let H b e the subset of G comp osed of all con tin uous functions. What theorems in calculus sho w that H and K are subgroups of G ? What theorem sho ws that K is a subset (and th us subgroup) of H ? Order Supp ose G is a m ultiplicativ e group. If G has an innite n um b er of PAGE 31 Chapter 2 Groups 23 elemen ts, w e sa y that o ( G ), the or der of G is innite. If G has n elemen ts, then o ( G ) = n Supp ose a 2 G and H = f a i : i 2 Z g H is an ab elian subgroup of G called the sub gr oup gener ate d by a W e dene the or der of the element a to b e the order of H i.e., the order of the subgroup generated b y a Let f : Z H b e the surjectiv e function dened b y f ( m ) = a m Note that f ( k + l ) = f ( k ) f ( l ) where the addition is in Z and the m ultiplication is in the group H W e come no w to the rst real theorem in group theory It sa ys that the elemen t a has nite order i f is not injectiv e, and in this case, the order of a is the smallest p ositiv e in teger n with a n = e Theorem Supp ose a is an elemen t of a m ultiplicativ e group G and H = f a i : i 2 Z g If 9 distinct in tegers i and j with a i = a j then a has some nite order n In this case H has n distinct elemen ts, H = f a 0 ; a 1 ; : : : ; a n 1 g and a m = e i n j m In particular, the order of a is the smallest p ositiv e in teger n with a n = e and f 1 ( e ) = n Z Pro of Supp ose j < i and a i = a j Then a i j = e and th us 9 a smallest p ositiv e in teger n with a n = e This implies that the elemen ts of f a 0 ; a 1 ; :::; a n 1 g are distinct, and w e m ust sho w they are all of H If m 2 Z the Euclidean algorithm states that 9 in tegers q and r with 0 r < n and m = nq + r Th us a m = a nq a r = a r and so H = f a 0 ; a 1 ; :::; a n 1 g and a m = e i n j m Later in this c hapter w e will see that f is a homomorphism from an additiv e group to a m ultiplicativ e group and that, in additiv e notation, H is isomorphic to Z or Z n Exercise W rite out this theorem for G an additiv e group. T o b egin, supp ose a is an elemen t of an additiv e group G and H = f ai : i 2 Z g Exercise Sho w that if G is a nite group of ev en order, then G has an o dd n um b er of elemen ts of order 2. Note that e is the only elemen t of order 1. Denition A group G is cyclic if 9 an elemen t of G whic h generates G Theorem If G is cyclic and H is a subgroup of G then H is cyclic. Pro of Supp ose G = f a i : i 2 Z g is a cyclic group and H is a subgroup of G If H = e then H is cyclic, so supp ose H 6 = e No w there is a smallest p ositiv e in teger m with a m 2 H If t is an in teger with a t 2 H then b y the Euclidean algorithm, m divides t and th us a m generates H Note that in the case G has nite order n i.e., G = f a 0 ; a 1 ; : : : ; a n 1 g then a n = e 2 H and th us the p ositiv e in teger m divides n In either case, w e ha v e a clear picture of the subgroups of G: Also note that this theorem w as pro v ed on page 15 for the additiv e group Z PAGE 32 24 Groups Chapter 2 Cosets Supp ose H is a subgroup of a group G It will b e sho wn b elo w that H partitions G in to righ t cosets. It also partitions G in to left cosets, and in general these partitions are distinct. Theorem If H is a subgroup of a m ultiplicativ e group G then a b dened b y a b i a b 1 2 H is an equiv alence relation. If a 2 G cl ( a ) = f b 2 G : a b g = f h a : h 2 H g = H a: Note that a b 1 2 H i b a 1 2 H If H is a subgroup of an additiv e group G then a b dened b y a b i ( a b ) 2 H is an equiv alence relation. If a 2 G cl ( a ) = f b 2 G : a b g = f h + a : h 2 H g = H + a: Note that ( a b ) 2 H i ( b a ) 2 H Denition These equiv alence classes are called right c osets If the relation is dened b y a b i b 1 a 2 H then the equiv alence classes are cl ( a ) = aH and they are called left c osets H is a left and righ t coset. If G is ab elian, there is no distinction b et w een righ t and left cosets. Note that b 1 a 2 H i a 1 b 2 H In the theorem ab o v e, H is used to dene an equiv alence relation on G and th us a partition of G W e no w do the same thing a dieren t w a y W e dene the righ t cosets directly and sho w they form a partition of G: Y ou migh t nd this easier. Theorem Supp ose H is a subgroup of a m ultiplicativ e group G If a 2 G dene the righ t coset con taining a to b e H a = f h a : h 2 H g : Then the follo wing hold. 1) H a = H i a 2 H 2) If b 2 H a; then H b = H a; i.e., if h 2 H then H ( h a ) = ( H h ) a = H a 3) If H c \ H a 6 = ; ; then H c = H a 4) The righ t cosets form a partition of G i.e., eac h a in G b elongs to one and only one righ t coset. 5) Elemen ts a and b b elong to the same righ t coset i a b 1 2 H i b a 1 2 H Pro of There is no b etter w a y to dev elop facilit y with cosets than to pro v e this theorem. Also write this theorem for G an additiv e group. Theorem Supp ose H is a subgroup of a m ultiplicativ e group G PAGE 33 Chapter 2 Groups 25 1) An y t w o righ t cosets ha v e the same n um b er of elemen ts. That is, if a; b 2 G f : H a H b dened b y f ( h a ) = h b is a bijection. Also an y t w o left cosets ha v e the same n um b er of elemen ts. Since H is a righ t and left coset, an y t w o cosets ha v e the same n um b er of elemen ts. 2) G has the same n um b er of righ t cosets as left cosets. The function F dened b y F ( H a ) = a 1 H is a bijection from the collection of righ t cosets to the left cosets. The n um b er of righ t (or left) cosets is called the index of H in G 3) If G is nite, o ( H ) (index of H ) = o ( G ) and so o ( H ) j o ( G ) : In other w ords, o ( G ) =o ( H ) = the n um b er of righ t cosets = the n um b er of left cosets. 4) If G is nite, and a 2 G then o ( a ) j o ( G ) : (Pro of: The order of a is the order of the subgroup generated b y a and b y 3) this divides the order of G .) 5) If G has prime order, then G is cyclic, and an y elemen t (except e ) is a generator. (Pro of: Supp ose o ( G ) = p and a 2 G a 6 = e: Then o ( a ) j p and th us o ( a ) = p: ) 6) If o ( G ) = n and a 2 G then a n = e (Pro of: a o ( a ) = e and n = o ( a ) ( o ( G ) =o ( a )) : ) Exercisesi) Supp ose G is a cyclic group of order 4, G = f e; a; a 2 ; a 3 g with a 4 = e Find the order of eac h elemen t of G: Find all the subgroups of G ii) Supp ose G is the additiv e group Z and H = 3 Z : Find the cosets of H iii) Think of a circle as the in terv al [0 ; 1] with end p oin ts iden tied. Supp ose G = R under addition and H = Z : Sho w that the collection of all the cosets of H can b e though t of as a circle. iv) Let G = R 2 under addition, and H b e the subgroup dened b y H = f ( a; 2 a ) : a 2 R g : Find the cosets of H : (See the last exercise on p 5.) Normal Subgroups W e w ould lik e to mak e a group out of the collection of cosets of a subgroup H In PAGE 34 26 Groups Chapter 2 general, there is no natural w a y to do that. Ho w ev er, it is easy to do in case H is a normal subgroup, whic h is describ ed b elo w. Theorem If H is a subgroup of a group G then the follo wing are equiv alen t. 1) If a 2 G then aH a 1 = H 2) If a 2 G then aH a 1 H 3) If a 2 G then aH = H a 4) Ev ery righ t coset is a left coset, i.e., if a 2 G 9 b 2 G with H a = bH Pro of 1) ) 2) is ob vious. Supp ose 2) is true and sho w 3). W e ha v e ( aH a 1 ) a H a so aH H a Also a ( a 1 H a ) aH so H a aH : Th us aH = H a 3) ) 4) is ob vious. Supp ose 4) is true and sho w 3). H a = bH con tains a so bH = aH b ecause a coset is an equiv alence class. Th us aH = H a Finally supp ose 3) is true and sho w 1). Multiply aH = H a on the righ t b y a 1 Denition If H satises an y of the four conditions ab o v e, then H is said to b e a normal subgroup of G: (This concept go es bac k to Ev ariste Galois in 1831.) Note F or an y group G G and e are normal subgroups. If G is an ab elian group, then ev ery subgroup of G is normal. Exercise Sho w that if H is a subgroup of G with index 2, then H is normal. Exercise Sho w the in tersection of a collection of normal subgroups of G is a normal subgroup of G: Sho w the union of a monotonic collection of normal subgroups of G is a normal subgroup of G Exercise Let A R 2 b e the square with v ertices ( 1 ; 1) ; (1 ; 1) ; (1 ; 1), and ( 1 ; 1), and G b e the collection of all \isometries" of A on to itself. These are bijections of A on to itself whic h preserv e distance and angles, i.e., whic h preserv e dot pro duct. Sho w that with m ultiplication dened as comp osition, G is a m ultiplicativ e group. Sho w that G has four rotations, t w o rerections ab out the axes, and t w o rerections ab out the diagonals, for a total of eigh t elemen ts. Sho w the collection of rotations is a cyclic subgroup of order four whic h is a normal subgroup of G Sho w that the rerection ab out the x -axis together with the iden tit y form a cyclic subgroup of order t w o whic h is not a normal subgroup of G Find the four righ t cosets of this subgroup. Finally nd the four left cosets of this subgroup. PAGE 35 Chapter 2 Groups 27 Quotien t Groups Supp ose N is a normal subgroup of G and C and D are cosets. W e wish to dene a coset E whic h is the pro duct of C and D If c 2 C and d 2 D dene E to b e the coset con taining c d i.e., E = N ( c d ). The coset E do es not dep end up on the c hoice of c and d This is made precise in the next theorem, whic h is quite easy Theorem Supp ose G is a m ultiplicativ e group, N is a normal subgroup, and G= N is the collection of all cosets. Then ( N a ) ( N b ) = N ( a b ) is a w ell dened m ultiplication (binary op eration) on G= N and with this m ultiplication, G= N is a group. Its iden tit y is N and ( N a ) 1 = ( N a 1 ). F urthermore, if G is nite, o ( G= N ) = o ( G ) =o ( N ). Pro of Multiplication of elemen ts in G= N is m ultiplication of subsets in G ( N a ) ( N b ) = N ( aN ) b = N ( N a ) b = N ( a b ) : Once m ultiplication is w ell dened, the group axioms are immediate. Exercise W rite out the ab o v e theorem for G an additiv e group. In the additiv e ab elian group R = Z ; determine those elemen ts of nite order. Example Supp ose G = Z under +, n > 1, and N = n Z Z n the gr oup of inte gers mo d n is dened b y Z n = Z =n Z : If a is an in teger, the coset a + n Z is denoted b y [ a ]. Note that [ a ] + [ b ] = [ a + b ], [ a ] = [ a ], and [ a ] = [ a + nl ] for an y in teger l An y additiv e ab elian group has a scalar m ultiplication o v er Z and in this case it is just [ a ] m = [ am ]. Note that [ a ] = [ r ] where r is the remainder of a divided b y n and th us the distinct elemen ts of Z n are [0] ; [1] ; :::; [ n 1]. Also Z n is cyclic b ecause eac h of [1] and [ 1] = [ n 1] is a generator. W e already kno w that if p is a prime, an y non-zero elemen t of Z p is a generator, b ecause Z p has p elemen ts. Theorem If n > 1 and a is an y in teger, then [ a ] is a generator of Z n i ( a; n ) = 1. Pro of The elemen t [ a ] is a generator i the subgroup generated b y [ a ] con tains [1] i 9 an in teger k suc h that [ a ] k = [1] i 9 in tegers k and l suc h that ak + nl = 1. Exercise Sho w that a p ositiv e in teger is divisible b y 3 i the sum of its digits is divisible b y 3. Note that [10] = [1] in Z 3 : (See the fth exercise on page 18.) Homomorphisms Homomorphisms are functions b et w een groups that comm ute with the group operations. It follo ws that they honor iden tities and in v erses. In this section w e list PAGE 36 28 Groups Chapter 2 the basic prop erties. Prop erties 11), 12), and 13) sho w the connections b et w een coset groups and homomorphisms, and should b e considered as the cornerstones of abstract algebra. As alw a ys, the studen t should rewrite the material in additiv e notation. Denition If G and G are m ultiplicativ e groups, a function f : G G is a homomorphism if, for all a; b 2 G f ( a b ) = f ( a ) f ( b ). On the left side, the group op eration is in G while on the righ t side it is in G The kernel of f is dened b y k er ( f ) = f 1 ( e ) = f a 2 G : f ( a ) = e g In other w ords, the k ernel is the set of solutions to the equation f ( x ) = e (If G is an additiv e group, k er ( f ) = f 1 (0 ) : ) Examples The constan t map f : G G dened b y f ( a ) = e is a homomorphism. If H is a subgroup of G the inclusion i : H G is a homomorphism. The function f : Z Z dened b y f ( t ) = 2 t is a homomorphism of additiv e groups, while the function dened b y f ( t ) = t + 2 is not a homomorphism. The function h : Z R 0 dened b y h ( t ) = 2 t is a homomorphism from an additiv e group to a m ultiplicativ e group. W e no w catalog the basic prop erties of homomorphisms. These will b e helpful later on in the study of ring homomorphisms and mo dule homomorphisms. Theorem Supp ose G and G are groups and f : G G is a homomorphism. 1) f ( e ) = e 2) f ( a 1 ) = f ( a ) 1 : The rst in v erse is in G and the second is in G 3) f is injectiv e k er( f ) = e 4) If H is a subgroup of G f ( H ) is a subgroup of G: In particular, image( f ) is a subgroup of G 5) If H is a subgroup of G f 1 ( H ) is a subgroup of G: F urthermore, if H is normal in G then f 1 ( H ) is normal in G 6) The k ernel of f is a normal subgroup of G 7) If g 2 G f 1 ( g ) is v oid or is a coset of k er( f ), i.e., if f ( g ) = g then f 1 ( g ) = N g where N = k er( f ). In other w ords, if the equation f ( x ) = g has a PAGE 37 Chapter 2 Groups 29 solution, then the set of all solutions is a coset of N = k er( f ). This is a k ey fact whic h is used routinely in topics suc h as systems of equations and linear dieren tial equations. 8) The comp osition of homomorphisms is a homomorphism, i.e., if h : G = G is a homomorphism, then h f : G = G is a homomorphism. 9) If f : G G is a bijection, then the function f 1 : G G is a homomorphism. In this case, f is called an isomorphism and w e write G G: In the case G = G f is also called an automorphism 10) Isomorphisms preserv e all algebraic prop erties. F or example, if f is an isomorphism and H G is a subset, then H is a subgroup of G i f ( H ) is a subgroup of G H is normal in G i f ( H ) is normal in G G is cyclic i G is cyclic, etc. Of course, this is somewhat of a cop-out, b ecause an algebraic prop ert y is one that, b y denition, is preserv ed under isomorphisms. 11) Supp ose H is a normal subgroup of G Then : G G=H dened b y ( a ) = H a is a surjectiv e homomorphism with k ernel H : F urthermore, if f : G G is a surjectiv e homomorphism with k ernel H then G=H G (see b elo w). 12) Supp ose H is a normal subgroup of G If H k er ( f ), then f : G=H G dened b y f ( H a ) = f ( a ) is a w ell-dened homomorphism making the follo wing diagram comm ute. G G G=H f ? > f Th us dening a homomorphism on a quotien t group is the same as dening a homomorphism on the n umerator whic h sends the denominator to e The image of f is the image of f and the k ernel of f is k er( f ) =H Th us if H = k er ( f ), f is injectiv e, and th us G=H image( f ). 13) Giv en an y group homomorphism f domain( f ) = k er ( f ) image( f ) : This is the fundamen tal connection b et w een quotien t groups and homomorphisms. PAGE 38 30 Groups Chapter 2 14) Supp ose K is a group. Then K is an innite cycle group i K is isomorphic to the in tegers under addition, i.e., K Z K is a cyclic group of order n i K Z n Pro of of 14) Supp ose G = K is generated b y some elemen t a Then f : Z K dened b y f ( m ) = a m is a homomorphism from an additiv e group to a m ultiplicativ e group. If o ( a ) is innite, f is an isomorphism. If o ( a ) = n k er( f ) = n Z and f : Z n K is an isomorphism. Exercise If a is an elemen t of a group G there is alw a ys a homomorphism from Z to G whic h sends 1 to a When is there a homomorphism from Z n to G whic h sends [1] to a ? What are the homomorphisms from Z 2 to Z 6 ? What are the homomorphisms from Z 4 to Z 8 ? Exercise Supp ose G is a group and g is an elemen t of G g 6 = e 1) Under what conditions on g is there a homomorphism f : Z 7 G with f ([1]) = g ? 2) Under what conditions on g is there a homomorphism f : Z 15 G with f ([1]) = g ? 3) Under what conditions on G is there an injectiv e homomorphism f : Z 15 G ? 4) Under what conditions on G is there a surjectiv e homomorphism f : Z 15 G ? Exercise W e kno w ev ery nite group of prime order is cyclic and th us ab elian. Sho w that ev ery group of order four is ab elian. Exercise Let G = f h : [0 ; 1] R : h has an innite n um b er of deriv ativ es g Then G is a group under addition. Dene f : G G b y f ( h ) = dh dt = h 0 Sho w f is a homomorphism and nd its k ernel and image. Let g : [0 ; 1] R b e dened b y g ( t ) = t 3 3 t + 4 : Find f 1 ( g ) and sho w it is a coset of k er ( f ). Exercise Let G b e as ab o v e and g 2 G Dene f : G G b y f ( h ) = h 00 + 5 h 0 + 6 t 2 h Then f is a group homomorphism and the dieren tial equation h 00 + 5 h 0 + 6 t 2 h = g has a solution i g lies in the image of f No w supp ose this equation has a solution and S G is the set of all solutions. F or whic h subgroup H of G is S an H -coset? Chapter 3 Rings 39 generally if a 1 ; a 2 ; :::; a n are units, then their pro duct is a unit with ( a 1 a 2 a n ) 1 = a 1 n a 1 n 1 a 1 1 The set of all units of R forms a m ultiplicativ e group denoted b y R : Finally if a is a unit, ( a ) is a unit and ( a ) 1 = ( a 1 ). In order for a to b e a unit, it m ust ha v e a t w o-sided in v erse. It suces to require a left in v erse and a righ t in v erse, as sho wn in the next theorem. Theorem Supp ose a 2 R and 9 elemen ts b and c with b a = a c = 1 Then b = c and so a is a unit with a 1 = b = c Pro of b = b 1 = b ( a c ) = ( b a ) c = 1 c = c Corollary In v erses are unique. Domains and Fields In order to dene these t w o t yp es of rings, w e rst consider the concept of zero divisor. Denition Supp ose R is a comm utativ e ring. An elemen t a 2 R is called a zer o divisor pro vided it is non-zero and 9 a non-zero elemen t b with a b = 0 Note that if a is a unit, it cannot b e a zero divisor. Theorem Supp ose R is a comm utativ e ring and a 2 ( R 0 ) is not a zero divisor. Then ( a b = a c ) ) b = c: In other w ords, m ultiplication b y a is an injectiv e map from R to R : It is surjectiv e i a is a unit. Denition A domain (or inte gr al domain ) is a comm utativ e ring suc h that, if a 6 = 0 a is not a zero divisor. A eld is a comm utativ e ring suc h that, if a 6 = 0 a is a unit. In other w ords, R is a eld if it is comm utativ e and its non-zero elemen ts form a group under m ultiplication. Theorem A eld is a domain. A nite domain is a eld. Pro of A eld is a domain b ecause a unit cannot b e a zero divisor. Supp ose R is a nite domain and a 6 = 0 Then f : R R dened b y f ( b ) = a b is injectiv e and, b y the pigeonhole principle, f is surjectiv e. Th us a is a unit and so R is a eld. Chapter 3 Rings 41 [ a ][ b ] = [0] ; and th us [ a ] is a zero divisor and 1) is false. Exercise List the units and their in v erses for Z 7 and Z 12 Sho w that ( Z 7 ) is a cyclic group but ( Z 12 ) is not. Sho w that in Z 12 the equation x 2 = 1 has four solutions. Finally sho w that if R is a domain, x 2 = 1 can ha v e at most t w o solutions in R (see the rst theorem on page 46). Subrings Supp ose S is a subset of a ring R The statemen t that S is a subring of R means that S is a subgroup of the group R 1 2 S and ( a; b 2 S ) a b 2 S ). Then clearly S is a ring and has the same m ultiplicativ e iden tit y as R Note that Z is a subring of Q Q is a subring of R and R is a subring of C Subrings do not pla y a role analogous to subgroups. That role is pla y ed b y ideals, and an ideal is nev er a subring (unless it is the en tire ring). Note that if S is a subring of R and s 2 S then s ma y b e a unit in R but not in S Note also that Z and Z n ha v e no prop er subrings, and th us o ccup y a sp ecial place in ring theory as w ell as in group theory Ideals and Quotien t Rings Ideals in ring theory pla y a role analagous to normal subgroups in group theory Denition A subset I of a ring R is a 8><>: leftrigh t 2 sided 9>=>; ideal pro vided it is a subgroup of the additiv e group R and if a 2 R and b 2 I then 8><>: a b 2 I b a 2 I a b and b a 2 I 9>=>; The w ord \ideal means \2-sided ideal". Of course, if R is comm utativ e, ev ery righ t or left ideal is an ideal. Theorem Supp ose R is a ring. 1) R and 0 are ideals of R : These are called the impr op er ideals. 2) If f I t g t 2 T is a collection of righ t (left, 2-sided) ideals of R then \ t 2 T I t is a righ t (left, 2-sided) ideal of R : (See page 22.) PAGE 50 42 Rings Chapter 3 3) F urthermore, if the collection is monotonic, then [ t 2 T I t is a righ t (left, 2-sided) ideal of R 4) If a 2 R ; I = aR is a righ t ideal. Th us if R is comm utativ e, aR is an ideal, called a princip al ide al Th us ev ery subgroup of Z is a principal ideal, b ecause it is of the form n Z 5) If R is a comm utativ e ring and I R is an ideal, then the follo wing are equiv alen t. i) I = R ii) I con tains some unit u iii) I con tains 1 Exercise Supp ose R is a comm utativ e ring. Sho w that R is a eld i R con tains no prop er ideals. The follo wing theorem is just an observ ation, but it is in some sense the b eginning of ring theory Theorem Supp ose R is a ring and I R is an ideal, I 6 = R Since I is a normal subgroup of the additiv e group R R =I is an additiv e ab elian group. Multiplication of cosets dened b y ( a + I ) ( b + I ) = ( ab + I ) is w ell-dened and mak es R =I a ring. Pro of ( a + I ) ( b + I ) = a b + aI + I b + I I a b + I Th us m ultiplication is w ell dened, and the ring axioms are easily v eried. The m ultiplicativ e iden tit y is (1 + I ). Observ ation If R = Z n > 1, and I = n Z ; the ring structure on Z n = Z =n Z is the same as the one previously dened. Homomorphisms Denition Supp ose R and R are rings. A function f : R R is a ring homomorphism pro vided 1) f is a group homomorphism 2) f (1 R ) = 1 R and 3) if a; b 2 R then f ( a b ) = f ( a ) f ( b ). (On the left, m ultiplication PAGE 51 Chapter 3 Rings 43 is in R while on the righ t m ultiplication is in R .) The kernel of f is the k ernel of f considered as a group homomorphism, namely k er ( f ) = f 1 (0 ). Here is a list of the basic prop erties of ring homomorphisms. Muc h of this w ork has already b een done b y the theorem in group theory on page 28. Theorem Supp ose eac h of R and R is a ring. 1) The iden tit y map I R : R R is a ring homomorphism. 2) The zero map from R to R is not a ring homomorphism (b ecause it do es not send 1 R to 1 R ). 3) The comp osition of ring homomorphisms is a ring homomorphism. 4) If f : R R is a bijection whic h is a ring homomorphism, then f 1 : R R is a ring homomorphism. Suc h an f is called a ring isomorphism In the case R = R f is also called a ring automorphism 5) The image of a ring homomorphism is a subring of the range. 6) The k ernel of a ring homomorphism is an ideal of the domain. In fact, if f : R R is a homomorphism and I R is an ideal, then f 1 ( I ) is an ideal of R 7) Supp ose I is an ideal of R I 6 = R and : R R =I is the natural pro jection, ( a ) = ( a + I ) : Then is a surjectiv e ring homomorphism with k ernel I F urthermore, if f : R R is a surjectiv e ring homomorphism with k ernel I then R =I R (see b elo w). 8) F rom no w on the w ord \homomorphism" means \ring homomorphism". Supp ose f : R R is a homomorphism and I is an ideal of R I 6 = R If I k er ( f ) ; then f : R =I R dened b y f ( a + I ) = f ( a ) PAGE 52 44 Rings Chapter 3 is a w ell-dened homomorphism making the follo wing diagram comm ute. R R R =I f ? > f Th us dening a homomorphism on a quotien t ring is the same as dening a homomorphism on the n umerator whic h sends the denominator to zero. The image of f is the image of f and the k ernel of f is k er( f ) =I : Th us if I = k er ( f ), f is injectiv e, and so R =I image ( f ). Pro of W e kno w all this on the group lev el, and it is only necessary to c hec k that f is a ring homomorphism, whic h is ob vious. 9) Giv en an y ring homomorphism f ; domain( f ) = k er ( f ) image( f ). Exercise Find a ring R with a prop er ideal I and an elemen t b suc h that b is not a unit in R but ( b + I ) is a unit in R =I Exercise Sho w that if u is a unit in a ring R then conjugation b y u is an automorphism on R : That is, sho w that f : R R dened b y f ( a ) = u 1 a u is a ring homomorphism whic h is an isomorphism. Exercise Supp ose T is a non-v oid set, R is a ring, and R T is the collection of all functions f : T R Dene addition and m ultiplication on R T p oin t-wise. This means if f and g are functions from T to R then ( f + g )( t ) = f ( t ) + g ( t ) and ( f g )( t ) = f ( t ) g ( t ). Sho w that under these op erations R T is a ring. Supp ose S is a non-v oid set and : S T is a function. If f : T R is a function, dene a function ( f ) : S R b y ( f ) = f : Sho w : R T R S is a ring homomorphism. Exercise No w consider the case T = [0 ; 1] and R = R Let A R [0 ; 1] b e the collection of all C 1 functions, i.e., A = f f : [0 ; 1] R : f has an innite n um b er of deriv ativ es g Sho w A is a ring. Notice that m uc h of the w ork has b een done in the previous exercise. It is only necessary to sho w that A is a subring of the ring R [0 ; 1] PAGE 53 Chapter 3 Rings 45 P olynomial Rings In calculus, w e consider real functions f whic h are p olynomials, f ( x ) = a 0 + a 1 x + + a n x n The sum and pro duct of p olynomials are again p olynomials, and it is easy to see that the collection of p olynomial functions forms a comm utativ e ring. W e can do the same thing formally in a purely algebraic setting. Denition Supp ose R is a comm utativ e ring and x is a \v ariable" or \sym b ol". The p olynomial ring R [ x ] is the collection of all p olynomials f = a 0 + a 1 x + + a n x n where a i 2 R Under the ob vious addition and m ultiplication, R [ x ] is a comm utativ e ring. The de gr e e of a non-zero p olynomial f is the largest in teger n suc h that a n 6 = 0 and is denoted b y n = deg ( f ) : If the top term a n = 1 then f is said to b e monic T o b e more formal, think of a p olynomial a 0 + a 1 x + as an innite sequence ( a 0 ; a 1 ; ::: ) suc h that eac h a i 2 R and only a nite n um b er are non-zero. Then ( a 0 ; a 1 ; ::: ) + ( b 0 ; b 1 ; ::: ) = ( a 0 + b 0 ; a 1 + b 1 ; ::: ) and ( a 0 ; a 1 ; ::: ) ( b 0 ; b 1 ; ::: ) = ( a 0 b 0 ; a 0 b 1 + a 1 b 0 ; a 0 b 2 + a 1 b 1 + a 2 b 0 ; ::: ). Note that on the righ t, the ring m ultiplication a b is written simply as ab as is often done for con v enience. Theorem If R is a domain, R [ x ] is also a domain. Pro of Supp ose f and g are non-zero p olynomials. Then deg ( f )+deg ( g ) = deg ( f g ) and th us f g is not 0 Another w a y to pro v e this theorem is to lo ok at the b ottom terms instead of the top terms. Let a i x i and b j x j b e the rst non-zero terms of f and g : Then a i b j x i + j is the rst non-zero term of f g Theorem (The Division Algorithm) Supp ose R is a comm utativ e ring, f 2 R [ x ] has degree 1 and its top co ecien t is a unit in R (If R is a eld, the top co ecien t of f will alw a ys b e a unit.) Then for an y g 2 R [ x ], 9 h; r 2 R [ x ] suc h that g = f h + r with r = 0 or deg ( r ) < deg ( f ). Pro of This theorem states the existence and uniqueness of p olynomials h and r W e outline the pro of of existence and lea v e uniqueness as an exercise. Supp ose f = a 0 + a 1 x + + a m x m where m 1 and a m is a unit in R F or an y g with deg( g ) < m set h = 0 and r = g F or the general case, the idea is to divide f in to g un til the remainder has degree less than m The pro of is b y induction on the degree of g Supp ose n m and the result holds for an y p olynomial of degree less than 50 Rings Chapter 3 The Chinese Remainder Theorem The natural map from Z to Z m Z n is a group homomorphism and also a ring homomorphism. If m and n are relativ ely prime, this map is surjectiv e with k ernel mn Z and th us Z mn and Z m Z n are isomorphic as groups and as rings. The next theorem is a classical generalization of this. (See exercise three on page 35.) Theorem Supp ose n 1 ; :::; n t are in tegers, eac h n i > 1, and ( n i ; n j ) = 1 for all i 6 = j Let f i : Z Z n i b e dened b y f i ( a ) = [ a ]. (Note that the brac k et sym b ol is used am biguously .) Then the ring homomorphism f = ( f 1 ; ::; f t ) : Z Z n 1 Z n t is surjectiv e. F urthermore, the k ernel of f is n Z where n = n 1 n 2 n t Th us Z n and Z n 1 Z n t are isomorphic as rings, and th us also as groups. Pro of W e wish to sho w that the order of f (1) is n and th us f (1) is a group generator, and th us f is surjectiv e. The elemen t f (1) m = ([1] ; ::; [1]) m = ([ m ] ; ::; [ m ]) is zero i m is a m ultiple of eac h of n 1 ; ::; n t Since their least common m ultiple is n the order of f (1) is n: (See the fourth exercise on page 36 for the case t = 3.) Exercise Sho w that if a is an in teger and p is a prime, then [ a ] = [ a p ] in Z p (F ermat's Little Theorem). Use this and the Chinese Remainder Theorem to sho w that if b is a p ositiv e in teger, it has the same last digit as b 5 Characteristic The follo wing theorem is just an observ ation, but it sho ws that in ring theory the ring of in tegers is a \cornerstone". Theorem If R is a ring, there is one and only one ring homomorphism f : Z R It is giv en b y f ( m ) = m 1 = m Th us the subgroup of R generated b y 1 is a subring of R isomorphic to Z or isomorphic to Z n for some p ositiv e in teger n Denition Supp ose R is a ring and f : Z R is the natural ring homomorphism f ( m ) = m 1 = m The non-negativ e in teger n with k er ( f ) = n Z is called the char acteristic of R : Th us f is injectiv e i R has c haracteristic 0 i 1 has innite order. If f is not injectiv e, the c haracteristic of R is the order of 1 It is an in teresting fact that, if R is a domain, all the non-zero elemen ts of R ha v e the same order. (See page 23 for the denition of order.) PAGE 59 Chapter 3 Rings 51 Theorem Supp ose R is a domain. If R has c haracteristic 0, then eac h non-zero a 2 R has innite order. If R has nite c haracteristic n then n is a prime and eac h non-zero a 2 R has order n Pro of Supp ose R has c haracteristic 0, a is a non-zero elemen t of R and m is a p ositiv e in teger. Then ma = m a cannot b e 0 b ecause m ; a 6 = 0 and R is a domain. Th us o ( a ) = 1 No w supp ose R has c haracteristic n Then R con tains Z n as a subring, and th us Z n is a domain and n is a prime. If a is a non-zero elemen t of R na = n a = 0 a = 0 and th us o ( a ) j n and th us o ( a ) = n Exercise Sho w that if F is a eld of c haracteristic 0, F con tains Q as a subring. That is, sho w that the injectiv e homomorphism f : Z F extends to an injectiv e homomorphism f : Q F Bo olean Rings This section is not used elsewhere in this b o ok. Ho w ev er it ts easily here, and is included for reference. Denition A ring R is a Bo ole an ring if for eac h a 2 R a 2 = a i.e., eac h elemen t of R is an idemp oten t. Theorem Supp ose R is a Bo olean ring. 1) R has c haracteristic 2. If a 2 R ; 2 a = a + a = 0 ; and so a = a Pro of ( a + a ) = ( a + a ) 2 = a 2 + 2 a 2 + a 2 = 4 a: Th us 2 a = 0 : 2) R is comm utativ e. Pro of ( a + b ) = ( a + b ) 2 = a 2 + ( a b ) + ( b a ) + b 2 = a + ( a b ) ( b a ) + b: Th us a b = b a 3) If R is a domain, R Z 2 Pro of Supp ose a 6 = 0 : Then a (1 a ) = 0 and so a = 1 4) The image of a Bo olean ring is a Bo olean ring. That is, if I is an ideal of R with I 6 = R then ev ery elemen t of R =I is idemp oten t and th us R =I is a Bo olean ring. It follo ws from 3) that R =I is a domain i R =I is a eld i R =I Z 2 : (In the language of Chapter 6, I is a prime ideal i I is a maximal ideal i R =I Z 2 ). PAGE 60 52 Rings Chapter 3 Supp ose X is a non-v oid set. If a is a subset of X let a 0 = ( X a ) b e a complemen t of a in X No w supp ose R is a non-v oid collection of subsets of X Consider the follo wing prop erties whic h the collection R ma y p ossess. 1) a 2 R ) a 0 2 R 2) a; b 2 R ) ( a \ b ) 2 R : 3) a; b 2 R ) ( a [ b ) 2 R : 4) ; 2 R and X 2 R Theorem If 1) and 2) are satised, then 3) and 4) are satised. In this case, R is called a Bo ole an algebr a of sets Pro of Supp ose 1) and 2) are true, and a; b 2 R Then a [ b = ( a 0 \ b 0 ) 0 b elongs to R and so 3) is true. Since R is non-v oid, it con tains some elemen t a Then ; = a \ a 0 and X = a [ a 0 b elong to R and so 4) is true. Theorem Supp ose R is a Bo olean algebra of sets. Dene an addition on R b y a + b = ( a [ b ) ( a \ b ). Under this addition, R is an ab elian group with 0 = ; and a = a Dene a m ultiplication on R b y a b = a \ b Under this m ultiplication R b ecomes a Bo olean ring with 1 = X Exercise Let X = f 1 ; 2 ; :::; n g and let R b e the Bo olean ring of all subsets of X Note that o ( R ) = 2 n Dene f i : R Z 2 b y f i ( a ) = [1] i i 2 a Sho w eac h f i is a homomorphism and th us f = ( f 1 ; :::; f n ) : R Z 2 Z 2 Z 2 is a ring homomorphism. Sho w f is an isomorphism. (See exercises 1) and 4) on page 12.) Exercise Use the last exercise on page 49 to sho w that an y nite Bo olean ring is isomorphic to Z 2 Z 2 Z 2 ; and th us also to the Bo olean ring of subsets ab o v e. Note Supp ose R is a Bo olean ring. It is a classical theorem that 9 a Bo olean algebra of sets whose Bo olean ring is isomorphic to R So let's just supp ose R is a Bo olean algebra of sets whic h is a Bo olean ring with addition and m ultiplication dened as ab o v e. No w dene a b = a [ b and a ^ b = a \ b These op erations cup and cap are asso ciativ e, comm utativ e, ha v e iden tit y elemen ts, and eac h distributes o v er the other. With these t w o op erations (along with complemen t), R is called a Bo ole an algebr a R is not a group under cup or cap. An yw a y it is a classical fact that, if y ou ha v e a Bo olean ring (algebra), y ou ha v e a Bo olean algebra (ring). The adv an tage of the algebra is that it is symmetric in cup and cap. The adv an tage of the ring viewp oin t is that y ou can dra w from the ric h theory of comm utativ e rings. PAGE 61 Chapter 4 Matrices and Matrix Rings W e rst consider matrices in full generalit y i.e., o v er an arbitrary ring R Ho w ev er, after the rst few pages, it will b e assumed that R is comm utativ e. The topics, suc h as in v ertible matrices, transp ose, elemen tary matrices, systems of equations, and determinan t, are all classical. The highligh t of the c hapter is the theorem that a square matrix is a unit in the matrix ring i its determinan t is a unit in the ring. This c hapter concludes with the theorem that similar matrices ha v e the same determinan t, trace, and c haracteristic p olynomial. This will b e used in the next c hapter to sho w that an endomorphism on a nitely generated v ector space has a w ell-dened determinan t, trace, and c haracteristic p olynomial. Denition Supp ose R is a ring and m and n are p ositiv e in tegers. Let R m;n b e the collection of all m n matrices A = ( a i;j ) = 0BB@ a 1 ; 1 : : : a 1 ;n ... ... a m; 1 : : : a m;n 1CCA where eac h en try a i;j 2 R : A matrix ma y b e view ed as m n -dimensional ro w v ectors or as n m -dimensional column v ectors. A matrix is said to b e squar e if it has the same n um b er of ro ws as columns. Square matrices are so imp ortan t that they ha v e a sp ecial notation, R n = R n;n R n is dened to b e the additiv e ab elian group R R R T o emphasize that R n do es not ha v e a ring structure, w e use the \sum" notation, R n = R R R Our con v en tion is to write elemen ts of R n as column v ectors, i.e., to iden tify R n with R n; 1 If the elemen ts of R n are written as ro w v ectors, R n is iden tied with R 1 ;n 53 56 Matrices Chapter 4 T ransp ose Notation F or the r emainder of this chapter on matric es, supp ose R is a c ommutative ring Of course, for n > 1 ; R n is non-comm utativ e. T ransp ose is a function from R m;n to R n;m If A 2 R m;n ; A t 2 R n;m is the matrix whose ( i; j ) term is the ( j; i ) term of A So ro w i (column i ) of A b ecomes column i (ro w i ) of A t : If A is an n -dimensional ro w v ector, then A t is an n -dimensional column v ector. If A is a square matrix, A t is also square. Theorem 1) ( A t ) t = A 2) ( A + B ) t = A t + B t 3) If c 2 R ; ( Ac ) t = A t c 4) ( AB ) t = B t A t 5) If A 2 R n then A is in v ertible i A t is in v ertible. In this case ( A 1 ) t = ( A t ) 1 Pro of of 5) Supp ose A is in v ertible. Then I = I t = ( AA 1 ) t = ( A 1 ) t A t Exercise Characterize those in v ertible matrices A 2 R 2 whic h ha v e A 1 = A t Sho w that they form a subgroup of GL 2 ( R ). T riangular Matrices If A 2 R n then A is upp er (lower) triangular pro vided a i;j = 0 for all i > j (all j > i ). A is strictly upp er (lower) triangular pro vided a i;j = 0 for all i j (all j i ). A is diagonal if it is upp er and lo w er triangular, i.e., a i;j = 0 for all i 6 = j: Note that if A is upp er (lo w er) triangular, then A t is lo w er (upp er) triangular. Theorem If A 2 R n is strictly upp er (or lo w er) triangular, then A n = 0 Pro of The w a y to understand this is just m ultiply it out for n = 2 and n = 3 : The geometry of this theorem will b ecome transparen t later in Chapter 5 when the matrix A denes an R -mo dule endomorphism on R n (see page 93). Denition If T is an y ring, an elemen t t 2 T is said to b e nilp otent pro vided 9 n suc h that t n = 0. In this case, (1 t ) is a unit with in v erse 1 + t + t 2 + + t n 1 Th us if T = R n and B is a nilp oten t matrix, I B is in v ertible. Chapter 4 Matrices 63 W rite out the determinan t of A expanding b y the rst column and also expanding b y the second ro w. Theorem If A is an upp er or lo w er triangular matrix, j A j is the pro duct of the diagonal elemen ts. If A is an elemen tary matrix of t yp e 2, j A j = 1. If A is an elemen tary matrix of t yp e 3, j A j = 1. Pro of W e will pro v e the rst statemen t for upp er triangular matrices. If A 2 R 2 is an upp er triangular matrix, then its determinan t is the pro duct of the diagonal elemen ts. Supp ose n > 2 and the theorem is true for matrices in R n 1 Supp ose A 2 R n is upp er triangular. The result follo ws b y expanding b y the rst column. An elemen tary matrix of t yp e 3 is a sp ecial t yp e of upp er or lo w er triangular matrix, so its determinan t is 1. An elemen tary matrix of t yp e 2 is obtained from the iden tit y matrix b y in terc hanging t w o ro ws or columns, and th us has determinan t 1. Theorem (Determinan t b y blo c ks) Supp ose A 2 R n ; B 2 R n;m and D 2 R m Then the determinan t of A B O D is j A jj D j Pro of Expand b y the rst column and use induction on n The follo wing remark able theorem tak es some w ork to pro v e. W e assume it here without pro of. (F or the pro of, see page 130 of the App endix.) Theorem The determinan t of the pro duct is the pro duct of the determinan ts, i.e., if A; B 2 R n ; j AB j = j A jj B j Th us j AB j = j B A j and if C is in v ertible, j C 1 AC j = j AC C 1 j = j A j Corollary If A is a unit in R n ; then j A j is a unit in R and j A 1 j = j A j 1 Pro of 1 = j I j = j AA 1 j = j A jj A 1 j : One of the ma jor goals of this c hapter is to pro v e the con v erse of the preceding corollary Classical adjoin t Supp ose R is a comm utativ e ring and A 2 R n The classic al adjoint of A is ( C i;j ) t i.e., the matrix whose ( j; i ) term is the ( i; j ) cofactor. Before PAGE 72 64 Matrices Chapter 4 w e consider the general case, let's examine 2 2 matrices. If A = a b c d then ( C i;j ) = d c b a and so ( C i;j ) t = d b c a Then A ( C i;j ) t = ( C i;j ) t A = j A j 0 0 j A j = j A j I Th us if j A j is a unit in R ; A is in v ertible and A 1 = j A j 1 ( C i;j ) t In particular, if j A j = 1 ; A 1 = d b c a Here is the general case. Theorem If R is comm utativ e and A 2 R n then A ( C i;j ) t = ( C i;j ) t A = j A j I Pro of W e m ust sho w that the diagonal elemen ts of the pro duct A ( C i;j ) t are all j A j and the other elemen ts are 0 The ( s; s ) term is the dot pro duct of ro w s of A with ro w s of ( C i;j ) and is th us j A j (computed b y expansion b y ro w s ). F or s 6 = t the ( s; t ) term is the dot pro duct of ro w s of A with ro w t of ( C i;j ). Since this is the determinan t of a matrix with ro w s = ro w t the ( s; t ) term is 0 The pro of that ( C i;j ) t A = j A j I is similar and is left as an exercise. W e are no w ready for one of the most b eautiful and useful theorems in all of mathematics.Theorem Supp ose R is a comm utativ e ring and A 2 R n Then A is a unit in R n i j A j is a unit in R (Th us if R is a eld, A is in v ertible i j A j 6 = 0 .) If A is in v ertible, then A 1 = j A j 1 ( C i;j ) t Th us if j A j = 1 ; A 1 = ( C i;j ) t the classical adjoin t of A Pro of This follo ws immediately from the preceding theorem. Exercise Sho w that an y righ t in v erse of A is also a left in v erse. That is, supp ose A; B 2 R n and AB = I : Sho w A is in v ertible with A 1 = B and th us B A = I Similarit y Supp ose A; B 2 R n B is said to b e similar to A if 9 an in v ertible C 2 R n suc h that B = C 1 AC ; i.e., B is similar to A i B is a c onjugate of A Theorem B is similar to B PAGE 73 Chapter 4 Matrices 65 B is similar to A i A is similar to B If D is similar to B and B is similar to A then D is similar to A \Similarit y" is an equiv alence relation on R n Pro of This is a go o d exercise using the denition. Theorem Supp ose A and B are similar. Then j A j = j B j and th us A is in v ertible i B is in v ertible. Pro of Supp ose B = C 1 AC : Then j B j = j C 1 AC j = j AC C 1 j = j A j T race Supp ose A = ( a i;j ) 2 R n Then the tr ac e is dened b y trace ( A ) = a 1 ; 1 + a 2 ; 2 + + a n;n : That is, the trace of A is the sum of its diagonal terms. One of the most useful prop erties of trace is trace ( AB ) = trace( B A ) whenev er AB and B A are dened. F or example, supp ose A = ( a 1 ; a 2 ; :::; a n ) and B = ( b 1 ; b 2 ; :::; b n ) t Then AB is the scalar a 1 b 1 + + a n b n while B A is the n n matrix ( b i a j ). Note that trace ( AB ) = trace( B A ) : Here is the theorem in full generalit y Theorem Supp ose A 2 R m;n and B 2 R n;m Then AB and B A are square matrices with trace ( AB ) = trace( B A ). Pro of This pro of in v olv es a c hange in the order of summation. By denition, trace( AB ) = X 1 i m a i; 1 b 1 ;i + + a i;n b n;i = X 1 i m 1 j n a i;j b j;i = X 1 j n b j; 1 a 1 ;j + + b j;m a m;j = trace( B A ). Theorem If A; B 2 R n ; trace( A + B ) = trace( A ) + trace( B ) and trace( AB ) = trace( B A ). Pro of The rst part of the theorem is immediate, and the second part is a sp ecial case of the previous theorem. Theorem If A and B are similar, then trace( A ) = trace( B ). Pro of trace ( B ) = trace( C 1 AC ) = trace( AC C 1 ) = trace ( A ). Chapter 5 Linear Algebra The exalted p osition held b y linear algebra is based up on the sub ject's ubiquitous utilit y and ease of application. The basic theory is dev elop ed here in full generalit y i.e., mo dules are dened o v er an arbitrary ring R and not just o v er a eld. The elemen tary facts ab out cosets, quotien ts, and homomorphisms follo w the same pattern as in the c hapters on groups and rings. W e giv e a simple pro of that if R is a comm utativ e ring and f : R n R n is a surjectiv e R -mo dule homomorphism, then f is an isomorphism. This sho ws that nitely generated free R -mo dules ha v e a w ell dened dimension, and simplies some of the dev elopmen t of linear algebra. It is in this c hapter that the concepts ab out functions, solutions of equations, matrices, and generating sets come together in one unied theory After the general theory w e restrict our atten tion to v ector spaces, i.e., mo dules o v er a eld. The k ey theorem is that an y v ector space V has a free basis, and th us if V is nitely generated, it has a w ell dened dimension, and incredible as it ma y seem, this single in teger determines V up to isomorphism. Also an y endomorphism f : V V ma y b e represen ted b y a matrix, and an y c hange of basis corresp onds to conjugation of that matrix. One of the goals in linear algebra is to select a basis so that the matrix represen ting f has a simple form. F or example, if f is not injectiv e, then f ma y b e represen ted b y a matrix whose rst column is zero. As another example, if f is nilp oten t, then f ma y b e represen ted b y a strictly upp er triangular matrix. The theorem on Jordan canonical form is not pro v ed in this c hapter, and should not b e considered part of this c hapter. It is stated here in full generalit y only for reference and completeness. The pro of is giv en in the App endix. This c hapter concludes with the study of real inner pro duct spaces, and with the b eautiful theory relating orthogonal matrices and symmetric matrices. 67 Chapter 5 Linear Algebra 83 Pro of Supp ose U M is a nite generating set and S is a basis. Then an y elemen t of U is a nite linear com bination of elemen ts of S and th us S is nite. Theorem Supp ose M is a f.g. mo dule. If M has a basis, that basis is nite and an y other basis has the same n um b er of elemen ts. This n um b er is denoted b y dim( M ), the dimension of M : (By con v en tion, 0 is a free mo dule of dimension 0.) Pro of By the previous lemma, an y basis for M m ust b e nite. M has a basis of n elemen ts i M R n : The result follo ws b ecause R n R m i n = m Change of Basis Before c hanging basis, w e recall what a basis is. Previously w e dened generating, indep endence, and basis for sequences, not for collections. F or the concept of generating it matters not whether y ou use sequences or collections, but for indep endence and basis, y ou m ust use sequences. Consider the columns of the real matrix A = 2 3 2 1 4 1 If w e consider the column v ectors of A as a collection, there are only t w o of them, y et w e certainly don't wish to sa y the columns of A form a basis for R 2 In a set or collection, there is no concept of rep etition. In order to mak e sense, w e m ust consider the columns of A as an ordered triple of v ectors, and this sequence is dep enden t. In the denition of basis on page 78, basis is dened for sequences, not for sets or collections. Tw o sequences cannot b egin to b e equal unless they ha v e the same index set. Here w e follo w the classical con v en tion that an index set with n elemen ts will b e f 1 ; 2 ; ::; n g and th us a basis for M with n elemen ts is a sequence S = f u 1 ; ::; u n g or if y ou wish, S = ( u 1 ; ::; u n ) 2 M n Supp ose M is an R -mo dule with a basis of n elemen ts. Recall there is a bijection : Hom R ( R n ; M ) M n dened b y ( h ) = ( h ( e 1 ) ; ::; h ( e n )) : No w h : R n M is an isomorphism i ( h ) is a basis for M Summary The p oin t of all this is that selecting a basis of n elemen ts for M is the same as selecting an isomorphism from R n to M and from this viewp oin t, c hange of basis can b e displa y ed b y the diagram b elo w. Endomorphisms on R n are represen ted b y square matrices, and th us ha v e a determinan t and trace. No w supp ose M is a f.g. free mo dule and f : M M is a homomorphism. In order to represen t f b y a matrix, w e m ust select a basis for M (i.e., an isomorphism with R n ). W e will sho w that this matrix is w ell dened up to similarit y and th us the determinan t, trace, and c haracteristic p olynomial of f are w ell-dened. 90 Linear Algebra Chapter 5 com bination of those v ectors. Th us the image of f is the submo dule of F m generated b y the columns of A and its dimension is the column rank of A This dimension is the same as the dimension of the image of g f h : F n F m where h is an y automorphism on F n and g is an y automorphism on F m This pro v es the theorem for column rank. The theorem for ro w rank follo ws using transp ose. Theorem If A 2 F m;n the ro w rank and the column rank of A are equal. This n um b er is called the r ank of A and is min f m; n g Pro of By the theorem ab o v e, elemen tary ro w and column op erations c hange neither the ro w rank nor the column rank. By ro w and column op erations, A ma y b e c hanged to a matrix H where h 1 ; 1 = = h t;t = 1 and all other en tries are 0 (see the rst exercise on page 59). Th us ro w rank = t = column rank. Exercise Supp ose A has rank t Sho w that it is p ossible to select t ro ws and t columns of A suc h that the determined t t matrix is in v ertible. Sho w that the rank of A is the largest in teger t suc h that this is p ossible. Exercise Supp ose A 2 F m;n has rank t What is the dimension of the solution set of AX = 0 ? Denition If N and M are nite dimensional v ector spaces and f : N M is a linear transformation, the r ank of f is the dimension of the image of f If f : F n F m is giv en b y a matrix A then the rank of f is the same as the rank of the matrix A Geometric In terpretation of Determinan t Supp ose V R n is some nice subset. F or example, if n = 2, V migh t b e the in terior of a square or circle. There is a concept of the n -dimensional v olume of V F or n = 1, it is length. F or n = 2, it is area, and for n = 3 it is \ordinary v olume". Supp ose A 2 R n and f : R n R n is the homomorphism giv en b y A The v olume of V do es not c hange under translation, i.e., V and V + p ha v e the same v olume. Th us f ( V ) and f ( V + p ) = f ( V ) + f ( p ) ha v e the same v olume. In street language, the next theorem sa ys that \ f m ultiplies v olume b y the absolute v alue of its determinan t". Theorem The n -dimensional v olume of f ( V ) is j A j (the n -dimensional v olume of V ). Th us if j A j = 1 ; f preserv es v olume. 102 Linear Algebra Chapter 5 Theorem Supp ose eac h of U and V is an n -dimensional IPS. Then 9 an isometry f : U V : In particular, U is isometric to R n with its standard inner pro duct. Pro of There exist orthonormal bases f u 1 ; ::; u n g for U and f v 1 ; ::; v n g for V By the rst theorem on page 79, there exists a homomorphism f : U V with f ( u i ) = v i ; and b y the previous theorem, f is an isometry Exercise Let f : R 3 R b e the homomorphism dened b y the matrix (2,1,3). Find a linear transformation h : R 2 R 3 whic h giv es an isometry from R 2 to k er ( f ). Orthogonal Matrices As noted earlier, linear algebra is not so m uc h the study of v ector spaces as it is the study of endomorphisms. W e no w wish to study isometries from R n to R n W e kno w from a theorem on page 90 that an endomorphism preserv es v olume i its determinan t is 1. Isometries preserv e inner pro duct, and th us preserv e angle and distance, and so certainly preserv e v olume. Theorem Supp ose A 2 R n and f : R n R n is the homomorphism dened b y f ( B ) = AB Then the follo wing are equiv alen t. 1) The columns of A form an orthonormal basis for R n ; i.e., A t A = I 2) The ro ws of A form an orthonormal basis for R n ; i.e., AA t = I 3) f is an isometry Pro of A left in v erse of a matrix is also a righ t in v erse (see the exercise on page 64). Th us 1) and 2) are equiv alen t b ecause eac h of them sa ys A is in v ertible with A 1 = A t No w f e 1 ; ::; e n g is the canonical orthonormal basis for R n and f ( e i ) is column i of A: Th us b y the previous section, 1) and 3) are equiv alen t. Denition If A 2 R n satises these three conditions, A is said to b e ortho gonal The set of all suc h A is denoted b y O ( n ), and is called the ortho gonal gr oup Theorem 1) If A is orthogonal, j A j = 1. 2) If A is orthogonal, A 1 is orthogonal. If A and C are orthogonal, AC is orthogonal. Th us O ( n ) is a m ultiplicativ e subgroup of GL n ( R ). 106 Linear Algebra Chapter 5 Exercise Let A = 2 2 2 2 Find an orthogonal C suc h that C 1 AC is diagonal. Do the same for A = 2 1 1 2 Exercise Supp ose A; D 2 R n are symmetric. Under what conditions are A and D similar? Sho w that, if A and D are similar, 9 an orthogonal C suc h that D = C 1 AC Exercise Supp ose V is an n -dimensional real v ector space. W e kno w that V is isomorphic to R n Supp ose f and g are isomorphisms from V to R n and A is a subset of V Sho w that f ( A ) is an op en subset of R n i g ( A ) is an op en subset of R n This sho ws that V an algebraic ob ject, has a go d-giv en top ology Of course, if V has an inner pro duct, it automatically has a metric, and this metric will determine that same top ology Finally supp ose V and W are nite-dimensional real v ector spaces and h : V W is a linear transformation. Sho w that h is con tin uous. Exercise Dene E : C n C n b y E ( A ) = e A = I + A + (1 = 2!) A 2 + This series con v erges and th us E is a w ell dened function. If AB = B A then E ( A + B ) = E ( A ) E ( B ). Since A and A comm ute, I = E (0 ) = E ( A A ) = E ( A ) E ( A ), and th us E ( A ) is in v ertible with E ( A ) 1 = E ( A ). F urthermore E ( A t ) = E ( A ) t and if C is in v ertible, E ( C 1 AC ) = C 1 E ( A ) C No w use the results of this section to pro v e the statemen ts b elo w. (F or part 1, assume the Jordan form, i.e., assume an y A 2 C n is similar to a lo w er triangular matrix.) 1) If A 2 C n then j e A j = e trace ( A ) Th us if A 2 R n j e A j = 1 i trace( A ) = 0. 2) 9 a non-zero matrix N 2 R 2 with e N = I 3) If N 2 R n is symmetric, then e N = I i N = 0 4) If A 2 R n and A t = A then e A 2 O ( n ). PAGE 115 Chapter 6 App endix The v e previous c hapters w ere designed for a y ear undergraduate course in algebra. In this app endix, enough material is added to form a basic rst y ear graduate course. Tw o of the main goals are to c haracterize nitely generated ab elian groups and to pro v e the Jordan canonical form. The st yle is the same as b efore, i.e., ev erything is righ t do wn to the n ub. The organization is mostly a linearly ordered sequence except for the last t w o sections on determinan ts and dual spaces. These are indep enden t sections added on at the end. Supp ose R is a comm utativ e ring. An R -mo dule M is said to b e cyclic if it can b e generated b y one elemen t, i.e., M R =I where I is an ideal of R The basic theorem of this c hapter is that if R is a Euclidean domain and M is a nitely generated R -mo dule, then M is the sum of cyclic mo dules. Th us if M is torsion free, it is a free R -mo dule. Since Z is a Euclidean domain, nitely generated ab elian groups are the sums of cyclic groups { one of the jew els of abstract algebra. No w supp ose F is a eld and V is a nitely generated F -mo dule. If T : V V is a linear transformation, then V b ecomes an F [ x ]-mo dule b y dening v x = T ( v ). No w F [ x ] is a Euclidean domain and so V F [ x ] is the sum of cyclic mo dules. This classical and v ery p o w erful tec hnique allo ws an easy pro of of the canonical forms. There is a basis for V so that the matrix represen ting T is in Rational canonical form. If the c haracteristic p olynomial of T factors in to the pro duct of linear p olynomials, then there is a basis for V so that the matrix represen ting T is in Jordan canonical form. This alw a ys holds if F = C A matrix in Jordan form is a lo w er triangular matrix with the eigen v alues of T displa y ed on the diagonal, so this is a p o w erful concept. In the c hapter on matrices, it is stated without pro of that the determinan t of the pro duct is the pro duct of the determinan ts. A pro of of this, whic h dep ends up on the classication of certain t yp es of alternating m ultilinear forms, is giv en in this c hapter. The nal section giv es the fundamen tals of dual spaces. 107 PAGE 116 108 App endix Chapter 6 The Chinese Remainder Theorem On page 50 in the c hapter on rings, the Chinese Remainder Theorem w as pro v ed for the ring of in tegers. In this section this classical topic is presen ted in full generalit y Surprisingly the theorem holds ev en for non-comm utativ e rings. Denition Supp ose R is a ring and A 1 ; A 2 ; :::; A m are ideals of R Then the sum A 1 + A 2 + + A m is the set of all a 1 + a 2 + + a m with a i 2 A i The pr o duct A 1 A 2 A m is the set of all nite sums of elemen ts a 1 a 2 a m with a i 2 A i Note that the sum and pro duct of ideals are ideals and A 1 A 2 A m ( A 1 \ A 2 \ \ A m ). Denition Ideals A and B of R are said to b e c omaximal if A + B = R Theorem If A and B are ideals of a ring R then the follo wing are equiv alen t. 1) A and B are comaximal. 2) 9 a 2 A and b 2 B with a + b = 1 3) ( A ) = R =B where : R R =B is the pro jection. Theorem If A 1 ; A 2 ; :::; A m and B are ideals of R with A i and B comaximal for eac h i then A 1 A 2 A m and B are comaximal. Th us A 1 \ A 2 \ \ A m and B are comaximal. Pro of Consider : R R =B Then ( A 1 A 2 A m ) = ( A 1 ) ( A 2 ) ( A m ) = ( R =B )( R =B ) ( R =B ) = R =B Chinese Remainder Theorem Supp ose A 1 ; A 2 ; :::; A n are pairwise comaximal ideals of R with eac h A i 6 = R Then the natural map : R R = A 1 R = A 2 R = A n is a surjectiv e ring homomorphism with k ernel A 1 \ A 2 \ \ A n Pro of There exists a i 2 A i and b i 2 A 1 A 2 A i 1 A i +1 A n with a i + b i = 1 Note that ( b i ) = (0 ; ::; 0 ; 1 i ; 0 ; ::; 0). If ( r 1 + A 1 ; r 2 + A 2 ; :::; r n + A n ) is an elemen t of the range, it is the image of r 1 b 1 + r 2 b 2 + + r n b n = r 1 (1 a 1 ) + r 2 (1 a 2 ) + + r n (1 a n ). Theorem If R is comm utativ e and A 1 ; A 2 ; :::; A n are pairwise comaximal ideals of R then A 1 A 2 A n = A 1 \ A 2 \ \ A n Pro of for n = 2. Sho w A 1 \ A 2 A 1 A 2 9 a 1 2 A 1 and a 2 2 A 2 with a 1 + a 2 = 1 If c 2 A 1 \ A 2 then c = c ( a 1 + a 2 ) 2 A 1 A 2 PAGE 117 Chapter 6 App endix 109 Prime and Maximal Ideals and UFD s In the rst c hapter on bac kground material, it w as sho wn that Z is a unique factorization domain. Here it will b e sho wn that this prop ert y holds for an y principle ideal domain. Later on it will b e sho wn that ev ery Euclidean domain is a principle ideal domain. Th us ev ery Euclidean domain is a unique factorization domain. Denition Supp ose R is a comm utativ e ring and I R is an ideal. I is prime means I 6 = R and if a; b 2 R ha v e ab 2 I then a or b 2 I I is maximal means I 6 = R and there are no ideals prop erly b et w een I and R Theorem 0 is a prime ideal of R i R is 0 is a maximal ideal of R i R is Theorem Supp ose J R is an ideal, J 6 = R J is a prime ideal i R =J is J is a maximal ideal i R =J is Corollary Maximal ideals are prime. Pro of Ev ery eld is a domain. Theorem If a 2 R is not a unit, then 9 a maximal ideal I of R with a 2 I Pro of This is a classical application of the Hausdor Maximalit y Principle. Consider f J : J is an ideal of R con taining a with J 6 = R g This collection con tains a maximal monotonic collection f V t g t 2 T The ideal V = [ t 2 T V t do es not con tain 1 and th us is not equal to R Therefore V is equal to some V t and is a maximal ideal con taining a Note T o prop erly appreciate this pro of, the studen t should w ork the exercise in group theory at the end of this section (see page 114). Denition Supp ose R is a domain and a; b 2 R Then w e sa y a b i there exists a unit u with au = b: Note that is an equiv alence relation. If a b then a PAGE 118 110 App endix Chapter 6 and b are said to b e asso ciates Examples If R is a domain, the asso ciates of 1 are the units of R while the only asso ciate of 0 is 0 itself. If n 2 Z is not zero, then its asso ciates are n and n If F is a eld and g 2 F [ x ] is a non-zero p olynomial, then the asso ciates of g are all cg where c is a non-zero constan t. The follo wing theorem is elemen tary but it sho ws ho w asso ciates t in to the sc heme of things. An elemen t a divides b ( a j b ) if 9 c 2 R with ac = b Theorem Supp ose R is a domain and a; b 2 ( R 0 ). Then the follo wing are equiv alen t. 1) a b 2) a j b and b j a 3) aR = bR P arts 1) and 3) ab o v e sho w there is a bijection from the asso ciate classes of R to the principal ideals of R Th us if R is a PID, there is a bijection from the asso ciate classes of R to the ideals of R : If an elemen t of a domain generates a non-zero prime ideal, it is called a prime elemen t. Denition Supp ose R is a domain and a 2 R is a non-zero non-unit. 1) a is irr e ducible if it do es not factor, i.e., a = bc ) b or c is a unit. 2) a is prime if it generates a prime ideal, i.e., a j bc ) a j b or a j c Note If a is a prime and a j c 1 c 2 c n then a j c i for some i This follo ws from the denition and induction on n: If eac h c j is irreducible, then a c i for some i Note If a b then a is irreducible (prime) i b is irreducible (prime). In other w ords, if a is irreducible (prime) and u is a unit, then au is irreducible (prime). Note a is prime ) a is irreducible. This is immediate from the denitions. Theorem F actorization in to primes is unique up to order and asso ciates, i.e., if d = b 1 b 2 b n = c 1 c 2 c m with eac h b i and eac h c i prime, then n = m and for some p erm utation of the indices, b i and c ( i ) are asso ciates for ev ery i Note also 9 a unit u and primes p 1 ; p 2 ; : : : ; p t where no t w o are asso ciates and du = p s 1 1 p s 2 2 p s t t PAGE 119 Chapter 6 App endix 111 Pro of This follo ws from the notes ab o v e. Denition R is a factorization domain (FD) means that R is a domain and if a is a non-zero non-unit elemen t of R then a factors in to a nite pro duct of irreducibles. Denition R is a unique factorization domain (UFD) means R is a FD in whic h factorization is unique (up to order and asso ciates). Theorem If R is a UFD and a is a non-zero non-unit of R then a is irreducible a is prime. Th us in a UFD, elemen ts factor as the pro duct of primes. Pro of Supp ose R is a UFD, a is an irreducible elemen t of R and a j bc If either b or c is a unit or is zero, then a divides one of them, so supp ose eac h of b and c is a non-zero non-unit elemen t of R There exists an elemen t d with ad = bc Eac h of b and c factors as the pro duct of irreducibles and the pro duct of these pro ducts is the factorization of bc It follo ws from the uniqueness of the factorization of ad = bc that one of these irreducibles is an asso ciate of a and th us a j b or a j c Therefore the elemen t a is a prime. Theorem Supp ose R is a FD. Then the follo wing are equiv alen t. 1) R is a UFD. 2) Ev ery irreducible elemen t of R is prime, i.e., a irreducible a is prime. Pro of W e already kno w 1) ) 2). P art 2) ) 1) b ecause factorization in to primes is alw a ys unique. This is a rev ealing and useful theorem. If R is a FD, then R is a UFD i eac h irreducible elemen t generates a prime ideal. F ortunately principal ideal domains ha v e this prop ert y as seen in the next theorem. Theorem Supp ose R is a PID and a 2 R is non-zero non-unit. Then the follo wing are equiv alen t. 1) aR is a maximal ideal. 2) aR is a prime ideal, i.e., a is a prime elemen t. 3) a is irreducible. Pro of Ev ery maximal ideal is a prime ideal, so 1) ) 2). Ev ery prime elemen t is an irreducible elemen t, so 2) ) 3). No w supp ose a is irreducible and sho w aR is a maximal ideal. If I is an ideal con taining aR 9 b 2 R with I = bR Since b divides a the elemen t b is a unit or an asso ciate of a: This means I = R or I = aR PAGE 120 112 App endix Chapter 6 Our goal is to pro v e that a PID is a UFD. Using the t w o theorems ab o v e, it only remains to sho w that a PID is a FD. The pro of will not require that ideals b e principally generated, but only that they b e nitely generated. This turns out to b e equiv alen t to the prop ert y that an y collection of ideals has a \maximal" elemen t. W e shall see b elo w that this is a useful concept whic h ts naturally in to the study of unique factorization domains. Theorem Supp ose R is a comm utativ e ring. Then the follo wing are equiv alen t. 1) If I R is an ideal, 9 a nite set f a 1 ; a 2 ; :::; a n g R suc h that I = a 1 R + a 2 R + + a n R ; i.e., eac h ideal of R is nitely generated. 2) An y non-v oid collection of ideals of R con tains an ideal I whic h is maximal in the collection. This means if J is an ideal in the collection with J I then J = I (The ideal I is maximal only in the sense describ ed. It need not con tain all the ideals of the collection, nor need it b e a maximal ideal of the ring R .) 3) If I 1 I 2 I 3 ::: is a monotonic sequence of ideals, 9 t 0 1 suc h that I t = I t 0 for all t t 0 Pro of Supp ose 1) is true and sho w 3). The ideal I = I 1 [ I 2 [ : : : is nitely generated and 9 t 0 1 suc h that I t 0 con tains those generators. Th us 3) is true. No w supp ose 2) is true and sho w 1). Let I b e an ideal of R and consider the collection of all nitely generated ideals con tained in I By 2) there is a maximal one, and it m ust b e I itself, and th us 1) is true. W e no w ha v e 2) ) 1) ) 3), so supp ose 2) is false and sho w 3) is false. So there is a collection of ideals of R suc h that an y ideal in the collection is prop erly con tained in another ideal of the collection. Th us it is p ossible to construct a sequence of ideals I 1 I 2 I 3 : : : with eac h prop erly con tained in the next, and therefore 3) is false. (Actually this construction requires the Hausdor Maximalit y Principle or some form of the Axiom of Choice, but w e slide o v er that.) Denition If R satises these prop erties, R is said to b e No etherian or it is said to satisfy the asc ending chain c ondition This prop ert y is satised b y man y of the classical rings in mathematics. Ha ving three denitions mak es this prop ert y useful and easy to use. F or example, see the next theorem. Theorem A No etherian domain is a FD. In particular, a PID is a FD. Pro of Supp ose there is a non-zero non-unit elemen t that do es not factor as the nite pro duct of irreducibles. Consider all ideals dR where d do es not factor. Since R is No etherian, 9 a maximal one cR : The elemen t c m ust b e reducible, i.e., c = ab where neither a nor b is a unit. Eac h of aR and bR prop erly con tains cR and so eac h PAGE 121 Chapter 6 App endix 113 of a and b factors as a nite pro duct of irreducibles. This giv es a nite factorization of c in to irreducibles, whic h is a con tradiction. Corollary A PID is a UFD. So Z is a UFD and if F is a eld, F [ x ] is a UFD. Y ou see the basic structure of UFD s is quite easy It tak es more w ork to pro v e the follo wing theorems, whic h are stated here only for reference. Theorem If R is a UFD then R [ x 1 ; :::; x n ] is a UFD. Th us if F is a eld, F [ x 1 ; :::; x n ] is a UFD. (This theorem go es all the w a y bac k to Gauss.) If R is a PID, then the formal p o w er series R [[ x 1 ; :::; x n ]] is a UFD. Th us if F is a eld, F [[ x 1 ; :::; x n ]] is a UFD. (There is a UFD R where R [[ x ]] is not a UFD. See page 566 of Commutative A lgebr a b y N. Bourbaki.) Theorem Germs of analytic functions on C n form a UFD. Pro of See Theorem 6.6.2 of A n Intr o duction to Complex A nalysis in Sever al V ariables b y L. H ormander. Theorem Supp ose R is a comm utativ e ring. Then R is No etherian ) R [ x 1 ; :::; x n ] and R [[ x 1 ; :::; x n ]] are No etherian. (This is the famous Hilb ert Basis The or em .) Theorem If R is No etherian and I R is a prop er ideal, then R =I is No etherian. (This follo ws immediately from the denition. This and the previous theorem sho w that No etherian is a ubiquitous prop ert y in ring theory .) Domains With Non-unique F actorizations Next are presen ted t w o of the standard examples of No etherian domains that are not unique factorization domains. Exercise Let R = Z ( p 5) = f n + m p 5 : n; m 2 Z g Sho w that R is a subring of R whic h is not a UFD. In particular 2 2 = (1 p 5) ( 1 p 5 ) are t w o distinct irreducible factorizations of 4. Sho w R is isomorphic to Z [ x ] = ( x 2 5), where ( x 2 5) represen ts the ideal ( x 2 5) Z [ x ], and R = (2) is isomorphic to Z 2 [ x ] = ( x 2 [5]) = Z 2 [ x ] = ( x 2 + [1]) ; whic h is not a domain. Chapter 6 App endix 115 Denition Supp ose f : A B and g : B C are R -mo dule homomorphisms. The statemen t that 0 A f B g C 0 is a short exact se quenc e (s.e.s) means f is injectiv e, g is surjectiv e and f ( A ) = k er ( g ). The canonical split s.e.s. is A A C C where f = i 1 and g = 2 : A short exact sequence is said to split if 9 an isomorphism B A C suc h that the follo wing diagram comm utes. 0 A B C 0 A C f g i 1 2 Z Z Z Z Z Z ~ > ? W e no w restate the previous theorem in this terminology Theorem 1.1 A short exact sequence 0 A B C 0 splits i f ( A ) is a summand of B i B C has a splitting map. If C is a free R -mo dule, there is a splitting map and th us the sequence splits. Pro of W e kno w from the previous theorem f ( A ) is a summand of B i B C has a splitting map. Sho wing these prop erties are equiv alen t to the splitting of the sequence is a go o d exercise in the art of diagram c hasing. No w supp ose C has a free basis T C and g : B C is surjectiv e. There exists a function h : T B suc h that g h ( c ) = c for eac h c 2 T The function h extends to a homomorphism from C to B whic h is a righ t in v erse of g Theorem 2 If R is a domain, then the follo wing are equiv alen t. 1) R is a PID. 2) Ev ery submo dule of R R is a free R -mo dule of dimension 1. This theorem restates the ring prop ert y of PID as a mo dule prop ert y Although this theorem is transparen t, 1) ) 2) is a precursor to the follo wing classical result. Theorem 3 If R is a PID and A R n is a submo dule, then A is a free R -mo dule of dimension n: Th us subgroups of Z n are free Z -mo dules of dimension n Pro of F rom the previous theorem w e kno w this is true for n = 1. Supp ose n > 1 and the theorem is true for submo dules of R n 1 Supp ose A R n is a submo dule. Chapter 6 App endix 117 Theorem 1 If R is a Euclidean domain, then R is a PID and th us a UFD. Pro of If I is a non-zero ideal, then 9 b 2 I 0 satisfying ( b ) ( a ) 8 a 2 I 0 Then b generates I b ecause if a 2 I 0 9 q ; r with a = bq + r No w r 2 I and r 6 = 0 ) ( r ) < ( b ) whic h is imp ossible. Th us r = 0 and a 2 bR so I = bR Theorem 2 If R is a Euclidean domain and a; b 2 R 0 then (1 ) is the smallest in teger in the image of a is a unit in R i ( a ) = (1 ). a and b are asso ciates ) ( a ) = ( b ). Pro of This is a go o d exercise. Ho w ev er it is unnecessary for Theorem 3 b elo w. The follo wing remark able theorem is the foundation for the results of this section. Theorem 3 If R is a Euclidean domain and ( a i;j ) 2 R n;t is a non-zero matrix, then b y elemen tary ro w and column op erations ( a i;j ) can b e transformed to 0BBBBBBBBBBB@ d 1 0 0 0 d 2 ... . d m 0 0 0 1CCCCCCCCCCCA where eac h d i 6 = 0 ; and d i j d i +1 for 1 i < m Also d 1 generates the ideal of R generated b y the en tries of ( a i;j ). Pro of Let I R b e the ideal generated b y the elemen ts of the matrix A = ( a i;j ). If E 2 R n then the ideal J generated b y the elemen ts of E A has J I If E is in v ertible, then J = I In the same manner, if E 2 R t is in v ertible and J is the ideal generated b y the elemen ts of AE then J = I This means that ro w and column op erations on A do not c hange the ideal I Since R is a PID, there is an elemen t d 1 with I = d 1 R and this will turn out to b e the d 1 displa y ed in the theorem. The matrix ( a i;j ) has at least one non-zero elemen t d with ( d ) a mimin um. Ho w ev er, ro w and column op erations on ( a i;j ) ma y pro duce elemen ts with smaller Chapter 6 App endix 119 U ( a i;j ) V = 0BBBBBBBBB@ d 1 0 0 0 d 2 0 ... 0 . d t 0 0 1CCCCCCCCCA with d i j d i +1 Since c hanging the isomorphisms R t A and B R n corresp onds to c hanging the bases f v 1 ; v 2 ; :::; v t g and f w 1 ; w 2 ; :::; w n g ; the theorem follo ws. Theorem 5 If R is a Euclidean domain and M is a nitely generated R -mo dule, then M R =d 1 R =d 2 R =d t R m where eac h d i 6 = 0 and d i j d i +1 for 1 i < t Pro of By h yp othesis 9 a nitely generated free mo dule B and a surjectiv e homomorphism B M 0. Let A b e the k ernel, so 0 A B M 0 is a s.e.s. and B = A M : The result no w follo ws from the previous theorem. The w a y Theorem 5 is stated, some or all of the elemen ts d i ma y b e units, and for suc h d i ; R =d i = 0 If w e assume that no d i is a unit, then the elemen ts d 1 ; d 2 ; :::; d t are called invariant factors They are unique up to asso ciates, but w e do not b other with that here. If R = Z and w e select the d i to b e p ositiv e, they are unique. If R = F [ x ] and w e select the d i to b e monic, then they are unique. The splitting in Theorem 5 is not the ultimate b ecause the mo dules R =d i ma y split in to the sum of other cyclic mo dules. T o pro v e this w e need the follo wing Lemma. Lemma Supp ose R is a PID and b and c are non-zero non-unit elemen ts of R Supp ose b and c are relativ ely prime, i.e., there is no prime common to their prime factorizations. Then bR and cR are comaximal ideals. (See p 108 for comaximal.) Pro of There exists an a 2 R with aR = bR + cR : Since a j b and a j c a is a unit, so R = bR + cR Theorem 6 Supp ose R is a PID and d is a non-zero non-unit elemen t of R Assume d = p s 1 1 p s 2 2 p s t t is the prime factorization of d (see b ottom of p 110). Then the natural map R =d R =p s 1 1 R =p s t t is an isomorphism of R -mo dules. (The elemen ts p s i i are called elementary divisors of R =d .) Pro of If i 6 = j p s i i and p s j j are relativ ely prime. By the Lemma ab o v e, they are PAGE 128 120 App endix Chapter 6 comaximal and th us b y the Chinese Remainder Theorem, the natural map is a ring isomorphism (page 108). Since the natural map is also an R -mo dule homomorphism, it is an R -mo dule isomorphism. This theorem carries the splitting as far as it can go, as seen b y the next exercise. Exercise Supp ose R is a PID, p 2 R is a prime elemen t, and s 1. Then the R -mo dule R =p s has no prop er submo dule whic h is a summand. T orsion Submo dules This will giv e a little more p ersp ectiv e to this section. Denition Supp ose M is a mo dule o v er a domain R An elemen t m 2 M is said to b e a torsion element if 9 r 2 R with r 6 = 0 and mr = 0 This is the same as sa ying m is dep enden t. If R = Z it is the same as sa ying m has nite order. Denote b y T ( M ) the set of all torsion elemen ts of M If T ( M ) = 0 w e sa y that M is torsion free.Theorem 7 Supp ose M is a mo dule o v er a domain R Then T ( M ) is a submo dule of M and M =T ( M ) is torsion free. Pro of This is a simple exercise. Theorem 8 Supp ose R is a Euclidean domain and M is a nitely generated R -mo dule whic h is torsion free. Then M is a free R -mo dule, i.e., M R m Pro of This follo ws immediately from Theorem 5. Theorem 9 Supp ose R is a Euclidean domain and M is a nitely generated R -mo dule. Then the follo wing s.e.s. splits. 0 T ( M ) M M =T ( M ) 0 Pro of By Theorem 7, M =T ( M ) is torsion free. By Theorem 8, M =T ( M ) is a free R -mo dule, and th us there is a splitting map. Of course this theorem is transparen t an yw a y b ecause Theorem 5 giv es a splitting of M in to a torsion part and a free part. PAGE 129 Chapter 6 App endix 121 Note It follo ws from Theorem 9 that 9 a free submo dule V of M suc h that T ( M ) V = M The rst summand T ( M ) is unique, but the complemen tary summand V is not unique. V dep ends up on the splitting map and is unique only up to isomorphism. T o complete this section, here are t w o more theorems that follo w from the w ork w e ha v e done. Theorem 10 Supp ose T is a domain and T is the m ultiplicativ e group of units of T If G is a nite subgroup of T then G is a cyclic group. Th us if F is a nite eld, the m ultiplicativ e group F is cyclic. Th us if p is a prime, ( Z p ) is cyclic. Pro of This is a corollary to Theorem 5 with R = Z The m ultiplicativ e group G is isomorphic to an additiv e group Z =d 1 Z =d 2 Z =d t where eac h d i > 1 and d i j d i +1 for 1 i < t Ev ery u in the additiv e group has the prop ert y that ud t = 0 So ev ery g 2 G is a solution to x d t 1 = 0 If t > 1, the equation will ha v e degree less than the n um b er of ro ots, whic h is imp ossible. Th us t = 1 and so G is cyclic. Exercise F or whic h primes p and q is the group of units ( Z p Z q ) a cyclic group? W e kno w from Exercise 2) on page 59 that an in v ertible matrix o v er a eld is the pro duct of elemen tary matrices. This result also holds for an y in v ertible matrix o v er a Euclidean domain. Theorem 11 Supp ose R is a Euclidean domain and A 2 R n is a matrix with non-zero determinan t. Then b y elemen tary ro w and column op erations, A ma y b e transformed to a diagonal matrix 0BBBB@ d 1 0 d 2 . 0 d n 1CCCCA where eac h d i 6 = 0 and d i j d i +1 for 1 i < n Also d 1 generates the ideal generated b y the en tries of A F urthermore A is in v ertible i eac h d i is a unit. Th us if A is in v ertible, A is the pro duct of elemen tary matrices. 126 App endix Chapter 6 The c haracteristic p olynomial of T is p = ( x 1 ) s 1 ( x r ) s r and p ( T ) = 0 This is called the Jor dan c anonic al form for T : Note that the i need not b e distinct. Note A diagonal matrix is in Rational canonical form and in Jordan canonical form. This is the case where eac h blo c k is one b y one. Of course a diagonal matrix is ab out as canonical as y ou can get. Note also that if a matrix is in Jordan form, its trace is the sum of the eigen v alues and its determinan t is the pro duct of the eigen v alues. Finally this section is lo osely written, so it is imp ortan t to use the transp ose principle to write three other v ersions of the last t w o theorems. Exercise Supp ose F is a eld of c haracteristic 0 and T 2 F n has trace( T i ) = 0 for 0 < i n Sho w T is nilp oten t. Let p 2 F [ x ] b e the c haracteristic p olynomial of T The p olynomial p ma y not factor in to linears in F [ x ], and th us T ma y ha v e no conjugate in F n whic h is in Jordan form. Ho w ev er this exercise can still b e w ork ed using Jordan form. This is based on the fact that there exists a eld F con taining F as a subeld, suc h that p factors in to linears in F [ x ]. This fact is not pro v ed in this b o ok, but it is assumed for this exercise. So 9 an in v ertible matrix U 2 F n so that U 1 T U is in Jordan form, and of course, T is nilp oten t i U 1 T U is nilp oten t. The p oin t is that it sucies to consider the case where T is in Jordan form, and to sho w the diagonal elemen ts are all zero. So supp ose T is in Jordan form and trace ( T i ) = 0 for 1 i n Th us trace ( p ( T )) = a 0 n where a 0 is the constan t term of p ( x ). W e kno w p ( T ) = 0 and th us trace ( p ( T )) = 0 and th us a 0 n = 0 Since the eld has c haracteristic 0, a 0 = 0 and so 0 is an eigen v alue of T This means that one blo c k of T is a strictly lo w er triangular matrix. Remo ving this blo c k lea v es a smaller matrix whic h still satises the h yp othesis, and the result follo ws b y induction on the size of T This exercise illustrates the p o w er and facilit y of Jordan form. It also has a cute corollary Corollary Supp ose F is a eld of c haracteristic 0, n 1, and ( 1 ; 2 ; ::; n ) 2 F n satises i1 + i2 + + in = 0 for eac h 1 i n: Then i = 0 for 1 i n Minimal p olynomials T o conclude this section here are a few commen ts on the minimal p olynomial of a linear transformation. This part should b e studied only if y ou need it. Supp ose V is an n -dimensional v ector space o v er a eld F and T : V V is a linear transformation. As b efore w e mak e V a mo dule o v er F [ x ] with T ( v ) = v x
Applied Sciences HND Unit: Mathematics for Science 1 UNIT DESCRIPTION: This unit is designed to provide the basic mathematical skills required when studying HN Science. Prior to commencement of this unit it is recommended that you will have a background in basic mathematics as well as have good numerical skills.
Linear Algebra and Its Applications - Study Guide - 4th edition Summary: An integral part of this text, the Study Guide incorporates detailed solutions to every third odd-numbered exercise, as well as solutions to every odd-numbered writing exercise for which the main text only provides a hint. PAPERBACK Good 0321388836 Clean, good binding, no marks or notes. Quick shipping. $1992 +$3.99 s/h Good text book recycle ny malone, NY 2011-03-28 Paperback Good We ship everyday and offer PRIORITY SHIPPING. $191117 +$3.99 s/h VeryGood BookCellar-NH Nashua, NH 032138883622.80 +$3.99 s/h Acceptable TEXTBOOKFETCHER! Cortland, NY 0321388836 This is a used item. $2323
Through this course students develop various concepts of Algebra. Students will solve linear, quadratic, rational, and radical equations; graph linear equations and inequalities in one variable; graph linear equations in two variables; solve and graph systems of linear equations and inequalities in two variables; simplify rational expressions; simplify expressions containing rational exponents; simplify complex numbers.
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Chapter Zero Fundamental Notions of Abstract Mathematics 9780201437249 ISBN: 0201437244 Edition: 2 Pub Date: 2000 Publisher: Addison-Wesley Summary: Chapter Zero is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach in which students construct their own understandings. However, while students are called upon to write their own proofs, they are also encouraged to work in groups. There are few finished proofs contained in the text, but the author offers proof ske...tches and helpful technique tips to help students as they develop their proof writing skills. This book is most successful in a small, seminar style class. Schumacher, Carol is the author of Chapter Zero Fundamental Notions of Abstract Mathematics, published 2000 under ISBN 9780201437249 and 0201437244. Five hundred forty eight Chapter Zero Fundamental Notions of Abstract Mathematics textbooks are available for sale on ValoreBooks.com, one hundred sixteen used from the cheapest price of $39.80, or buy new starting at $95.33.[read more] Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). 018 Item is intact, but may show shelf wear. Pages may include notes and highlighting. May or may... [more]RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks).018 Item is intact, but may show shelf wear. Pages may include notes and highlighting. May or may... [more] book is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach [more] This book is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach in which readers construct thei
Mathematical Methods for Physicists This best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital ...Show synopsisThis best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital addition to the bookshelf of any serious student of physics or research professional in the field. The authors have put considerable effort into revamping this new edition. * Updates the leading graduate-level text in mathematical physics * Provides comprehensive coverage of the mathematics necessary for advanced study in physics and engineering * Focuses on problem-solving skills and offers a vast array of exercises * Clearly illustrates and proves mathematical relations New in the Sixth Edition: * Updated content throughout, based on users' feedback * More advanced sections, including differential forms and the elegant forms of Maxwell's equations * A new chapter on probability and statistics * More elementary sections have been deleted Page edges are slightly dirty6. Hardcover Description:Good. 1966 Hardcover. Former Library book. Shows some signs of...Good. 1966 Hardcover Mathematical Methods for Physicists The book covers a very large range of mathematical issues. Some topics are well developed, like the ones covering complex analysis, while others, like the group theory, are much concise (in my opinion). In general, the book offers a good introduction to several topics, not only for the physicists but for the math and enginnering students
A week's worth of teaching on the Binomial Theorem. Lesson examples and a plethora of worksheets included. Learners find coefficients of specific terms within binomial expansions using notation of factorials and then apply these skills in using the Binomial Theorem to find solutions to practical applications. In this video on the Binomial Theorem, Sal tries to give an intuition behind why combinations are part of its definition. By looking at the expansion of (a + b)3 and carefully looking at where each value originates from, one can see how we are really asking a question about combinatorics. A comprehensive lesson plan that explores and researches Pascal's triangle and relates its properties to the Binomial Theorem through a variety of lessons. Have the class practice expanding polynomials using the theorem. A few other formulas and functions related to this theorem will be explored. Sal shows two ways to quickly calculate the coefficients of a binomial expansion. With the first method, he shows the relationship between PascalÕs triangle and the coefficients, and in the second method, he shows an even faster way for one to write the coefficients without calculating previous rows of coefficients. Continuing his discussion of the Poisson Distribution (or Process) from the previous video, Sal takes students through the derivation of the traffic problem he had begun. The math gets gritty in this video as Sal takes out the graphic calculator to solve the problem. In this sequences and series worksheet, 11th graders solve and complete 9 different problems that include various sequences and series. First, they evaluate each given expression using the binomial theorem. Then, students use Pascal's Triangle to expand each binomial. In this Algebra II worksheet, 11th graders apply the binomial theorem to expand a binomial and determine a specific term of the expansion. The one page worksheet contains four problems. Answers are provided. Students examine the patterns in Pascal's Triangle. In this recognizing lesson, students view a model of Pascal's Triangle and describe the patterns of the multiples. Students identify the shapes that are made within Pascal's Triangle.
More About This Textbook Overview From its history as an elegant but abstract area of mathematics, algebraic number theory now takes its place as a useful and accessible study with important real-world practicality. Unique among algebraic number theory texts, this important work offers a wealth of applications to cryptography, including factoring, primality-testing, and public-key cryptosystems. A follow-up to Dr. Mollin's popular Fundamental Number Theory with Applications, Algebraic Number Theory provides a global approach to the subject that selectively avoids local theory. Instead, it carefully leads the student through each topic from the level of the algebraic integer, to the arithmetic of number fields, to ideal theory, and closes with reciprocity laws. In each chapter the author includes a section on a cryptographic application of the ideas presented, effectively demonstrating the pragmatic side of theory. In this way Algebraic Number Theory provides a comprehensible yet thorough treatment of the material. Written for upper-level undergraduate and graduate courses in algebraic number theory, this one-of-a-kind text brings the subject matter to life with historical background and real-world practicality. It easily serves as the basis for a range of courses, from bare-bones algebraic number theory, to a course rich with cryptography applications, to a course using the basic theory to prove Fermat's Last Theorem for regular primes. Its offering of over 430 exercises with odd-numbered solutions provided in the back of the book and, even-numbered solutions available a separate manual makes this the ideal text for both students and instructors. Editorial Reviews Booknews Provides a global approach to the subject, leading readers through algebraic numbers, the arithmetic of field numbers, ideal theory, ideal decomposition in extension fields, and reciprocity laws. Each chapter includes a section on a cryptographic application of ideas presented, demonstrating the pragmatic side of theory. Engages readers by offering a historical perspective through the lives of mathematicians who played key roles in developing algebraic number theory. Includes chapter exercises, and background appendices. For a course for senior undergraduate and beginning graduate students
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first textbook on mathematical methods focusing on techniques for optical science and engineering, this text is ideal for upper division undergraduate and graduate students in optical physics. Containing detailed sections on the basic theory, the textbook places strong emphasis on connecting the abstract mathematical concepts to the optical systems to which they are applied. It covers many topics which usually only appear in more specialized books, such as Zernike polynomials, wavelet and fractional Fourier transforms, vector spherical harmonics, the z-transform, and the angular spectrum representation. Most chapters end by showing how the techniques covered can be used to solve an optical problem. Essay problems based on research publications and numerous exercises help to further strengthen the connection between the theory and its applications. less
Find a Collingdale can provide that. A continuation of Algebra 1 (see course description). Use of irrational numbers, imaginary numbers, quadratic equations, graphing, systems of linear equations, absolute values, and various other topics. May be combined with some basic geometry.
Mettawa, IL Physics is defined less by what topics are included than by what is excluded. Excluded are notions of continuity upon which calculus is built. Consequently, discrete math is described as "non-calculus" math
Editorial Reviews From the Back CoverThis book provides an introduction both to real analysis and to a range of important applications that require this material. More than half the book is a series of essentially independent chapters covering topics from Fourier series and polynomial approximation to discrete dynamical systems and convex optimization. Studying these applications can, we believe, both improve understanding of real analysis and prepare for more intensive work in each topic. There is enough material to allow a choice of applications and to support courses at a variety of levels. The first part of the book covers the basic machinery of real analysis, focusing on that part needed to treat the applications. This material is organized to allow a streamlined approach that gets to the applications quickly, or a more wide-ranging introduction. To this end, certain sections have been marked as enrichment topics or as advanced topics to suggest that they might be omitted. It is our intent that the instructor will choose topics judiciously in order to leave sufficient time for material in the second part of the book. A quick look at the table of contents should convince the reader that applications are more than a passing fancy in this book. Material has been chosen from both classical and modern topics of interest in applied mathematics and related fields. Our goal is to discuss the theoretical underpinnings of these applied areas concentrating on the role of fundamental principles of analysis. This is not a methods course, although some familiarity with the computational or methods-oriented aspects of these topics may help the student appreciate how the topics are developed. In each application, we have attempted to get to a number of substantial results and to show how these results depend on the fundamental ideas of real analysis. In particular, the notions of limit and approximation are two sides of the same coin, and this interplay is central to the whole book. We emphasize the role of normed vector spaces in analysis, as they provide a natural framework for most of the applications. This begins early with a separate treatment of Rn. Normed vector spaces are introduced to study completeness and limits of functions. There is a separate chapter on metric spaces that we use as an opportunity to put in a few more sophisticated ideas. This format allows its omission, if need be. The basic ideas of calculus are covered carefully, as this level of rigour is not generally possible in a first calculus course. One could spend a whole semester doing this material, which forms the basis of many standard analysis courses today. When we have taught a course from these notes, however, we have often chosen to omit topics such as the basics of differentiation and integration on tile grounds that these topics have been covered adequately for many students. The goal of getting further into the applications chapters may make it worth cutting here. We have treated only tangentially some topics commonly covered in real analysis texts, such as multivariate calculus or a brief development of the Lebesgue integral. To cover this material in an accessible way would have left no time, even in a one-year course, for the real goal of the book. Nevertheless, we deal throughout with functions on domains in Rn, and we do manage to deal with issues of higher dimensions without differentiability. For example, the chapter on convexity and optimization yields some deep results on "nonsmooth" analysis that contain the standard differentiable results such as Lagrange multipliers. This is possible because the subject is based on directional derivatives, an essentially one-variable idea. Ideas from multivariate calculus appear once or twice in the advanced sections, such as the use of Green's Theorem in the section on the isoperimetric inequality. Not covering measure theory was another conscious decision to keep the material accessible and to keep the size of the book under control. True, we do make use of the L2 norm and do mention the LP spaces because these are important ideas. We feel, however, that the basics of Fourier series, approximation theory, and even wavelets can be developed while keeping measure theory to a minimum. Of course, this does not mean we think that the subject is unimportant. Rather we wished to aim the book at an undergraduate audience. To deal partially with some of the issues that arise here, we have included a section on metric space completion. This allows a treatment of LP spaces as complete spaces of bona fide functions, by means of the Daniell jntegral. This is certainly an enrichment topic, which can be used to motivate the need for measure theory and to satisfy curious students. This book began in 1984 when the first author wrote a short set of course notes (120 pages) for a real analysis class at the University of Waterloo designed for students who came primarily from applied math and computer science. The idea was to get to the basic results of analysis quickly and then illustrate their role in a variety of applications. At that time, the applications were limited to polynomial approximation, Newton's method, differential equations, and Fourier series. A plan evolved to expand these notes into a textbook suitable for one semester or a year-long course. We expanded both the theoretical section and the choice of applications in order to make the text more flexible. As a consequence, the text is not uniformly difficult. The material is arranged by topic, and generally each chapter gets more difficult as one progresses through it. The instructor can choose to omit some more difficult topics in the chapters on abstract analysis if they will not be needed later. We provide a flow chart indicating the topics in abstract analysis required for each part of the applications chapters. For example, the chapter on limits of functions begins with the basic notion of uniform convergence and the fundamental result that the uniform limit of continuous functions is continuous. It ends with much more difficult material, such as the Arzela-Ascoli Theorem. Even if one plans to do the chapter on differential equations, it is possible to stop before the last section on Peano's Theorem, where the Arzela-Ascoli Theorem is needed. So both topics can be conveniently omitted. Although one cannot proceed linearly through the text, we hope there is some compensation in demonstrating that, even at a high level, there is a continued interplay between theory and application. The background assumed for using this text is decent courses in both calculus and linear algebra. What we expect is outlined in the background chapter. A student should have a reasonable working knowledge of differential and integral calculus. Multivariable calculus is an asset because of the increased level of sophistication and the incorporation of linear algebra; it is not essential. We certainly expect that the student is used to working with exponentials, logarithms, and trigonometric functions. Linear algebra is needed because we treat Rn, C(X), and L2 (-π, π) as vector spaces. We develop the notion of norms on vector spaces as an important tool for measuring convergence. As such, the reader should be comfortable with the notion of a basis in finite-dimensional spaces. Familiarity with linear transformations is also sometimes useful. A course that introduces the student to proofs would also be an asset. Although we have attempted to address this in the background chapter (Chapter 1), we have no illusions that this text would be easy for a student having no prior experience with writing proofs. While this background is in principle enough for the whole book, sections marked with a • require additional mathematical maturity or are not central to the main development, and sections marked with a * are more difficult yet. By and large, the various applications are independent of each other. However, there are references to material in other chapters. For example, in the wavelets chapter (Chapter 15), it seems essential to make comparisons with the classical approximation results for Fourier series and for polynomials. It is also possible to use an application chapter on its own for a student seminar or other topics course. We have included several modern topics of interest in addition to the classical subjects of applied mathematics. The chapter on discrete dynamical systems (Chapter 11) introduces the notions of chaos and fractals and develops a number of examples. The chapter on wavelets (Chapter 15) illustrates the ideas with the Haar wavelet. It continues with a construction of wavelets of compact support, and gives a complete treatment of a somewhat easier continuous wavelet. In the final chapter (Chapter 16), we study convex optimization and convex programming. Both of these latter chapters require more linear algebra than the others. We would like to thank various people who worked with early versions of this book for their helpful comments; in particular, Robert Andre, John Baker, Brian Forrest, John Holbrook, David Seigel, and Frank Zorzitto. We also thank various people who offered us assistance in various ways, including Jon Borwein, Stephen Krantz, Justin Peters, and Ed Vrscay. We also thank our student Masoud Kamgarpour for working through parts of the book. We would particularly like to thank the students in various classes, at the University of Waterloo and at the University of Nebraska, where early versions of the text were used. Most Helpful Customer Reviews I'm now almost a third year in graduate school and searched Amazon for this book, as it was my favorite analysis textbook as an undergraduate, and I thought of buying it for reference to go with the solutions that I wrote (and saved) as an undergraduate. I was shocked to find such poor reviews of such a well written text. I found the book very readable, the examples helpful, and most of all, the exercises very interesting and fun to solve. Do not be put off by the previous critiques. This is an excellent book and it is the first book in analysis that enjoyed learning from. I am currently a graduate student, and we are using this book in my first-year graduate course in analysis. To be quite honest, I find this book utterly useless, except for looking up homework problems that are so hard you are forced to look elsewhere just to learn how to solve them! The authors spend way too little time building up the theory and just expect their readers to be able to follow what they're doing with very few examples (or ones too complicated to really illustrate what's going on), and then give problems where even the easier ones can seem near impossible. This book makes more sense as a graduate text, certainly, especially if you've already had analysis; in that case, then you may only need to see the major theorems as a refresher and then you can start right on the challenging problems. However, if you're an undergrad and this is your first exposure to analysis, go elsewhere, please! My fellow grad students and I have gotten so frustrated over this book and its problems, and we've all had analysis before! If you've never had analysis before, I would suggest Bartle/Sherbert's Intro. to Real Analysis; they spend a good amount of time with examples and what I call "warm-up" homework problems to get you used to the concepts, followed by some doozies (and, yes; selected answers and hints are in the back!). If you're very strong in math, then perhaps Rudin's "Principles of Mathematical Analysis" may be more up your alley (aka Baby Rudin). Best of luck to you! I've been using this text for two semesters. I have to say this book is too advanced for starters, especially after chapter 6. My biggest complaint is the author does not provide enough examples to illustrate the theorems. A majority of the sections usually go like: 1. proposition of a theorem 2. proof 3. major theorem. 4. proof 5. corollary 6. proof 7. tons of hard problems left to homework. I would suggest the author give more examples when showing off those hard theorems. It could be better if the author also provides solutions to (at least) half of the exercises at the end of each section. Remember your readers are not academic conference colleagues, but first-time undergrad students. We learn things from examples! A well balanced book! The first solid analysis course, with proofs, is central in the offerings of any math.-dept.;-- and yet, the new books that hit the market don't always hit the mark: The balance between theory and applications, --between technical profs and intuitive ideas,--between classical and modern subjects, and between real life exercises vs. the ones that drill a new concept. The Davidson-Donsig book is outstanding, and it does hit the mark. The writing is both systematic and engaged.- Refreshing! Novel: includes wavelets, approximation theory, discrete dynamics, differential equations, Fourier analysis, and wave mechanics.
This video describes how to set GeoGebra up for demonstration in the classroom. This includes changing the default view, increasing the font size, turning on and off the grid and axes and much more. (Duration: 4 mins) This video provides details of how to move and scale the axes in GeoGebra. This may be useful if you want to centre the graph of a function or if the graph of a function doesn't fully fit onscreen. (Duration: 3 mins) Using the Calculator Notes on Using the TI-Nspire Calculator You might like to view some of the "Quick Look" videos on the TI-Nspire website. In the space of a minute or two they capture very succinctly key topics such as graphs, geometry and calculus – with about a dozen short videos ready to play. You can view the videos here: (Note: These calculators are prohibited for use in the State Examinations in Ireland)
This book is an introduction to the study of fundamental inequalities such as the arithmetic mean-geometric mean inequality, the Cauchy-Schwarz inequality, the Chebyshev inequality, the rearrangement inequality, and the inequalities for convex and concave functions. The emphasis is on the use of these inequalities for solving problems. The book's special feature is a chapter on the geometrical inequalities that studies relations between various geometrical measures. It contains more than 300 problems, many of which are applications of inequalities. A large number of problems are taken from the International Mathematical Olympiads (IMO) and many national olympiads from countries across the world. The book should be very useful for students participating in mathematical contests. It should also help graduate students consolidate their knowledge of inequalities by way of applications. A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.
Calculus I Calculus I This is a free online course offered by the Saylor Foundation.'... More This is a free online course offered by the Saylor Foundation. ' as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt where t is elapsed time and X0 is the initial deposit. But with compounded interest, now things get complicated for algebra as the rate R is now itself a function of time with Y = X0+ R(t)t. Now we have a rate of change which itself is changing. Calculus "to the rescue," as Isaac Newton introduced the world to mathematics specifically designed to handle "those things that change." Calculus is among the most important and useful developments of human thought. Even though it is over 300 years old, it is still considered the beginning and cornerstone of modern mathematics. It is a wonderful, beautiful, and useful set of ideas and techniques. You will see the fundamental ideas of this course over and over again in future courses in mathematics as well as in all of the sciences, including physical, biological, social, economic, and engineering. However, calculus is an intellectual step up from your previous mathematics courses. Many of the ideas you will learn in this course are more carefully defined and have both a functional and a graphical meaning. Some of the algorithms are quite complicated, and in many cases, you will need to make a decision as to which appropriate algorithm to use. Calculus offers a huge variety of applications and many of them will be saved for future courses you might take. This course is divided into four learning sections, or units, plus a reference section, or Appendix. The course begins with a unit that provides a review of algebra specifically designed to help and prepare for the study of calculus. The second unit discusses functions, graphs, limits, and continuity. Understanding "limits" could not be more important as that topic really begins the study of calculus. The third unit will introduce and explain derivatives. With derivatives we are now ready to handle all those "things that change" mentioned above. The fourth unit makes "visual sense" of derivatives by discussing derivatives and graphs. Finally, the fifth unit provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over.'
Microsoft Mathematics provides a set of mathematical tools that help students get school work done quickly and easily. With Microsoft Mathematics, students can learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra, algebra, trigonometry, physics, chemistry, and calculus. Microsoft Mathematics includes a full-featured graphing calculator that's designed to work just like a handheld calculator. Additional math tools help you evaluate triangles, convert from one system of units to another, and solve systems of equations.
Summary: Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's BASIC MATHEMATICS FOR COLLEGE STUDENTS, Fourth Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math students, mathematics is like a foreign language. They have difficulty translating the words, their meanings, and how they apply to problem solving. Emphasizing the "language of mathematics," the text's fu...show morelly integrated learning process is designed to expand students' reasoning abilities and teach them how to read, write, and think mathematically. It blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills95 +$3.99 s/h Acceptable Borgasorus Books, Inc. MO Wentzville, MO PAPERBACK Fair 053873408618
Synopses & Reviews Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments. Publisher Comments: Get a better grade with PRECALCULUS and accompanying technology! With a focus on teaching the essentials, this mathematics text provides you with the fundamentals necessary to be successful in this course and your future calculus course. Exercises and examples are presented the way that you will encounter them in calculus so that you are truly prepared for your next course. Tools found throughout the text, such as exercises, calculus connections, and true and false questions also help you master difficult concepts. Synopsis: About the Author J. Douglas Faires is a Emeritus Professor of Mathematics at Youngstown State University, where he received his undergraduate degree. His masters and doctoral degrees were awarded by the University of South Carolina. His mathematical interests include analysis, numerical analysis, mathematics history, and problems solving. Dr. Faires has won numerous awards, including the Outstanding College-University Teacher of Mathematics by the Ohio Section of MAA and five Distinguished Faculty awards from Youngstown State University, which also awarded him an Honorary Doctor of Science award in 2006. Faires served on the Council of Pi Mu Epsilon for nearly two decades, including a term as President, was the Co-Director of the American Mathematics Competitions AMC-10 and AMC-12 examinations for 8 years, and has been a long-term judge for the COMAP International Contest in Mathematical Modeling. He has authored or co-authored more than 20 books, including recent MAA publications to assist young students with mathematical problem solving. Jim DeFranza is a Professor of Mathematics at St. Lawrence University. His research interests are in analysis, sequence space theory and summability. He received his PhD from Kent State University. "Synopsis" by Netread,
How to Solve Maths Problems How to Solve Maths Problems Today, we are going to see how math homework help available online helps to solve our geometry, statistics and algebra problem. Math is a very vast subject and it plays an important role in our daily life. It is the study of space, relation, structure, change, and many other topics of pattern. It becomes an essential part in all the areas which includes engineering, social science, medicine, natural science etc. free online math help solvers or various homework helper are available for all grade, college and university level students to provide math homework help. Equations are of many types like Linear equation, Quadratic equation, Differential equations etc As sometimes the equation to solve becomes so complicated that it becomes very difficult to solve them. As equation may contain radicals, rational numbers and various complicated algebraic expressions also. Know More About : Planar Graph Coloring Tutorcircle.com Page No. : 1/4 So for solving equations, various equation solvers are available which helps to solve such problems in an easier way. The equation solver lets you solve an equation or system of equations. Algebra is a very vast concept or we can say it as a subject to study. As it plays an important role in almost every section not only in maths but even in other areas also like in economics, mechanical engineering, electrical or electronic applications etc. Algebra is a simple language, used to form mathematical models for real--world situations and to solve this problem that we are not able to solve by using basic arithmetic. An algebraic equation has one or more algebraic terms in a phrase. Algebra comes with Logarithms, Linear equations, inequalities, Geometry, Statistics etc. As we move towards our college education, mathematics becomes tougher and complex. We need a daily practice to gain a skill to solve our mathematics problems. We can solve our math problems online by using various homework solvers and helpers. By taking Geometry Homework help tool, we can solve the complicated problems in a faster way.. Statistics homework help tool helps to solve problems related to statistics and probability. The algebra homework help tool allows you to expand, factor or simplify virtually any expression you choose. Read More About : X Coordinate Of A Point Tutorcircle.com Page No. : 2/4 It also comes with commands which helps us to split fractions into partial fractions and combining several fractions into one. We can use Linear Equation Calculator available over the internet to calculate or solve our linear equation homework. Two online calculators are available to solve the linear equations. The calculators are available for 2 by 2 or 3 by 3 systems. Linear Inequality is also a part of Algebra, as various algebra tutors or equation solvers are available to solve the equation. It generates answer on one click by user on solve option and also shows the student how to arrive at the answer. Tutorcircle.com Page No. : 3/4 Page No. :
The Math Zone The Math Zone is located in the new, state-of-the-art Math Learning Center in North Hall, directly adjacent to Echlin Hall. An Innovation in Learning Mathematics The Math Zone provides a non-traditional education experience in mathematics that is targeted to each student's unique learning style. The Math Zone is located in the new, state-of-the-art Math Learning Center in North Hall, directly adjacent to Echlin Hall, and is a part of the Department of Mathematics. The Math Zone seeks to create an individualized educational experience where students learn with the support of math instructors and tutors. In addition to the guidance provided by dedicated faculty, your mastery will be supplemented by a learning platform that quickly adapts to areas where students demonstrate mastery of topics and moves on to deliver precise instruction and support to subjects where a student is having difficulty. Our goal is student success The faculty are there not only to answer the math questions students have, but to monitor student progress and to guide them in the mathematics learning process. The Math Zone is an innovation in mathematics education, merging e-learning technology with traditional teaching methods to enhance student success. Important Details for Students to Know If you are taking a math class run through the Math Zone, and have any questions regarding your class, please contact the Math Zone Coordinator. Use the the list below to select the area for which you need more information.
Practical Approach to Arithmetic And Algebra for College Students 9780759352230 ISBN: 0759352232 Publisher: CENGAGE Learning Custom Publishing Summary: Dr. Jerry Kornbluth earned his B.S. at Bowling Green State University and his MBA and PhD from Hofstra University. He retired from Nassau Community College after 35 years of teaching math and statistics, reaching rank of full professor in 1979 and awarded the rank of Professor Emeritus in 2001. Kornbluth also was an adjunct professor at La Guarida Community College and Queensborough Community College for 32 years. He... has worked with school districts throughout New York City and Long Island as well as federal agencies as a consultant on statistics related projects. Kornbluth is currently the Assistant Vice President of Academics at Interboro Institute. Green, Edward L. is the author of Practical Approach to Arithmetic And Algebra for College Students, published under ISBN 9780759352230 and 0759352232. Two hundred twenty eight Practical Approach to Arithmetic And Algebra for College Students textbooks are available for sale on ValoreBooks.com, three used from the cheapest price of $18.98, or buy new starting at $56.30.[read more] A Practical Approach to Arithmetic and Algebra for College Students covers basic arithmetic concepts involving critical thinking and its application to verbal problems. Conce [more] A Practical Approach to Arithmetic and Algebra for College Students covers basic arithmetic concepts involving critical thinking and its application to verbal problems. Concepts covered include, but are not limited to: decimals, fractions, percents.[less]
More About This Textbook Overview Think of it as portable office hours! The Interactive Video Skillbuilder CD-ROM contains video instruction covering each chapter of the text and features a 10-question Web quiz per section (the results of which can be emailed to the instructor), a test for each chapter, with answers, and MathCue tutorial and quizzing software. New to this CD is English/Spanish closed caption translations that can be selected to display along with the video instruction. Related Subjects Meet the Author Lewis Hirsch (Ph.D., Pennsylvania State University) currently teaches in the mathematics department at Rutgers University. Dr. Hirsch teaches both developmental mathematics and higher level courses such as college algebra and pre-calculus. His experiences in the classroom make him committed to properly preparing students in lower-level courses so they can succeed in for credit courses, and this is reflected in the way he writes his textbooks. Dr. Arthur Goodman (Ph.D., Yeshiva University) currently teaches in the mathematics department at Queens College of the City University of New York. Dr. Goodman takes great pride in the mathematical accuracy and in depth explanation in all of his
Poems sometimes have interesting connections to mathematics, either because of mathematical imagery in the poem or because of mathematics in the structure of the poem (word or syllable counts or shape). This article explores some of these connections In this article we use Java applets to interactively explore some of the classical results on approximation using Chebyshev polynomials. We also discuss an active research area that uses the Chebyshev polynomials. The Chebyshev equioscillation theorem is of great importance in polynomial approximation, but is usually omitted from the undergraduate numerical analysis course. Our aim in this paper is to use applets to motivate and illustrate the proof. Our purpose in this descriptive, exploratory study is to report on our efforts in conducting a cyber tutoring project that combined two curricular domains ---- mathematics education and engineering education. This Developers' Area article presents a five-step procedure to help authors construct mathlets in Excel. The procedure is demonstrated with a mathlet that shows the tangent line to the graph of a function.
Trigonometría plana y esférica, con aplicaciones en navegación Written for students, this book is devoted to the study of planar and spherical trigonometry. The book provides a reference manual for setting out the most important contents of plane and spherical trigonometry. Topics covered include trigonometric functions, navigation applications, spherical surface, spherical triangles and applications of spherical trigonometry in navigation. MATLAB is used to solve application examples in the book. In addition, a supplemental set of MATLAB code files are available for download.
What is Numerical Analysis? Numerical analysis addresses two problems: "how do we find an approximate numerical solution to a given mathematics problem?" and "how do we bound the error in our approximations?" The type of problems that are most amenable to approximation are those involving (at least piecewise) continuous functions, so many of the the main tools of numerical methods come from Calculus. The problems we will look at involve approximating functions, their zeros and integrals; solutions to differential equations; and solutions to systems of equations. Here is the catalog description of the course: Though the title of our course is numerical analysis, not numerical methods, we have traditionally taught it as if it were a methods course. This means the emphasis is more on the selection and applications of methods than on the mathematical theory required to derive them. This is not a proof course, but will test your memory of first year Calculus (251-252). We will quickly review any required Linear Algebra (310) as it is needed. The prerequisites also include CSCI 221. That is because we will program these methods, in Maple, to better understand their use. Why Maple? Because it will allow us to use the full power of an algebraic operating system and graphical interface, which still doing numerical approximations where appropriate. It is a bit awkward to program in, but we will learn as we go. (Our text is also strongly linked to the use of Maple.) What textbook will we use? The text for this course is: Numerical Methods by J. Douglas Faires and Richard Burdon, 3rd edition. Publisher-Thompson and Brooks/Cole. ISBN: 0-534-40761-7. We will cover much of chapters one through eight (or nine). This is a very standard text (well, actually their Numerical Analysis is very standard and Numerical Methods covers roughly the same material without the emphasis on deriving all of the methods.) How do we succeed? The suggestions are the same for most any mathematics class: Come to class and do the homework Always seek to understand, not just memorize. Study with a friend. Make a new friend if necessary! Use other books as references. Get familiar with the library. Stay ahead of the class in the text. Come by my office. Do not hesitate because I look busy. I rarely just sit around, but you are my top priority. Do not fall behind. Do the homework (maybe extra even). I want you to succeed and will gladly help you, but you must start by working on the homework and reading the text. Dr. L. Kolitsch, Mr. Eskew and many of the other faculty member might also be able to answer some questions (if they have time). How will we determine the grade? Homework will be assigned daily but sporadically collected. At the beginning of the following class period I may ask students to present parts of the homework as "board problems." I expect you to use your homework as notes. We will have three or four, fifty minute tests and a comprehensive final. The homework should give you an idea what the test question will look like. The course grade will probably be determined as follows:
INTERMEDIATE ALGEBRA COURSE DESCRIPTION: Intermediate Algebra expands on the mathematical content of Basic Algebra. Emphasis will be on functions and algebraic solutions to various types of problems. Abstract thinking skills (including some proofs, and the notion of 'generality of a statement') will be introduced and cultivated. COURSE OBJECTIVES: After completing the course, students will be able to: Be conversant with a number of mathematical topics (see the Course Description for a list of these topics) Have enough computational skill with each topic that they will be able to correctly apply that skill whenever such skill is required in a subsequent mathematics course Come away with an understanding and appreciation of where the topics arise in real world applications
A tutorial that explores the nature of rational numbers, the significance of the minimum and maximum integers in a rational number system, and the meaning of overflow. Includes a Java applet that opens in a separate window, for use alongside the tutorial. From a Computer Science course at the University of Utah, and the book Introduction to Scientific Programming. Computational Problem Solving Using: Maple and C; Mathematica and C.
Integers, Polynomials, and Rings: A Course in Algebra This book began life as a set of notes that I developed for a course at the University of Washington entitled Introduction to Modern Algebra for Tea- ...Show synopsisThis book began life as a set of notes that I developed for a course at the University of Washington entitled Introduction to Modern Algebra for Tea- ers. Originally conceived as a text for future secondary-school mathematics teachers, it has developed into a book that could serve well as a text in an - dergraduatecourseinabstractalgebraoracoursedesignedasanintroduction to higher mathematics. This book di?ers from many undergraduate algebra texts in fundamental ways; the reasons lie in the book's origin and the goals I set for the course. The course is a two-quarter sequence required of students intending to f- ?ll the requirements of the teacher preparation option for our B.A. degree in mathematics, or of the teacher preparation minor. It is required as well of those intending to matriculate in our university's Master's in Teaching p- gram for secondary mathematics teachers. This is the principal course they take involving abstraction and proof, and they come to it with perhaps as little background as a year of calculus and a quarter of linear algebra. The mathematical ability of the students varies widely, as does their level of ma- ematical interestNew. This item is printed on demand. This introduction to...New. This item is printed on demand. This introduction to modern algebra differs from texts in this area in fundamental ways. The author's primary goal is to have the reader learn to work with mathematics through reading, writing, speaking, and listening. The
An Intermediate Course in Algebra: An Interactive Approach Established authors Alison Warr, Cathy Curtis, and Penny Slingerland have produced class-tested first editions written to speak to the challenge of ...Show synopsisEstablished authors Alison Warr, Cathy Curtis, and Penny Slingerland have produced class-tested first editions written to speak to the challenge of the NCTM and AMATYC Standards and technology integration in the classroom, the authors use the following means to address the standards: * Numerical, Graphical, and Algebraic Models: The exposition usually begins by synthesizing the concepts discovered in the activity sets. Concepts are presented through non trivial applications. Mathematical models are presented numerically, graphically, and algebraically with an emphasis on the connection among the three representations. * Guided Discovery Activities: Students discover concepts and strategies through guided activities. Co-operative learning teams are used to facilitate understanding and improve communication. * Problems Solving: Students are expected to communicate their ideas, processes, and understanding orally and in writing. * Collaborative Learning: The problem sets include skill building problems, conceptual questions, and applications. Answers to problems, which can be verified or checked, are usually omitted. In-depth application problems, appropriate for cooperative learning teams, are included.Hide synopsis Description:Good. Spiral. May include moderately worn cover, writing,...Good. Spiral. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780534436736735 Some pages contain highlighting or writing, otherwise just fine
The basic concepts behind sizing systems currently used in the manufacture of ready-to-wear garments were originally developed in the nineteenth century. These systems are frequently based on outdated anthropometric data, they lack standard labelling... What you'll find here is a fascinating compendium of fundamental problem formulations of analog design centering and sizing. This essential work provides a differentiated knowledge about the tasks of analog design centering and sizing. In particular,... Expert teachers share a wealth of classroom-tested lessons that help students understand why and how to measure, focusing on concepts important to the middle school math curriculum, including length, area, volume, ratio and rates, similarity, and... Expert teachers share a wealth of classroom-tested lessons that help students understand why and how to measure length, area, capacity, weight, time, and temperature. The book provides engaging real-world contexts to help students understand what it... Expert teachers share a wealth of classroom-tested lessons that help students understand why and how to measure length, area, volume, weight, time, temperature, and angles. The book provides engaging real-world contexts to help students understand Many people have had the frustrating experience of ordering shoes online, only to find that the shoes do not fit their feet correctly. The shoes may be the same size and width that the buyer has always wears,...
Precalculus Functions And Graphs 9780495108375 ISBN: 0495108375 Edition: 11 Pub Date: 2007 Publisher: Thomson Learning Summary: Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text popular among students year after year. This latest edition of Swokowski and Cole's PRECALCULUS: FUNCTIONS AND GRAPHS retains these features. The problems have been consistently praised for being at just the right level for precalculus students like you. The book also p...rovides calculator examples, including specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics. Swokowski, Earl W. is the author of Precalculus Functions And Graphs, published 2007 under ISBN 9780495108375 and 0495108375. Two hundred fifty one Precalculus Functions And Graphs textbooks are available for sale on ValoreBooks.com, one hundred fifteen used from the cheapest price of $4.20, or buy new starting at $46.66
Nature of Mathematics Written for liberal arts students and based on the belief that learning to solve problems is the principal reason for studying mathematics, Karl ...Show synopsisWritten for liberal arts students and based on the belief that learning to solve problems is the principal reason for studying mathematics, Karl Smith introduces students to Polya's problem-solving techniques and shows them how to use these techniques to solve unfamiliar problems that they encounter in their own lives. Through the emphasis on problem solving and estimation, along with numerous in-text study aids, students are assisted in understanding the concepts and mastering the techniques. In addition to the problem-solving emphasis, "The Nature Of Mathematics, 12e, International Edition" is renowned for its clear writing, coverage of historical topics, selection of topics, level, and excellent applications problems. Smith includes material on such practical real-world topics as finances (e.g. amortization, installment buying, annuities) and voting and apportionment. With the help of this text, thousands of students have 'experienced' mathematics rather than just do problems-and benefited from a writing style that boosts their confidence and fosters their ability to use mathematics effectively in their everyday lives.Hide synopsis Description:Very Good. Very good hardcover, no dj. Pages are clean, crisp...Very Good. Very good hardcover, no dj. Pages are clean, crisp and unmarked; edges lightly foxed. Cover shows light wear and some rubbingThe title of this book does not indicate that it is the teacher's edition, neither is this reflected in the photo. If you read the entire description it is confusing. It does state that the teacher's edition is the same as the student edition, but does not in any way indicate that you are actually purchasing the teacher's
Math 127: Precalculus - Syllabus Typical Syllabus for Math 127: Precalculus at Lycoming College Note: This is a typical syllabus for this course. Each semester a semester-specific syllabus will be distributed, which will spell out --in addition to the topics-- administrative details such as information about grading, homework, tests, labs, etc. Precalculus attempts to prepare students for calculus in two general ways: (1) by remedying the mathematical prerequisites in the algebra and geometry of the elementary functions, and (2) by introducing students to Maple, a computer algebra system that will be used throughout the calculus curriculum. Exponential functions and logarithms (the laws of exponents and logarithms, solving exponential and logarithmic equations, applications to population, radioactive decay, Newton's law of heat transfer, etc) Circular functions and their inverses (definitions of the six circular functions, special values, modeling with the sine and cosine, right triangle trigonometry, trigonometric identities, solving trigonometric equations, the inverse trigonometric functions) Systems of 2 non-linear equations in two unknowns, especially pairs of linked quadratic equations. (This is a skill needed in calculus 3.) The Maple system provides a typical example of the capabilities of a computer algebra system. At the Precalculus level, it can be used to plot the graphs of a great variety of polynomials, rational functions and conics, as well as to elegantly solve modeling problems involving the sine and the exponential. Geometer's Sketchpad is used, where possible, to illustrate certain properties of the circular functions.
This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2013 makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy... more... During the springs of 2011 and 2012, the author was invited by Peking University to give an advanced undergraduate algebra course (once a week over two months each year). This book was written during and for that course. By no way does it claim to be too exhaustive. It was originally intended as a brief introduction to algebra for an extremely pleasant... more... The book focuses on advanced computer algebra methods and special functions that have striking applications in the context of quantum field theory. It presents the state of the art and new methods for (infinite) multiple sums, multiple integrals, in particular Feynman integrals, difference and differential equations in the format of survey articles.... more... Get up-to-speed on the functionality of your TI-84 Plus calculator Completely revised to cover the latest updates to the TI-84 Plus calculators, this bestselling guide will help you become the most savvy TI-84 Plus user in the classroom! Exploring the standard device, the updated device with USB plug and upgraded memory (the TI-84 Plus Silver Edition),... more... Algebra: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using Algebra. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all the key areas of algebra including elementary operations, linear equations,... more... This first book in the series will describe the Net Generation as visual learners who thrive when surrounded with new technologies and whose needs can be met with the technological innovations. These new learners seek novel ways of studying, such as collaborating with peers, multitasking, as well as use of multimedia, the Internet, and other Information... more...
Mathematics Methods for Elementary and Middle School Teachers Provides techniques for modern mathematics teaching. This title integrates technology with experience, research and standards. It highlights Praxis ...Show synopsisProvides techniques for modern mathematics teaching. This title integrates technology with experience, research and standards. It highlights Praxis II[trademark]-Style Test Questions. It gives you these questions, which will help you prepare for the test used for teacher certification Mathematics Methods for Elementary and Middle School...Good. Mathematics Methods for Elementary and Middle School Teachers
Algebra 2A/B Syllabus This course is an extension of Algebra I and provides further development of the concept of a function. Topics include: (1) relations, functions, equations and inequalities; (2) conic sections; (3) polynomials; (4) algebraic fractions; (5) logarithmic and exponential functions; (6) sequences and series; and (7) counting principles and probability, and as time allows, (8) Trigonometric Functions and Relations. Be reminded, Algebra 2 is NOT an introductory course and topics learned in Algebra 1 will not be rehashed but rather built upon. And, because all Indiana state universities and colleges require Algebra 2 for admission, this Algebra 2 course is taught at the college preparatory level and taught for proficiency on university admission tests. Students will work to their potential and ultimately benefit from taking this class as borne out from recent SAT scores. Those who score well in algebera 2 do well on college admissions tests. State Standards: STANDARD 1 — RELATIONS AND FUNCTIONS STANDARD 2 — LINEAR and ABSOLUTE VALUE EQUATIONS and INEQUALITIES STANDARD 3 — QUADRATIC EQUATIONS and FUNCTIONS STANDARD 4 — CONIC SECTIONS STANDARD 5 — POLYNOMIALS STANDARD 6 — ALGEBRAIC FRACTIONS STANDARD 7 — LOGARITHMIC and EXPONENTIAL FUNCTIONS STANDARD 8 — SEQUENCES and SERIES STANDARD 9 — COUNTING PRINCIPLES and PROBABILITY STANDARD10 — MATHEMATICAL REASONING and PROBLEM SOLVING Big Ideas: Quarter 1: 1. Multiple equations can be used to simultaneously model and solve a situation with several variables. 2. Equations, tables, graphs, and words can all be utilized interchangeably to represent the same event. 3. Equations can provide a model that can be used to analyze, draw conclusions and make predictions. 4. Equations and inequalities allow us to graphically or algebraically represent and solve problems. Quarter 2: 1. Complex numbers provide a means of evaluating expressions that have no real solution. 2. Quadratic expressions allow us to model real world situations that do not follow a linear pattern. 3. An understanding of the factors of a polynomial can be used to illustrate characteristics of graphs and solutions of equations. 4. The shape and features of a graph provide valuable information about its corresponding equation. Quarter 3: 1. Algebraic fractions can be manipulated in a similar manner to simple numerical fractions. 2. Exponentials and logarithms are inverse operations that can be manipulated and rewritten. 3. The characteristics of exponential and logarithmic functions and their representations are useful in solving real-world problems. 4. Fractional equations are useful in solving problems involving direct and inverse variation. Quarter 4: 1. Basic mathematical operations can be performed on given functions resulting an infinite number of possible new functions. 2. Counting methods can be used to determine possible outcomes as well as the likelihood of an event occurring. 3. Circles, ellipses, parabolas, and hyperbolas can be found from cross sections of a double-napped cone, but have distinctly different characteristics and equations. 4. Numerical patterns can be quantified and used to make predictions. 5.Graphing calculators – provided in class, will be used to aid in exploring Algebra; however graphers will not be used as a "crutch" for applications that students cannot perform on their own. 6. iPads will be used by permission only. 7. A positive attitude – "If there were one word that could be used to describe a successful person, that one word would be ATTITUDE." Bart Starr Assignments/Make-up Work: Homework will be assigned on a daily basis and generally due the following day (at the beginning of class). Though assignments are generally not graded, I retain the right to grade homework for completeness and effort, and at times for accuracy. Homework checks/pop quizzes will be given regularly. Problems from recent previous assignments will be given to the student at the beginning of class the next day, to complete in class with use of notes/binder/homework within a reasonable amount of time. Adequate time is allowed to complete the homework check provided the student has completed his or her assignment. I expect every problem in each homework assignment to be given an honest attempt by you. Each problem should have logical work showneven though you are convinced that your work or answers are wrong. Anything less than an honest attempt is considered incomplete. A zero will be given for a homework assignment turned in with answers only, unless specified by your teacher. Students will assume the responsibility to find out what work and notes you missed while absent and that you complete those tasks promptly. After your return to school, you will be allowed one day per day absent to complete all missing material. If you miss a homework check, then expect to turn in the appropriate assignment in its place. If you are absent the class before a test/quiz, expect to take the test/quiz on the regularly schedule day (this means you must review on your own). If you are absent the day of a test/quiz, expect to take the test/quiz on the 1st day you returnFinally, you will receive at most, only half-credit for any late paper. Classroom Policies: 1. Be respectful at all times (towards me, other faculty, and your classmates). I will respect you. 2. Be on time and prepared (have sharpened pencil, paper, homework, and calculator ready to go when the bell rings!) 3. Be responsible for all material discussed while you were absent on a field trip and upon return the next day, have completed all assignments due. PLAN AHEAD FOR FIELD TRIPS!!! 4. Be on task and use study/homework time to work exclusively on your math assignment. When your math assignment is completed you may then use that extra time to work on another assignment. 5. Be honest with your work on all assignments, projects, tests and quizzes whether graded or not. If you are found to have cheated, you will receive a zero for that grade, your CPG will be docked 5 points, and you will be referred for disciplinary action and a Friday night school. 6. Absolutely NO CELL PHONES, food, drinks (other than water), or backpacks/trapper keeper/bag in the classroom. 7. All Manchester Jr/Sr High School rules apply. Possible Consequences : 1. Verbal warning 2. A meeting with Ms. Stone 3. Parents called 4. Loss of CPG points (see CPG contract) 5. Written referral Any serious infraction will be referred to the office immediately for possible ISS or FNS. Grading: Your individual midterm and semester grades depend on several different scores: 1. Tests (a major portion) 2. Announced Quizzes 3. Frequent Pop Homework checks 4. Homework—I reserve the right to grade any assignment. 5. CPG (a score out of 25 points)—recorded at end of semester 6. Extra-credit, recorded at the end of thesemester Make-ups: 1. If you miss an ANNOUNCED quiz, your grade will be recorded as 0 out of 0. It does not count for or against your grade. 2. If you miss a POP homework check, then I will grade the homework assignment pertaining to the checked material. 34. For all others absences, you have one day for every day you missed to complete the outstanding assignments, tests or homework check. After such time, if the work is still missing, the grade will be recorded as a zero! Grading Scale: (in percents) You will be expected to keep track of your own point totals. An assignment sheet is provided for that purpose. Any "rounding" will be done on an individual basis. Attitude, effort, class participation, etc. will be considered in any rounding situation. .5% and above is not necessarily cause for me to automatically round up! Semester Average = 80% from your total percent in a semester (including any extra credit and CPG) prior to the final exam + 20 % from the final exam percent. If a midterm grade is given, it will be based on your total points earned divided by the total points possible. Extra credit and CPG are added in when determining the semester grade. Extra Credit: There will be minimal opportunities for extra credit. The best way to get extra credit is extra effort. Show me that you are working to your potential. Be an active participator in class. It won't go unnoticed. Electronic device Policy IPADs, IPODs, laptops, or other electronic devices should remain unseen and unheard unless I give you permission to use them. In that case they should be used only for the assigned task. If a student is seen using an electronic device when not permitted or if the device is heard disrupting class, it will be confiscated* and taken to the office. The student will receive a Friday Night School. If any devices are seen or heard before, during, or after a homework check, quiz, or test, the student will be assumed to be cheating. That student will be given a 0 on the given assignment. HELP: My job is to help you reach your potential. Your success in this class is vitally important to me. I am often available for extra assistance before each school day (7:45 AM, after each school day until 3:30 PM or later by appointment, and at home, via phone. Students are encouraged to call me at home (982-2949, prior to 8 pm) for extra help with their homework. Further assistance is available from peer tutors in the study hall , via the internet at the website listed on the front of your textbook. After school tutoring is available to all free of charge from 3:30-4:30 PM M-TH.There is also FREE homework assistance through Rose Hulman Institute at 1-877-ASK-ROSE (toll free) every Sunday – Thursday evenings, 7-10 PM, excluding any Rose Hulman breaks. Homework Policy Homework Helpline: 877-ASK-ROSE 1. Homework will be given on a nearly daily basis. I intend to allow study time at the end of each period so that you may begin the assignment under my supervision. You will use that time for that purpose only. BUT, expect to spend 5-30 minutes, on average, outside of class to complete each assignment. 2. You will complete in PENCIL ONLY all assignments, quizzes, and tests. You will receive a zero for graded material done in pen after the first week in a course. Ask to borrow a pencil before the bell rings as needed. 3. I expect every problem in each homework assignment to be given an honest attempt by you. Each problem should have logical work shown even though you are convinced that your work or answers are wrong. Anything less than an honest attempt is considered incomplete. A zero and a homework check will be given for an assignment turned in with answers only unless specified by your teacher. 4. You will follow the format as given in class for your homework. Always copy the original problem, with the exception of "story" problems. 5. Expect pop quizzes over your assignments. I will periodically have you transfer your homework solutions or notes over to another paper for a pop quiz. Other pop quizzes will be given orally or be handed out. You WILL be allowed to use your notebooks (notes and homework papers) on all Pop Quizzes (only). 7. You will assume the responsibility to find out what work and notes you missed while absent and that you complete those tasks promptly. After your return to school, you will be allowed one day per day absent to complete all missing tests, quizzes, and homework. After that timeline, all incomplete assignments will be recorded as a zero. Have your parent call the office (982-2196) to collect your assignments even if only for one day. It is best to stay current with all assignments even while ill as your condition allows. 8. You are responsible for all material discussed while you are gone on a field trip and upon return, you will have completed all assignments due for the day that you missed and for the day that you return. Remember you are considered PRESENT while attending a field trip. I want this class to challenge you, to satisfy your intellectual curiosity, and to fulfill your academic needs. You WILL have to work hard, but I can guarantee you that I will be working even harder to see you meet your goals. Let's make this a great school year by working together toward the common goals of academic success and maturity.
An application for math plot.Can be used arithmetic operations, trigonometric functions (angles measured in radians), decimal, natural logarithms, the logarithm to an arbitrary ground, whole and fractional parts of
Intermediate Algebra - 3rd edition Summary: KEY BENEFIT:Intermediate Algebra, Third Edition, by Tom Carson, addresses two fundamental issues-individual learning styles and student comprehension of key mathematical concepts-to meet the needs of today's students and instructors.Carson's Study System, presented in the ldquo;To the Studentrdquo; section at the front of the text, adapts to the way each student learns to ensure their success in this and future courses. The consistent emphasis on thebig picture of algebra, with pedag...show moreogy and support that helps students put each new concept into proper context, encourages conceptual understanding. KEY TOPICS: Real Numbers and Expressions; Linear Equations and Inequalities in One Variable; Equations and Inequalities in Two Variables and Functions; Systems of Linear Equations and Inequalities; Exponents, Polynomials, and Polynomial Functions; Factoring; Rational Expressions and Equations; Rational Exponents, Radicals, and Complex Numbers; Quadratic Equations and Functions; Exponential and Logarithmic Functions; Conic Sections MARKET:3143.2516
I am currently in my summer vacations. Next year I will star my undergraduate studies in mathematics. I used to be in mathematics competitions. Last year I got a silver medal in my countries national ... I really liked Teach Yourself Logic: A Study Guide by the user Peter Smith. It is a thorough guide how to teach yourself logic and set-theory from scratch up to any level with book recommendations for ... If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, then, for example, if $f'(x_0)=0$ and $f'$ is again differentiable at $x_0$ and $f''(x_0)\neq 0$, then $f$ has a maximum or minimum at $x_0$. ...I'm studying AS maths, and am trying to connect my thoughts around polynomials and the factor and remainder theorems. I understand the factor theorem and its application: it helps me find roots of a ... I've heard people refer to the Riemann integral as a "teaching integral" and in a sequence of an analysis course at my school (which is rarely offered) we discuss the mysterious Lebesgue integral. I ... I have heard about the field of Applied and Computational Mathematics, Scientific Computing and want to get some information. Is this a combination of computer science and mathematics? What subjects ... I was just wondering if anyone with some experience could recommend a book for one still in the early stages of their mathematical studies(first year). Maybe something related to Algebra or history of ... From Stokes theorem, it is easy to prove the folowing proposition: $\int\int_\vec{S}\vec{F}d\vec{S}=0$ if $\vec{F}=curl$ $\vec{G}$ for vector fields $\vec{F}$ and $\vec{G}$ and a closed parametrizedHaving taken an introductory course on algebraic geometry (without introducing schemes), the notion of schemes seems to be quite unrelated to all we've done there. What are the most important reasons ... First little background. I have master degree in mathematics. Then I decided to continue to study PhD level. After some years I cancel study (reason was in some things in my life). Now I am returning ... One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ... I know that this question does not admit a definitive answer. Surely it depends on how long and how difficult the book is, the reader's ability and background knowledge, etc. But I still want to ask ... When submitting to a journal for publication and suggesting referees expert in the field for peer-review, is it the done thing (or just polite) to contact the referees to ask if they would be happy to ... Today in Italy all students under 18 faced with a test used to establish the quality of the schools they are in. I read one of the question (a mathematical question, of course!) that make me thing for ... As a natural science student in university, you may encounter so many problems that might require a deep understanding in integrating skills and series calculation. But as many of the college students ... Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ... Next year i'll be going to UMCP and i'be been doing competition math from middle school and throughout high school. I've done some looking around, and other than the Putnam Competition, there don't ... Obviously the enumerating notation of sets like $\{ 1,2,3,4,5,\ldots \}$ is not unique because it is not clearly defined how to continue the dots "$\ldots$". For example the above defined set could ...
ISBN: 0072933224 / ISBN-13: 9780072933222 Beginning and Intermediate Algebra: The Language and Symbolism of Mathematics Intended for schools that want a single text covering the standard topics from Beginning and Intermediate Algebra. Topics are organized by using the ...Show synopsis graphical -- has proven popular with a broad cross section of students.Hide synopsis Beginning and Intermediate Algebra: The Language and Symbolism of Mathematics (McGraw-Hill Companies) – Hardcover (2007) by James W Hall Hardcover, McGraw-Hill Companies 2007 English 2nd ed. ISBN: 0072933224 ISBN-13: 9780072933222 ...Show more broad cross section of students. Use of a graphing calculator is assumed. BEGINNING AND INTERMEDIATE ALGEBRA: THE LANGUAGE AND SYMBOLISM OF MATHEMATICS is a reform-oriented Beginning and Intermediate Algebra: The Language and...Good. Beginning and Intermediate Algebra: The Language and Symbolism of Mathematics
This volume covers topics ranging from pure and applied mathematics to pedagogical issues in mathematics. There are papers in mathematical biology, differential equations, difference equations, dynamical systems, orthogonal polynomials, topology, calculus reform, algebra and numerical analysis. Most of the papers include recent results in the respective subjects. However, there are some papers of an expository nature.
9780972999 Mathematics for Engineers, Third Edition This comprehensive text offers an effective and easy-to-follow coverage of the fundamentals of applied mathematics and their role in engineering analysis. Each topic is presented in great detail and accompanied by a large number of thoroughly worked-out examples, as well as several related exercises. Engineering applications, ranging from mechanical and electrical to structural systems are greatly stressed and integrated throughout the book. Highlights include: A detailed coverage of elemental and basic laws leading to mathematical models of dynamic systems (Ch. 1). Derivation of the mathematical models of most physical systems under consideration to familiarize the reader with the proper laws and engineering terminology
Summary: Contemporary Mathematics for Business and Consumers is a 21-chapter educational adventure into today's business world and its associated mathematical procedures. The book is designed to provide solid mathematical preparation and foundation for students going on to business courses and careers. It begins with a business-oriented review of the basic operations, including whole numbers, fractions, and decimals. Once students have mastered these operations, they are intr...show moreoduced to the concept of basic equations and how they are used to solve business problems. From that point, each chapter presents a business math topic that utilizes the student's knowledge of these basic operations and equations. In keeping with the philosophy of ''practice makes perfect,'' the text contains over 2,000 realistic business math exercises--many with multiple steps and answers designed to prepare students to use math to make business decisions and develop critical-thinking and problem-solving skills. Many of the exercises in each chapter are written in a ''you are the manager'' format, to enhance student involvement. The exercises cover a full range of difficulty levels, from those designed for beginners to those requiring moderate to challenge-level skills. ...show less 0324304552 Has heavy shelf wear, but still a good reading copy. Includes CD-ROM. A portion of your purchase of this book will be donated to non-profit organizations. We are a tested and proven comp...show moreany with over 900,000 satisfied customers since 1997. Choose expedited shipping (if available) for much faster delivery. Delivery confirmation on all US orders. ...show less $3.70Our feedback rating says it all: Five star service and fast delivery! We have shipped four million items to happy customers, and have one MILLION unique items ready to ship today! $3.703.71 +$3.99 s/h Acceptable Sierra Nevada Books Reno, NV Ex-Library Book - will contain Library Markings. Biggest little used bookstore in the world65.00 +$3.99 s/h New bluehouse acton, MA Brand new. $96
Probability theory is a rapidly expanding field and is used in many areas of science and technology. Beginning from a basis of abstract analysis, this mathematics book develops the knowledge needed for advanced students to develop a complex understanding of probability. The first part of the book systematically presents concepts and results from analysis before embarking on the study of probability theory. The initial section will also be useful for those interested in topology, measure theory, real analysis and functional analysis. The second part of the book presents the concepts, methodology and fundamental results of probability theory. Exercises are included throughout the text, not just at the end, to teach each concept fully as it is explained, including presentations of interesting extensions of the theory. The complete and detailed nature of the book makes it ideal as a reference book or for self-study in probability and related fields. Covers a wide range of subjects including f-expansions, Fuk-Nagaev inequalities and Markov triples. Provides multiple clearly worked exercises with complete proofs. Guides readers through examples so they can understand and write research papers independently.
Calculus : Single and Multivariable - 5th edition Summary: Calculus teachers recognize Calculus as the leading resource among the ''reform'' projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The fifth edition uses all strands of the ''Rule of Four'' - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are ...show morenot fundamentally unique. Readers will also gain access to WileyPLUS, an online tool that allows for extensive drills and practice. Calculus teachers will build on their understanding in the field and discover new ways to present concepts to their students6420 +$3.99 s/h New newrecycleabook centerville, OH 0470089148 US
Site Feedback IIURL Information Site owners jhasenbank Jennifer Kosiak Robert H. Hoar Robert Hoar PPST Math Content Welcome to the IIURL's PPST Mathematics Site The UW System Institute for Innovation in Undergraduate Research and Learning (IIURL) has developed this site to share a collection of digital learning objects (LOs). These materials are designed to prepare educators for the Pre-Professional Skills Test in Mathematics by helping to develop a deeper understanding of the mathematics . According to published materials describing the PPST: The Pre-Professional Skills Test in Mathematics measures those mathematical skills and concepts that an educated adult might need. It focuses on the key concepts of mathematics and on the ability to solve problems and to reason in a quantitative context. Many of the problems require the integration of multiple skills to achieve a solution. The test questions are from four content categories: number and operations, algebra, geometry and measurement, and data analysis and probability. Computation is held to a minimum, and few technical words are used. Terms such as area, perimeter, ratio, integer, factor, and prime number are used because it is assumed that these are commonly encountered in the mathematics all examinees have studied. Figures are drawn as accurately as possible and lie in a plane unless otherwise noted. In this E-Study Guide you will find digital materials clustered into the following domain or content areas:
Equalculator is an equationcalculator. It is currently in the "pre-alpha" stage of development, so don't expect it to do miracles. You type in your equation, it will ask you for the variables and Voila!, it gives you the answer.. Chemical Equation Expert is an integrated tool for chemistry professionals and students. You\'ll find complicated work such as balancing chemical equations and related calculations so easy and even enjoyable! Key Feafures -1. An intelligent balancer Chemical EquAdvanced yet easy-to-use math calculator that immediately and precisely computes the result as you type a math expression. It allows multiples math expressions at same time. It also allows fractions and defining your own variables and functions. It has a beautiful user-interface that conveniently allows you to directly edit the expression or if you prefer use a visual keypad to enter new expressions. There is support for over 20 math functions. Smart Math Calculator is the easiest and most convenient calculator in the world with amazing features: -Type down any math expression and immediately observe the result. Quadratic equation has the form ax2 + bx + c = 0. It will generally have two solutions; that is, two different values of x that make the equation true. It can happen that both solutions are the same number, and it is possible that the solutions will be complex or imaginary numbers. To use this software, type in values for a, b, and c in the boxes below, and press the Solve for X button. You may enter positive, negative, or zero values, but the value of a cannot be zero.. In the summer 2003 I discovered the world of sundials and quick found that the community of sundial maker is the next great community after the telescope makers. Practically here is only once good dial design package called Shadows, so I decided to write my own powerful design program called SUNDI, that was first demonstrated in autumn 2003. SUNDI were written in Borland Delphi, the best programming language for Windows applications development by Ivan Krastev. The source code is not provided
This book takes the "complex variables" view of elliptic curves. It uses a particular number-theoretic problem to drive the discussion: the problem of characterizing congruent numbers. A congruent number (no relation to congruences modulo a number) is defined as a number that is the area of a right triangle with all sides rational. Thus 6 is a congruent number because it is the area of the familiar 3-4-5 right triangle. There is an equivalent formulation in terms of whether the elliptic curve y2 = x3 - n2x has rational solutions. This enables us to bring the powerful machinery of elliptic curves to bear on the problem, and the book develops that machinery and culminates with a proof of Tunnell's (almost complete) characterization of congruent numbers. I like the method of using a single difficult program to organize a book. I think it is not completely successful here, because the original problem drops out of view in the middle of the book, with many new concepts being introduced that are not clearly driven by it. The book travels though L and zeta funtions, elliptic functions, and modular functions and forms. Silverman and Tate's Rational Points on Elliptic Curves is a very different approach to elliptic curves, through abstract algebra and geometry. There is surprisingly little overlap between the two books, considering that they are introductions to the same subject. Koblitz is much faster-paced, and contains a lot of intricate arguments. It covers a much larger amount of material and requires more mathematical maturity (it is correctly placed in Springer's Graduate Texts series, while Silverman and Tate is in the Undergraduate Texts series). I like both books, but I think Silverman and Tate is a better introduction. Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis. Preface to the First Edition Preface to the Second Edition CHAPTER I From Congruent Numbers to Elliptic Curves 1. Congruent numbers 2. A certain cubic equation 3. Elliptic curves 4. Doubly periodic functions 5. The field of elliptic functions 6. Elliptic curves in Weierstrass form 7. The addition law 8. Points of finite order 9. Points over finite fields, and the congruent number problem CHAPTER II The Hasse-Weil L-Function of an Elliptic Curve 1. The congruence zeta-function 2. The zeta-function of En 3. Varying the prime p 4. The prototype: the Riemann zeta-function 5. The Hasse-Weil L-function and its functional equation 6. The critical value
Starting with maths Mathematics, as well as being a fascinating subject, underpins practically every aspect of modern life. Whether you're keeping tabs on a budget, tackling a DIY project or devising a formula for a spreadsheet, you'll need to understand maths. This Openings Access module introduces a range of key ideas to help you tackle everyday mathematical problems and also gently introduces you to OU study – ideal if you're a beginner or returning to study. You'll also try out learning online; the perfect way to gain the basic computing skills you'll need for the next step in your studies. This Openings Access module is only available if you live in Northern Ireland, Scotland or WalesIf you are particularly interested in maths, or you want to study a maths-based subject or one where maths will feature significantly, Starting with maths offers a friendly introduction. The Openings Access module will help you to feel more confident in using maths in a variety of different situations – at home, in work or in your other studies. There are four main themes developed in the module: improving your mathematical skills including using a calculator effectively developing problem-solving strategies so that you know what to do when you get stuck practising general study skills to help you become an effective learner learning how to use your computer for studying. Using mathematics in the real world is sometimes quite different to tackling a clearly-stated problem in a textbook. During the module, you will consider several real-life examples, including a case study based on a conservation issue, so that you can see the differences yourself, and feel more confident in using mathematics in your own life to solve problems and make decisions. Being able to communicate using mathematical ideas is important, whether you are reading the work of others or explaining your own solutions. The module will help you develop these skills, in particular using notation and language appropriately and writing good mathematical solutions that are easy to understand. As well as maths that is useful in everyday situations, such as using fractions, decimals and percentages, numbers, graphs, word formulas, geometry and handling data, this module also includes puzzles, bits of history and some mathematical ideas that are fascinating in their own right. You will find the module useful whether you are building up towards regular, structured study or are simply interested in finding out about mathematics and its place in our lives. In this module the mathematical ideas are emphasised more than the technological and scientific ones, although the skills are equally appropriate for anyone who intends to study technology or science. These ideas are explored further on the website where you can study two more chapters online. You will have the opportunity to gain skills such as working with podcasts, using online forums, using spreadsheets and searching the internet for information. This experience will provide you with a gentle introduction to using a computer to support your study, and will equip you with the basic computing skills you will need for the next step in your studies. You only need to use a computer for the last part of the module once you've completed the book, so if you don't currently have one you've plenty of time to make arrangements. You can use your own computer or one at a library or drop-in centre. Please note that you can study and pass this module if you don't have access to the internet and a computer. Outside the UK This Openings Access module is only available if you live in Northern Ireland, Scotland or Wales. If you live in England, the Channel Islands, the Isle of Man, or if you have a British Forces Post Office (BFPO) address, we offer a choice of three 30-credit Access modules: Teaching and assessment Support from your tutor You will have a tutor who will keep in touch by a combination of telephone, written correspondence and, if you want, email. There are no face-to-face tutorials; all tutorials are conducted between you and your tutor on the telephone. Your tutor will help you to plan your work and to think about the ideas explored in the module. Your tutor will also comment on and help you with your written work. At the end of the module you will discuss your progress with your tutor, and you will work together to review your learning. Course satisfaction survey Entry Like all Openings Access modules, this module is ideal if you're a beginner or returning to study, we will help you to develop your study skills and become a confident learner. You can study this module as an additional preparatory stage of your chosen qualification, or as a standalone module but whichever option you chose, you will receive all the preparation you need to be a successful university student. The study materials have been prepared with the needs of new learners in mind. No special knowledge or previous experience of studying is required. You will develop key study skills such as time management, note taking, reading for study purposes and reflection on your own learning support available to you. The fees and funding information provided here is based upon current details for year 1 August 2014 to 31 July 2015. This information was provided on 24/07/2014. What's included Module books, DVD, a calculator and a website where you can access the online resources. Digital copies (PDFs) of most study materials, and transcripts of the DVD can be found on the website. Transcripts are also available on the DVD itself, if it is accessed through a computer. You will need Access to a telephone (preferably a landline) for contact with your tutor; and the equipment to play and watch a video DVD e.g. a television and DVD player or a personal computer with DVD drive. You will need access to the internet and a computer to study the last two chapters of this module and if you wish to receive and send email and use our online services. If you have a disability Alternative questions will be provided for any assignments that depend on visual or aural material. Written transcripts of any audio components and Adobe Portable Document Format (PDF) versions of printed material are available. Some Adobe PDF components may not be available or fully accessible using a screen reader and mathematical, scientific or foreign language materials may be particularly difficult to read in this way. The study materials are available on audio in DAISY Digital Talking Book format. Other alternative formats of the study materials may be available in the future. Our Services for disabled students website has the latest information about availability. To do the online element of this module you will need to make use of a personal computer and the internet. If you use specialist hardware or software to assist you in operating a computer or have concerns about accessing the type of material outlined, talk to the Student Registration & Enquiry Service before registering about the support which can be given to meet your needs
Elements of Modern Algebra - 7th edition Summary: Helping to make the study of modern algebra more accessible, this text gradually introduces and develops concepts through helpful features that provide guidance on the techniques of proof construction and logic analysis. The text develops mathematical maturity for students by presenting the material in a theorem-proof format, with definitions and major results easily located through a user-friendly format. The treatment is rigorous and self-contained, in keeping with...show more the objectives of training the student in the techniques of algebra and of providing a bridge to higher-level mathematical23 +$3.99 s/h Acceptable Campus_Bookstore Fayetteville, AR Used - Acceptable Hardcover. TEXTBOOK ONLY! WATER DAMAGED! 7th Edition Not perfect, but still usable for class. Ships same or next day. Expedited shipping takes 2-3 business days; standard shipping ta...show morekes 4-14 business days. ...show less $11.23
Calculus Demystified - 03 edition Summary: Here's an innovative shortcut to gaining a more intuitive understanding of both differential and integral calculus. In Calculus Demystified an experienced teacher and author of more than 30 books puts all the math background you need inside and uses practical examples, real data, and a totally different approach to mastering calculus. With Calculus Demystified you ease into the subject one simple step at a time -- at your own speed. A user-friendly, accessible style ...show moreincorporating frequent reviews, assessments, and the actual application of ideas helps you to understand and retain all the important conceptsBookmans AZ Tucson, AZ 2002 Paperback Good Satisfaction 100% guaranteed
Math Program Updated July 2013 Catlin Gabel School Middle School Math Program Explanation July 2013 The purpose of this explanation is to clarify the flow of the Middle School math program. It reflects ongoing changes made in the math curriculum to better meet the needs of our students. In many ways, this fall, 2013-2014, will be a transition year for many students as we prepare to offer algebra over the course of two years in the Middle School. It includes course descriptions, information on how placement decisions are made, and indicates how the progression of math classes occurs. We believe building a strong algebra foundation is essential for success in subsequent mathematics courses. Algebra is one of the critical junctions in math curriculum because it demands the ability to think abstractly. It is the turning point from using arithmetic skills (concrete thinking) to solve basic math problems with specific numeric values, to creating generalized formulas for equations that use variables (abstract thinking). Abstract reasoning and analytical thinking involve building rules to represent functions by recognizing patterns and organizing data, reversibility and working backwards, and abstracting from computation. Students entering the Upper School as freshmen that do not demonstrate this foundation of knowledge will find themselves struggling when they take later courses. Therefore, all Middle School math courses focus on skill-building, problem-solving, and algebraic thinking, as we know these are the critical elements of a strong mathematics foundation. Courses this Fall- 2013-2014 Year One Algebra 2/Geometry Pre-Algebra Algebra 1 Year One Algebra 2/Geometry taken in Upper School Year Two, Algebra 2/Geometry Courses next Fall- 2014-2015B Year One Algebra 2/Geometry Pre-Algebra Algebra 1 Year One Algebra 2/Geometry taken in Middle School Year Two Algebra 2/Geometry What is Math 6? Math 6 is a course designed to solidify, strengthen, and deepen foundational math skills. In addition, the course emphasizes creative problem-solving strategies and generalizing patterns to push the growth of each student's abstract thinking and logical reasoning ability. The beginning of algebraic thinking is woven throughout the curriculum. The students are also introduced to computer programming during our gender-based grouping. What is Pre-Algebra? Pre-Algebra is a very broad term used to describe courses that prepare students for the study of algebra. This course includes topics that involve arithmetic review and the introduction of skills and concepts for algebra, geometry, and statistics. Our program emphasizes mastery of the facts and skills of mathematics and the development of abstract concepts, logical reasoning, and application of skills through problem-solving challenges. What is Algebra 1? Algebra courses vary greatly from one school to another in terms of their depth and rigor. At Catlin Gabel, our vigorous program emphasizes abstract thinking and logical reasoning. Topics covered in algebra include evaluation and simplification of algebraic expressions, solving and graphing linear equations, linear systems, operations with polynomials, radical and rational expressions, and factoring. Four dimensions of understanding are emphasized to maximize performance: skill in carrying out various algorithms; developing and using mathematics properties and relationships; applying mathematics in realistic situations; and representing or picturing mathematical concepts. What is Algebra 1A? This course was designed to allow students to complete the study of Algebra over two years to ensure a thorough understanding of the topics and to build a strong foundation for higher-level mathematics. Algebra 1A is the first of this two-year course. We believe the shift to algebra over the course of two years will allow for students to solidify conceptual understanding, as well as apply their learning through ongoing problem solving. What is Algebra 1B? Algebra 1B is the second year of this two-year course. How are math placements decided? We strive to place students in a class that will meet their individual needs as learners. Although there are courses and typical math paths, we recognize that students change, mature, develop more abstract reasoning and improve their critical thinking abilities, hence a student may be moved up or down over the course of the year to best serve their learning needs. In 6th grade, the first weeks of school are used to assess each student's basic math skills, abstract thinking, problem-solving ability, and spatial reasoning. Several placement assessments are used during this time, including 5th grade teacher input, pre-algebra readiness assessments, baseline exams, and teacher observations. The great majority of the students are placed in Math 6, a vigorous math curriculum reviewing and solidifying elementary math skills, deepening the conceptual understanding of these topics, extending these skills to more complex number sets, and exploring new topics. Students who demonstrate abstract thinking and mastery of elementary math will place into a Pre-Algebra course to best challenge them and meet their learning needs. Between 6th and 7th grades, students are further grouped. They are grouped by math proficiency as well as pace of learning. For our current students, placement into 7th grade math classes happens at the end of 6Between 7th and 8th grades, the groups continue along the same lines as 7Is there movement of students during the course of the year? If a teacher assesses a student is not able to meet the expectations of a course or is far exceeding the expectations of a course, the student will be moved to a more appropriate placement. What if I have questions about my child's math placement? Parents are always welcome to directly contact their child's math teacher, or Barbara, to discuss placement decisions and their child's math progress.
Friendly Introduction to Number Theory 9780131861374 ISBN: 0131861379 Edition: 3 Pub Date: 2005 Publisher: Prentice Hall Summary: Starting with nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh nota...tion and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Provides a new chapter that introduces the theory of continued fractions. Includes a new chapter on "Continued Fractions, Square Roots and Pell's Equation." Contains additional historical material, including material on Pell's equation and the Chinese Remainder Theorem. A useful reference for mathematics teachers. Silverman, Joseph is the author of Friendly Introduction to Number Theory, published 2005 under ISBN 9780131861374 and 0131861379. Two hundred forty one Friendly Introduction to Number Theory textbooks are available for sale on ValoreBooks.com, forty four used from the cheapest price of $4.16, or buy new starting at $35.16.[read more] Ships From:Dallas, TXShipping:StandardComments:109 Item is intact, but may show shelf wear. Pages may include notes and highlighting. May or may... [more]0131861379 Good to VERY GOOD physical shape and/or light to moderate markings/highlight! A great value priced book! Same or next day processing! Choose EXPEDITED for super fa [more] 0131861379
ISBN: 0801318661 / ISBN-13: 9780801318665 Elementary and Middle School Mathematics: Teaching Developmentally John A. Van de Walle has written a book that helps readers make sense of mathematics and become confident in their ability to teach mathematics to ...Show synopsis synopsis ...Show more Elementary and Middle School Mathematics: Teaching Developmentally This book gives you insight into teaching mathematics without memorization of algorithms. It explains how children learn math and apply their own algorithms to the learning process to understand math in depth
1111578478 9781111578473 Student Solutions Manual for Tussy/Gustafson's Elementary and Intermediate Algebra, 5th:Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
Hart, George W. & Henri Picciotto Zome Geometry: Hands-on Learning with Zome Models Publisher: 2001. New with no dust jacket Key Curriculum Paperback 0.8 x 10.9 x 8.5 Inches; 265 pages; This book uses more than 60 easy-to-follow activities and over 150 explorations to help students build spatial, conceptual, visualization, and geometric skills. They will explore a wide variety of geometry topics, including proportion, symmetry, area, volume, and coordinates. They will learn to prove there are only five Platonic solids and to analyze the thirteen Archimedean solids. They will also discover Euler's theorem and verify Descartes' theorem of angular deficit. They will have an opportunity to investigate space filling, duality, fractals, and the fourth dimension. Included is an eight-page color insert of full color images of various constructions. Answers and index are found at the back. Can be used with middle school students, but is intended for high school.; ISBN: 1559533854 Milliken Beginning Geometry 6-8 Publisher: Dayton N.D.. New with no dust jacket Milliken Publishing Company Paperback 4to 11" - 13" tall; 44 pages; This book clearly presents the theorems and principles of basic geometry, along with examples and exercises for practice. The concepts are explained in an easy-to-understand fashion to help students grasp geometry and form a solid foundation for advanced learning in mathematics. Each page introduces a new concept, along with a puzzle or riddle which reveals a fun fact. Thought-provoking exercises encourage students to enjoy working the pages while gaining valuable practice in geometry. Pages are reproducible and the answers are in the middle.; ISBN: 9780787700317 Vaughn, Steck Saxon Geometry 1st Edition Homeschool Kit w/ Solutions Manual Publisher: 2009. New with no dust jacket Saxon Publishers, Inc. Hardcover 1.9 x 11.2 x 8.5 Inches; Saxon Geometry includes all topics in a high school geometry course, presented through the familiar Saxon approach of incremental development and continual review. The homeschool kit includes the Student Textbook, with 120 Lessons, 12 Investigations, and 15 Labs, the Solutions Manual, with step-by-step solutions to every problem in the book, and the Homeschool Packet, which includes Test Forms and Test Answers. (publisher description); ISBN: 9781600329760 Weeks, Jeffrey R. Exploring the Shape of Space Activity Book / CD Bundle Publisher: 2001. New with no dust jacket Key Curriculum Press Paperback 0.43 x 10.84 x 8.4 Inches; 133 pages; This two-week unit was supports an animated video, The Shape of Space (not available here, but a digitized version is on the CD included), and takes students on a tour of the dimensions and possible shapes of space. This book includes not only paper-and-pencil games, but also computer games provided on a CD. On the simulated surface of a torus or Klein bottle, students can solve mazes, work jigsaw puzzles, and create or solve crossword puzzles and word searches. In competition with the computer or against each other, students can play tic-tac-toe or chess on these same wrap-around surfaces. CD requires a java-capable browser and works on Windows or Macintosh. Includes test, answers, and glossary.; ISBN: 1559534672
Elementary Algebra (17%). Questions in this content area are based on properties of exponents and square roots, evaluation of algebraic expressions through substitution, using variables to express functional relationships, understanding algebraic operations, and the solution of quadratic equation...
Developmental Mathematics-Text - 8th edition Summary: TheBittinger serieschanged the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed readers to understand the rationale behind each concept before practicing the associated skills and then moving on to the next topic. With this revision, Marv Bittinger continues to focus on building success through conceptual understanding, while also supporting readers with quality applications, exercises,...show more and new review and study materials to help students1731530 Item in good condition. Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!! $39.9641.2931 +$3.99 s/h Good newrecycleabook centerville, OH 0321731530 used book - free tracking number with every order. book may have some writing or highlighting, or used book stickers on front or back $59.86 +$3.99 s/h Good Campus_Bookstore Fayetteville, AR Used - Good TEXTBOOK ONLY130.79321731531
More About This Textbook Overview Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics. In this introduction to the modern theory of ideals, Professor Northcott assumes a sound background of mathematical theory but no previous knowledge of modern algebra. After a discussion of elementary ring theory, he deals with the properties of Noetherian rings and the algebraic and analytical theories of local rings. In order to give some idea of deeper applications of this theory the author has woven into the connected algebraic theory those results which play outstanding roles in the geometric
Thanks, Its mainly slopes and algebraic equations, those line graphs, y=mx+b etc. his final exam is on Monday and Tuesday. If you want I can send you images of his worksheets of what the teacher gave him. If by example, you mean questions that already have their solutions for the students to follow, then no... All there is in terms of that is a summary at the end of each worksheet that summarizes it all. The problem is that they learn something new each day and have a test for it the next day due to the curriculum being condensed into 10 days. Here's an image of one of his sheets, they all follow a similar criteria: Honestly I have no idea, things like algebra, y=mx+b/rise over run, functions, SOHCAHTOA (which is the one thing I know and can help him with) etc. Sorry it's been a while since I've been to high school, i'll be sure to ask him when he gets home today.
App Detail » Math Ref What's New - Fixes image display issue on iPhone and iPod Touches. App Description Math Ref is an award winning education app. Browse over 1,400 formulas, figures, and examples to help you with math, physics, chemistry and more. Use an expanding list of helpful tools such as a unit converter, quadratic solver, and triangle solver to perform common calculations. "Math Ref is just an awesomely useful app for students, teachers, and anyone else who works with math and needs to do a lot of calculations." - FindmySoft.com Features: -Tools: Ranging from Algebra to Physics -Favorites: Easily save equations in groups -Search: Easily find what you're looking for -Print Support: Print equations or groups -Editable Notes: Write what will help you best -Zoomable Equations: Tap any equation to make it zoomable -Copy as Text: Press and hold on Physical Constants to copy the values -Searchable Prime Numbers: Now easily search through the first 10,000 primes
Michael Sullivan'stime-tested approach focuses students on the fundamental skills they need for the course: preparingfor class, practicingwith homework, and reviewingthe concepts. The Enhanced with Graphing Utilities Serieshas evolved to meet today's course needs by integrating the usage of graphing calculator, active-learning, and technology in new ways to help students be successful in their course, as well as in their future endeavors. Author Biography Michael Sullivan, Emeritus Professor of Mathematics at Chicago State University, received a Ph.D. in mathematics from the Illinois Institute of Technology. Mike taught at Chicago State for 35 years before recently retiring. He is a native of Chicago's South Side and divides his time between a home in Oak Lawn IL and a condo in Naples FL. Mike is a member of the American Mathematical Society and the Mathematical Association of America. He is a past president of the Text and Academic Authors Association and is currently Treasurer of its Foundation. He is a member of the TAA Council of Fellows and was awarded the TAA Mike Keedy award in 1997 and the Lifetime Achievement Award in 2007. In addition, he represents TAA on the Authors Coalition of America. Mike has been writing textbooks for more than 35 years and currently has 15 books in print, twelve with Pearson Education. When not writing, he enjoys tennis, golf, gardening, and travel. Mike has four children: Kathleen, who teaches college mathematics; Michael III, who also teaches college mathematics, and who is his coauthor on two precalculus series; Dan, who is a sales director for Pearson Education; and Colleen, who teaches middle-school and secondary school mathematics. Twelve grandchildren round out the family. Mike Sullivan III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of three children and an avid golfer who tries to spend as much of his limited free time as possible on the golf course.
This eBook introduces the significant scientific notation of the very large, the intermediate and the very small in terms of numbers and algebra through an exploration of indices, the rules of indices and standard index formThis eBook introduces the subject of measures and measurement, and looks at both metric and imperial units of measurement, the process and accuracy of reading scales, limits on the accuracy of measurements and compound measurements. transformation as it relates to translations, reflections, rotations and enlargements either as individual operations or composite operations. In this eBook we illustrate each of these translations using right-angled triangles, but the principles developed extend to all 2D shapes as well as to 3D shapes using extensions. This eBook introduces the subjects of angles, bearings and scale drawings. To angles as it relates to angular turn about a point, angles in polygons, angle facts and inter-relational aspects with parallel as well as crossing lines, to bearings as they relate to navigation and scale drawings as an aspect of technical drawing. Loci, Constructions and 3D Co-ordinates is an introductory text on loci and their characteristics, constructing triangles, the bisector of a line and the perpendicular bisector of a line as well as using 3D co-ordinates This eBook introduces 2D planar and 3D solids (shapes) and their corners (vertices), faces, edges, lines and planes of symmetry and order of rotational symmetry. It introduces the student to regular and irregular polygons, quadrilaterals, triangles, circles, nets of solids, cuboids, prisms and cylinders including formulating the algebra that describes their various areas, perimeters and volumes humorous account of the life of a teacher is Mary Brackenbury's first venture into the world of creative non-fiction. The author has cleverly compressed her teaching career into a single school year
Out of Stock Description of Lifepac Math: Grade 9: Complete SetThe Grade 9 Math Complete Set includes 10 student workbooks for a full year of study plus the comprehensive Teacher's Guide. Topics covered include the following: Variables and Numbers Solving Equations Problem Analysis and Solution Polynomials Algebraic Factors Algebraic Fractions Radical Expressions Graphing Systems Quadratic Equations and Review King James version Product: Lifepac Math: Grade 9: Complete Set Vendor: Alpha Omega Media Type: Book Minimum Age: 13 Maximum Age: 14 Minimum Grade: 9th Grade Maximum Grade: 9th Grade Weight: 6.55 pounds Length: 3.5 inches Width: 11.375 inches Height: 9.25 inches Vendor Part Number: MAT 0915 Subject: Math Curriculum Name: Lifepac/Alpha Omega Learning Style: Auditory, Kinesthetic, Visual Teaching Method: Traditional There are currently no reviews for Lifepac Math: Grade 9: Complete Set.
Edexcel GCSE Modular Mathematics 2007 Higher Unit 2 Student Book Matched to the September 2007 specification (2381), this book features extra multiple-choice practice questions to prepare students for the new unit ...Show synopsisMatched to the September 2007 specification (2381), this book features extra multiple-choice practice questions to prepare students for the new unit 2: stage 1 exam. An array of exercises gives students plenty of opportunities to practice skills and techniques, and comprehensive worked examples show exactly how to approach questions. It includes new examination practice papers and answers for the new Unit 2 exam papers: Stage 1 and 2 Edexcel GCSE Mathematics for 2006.. Full colour
GeoGebra math software – a review [Updated, May 2014: GeoGebra is still under active development and is now in version 4.4. The following tutorial is still relevant and it will get you started. The key features have not changed much, but there are now several more options and many examples on GeoGebraTube. See latest release details. I'm using JSXGraph now on IntMath, as it is more mobile-friendly than GeoGebra.] GeoGeobra is an intelligent graphing software that allows the user to interactively explore Cartesian & Euclidean geometry – as well as calculus. Best of all – it's a free offering! GeoGebra is a free and multi-platform dynamic mathematics software for schools that joins geometry, algebra and calculus. Building an Interactive Document in GeoGebra Let's go through the process of creating a document in GeoGebra. Our document will allow the end-user to explore the changing slope of a polynomial as x changes value. First, we enter the function at the bottom of the screen, using the carat "^" for powers of x: The program converts the function display (see under "Free objects") so that it is more easily read by a human. I have scaled the y-axis by clicking on the "Move" tool (the one on the far top right) and simply dragging the axis to the desired scale. Next, we are going to add a tangent line to our curve. We add a new point on the function using the "New point" tool: Note the other tools that are available on this tool item: Intersection of 2 two curves Midpoint between 2 points Now we choose the "Tangent" tool as follows: Note the other tools that are available on this drop-down. You can construct: Perpendicular line through a point Parallel line through a point Perpendicular bisector Bisector of an angle Diameter line of a conic section We now have a tangent line. The exploratory activity we can do now is to drag the point "A" to any position on the curve and the tangent line follows along. Even better, we can get a readout of the actual slope as we move around the curve, by typing in: s = Slope[t] Rather than just giving a numerical value for the slope, it actually gives a triangle with base length 1 unit, indicating more clearly what a slope at a point really means. The result for one part of the curve is as follows: Visual Calculus Let's now add the curve of the first derivative to our existing plot. (Of course, we expect the first derivative curve to be a parabola, since it will be a polynomial of degree 2). We achieve this by entering: Derivative[f] The green curve is the first derivative curve (a parabola, as expected): We can trace the locus of a point (B) moving on the first derivative curve, as follows: You can use GeoGebra to examine critical points like local maximums and minimums on the curve and the point of inflection (point A) illustrated above. Other Tools GeoGebra is a feature-rich offering. The other tools available in GeoGebra that I have not already mentioned include: Rotate an object around a point Draw line segments Draw vectors Draw polygons (including regular polygons) Construct various circles, arcs and sectors Angles, distances and areas You can add text and images You can zoom in and out on objects Split Functions GeoGebra will draw piece-wise functions (with a little coaxing). You can achieve the following (with a vector thrown in): Output I like the variety of output options. You can either: Save your file for later use (it will have a .GGB extension) Save the graph as an image in PNG, EPS, SVG or EMF format Save the graph to the clipboard (for manipulation in an image editing program or for pasting into a document) Save as an interactive Web page, but this can only be uploaded to GeoGebra Tube (not to your hard drive) Euclidean Geometry GeoGebra allows you to easily create angles, polygons and conics. As you can see in the regular dodecagon above, GeoGebra allows you to measure angles, including internal angles. Gripes You need Java 1.4.2 or later to run GeoGebra. I've never been a big fan of Java-based software, including those based on Web plugins, because the Java run-time environment is a huge memory hog and it takes a very long time to load. Java is very bloated. The size of the associated files in Geogebra is quite crazy: GeoGebra-parabola.ggb (2 kB) This is the file that I created above (with the parabola and first derivative. Nice and small) javagiac-win32.jar (4281 kB – it's unclear what this does) geogebra_properties.jar (2045 kB) – why so big? A properties file should consist of a small amount of text, surely? The Flash files I formerly used on Interactive Mathematics were around 30 kB and the required Flash plugin was around 2 MB (but this does not get downloaded each time you access a Flash file, just once). These are not fair comparisons, of course. Geogebra is a standalone application which novices can use to produce math applets, whereas JSXGraph (and Flash) requires coding knowledge. Being Java-based, GeoGebra will run on any operating system and that is a big plus. Now more compatible There is now a HTML5 export option in GeoGebra (so it can be run on Web pages without java) and it will run on iOS and Android tablets now. Still waiting for 3D I've been looking forward to the official release of Geogebra 5 for a long time now. It has 3D graph capability. It has been in beta for years. Conclusion GeoGebra is an impressive geometry and calculus exploratory tool. I tend to use it as an exploratory tool when I need something quick. But file size and java permission problems have meant I don't develop Geogebra applets for the Interactive Mathematics site. GeoGebra is more intelligent than MS Math 4.0, which I reviewed earlier (it has 3D capability – see Microsoft Math 4.0), but the audience for each product is not exactly the same. Having the 2 products will give you some excellent tools for exploring mathematics. Hi, I see you mentioned the files and sizes of the output files. Just a tip here: If you choose export as .html and remove the tick in the "ggb & jar" box, you will only get the .html file. I think it still works!
Haven't taken it yet... I'm an agriculture major, and Calc 1 plus precalc from 20 years ago finishes off my math requirement. I'm hesitantly considering taking 2/3/linear though, just for the shits and giggles of it. Not sure I'm brave enough though. It's been a year, and I'm still trying to understand what I think are probably pretty basic concepts. Heh. On page 5 (by the page markings, not the PDF) he indicates he's using "billion" as "a million times a million", or 1012, rather than 109. I've heard of the "billion = 1012" convention; is that a standard British thing?
1848009127 9781848009127 A Topological Aperitif:This is a book of elementary geometric topology, in which geometry, frequently illustrated, guides calculation. The book starts with a wealth of examples, often subtle, of how to be mathematically certain whether two objects are the same from the point of view of topology.After introducing surfaces, such as the Klein bottle, the book explores the properties of polyhedra drawn on these surfaces. Even in the simplest case, of spherical polyhedra, there are good questions to be asked.There are many examples and exercises making this a useful textbook for a first undergraduate course in topology. For much of the book the prerequisites are slight, though, so anyone with curiosity and tenacity will be able to enjoy the book. As well as arousing curiosity, the book gives a firm geometrical foundation for further study."A Topological Aperitif provides a marvellous introduction to the subject, with many different tastes of ideas.Stephen Huggett and David Jordan have excellent credentials for explaining the beauty of this curiously austere but potentially enormously general form of geometry." Professor Sir Roger Penrose OM FRS, Mathematical Institute, Oxford, UK Back to top Rent A Topological Aperitif 2nd edition today, or search our site for Stephen textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Springer.
LAUSD - Venice HS - Algebra 1A Mr. Arambula Guide practice Practice and Due date Chapters and sections Stand. Page do even #s applications do even #s Pre-asssessment 1.1 Variables in Algebra 15 6-8 2-10 14-22 and 38-46 1.2 Exponents and Powers 2 12-13 2-10 14-22 and 38-46 1.3 Order of Operations 25.2 18-19 2-10 14-22 and 38-46 1.4 Equations and Inequalities 5 27-28 2-10 14-22 and 38-46 1.5 Tranlating words into mathematical sysmbols 5.15 33-35 2-10 14-22 and 38-46 Chapter Review 55-58 2,4,6,8,10 12,18,24,34,38 Test 1 Project chapter 1 2.1 The Real Number Line 1 68-70 2-10 14-22 and 38-46 2.2 Absolute value 2,3,24.3,25.1 74-75 2-10 14-22 and 38-46 2.3 Adding Real Numbers 1 81-82 2-10 14-22 and 38-46 2.4 Subtracting real numbers 1,2 89-90 2-10 14-22 and 38-46 2.5 Multiplication of Real Numbers 1,15 96-97 2-10 14-22 and 38-46 2.6 The Distributive Property 1,4 103-4 2-10 14-22 and 38-46 2.7 Combining like terms 4,15 110-1 2-10 14-22 and 38-46 2.8 Division of Real Numbers 1,2,15,17 116-7 2-10 14-22 and 38-46 Chapter Review 121-4 2,4,6,8,10 12,18,24,34,38 Test 2 Project chapter 2 3.1 Solving Equations Using Addition and Subtraction 135-6 2-10 14-22 and 38-46 5 3.2 Solving Equations Using Multiplication and 141-2 2-10 14-22 and 38-46 Division 2,5,15 3.3 Solving Multi-Step Equations 4,5,15 147-8 2-10 14-22 and 38-46 3.4 Solving Equations with Variables on Both Sides 4,5 154-5 2-10 14-22 and 38-46 3.5 More on linear functions 4,5 160-1 2-10 14-22 and 38-46 3.6 Solving Decimal Equations 5 166-8 2-10 14-22 and 38-46 3.7 Formulas and Functions 15 174-6 2-10 14-22 and 38-46 3.8 Rates, and Ratios 15 180-2 2-10 14-22 and 38-46 39.8 Percents 5 186-8 2-10 14-22 and 38-46 Chapter Review 189-92 2,4,6,8,10 12,18,24,26,30 Test 3 Project chapter 3 4.1 Coordinates and Scatter Plots 6 206-8 2-10 14-22 and 38-46 4.2 Graphing Linear Equations 6,7 213-4 2-10 14-22 and 38-46 4.3 Quick Graphs Using Intercepts 6,16,17 219-21 2-10 14-22 and 38-46 4.4 The Slope of a Line 6 225-6 2-10 14-22 and 38-46 4.5 Direct Variation 6,7 233-5 2-10 14-22 and 38-46 4.6 Quick Graphs Using Slope-Intercept Form 6 239-41 2-10 14-22 and 38-46 4.7 Solving Linear Equations Using Graphs 6,8 246,7 2-10 14-22 and 38-46 4.8 Functions and Relations 16,17,18 255,6 2-10 14-22 and 38-46 Chapter Review 259-62 2,4,6,8,10 12,18,24,26,30 Test 4 Project chapter 4 2,4,6,8,10 24-44 5.1 Writing Linear Equations in Slope-Intercept Form 272-4 2-10 14-22 and 38-46 7 5.2 Writing Linear Equations Given the Slope and a 281-3 2-10 14-22 and 38-46 Point 7,8 5.3 Writing Linear Equations Given Two Points 7 288-90 2-10 14-22 and 38-46 5.4 Fitting a Line to Data 7 294-5 2-10 14-22 and 38-46 5.5 Point-Slope Form of a Linear Equation 7,15 301-4 2-10 14-22 and 38-46 5.6 The Standard Form of a Linear Equation 7,8 309-11 2-10 14-22 and 38-46 Chapter Review 313-6 2,4,6,8,10 12,18,24,26,30 10/29/2012 1 LAUSD - Venice HS - Algebra 1A Mr. Arambula Test 5 Project chapter 5 6.1 Solving One-Step Linear Inequalities 3,5 326-7 2-10 14-22 and 38-46 6.2 Solving Multi-Step Linear Inequalities 3,5 333-4 2-10 14-22 and 38-46 6.3 Solving Compound Inequalities 4,5 339-40 2-10 14-22 and 38-46 6.4 Solving Absolute-Value Equations and Inequalities 345-6 2-10 14-22 and 38-46 5 6.5 Graphing Linear Inequalities in Two Variables 5,15 351-3 2-10 14-22 and 38-46 6.6 Stem-and-Leaf Plots and Mean, Median, and Mode 358-9 2-10 14-22 and 38-46 3 6.7 Box-and-Whisker Plots 3,15 364-5 2-10 14-22 and 38-46 6.8 Graphing linear inequalities in two variables 6 370-1 2-10 14-22 and 38-46 Chapter Review 3758 2,4,6,8,10 12,18,24,26,30 Test 6 Project chapter with your answer. Answers by themselves do not count. 10/29/2012 2 LAUSD - Venice HS - Algebra 1A Mr. Arambula end 3 LAUSD - Venice HS - Algebra 1B Mr. Arambula Guide practice Practice and Due date Chapters and sections Stand. Page do even #s applications do even #s Pre-asssessment 7.1 Solving Linear Systems by Graphing 9 392-4 2-10 14-22 and 38-46 7.2 Solving Linear Systems by Substitution 9 399-401 2-10 14-22 and 38-46 7.3 Solving Linear Systems by Linear Combinations 9 405-7 2-10 14-22 and 38-46 9/14/2006 7.4 Applications of Linear Systems 9,15 412-4 2-10 14-22 and 38-46 7.5 Special Types of Linear Systems 9 420-22 2-10 14-22 and 38-46 7.6 Solving Systems of Linear Inequalities 9 427-9 2-10 14-22 and 38-46 Chapter Review 431-4 2,4,6,8,10 12,18,22,26,28 Test Project chapter 7 8.1 Multiplication Properties of Exponents 2 446-7 2-10 14-22 and 38-46 9/26/2006 8.2 Zero and Negative Exponents 2 452-3 2-10 14-22 and 38-46 9/27/2006 8.3 Division Properties of Exponents 16,17 458-9 2-10 14-22 and 38-46 9/28/2006 8.4 Scientific Notation 2 465-6 2-10 14-22 and 38-46 9/29/2006 8.5 Exponential Growth Functions 2 472-3 2-10 14-22 and 38-46 10/6/2006 8.6 Exponential Decay Functions 16 479-81 2-10 14-22 and 38-46 10/9/2006 8.7 Exponential Decay Functions 16 485-7 2-10 14-22 and 38-46 10/10/2006 Chapter Review 489-92 2,4,6,8,10 12,18,24,34,38 10/10/2006 Test Project chapter 8 9.1 Solving Quadratic Equations by Finding Square 502 2-10 14-22 and 38-46 Roots 2 9.2 Simplifying Radicals 2,15,23 508 2-10 14-22 and 38-46 10/13/2006 9.3 Graphing quadratic functions 2,15,24 514 2-10 14-22 and 38-46 9.4 Solving Quadratic Equations by Graphing 16,21,23 523-4 2-10 14-22 and 38-46 9.5 Solving Quadratic Equations by the Quadratic 529-30 2-10 14-22 and 38-46 Formula 21,23 9.6 Applications of the Discriminant 15,19,20,23 536-7 2-10 14-22 and 38-46 9.7 Graphing Quadratic Inequalities 22,23 543-4 2-10 14-22 and 38-46 9.8 Comparing Linear, Exponential, and Quadratic 550-2 2-10 14-22 and 38-46 Models 25.3 Chapter Review 553-6 2,4,6,8,10 12,18,24,26,30 Test Project chapter 9 10.1 Adding and Subtracting Polynomials 10 571-2 2-10 14-22 and 38-46 10.2 Multiplying Polynomials 10 578-9 2-10 14-22 and 38-46 10.3 Special Products of Polynomials 10 585-6 2-10 14-22 and 38-46 10.4 Solving Polynomial Equations in Factored Form 591-2 2-10 14-22 and 38-46 10,14 10.5 Factoring x2 + bx + c 11,14 599-600 2-10 14-22 and 38-46 10.6 Factoring ax2 + bx + c 11,14,15,23 606-7 2-11 14-22 and 38-47 10.7 Factoring Special Products 11,14,23 613 2-12 14-22 and 38-48 10.8 Factoring Using the Distributive Property 11 620-1 2-13 14-22 and 38-49 Chapter Review 623-6 2,4,6,8,10 12,18,24,26,30,34,40,46 Test Project chapter 10 11.1 Ratio and Proportion 13 636-8 2-10 14-22 and 38-46 11.2 Percents 13 642-4 2-10 14-22 and 38-46 11.3 Direct and Inverse Variation 10,12 649-50 2-10 14-22 and 38-46 11.4 Simplifying Rational Expressions 13 655-6 2-10 14-22 and 38-46 11.5 Multiplying and Dividing Rational Expressions 13 660-1 2-10 14-22 and 38-46 11.6 Adding and Subtracting Rational Expressions 13,15 667-8 2-10 14-22 and 38-46 11.7 Dividing Polynomials 13,15 674-5 2-10 14-22 and 38-46 11.8 Rational Equations and Functions 13 680 2-10 14-22 and 38-46 10/29/2012 4 LAUSD - Venice HS - Algebra 1B Mr. Arambula Chapter Review 681-4 2,4,6,8,10 12,18,28,34,38,44 Test Project chapter 11 12.1 Square-Root Functions 2,16,17 695-6 2-10 14-22 and 38-46 12.2 Operations with Radical Expressions 2 701-2 2-10 14-22 and 38-46 12.3 Solving Radical Equations 2 707-8 2-10 14-22 and 38-46 12.4 Completing the Square 2 713-4 2-10 14-22 and 38-46 12.5 The Pythagorean Theorem and Its Converse 14,19,23 719-20 2-10 14-22 and 38-46 12.6 The Distance and Midpoint Formulas 2,24.2 727-8 2-10 14-22 and 38-46 12.7 Trigonometric Ratios 2,25 733-5 2-10 14-22 and 38-46 12.8 Logical Reasoning: Proof 25.1 738-9 2-10 14-22 and 38-46 Chapter Review 747-50 2,4,6,8,10 12,18,24,26,30 Test Project chapter 12 Final California Content Standards for Algebra: 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. 11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. 14.0 Students solve a quadratic equation by factoring or completing the square19.0 Students know the quadratic formula and are familiar with its proof by completing the square. 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. 21.0 Students graph quadratic functions and know that their roots are the x- intercepts. 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. 24.0 Students use and know simple aspects of a logical argument: 24.2 Students identify the hypothesis and conclusion in logical deduction. 25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements: highlighter. Directions: 1. On the first page (top right) write your first and last name, section number, page number, period number, and subject matter (algebra or geometry). 2. Keep it clean, neat, and organize. 3. Label each one with its respective number. 4. Show the entire math process with your answer. Answers by themselves do not count. 10/29/2012 5 LAUSD - Venice HS - Algebra 1B Mr. Arambula 3. Label each one with its respective number. presentation+ work habits and cooperation + participation + tests + quizzes+ posters+others. 7-12 6 LAUSD - Venice HS - Algebra 1A Mr. Arambula Student Students Homework Due Chapters and sections Stand. text Assignments Helper date Pre-asssessment 1.1 Designing a patio 25 p8#4 P.1 #5-8 P.1 # 4-6 ####### 1.2 Lemonade,anyone? 25 1.3 Dinner with the stars 2, 25 P. 16 # 6 P5 # 1 and 5-10 P4 # 7-8 ####### 1.4 Working for the cia 1.5 Gauss' formula 2 P. 24 # 2-5 P9 # 1-5 P.7 # 1-5 ####### $8 an hour problem 2 P. 28 #13 p P11 #2 P8-9 #6-7 1.6 30 #20 1.7 The consultant problem 2, 25 P 11 # 1-3 P 10-11 # 5-6 ####### U.S. shirts 2, 25 P. 38-39 #1- P. 15 # 6-11 P. 12 # 4-8 ####### 1.8 7 Hot shirts P 42-3 # 1- P. 19 # 1-6 P. 13 # 1-5 ####### 1.9 2, 25 6 Comparing U.S. shirts and hot shirts P. 46-7 #2- P 21. 5-7 P. 15 # 3-7 ####### 1.10 26 Chapter Review Test 1 Poster 1 ####### 2.1 Left-Handed learners Making punch P. 59-61 # P. 25 #1-4 P. 19 # 1-4 ####### 2.2 15 2-6 2.3 Shadows and proportions 2.4 TV news ratings 2.5 Women at a university 2.6 Tipping in a restaurant 2.7 Taxes Deducted from your paychech Chapter Review Test 2 Poster 2 3.1 Collecting road tolls 3.2 Decorating the math lab 3.3 Earning sales commissions Rent a car from go-go car rentals, wreckem Rentals, and 3.4 good rents rents rentals 3.5 plastic containers 3.6 eEngeneering a high way 3.7 Shipwreck at the bottom of the sea 3.8 Engineering a highway Chapter Review Test 3 Poster 3 4.1 Up, up, and away! 4.2 Moving a sand Pile 4.3 Let's bowl! 4.4 Math magic 4.5 Numbers in your everyday life 4.6 Technology Reporter 4.7 Rules of sports Chapter Review Test 4 Poster 4 5.1 Widgets, dumbbells, and dumpsters 5.2 Selling Balloons 5.3 Recycling and saving 10/29/2012 7 LAUSD - Venice HS - Algebra 1A Mr. Arambula 5.4 Running in a marathon 5.5 Saving money 5.6 Spending money 5.7 The school play 5.8 Earning Interest Chapter Review Test 5 Poster 5 6.1 Mia's growing like a weed 6.2 Where do you buy your music 6.3 Stroop test 6.4 Jumping 6.5 Human chain; wrist experiment 6.6 Human chain; Shoulder experiment 6.7 making a quilt Chapter Review Test 6 PosterCarnegie. Algebra 1 (2006). Carnegie Learning TI-84 Plus graphing calculator, and laptop10/29/2012 8 LAUSD - Venice HS - Algebra 1A Mr. Arambula with Conduct Responsibility10/29/2012 9 LAUSD - Venice HS - Algebra 1A Mr. Arambula E-mail contact: mla2804@lausd.k12.ca.us Room 108 10/29/2012
series is to provide readers with a strong foundation in Algebra. Each book is designed to develop readers' critical thinking and problem-solving capabilities and prepare readers for subsequent Algebra courses as well as "service" math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. The Real Number System. Linear Equations and Inequalities in One Variable. Problem Solving. Linear Equations and Inequalities in Two Variables. Systems of Linear Equations and Inequalities. Exponents and Polynomials. Factoring Polynomials. Rational Expressions. Roots and Radicals. Quadratic Equations and Functions. For anyone interested in introductory and intermediate algebra and for the combined introductory and intermediate algebra.
Math 120 Rubric Unacceptable Needs Improvement Target Antiderivatives: Integration by Cannot recognize which function Cannot do all three levels as Can recognize when to apply this Parts is f(x) and which function is g(x) described in the target category; technique; can complete the and their derivatives; cannot can only complete elementary following problem types: complete a "level one" problem problems 1. No repetition problems 2. Repetitions 3. Tabular method Antiderivatives: Partial Fractions Cannot do these problems Cannot do all four levels as Can recognize when to apply this described in target category technique; can complete the following problem types: 1. Linear factors, non- repeated 2. Repeated linear factors 3. Irreducible quadratics 4. Repeated irreducible quadratics Antiderivatives: Trig Functions Cannot do these problems Can do only elementary functions Can find the antiderivatives of elementary and complex trig functions Indeterminate Forms Cannot do these problems Can do standard problems but has 0  trouble with non-standard ones Can do and indeterminate 0  forms and apply L'Hopital's rule; can convert non-standard indeterminate forms into standard forms and solve Evaluate Improper Integrals Cannot do Necessary Test or Struggles with L'Hopital's rule Apply Necessary Test to L'Hopital's Rule and/or finding antiderivatives determine if improper integrals are real numbers and find an antiderivative Limit of a Sequence Cannot find limit Can only find limits of basic Can find the limit of a sequence, functions, cannot apply Squeeze can apply the Squeeze Theorem Theorem or L'Hopital's rule and L'Hopital's Rule Infinite Series Cannot do tests for convergence Can do individual convergence Determines convergence or tests but cannot apply the correct divergence by applying the test to a given series appropriate convergence test Power Series Functions Cannot determine convergence Can do easier problems but not Given a power series, can interval challenging ones determine an interval for convergence Plane Curves Cannot analyze curves Can only do some tasks in the Can find slope, length, and target category and/or cannot determine concavity analyze all curves Polar Coordinates Cannot set up these problems May be able to set up arc length Can convert between polar and or area but cannot finish the rectangular coordinates; can find problem the distance between points in polar form; can sketch a graph of r  f ( ) ; can find arc length, area, and slope Vectors Cannot do these problems Cannot do cross product and/or Can do the following with vectors: cannot write the equation of a addition, scalar multiplication, dot plane or line product, and cross product; can find the magnitude in order to write the equation of a plane or line in 3-dimensional space Vector-Valued Functions Cannot do these problems Cannot do more than "level one" Can find limits and integrate and problems (polynomials in each differentiate vector-valued component) functions Functions of 2 Variables: Limits Cannot do these problems Cannot find the limit or cannot Knows the definition of f(x,y) and determine continuity of f(x,y) at a can apply it to find simple limits; point can determine if f(x,y) is continuous at a point Functions of 2 Variables: Partial Cannot do these problems Can do elementary but not more Can find partial derivatives of Derivatives complex functions functions of 2 variables Functions of 2 Variables: Cannot do these problems Can do elementary but not more Can find the directional derivative Directional Derivatives complex functions of a function at a given point Functions of 2 Variables: Tangent Cannot do these problems Can do elementary but not more Given a function and a point the Plane complex functions function passes through, can find the equation of a plane Functions of 2 Variables: Relative Cannot find any candidates for Cannot solve the system of Given a function of 2 variables, Extrema extrema equations to find all candidates can find candidates for relative and/or cannot do mechanics of extrema and can apply the 2nd nd the 2 partials test partials test to determine max, min, saddle points Functions of 2 Variables: Multiple Cannot do these problems Cannot set up the integrals and/or Can evaluate multiple integrals, Integrals make necessary changes as changing the order of integration described in target category or changing a variable to polar coordinates as needed
Problem Solving Approach to Mathematics for Elementary School Teachers, A (11th Edition) 9780321756664 ISBN: 0321756665 Edition: 11 Pub Date: 2012 Publisher: Addison Wesley Summary: This is the leading textbook for students learning how to teach mathematics to elementary school students, focusing on problem solving. It remains current with its discussion of standards in teaching today and it teaches students the value of professional development for their future careers. It encourages active learning and provides many exercises, study tools and opportunities for active learning. Students will ga...in valuable insight into how they can apply their mathematics and teaching knowledge in the classroom. We offer many high quality discounted mathematics textbooks to buy or rent by semester. Rick Billstein is the author of Problem Solving Approach to Mathematics for Elementary School Teachers, A (11th Edition), published 2012 under ISBN 9780321756664 and 0321756665. Seven hundred sixty four Problem Solving Approach to Mathematics for Elementary School Teachers, A (11th Edition) textbooks are available for sale on ValoreBooks.com, one hundred ninety seven used from the cheapest price of $87.89, or buy new starting at $127. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less]
Advanced Mathematical Concepts Precalculus With Applications 9780078608612 ISBN: 0078608619 Publisher: Glencoe/McGraw-Hill Summary: "Advanced Mathematical Concepts" on page T4 of the Teacher Wraparound Edition. "Advanced Mathe...matical Concepts" lessons develop mathematics using numerous examples, real-world applications, and an engaging narrative. Graphs, diagrams, and illustrations are used throughout to help students visualize concepts. Directions clearly indicate which problems may require the use of a graphing calculator. McGraw-Hill Staff is the author of Advanced Mathematical Concepts Precalculus With Applications, published under ISBN 9780078608612 and 0078608619. Six hundred thirty one Advanced Mathematical Concepts Precalculus With Applications textbooks are available for sale on ValoreBooks.com, five hundred twenty three used from the cheapest price of $6.19Book Leaves in 1 Business Day or Less! Leaves Same Day if Received by 2 pm EST! Cover is worn, mostly corners and binding. Binding is cracked, and it may be water damaged but [more] Book Leaves in 1 Business Day or Less! Leaves Same Day if Received by 2 pm EST! Cover is worn, mostly corners and binding. Binding is cracked, and it may be water damaged but still usable. Acceptable. MI 3E[less]
97805218030 Complex Variables for Scientists and Engineers This introduction to complex variable methods for scientists and engineers begins by carefully defining complex numbers and analytic functions and then offers accounts of complex integration, Taylor series, singularities, residues, and mappings. Both algebraic and geometric tools are employed to provide the greatest understanding, with many diagrams illustrating the concepts introduced. The book emphasizes the importance of understanding the use of methods, rather than on rigorous proofs. The book's devotion to applications of the material to physical problems will appeal to scientists. Example applications include detailed treatments of potential theory, hydrodynamics, electrostatics, gravitation and the uses of the Laplace transform for partial differential equations. With 300 stimulating exercises and solutions it will be highly suitable for students wishing to learn the elements of complex analysis in an applied context
Shipping prices may be approximate. Please verify cost before checkout. About the book: The main objective of this work is to develop a thorough understanding of the structure of graphs and the techniques used to analyze problems in graph theory. Fundamental graph algorithms are also included. Examples and over 600 exercises - at various levels of difficulty - guide students. Hardcover, ISBN 0132278286 Publisher: Pearson, 1995 Usually dispatched within 1-2 business days, Dispatched from North London; please allow 9-13 working days for delivery. Prompt and Friendly customer service. Hardcover, ISBN 0132278286 65 # Bookseller Notes Price 1. Better World Books via United States Hardcover, ISBN 0132278286 Publisher: Pearson Education (US), 1996 Very Good. US Edition. Great condition for a used book! Minimal wear. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!. Hardcover, ISBN 0132278286 Publisher: Prentice Hall, 1996 Used - Very Good. Great condition for a used book! Minimal wear. Shipped to over one million happy customers. Your purchase benefits world literacy! Hardcover, ISBN 0132278286 Publisher: Prentice Hall College Div32278286 Publisher: Prentice Hall College Div, 1995 Used - Good, Usually ships in 1-2 business days, This Book is in Good Condition. Clean Copy With Light Amount of Wear. 100% Guaranteed.
MATLAB, a software package for high-performance numerical computation and visualization, is one of the most widely used tools for science and engineering applications. Its broad appeal lies in its interactive environment with hundreds of built-in functions for technical computation, graphics, and animation. It also provides easy extensibility with its own high-level programming language. Getting Started with MATLAB 5: A Quick Introduction for Scientists and Engineers gets students started in MATLAB quickly and easily, in a few short hours. Chapters one and two provide a thorough introduction to the basics and five self-guided lessons. Remaining chapters cover useful and interesting elementary, advanced, and special MATLAB functions.
This calculator app made its first public demonstration last month at the 12th International Congress on Mathematical Education (ICME). Designed to help dyscalculics, Dyscalculator shows quantities in... More: lessons, discussions, ratings, reviews,... Kanakku is a Web 2.0 application that was designed to be used as a productivity utility. The application is a free online utility that is an advanced calculator with a spreadsheet user interface. Use... More: lessons, discussions, ratings, reviews,... A free web based math / scientific calculator specially designed for the education environment. It operates in a very similar way to the popular school calculators and so does not need re-learni... More: lessons, discussions, ratings, reviews,... MathPoint is a suite of math tools for students in grades 6 through 12 and college including color graphing, graphing calculator and interactive solving, and an open library for lessons and activit... More: lessons, discussions, ratings, reviews,... This TI Interactive file is best suited for teacher demonstration. This document allows the user to define a function f(x) and guess the derivative of this function. The slider bar's values range f... More: lessons, discussions, ratings, reviews,... Formulator Tarsia is an editor designed for teachers to create activities in the form of jigsaws, dominos, follow-me cards, etc. for later use in a class. It includes the equation editor for building ... More: lessons, discussions, ratings, reviews,... Geocadabra is dynamic geometry software that supports students learning 2D and 3D geometry, functions and curves (with analysis), and probability. The software was developed in Holland, and is avai... More: lessons, discussions, ratings, reviews,... GraphCalc is an all-in-one solution to everything from everyday arithmetic to statistical analysis, from betas to Booleans, from cubes to calculus, from decimals to derivatives. GraphCalc combines all... More: lessons, discussions, ratings, reviews,... A very powerful graphing program that is also especially easy to use. You can graph functions in two or more dimensions using different kinds of coordinates. You can make animations and save as movies... More: lessons, discussions, ratings, reviews,... This takes all the arithmetic out of matrix problems. With this web site, the student types in their matrix, and then gives the computer the the row-echlon commands - no silly mistakes. This helps th... More: lessons, discussions, ratings, reviews,... A user may enter math problems into the program and the output is a step-by-step solution. It's used primarily for solving expressions, relations, factoring, systems of relations, and other step-by-s... More: lessons, discussions, ratings, reviews,... Math Notepad is an editor for your day-to-day mathematical calculations. It is designed for engineers, teachers, and students. The application is a web-based notepad where you can enter editable expre
Math Connects: Concepts, Skills, and Problem Solving was written by the authorship team with the end results in mind. They looked at the content needed to be successful in Geometry and Algebra and backmapped the development of mathematical content, concepts, and procedures to PreK to ensure a solid foundation and seamless transition from grade level to grade level. The series is organized around the new NCTM Focal Points and is designed to meet most state standards. Math Connects focuses on three key areas of vocabulary to build mathematical literacy, intervention options aligned to RtI, and a comprehensive assessment system of diagnostic, formative, and summative assessments. Book Description:Glencoe/McGraw-Hill. Hardcover. Book Condition: New. 0078740428 new book, never used, Has slight shelf wear due to storage. Has school name and number in book.All books directly from Avitar Books, we never use a 3rd party. Will ship within 24 hours, Monday - Friday!. Bookseller Inventory # SKU0063
Tuba City High students use new technology to exercise their mind with math Tuba City High School calculus students use Texas Instruments graphing calculators to do their assigned math work. Rosanda Suetopka Thayer/NHO 2/12/2013 11:54:00 AM By Rosanda Suetopka Thayer Native students looking to help solve tribal environmental and natural resource problems and find new solutions to water and coal issues will likely find that a solid foundation in basic algebra will help. RayeLynn McCabe, a long-time math teacher at Tuba City High (TCH), advocates helping all students in college-prep or advanced placement programs at TCH to understand algebra, college algebra and even calculus. "I believe that algebra skills are the most important foundation for higher mathematics," McCabe said. "If a student is seeking a post-secondary education, then they will usually take a college algebra class or even calculus or beyond. The better their algebra foundation, the more they'll enjoy those classes. Some students will even find that if they really paid attention in high school, it will pay off in college classes. Our hope at TC High is that after they leave here, they will have the basic skills to be successful in whatever math classes they end up in. I always tell my own students, 'you get what you put into it.'" While some might have difficulty seeing how these advanced math classes have a practical application in everyday life, McCabe sees it differently. "It might seem difficult to see where analyzing polynomial functions would have practical applications in most people's lives but mathematics also requires you to use analytical skills as well as organizational skills. If you're careless you have a tendency to make more mistakes," McCabe said. "Mathematics even tests your patience. A lot of times, math gets a bad rap but when students understand and embrace those relationships as well as other benefits of using mathematics, they will have a much better appreciation of the subject." Algebra skills help with more complex university science classes but taking classes in trigonometry and calculus also contributes to other success in formula development and eventual mathematical probability outcome. McCabe is also using the Promethean Planet whiteboard to project large illustrations to help the entire class see the assigned problems. Sharing problem solutions and worksheets on a big screen creates a more engaging workstudy and round-table environment. "I am still exploring and integrating more ideas from the Promethean Planet website into my classroom work," McCabe said. "Both my students and I are taking baby steps in using this new technology to help us with our graphs, work problems and complex solutions. But it's exciting to have the use of this new technology to assist us." Students in the TCH calculus and advanced math classes also use school owned Texas Instrument graphing calculators. These graphing calculators are invaluable to McCabe teaching in algebra II, functions and calculus classes. It allows the students to be more hands-on and see the actual graphs for themselves.
This course involves the study of the real number system and its applications to personal and social issues in today's world. Students may be introduced to sets, logic, functions, algebraic concepts, geometry, probability, and statistics. Applications to music, art, politics, and business are other possible topics. The topics (and the homework) will be selected for relevance to the students' studies or interests. Subjects may include sets, systems of numeration, geometry, groups, and graph theory, as time allows. It's important to remember that learning mathematics is not a matter of just reading mathematics and writing mathematics: You have to DO mathematics. Goals • To provide students with an appreciation for and a better understanding of mathematical principles. • To develop critical thinking skills needed for problem solving, and to use them to identify and solve problems. The textbook materials are: The online tool "MyMathLab" (MML) for this textbook, ISBN 0-321-19991-X, is required. The textbook itself is optional, as most of the printed material, except for the end-of-section exercises is available online. If you are used to studying from a printed book, you may want to buy the book. On the other hand, if you are comfortable with studying at a computer, you may want to buy just the online tool, as it costs considerably less than the textbook. If you have a credit card or debit card, you can purchased this product online. Using and Understanding Mathematics, 5th edition, Bennett and Briggs, Pearson Addison Wesley, 2011, ISBN 0-321-65279-7 is the version of the textbook without MyMathLab. Using and Understanding Mathematics, 4th edition, Bennett and Briggs, Pearson Addison Wesley, 2011, ISBN 0-321-70895-4 is the version of the textbook bundled with MyMathLab. Reference Other MyMathLab: The course is being taught under MyMathLab, not under UNCP's Blackboard. The course code will be given to you in class or on Blackboard. It is in the form "russellxxxxx", with your professor's last name in lower case letters followed by some numbers. The course code for MAT1050-010-F13 is russell56143 and the course code for MAT1050-1050-800-F13 is russell59238. The access code is the very long number you will find under the pull-away strip inside the MyMathLab package. The web site is " You also need to know the zip code for UNC Pembroke: 28372. Grading Policy Your numerical grade for the course will be the result of your scores on the chapter tests during the semester, your score on the final examination, your grade on online quizzes, your grade on the online homework, and class participation. Class participation includes coming to class regularly, answering questions in class, putting answers to problems on the board in class, and refraining from disruptive behavior in class. Coming to class does not mean just showing up for class. You must be awake, paying attention to the discussion, and participating in the learning of mathematics. That is why we speak of "class participation." Remember that class includes time spent in the computer lab as well as time spent in the classroom. Your letter grade will be determined by the chart shown below in the section entitled "Final Grades". Your letter grade will be the highest letter listed below whose numerical equivalent is not greater than your numerical grade. Grade Components Name Weight Subject Tests during semester 30% MyMathLab is used for this part of the course. Most tests will be on two chapters and will be no longer than a 50-minute meeting. No makeup tests are given. The lowest test grade will be dropped. Online quizzes 20% MyMathLab is used for this part of the course. Online homework 30% MyMathLab is used for this part of the course. Final exam 20% MyMathLab is used for this part of the course. The final exam is held at a time and place determined by the university. The final exam will be in two parts, and both parts must be taken. Final Grades A: 92 B+: 88 C+: 78 D+: 68 F: <60 A-: 90 B: 82 C: 72 D: 62 B-: 80 C-: 70 D-: 60 Attendance Policy For most students, preparing well for class and regular attendance in class, as well as paying attention during class are all necessary parts of doing well in the course. In online classes, regular work on your assignments is also importment. Tell your professor, via email, when you will be away from the Internet for more than two or three days at a time. Student Conduct & Honor Code Student Conduct and Honor Code: Students are expected to act in a manner that promotes learning. The instructor will not allow disruptive behavior or rudeness in the classroom. Students are also expected to do their own work, and to refrain from helping other students to cheat. All students are expected to know and to abide by the UNCP Academic Code, which states that "Students have the responsibility to know and observe the UNCP Academic Honor Code. This code forbids cheating, plagiarism, abuse of academic materials, fabrication, or falsification of information, and complicity in academic dishonesty. Any special requirements or permission regarding academic honesty in this course will be provided to the students in writing at the beginning of the course, and are binding on the students. Academic evaluations in this course include a judgment that the student's work is free from academic dishonesty of any type; and grades in this course therefore should be and will be adversely affected by academic dishonesty. Students who violate the code can be dismissed from the University. The normal penalty for a first offense is an F in the course. Students are expected to report cases of academic dishonesty to the instructor." Cheating will not be tolerated, and any student who cheats or help another student to cheat will be punished severely.
16889184School Textbooks & Study Guides Publication Year: 04/09/2001 Subject 2: School Textbooks & Study Guides: Maths, Science & Technical Format: Paperback Age Level: 14-16 Series: Success Guides Educational Level: UK School Key Stage 4 Language: English ISBN: 9781858059884 Detailed item information Description This is a learning/revision tool designed to help students remember key information for GCSE maths higher. Each topic is presented on a double page spread using diagrams with integrated text, spider diagrams, mind maps and flow charts. It also contains graded GCSE-style questions. Key Features Publisher Letts Educational Date of Publication 04/09/2001 Language(s) English Format Paperback ISBN-10 1858059887 ISBN-13 9781858059884 Genre School Textbooks & Study Guides: Maths, Science & Technical Series Title Success Guides Publication Data Place of Publication London Country of Publication United Kingdom Imprint Letts Educational Out-of-print date 09/08/2008 Content Note diagrams and charts Dimensions Weight 357 g Width 211 mm Height 297 mm Pagination 112 Age Details Educational Level UK School Key Stage 4 Reading Age From 14 to 16 Interest Age 14-16
Description of Saxon Math 2: Complete Homeschool Kit by SaxonGeared specifically toward the homeschool classroom, Saxon Math 2 was designed to build the mathematical foundation necessary for students to transition successfully into higher-level math courses
University-Level Courses for Students in Grades 7-12 For over 20 years University-Level Courses at the Institute for Mathematics and Computer Science (IMACS) have been providing advanced middle and high school students an outlet for their academic talents and an incredible head start over their peers upon entering college. Our self-paced courses are praised by teachers and university professors for their technique and content and by former students for the profound advantage our courses gave them throughout their lives. What Will My Child Learn? IMACS offers two University-Level Courses, one in computer science and one in mathematics. University Computer Science The university-level computer science program at IMACS goes well beyond just learning to code. Our courses emphasize computational thinking by teaching students how to apply mathematical abstraction to concrete computer algorithms. Rather than focusing narrowly on ideas relevant only to a specific coding environment, University Computer Science develops problem-solving skills that can be applied to any programming situation. In addition to building a deep foundation in computer science, students who complete the entire University Computer Science curriculum gain significant experience in four diverse programming languages including Scheme, Haskell, Python and Java. IMACS recommends the following sequence of university-level computer science courses: AP® Computer Science: A comprehensive Java programming course leading to the Advanced Placement Computer Science exam. This course is qualified through the College Board's rigorous AP® Course Audit process. IMACS computer science students develop the skills and tenacity to solve large, technically complex problems by learning how to break them down into smaller, sensible pieces. Consequently, they have significantly outperformed their peers at universities such as MIT, Stanford, Caltech, Yale, Penn and Johns Hopkins! In fact, Ryan Newton, a graduate of our Computer Science program was accepted to the University of Indiana where he was exempted from all undergraduate computer science classes! University Mathematics The university-level mathematics program at IMACS is designed for middle and high school students with a natural talent for mathematics, a high degree of self-discipline, and a strong motivation to learn. University Mathematics courses are based on the IMACS Elements of Mathematics textbooks, which were developed by an international team of eminent mathematicians and mathematics educators to be both rigorous and engaging for young students. The curriculum begins with topics covered in the logic courses typically required of a college major in mathematics, engineering, computer science or philosophy. It then goes on to introduce more advanced techniques in mathematical logic and reasoning. IMACS students take University Mathematics courses in the following sequence: Logic for Mathematics I: An introduction to propositional logic, a branch of modern mathematics that provides the foundation for formal and rigorous mathematical proofs. Logic for Mathematics II: An introduction to predicate logic, a so-called "first-order logic" sufficient to formalize all of set theory, which provides the basic language in which most mathematical texts are written. Logic for Mathematics III: An introduction to axiomatic set theory, which plays a central role in modern mathematics and is fundamental to understanding math at its most sophisticated levels. Logic for Mathematics IV: After making the transition from demonstrations to paragraph proofs, conducts an in-depth study of relations, mappings and functions and considers the Axiom of Choice. Students who successfully complete the entire University Mathematics curriculum develop superior abstract reasoning abilities that make all future classes requiring critical thinking — from computer science to engineering to philosophy to pre-law — significantly easier. One of our graduates, Scott Caplan, completed our university-level classes in Real Analysis, Linear Algebra and Group Theory. He was heavily recruited by all of the top universities and finally settled on Yale. "In my first year at Yale everybody in my dorm was asking me questions about math proofs that seemed to me to be completely routine. Thank you IMACS for giving me such great preparation in college!" Are IMACS University-Level Course Right for my Child Many students enroll in our university-level courses because, regardless of age, they are ready for an academic challenge suited to their advanced abilities. Traditional schools often lack the resources or flexibility to meet the intellectual needs of these talented, young children. IMACS university-level courses provide an academic challenge worthy of the brightest students. Your child will truly take pride in his or her ability to solve advanced logic and math problems. When a student completes a course, IMACS provides an official transcript that reports the student's final grade along with a detailed description of the course content. For those who do well in our program, we provide strong, individually tailored letters of recommendations for college applications. In third grade Jennifer Hernandez began working her way through IMACS courses. While just a sophomore in high school she scored a perfect 800 on the SAT Math test. She eventually graduated valedictorian of her class and was accepted to MIT, Duke and a variety of other universities. The educational foundations students build at IMACS help them outperform the expectations of their parents, teachers and professors. For the rest of their lives, whether in school, at work or at play, IMACS graduates simply think better. Visit IMACS for a Free Evaluation Students in grade 7 and above who have mathematical talent and a genuine interest in mathematics or computer science may be ready for IMACS University-Level Courses.
books.google.com - A... High School First Course in Euclidean Plane Geometry A High School First Course in Euclidean Plane Geometry(Google eBook) A postulates of plane geometry and the most common theorems. It promotes the art and the skills of developing logical proofs. Most of the theorems are provided with detailed proofs. A large number of sample problems are presented throughout the book with detailed solutions. Practice problems are included at the end of each chapter and are presented in three groups: geometric construction problems, computational problems, and theorematical problems. The answers to the computational problems are included at the end of the book. Many of those problems are simplified classic engineering problems that can be solved by average students. The detailed solutions to all the problems in the book are contained in the Solutions Manual. A High School First Course in Euclidean Plane Geometry is the distillation of the author's experience in teaching geometry over many years in U.S. high schools and overseas. The book is best described in the introduction. The prologue offers a study guide to get the most benefits from the book. REVIEW by Bill Fredrick The best geometry textbook I have read during my thirty years of teaching. It is truly a first course at the high school level. I strongly recommend it to all geometry teachers and students. It is concise and to the point just as the author describes it in the introduction. The clarity and conciseness of the text are rare in geometry textbooks at the high school level. Many theorems in so many geometry textbooks are wrongly cited as postulates. They are actually theorems in their own right with complete proofs in this book. * The elegance and simplicity of the cover page mirror the content of the book. I haven t been teaching geometry for the past few years, but I read the book by sheer curiosity. One thing included in the book that I have not seen in other books is a unique Preface. The thorough instructions about how to study geometry will be valuable to all students, including home schoolers and other independent learners. If I am asked to teach high school geometry again I will follow those instructions. * The theorems are clearly proven following a new pedagogical approach. The proofs are followed by nicely detailed solutions illustrating how to solve proof problems using the theorems. * The problems at the end of the chapters are pedagogically laid out in construction, computational, and proof problems. The book is well illustrated with simplified real world problems. The engineering design problems are wonderful. Most of all, this is a plain geometry textbook without pork. I rate it an excellent geometry first course at the high school level. * According to the author, the book has two companions, a solutions manual which contains detailed solutions of all the problems in the book and a package of power point presentations for the geometric construction problems. Both of these would be excellent resources for all users of the book.
Algebra 1 9780078738227 ISBN: 0078738229 Pub Date: 2007 Publisher: McGraw-Hill Higher Education Summary: THE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! "Glencoe Algebra 1" is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scores on today's high-stakes assessments. McGraw-Hill Staff is the author of Algebra 1, published 2007 under ISBN 9780078738227 and 0078738229. Five hundred ...thirty seven Algebra 1 textbooks are available for sale on ValoreBooks.com, three hundred thirty six used from the cheapest price of $8.49, or buy new starting at $34
Elements Of Abstract Analysis - 1 edition Summary: While there are many books on functional analysis,Elementsof Abstract Analysistakes a very different approach. Unlike other books, it provides a comprehensive overview of the elementary concepts of analysis while preparing students to cross the threshold of functional analysis.The book is written specifically for final-year undergraduate students who should already be familiar with most of the mathematical structures discussed - for example, rings, linear spaces, and metric spaces - and with man...show morey of the principal analytical concepts - convergence, connectedness, continuity, compactness and completeness. It reviews the concepts at a slightly greater level of abstraction and enables students to understand their place within the broad framework of set-based mathematics.Carefully crafted, clearly written and precise, and with numerous exercises and examples, Elements of Abstract Analysis is a rigorous, self-contained introduction to functional analysis that will also serve as a text on abstract mathematics. ...show less New Book. Shipped from UK within 4 to 14 business days. Established seller since 2000. $47.24 +$3.99 s/h New PaperbackshopUS Secaucus, NJ New Book. Shipped from US within 4 to 14 business days. Established seller since 2000 $60.68 +$3.99 s/h VeryGood Herb Tandree Philosophy Books Stroud, Glos, 2001
Introduces engineering techniques and practices to high school students. This book is designed for a broad range of student abilities and does not require significant math or science prerequisites.'This is a free textbook offered by BookBoon.'This book is addressed to students in the fields of engineering and technology... see more This is a free textbook offered by BookBoon.'This book is addressed to students in the fields of engineering and technology as well as practicing engineers. It covers the fundamentals of commonly used optimization methods in engineering design. These include graphical optimization, linear and nonlinear programming, numerical optimization, and discrete optimization. Engineering examples have been used to build an understanding of how these methods can be applied. The material is presented roughly at senior undergraduate level. Readers are expected to have familiarity with linear algebra and multivariable calculus.' 'The book contains problems with worked solutions, called examples, and some additional problems for which the answers only... see more 'The book contains problems with worked solutions, called examples, and some additional problems for which the answers only are given, which cover the two Bookboon textbooks Control Engineering : An introduction with the use of Matlab and An Introduction to Nonlinearity in Control Systems. Although both books contain several worked examples it was felt that a reader would obtain further benefit from having additional ones to read and others to solve on their own. In both books a major emphasis was to show how the use of Matlab together with Simulink could avoid the tedium of doing some calculations, however, there are situations where this may not be possible, such as some student examinations. Thus in this book as well as working out in many cases the examples 'long hand', the solutions obtained using Matlab/Simulink are also given.' and

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