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9781133103479
Student Solutions Manual for Gustafson/Hughes' College Algebra, 11th: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. |
Modeling Digital Switching Circuits with Linear Algebra describes an approach for modeling digital information and circuitry that is an alternative to Boolean algebra. While the Boolean algebraic model has been wildly successful and is responsible for many advances in modern information technology, the approach described in this book offers new insight and different ways of solving problems. Modeling the bit as a vector instead of a scalar value in the set {0, 1} allows digital circuits to be characterized with transfer functions in the form of a linear transformation matrix. The use of...
Properly use negative numbers, units, inequalities, exponents, square roots, and absolute value
Round numbers and estimate answers
Solve problems with...
Tips for simplifying tricky basic math and pre-algebra operations
Whether you're a student preparing to take algebra or a parent who wants or needs to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you'll build necessary math skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations.
Explanations and practical examples that mirror today's teaching methods
Relevant cultural vernacular and references
Standard For...
With a substantial amount of new material, this best-selling handbook provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides readers from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition includes 20 new chapters that cover combinatorial matrix theory topics, numerical linear algebra topics, more applications of linear algebra, the use of Sage for linear algebra, and much more.
...
Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments.... problems as they are typically presented in algebra courses-and become prepared to solve those problems that were never discussed in class but always seem to find their way onto exams.
Annotations throughout the text clarify each problem and fill in...
The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a whole. The authors develop various applications, in particular to the study of Banach algebras where the numerical range provides an important link between the algebraic and metric structures.
...
The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence. The second, which assumes familiarity with the material in the first, introduces important classes of categories that have played a fundamental role in the subject's development and applications. In addition, after several chapters discussing specific categories, the book develops all the major concepts concerning Benabou's ideas of fibred...
Starting with the most basic notions, this text introduces all the key elements needed to read and understand current research in the field. The first part of the book focuses on core components, including subalgebras, congruences, lattices, direct and subdirect products, isomorphism theorems, clones, and free algebras. The second part covers topics that demonstrate the power and breadth of the subject, such as Jónsson's lemma, finitely and nonfinitely based algebras, primal and quasiprimal algebras, Murskiĭ's theorem, and directly representable varieties. Examples and exercises are...
Bringing the material up to date to reflect modern applications, this second edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. It offers a more complete and involved treatment of Galois theory, a more comprehensive section on Pollard's cubic factoring algorithm, and more detailed explanations of proofs to provide a sound understanding of challenging material. This edition also studies binary quadratic forms and compares the ideal and form class groups. The text includes convenient cross-referencing, a comprehensive index, and...
Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix. This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials. We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem. Then we present a careful and complete proof of...
Practice makes perfect—and helps deepen your understanding of algebra II by solving problems
1001 Algebra II Practice Problems For Dummies takes you beyond the instruction and guidance offered in Algebra II For Dummies, giving you 1001 opportunities to practice solving problems from the major topics in algebra II. Plus, an online component provides you with a collection of algebra problems presented in multiple choice format to further help you test your skills as you go.
Gives you a chance to practice and reinforce the skills you learn in Algebra II class
Helps you refine your...
Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the "bible of computer algebra", gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time...
The second volume of this comprehensive treatise focusses on Buchberger theory and its application to the algorithmic view of commutative algebra. In distinction to other works, the presentation here is based on the intrinsic linear algebra structure of Groebner bases, and thus elementary considerations lead easily to the state-of-the-art in issues of implementation. The same language describes the applications of Groebner technology to the central problems of commutative algebra. The book can be also used as a reference on elementary ideal theory and a source for the state-of-the-art in...
This collection of papers presents a series of in-depth examinations of a variety of advanced topics related to Boolean functions and expressions. The chapters are written by some of the most prominent experts in their respective fields and cover topics ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. Beyond the diversity of the questions raised and investigated in different chapters, a remarkable feature of the collection is the common thread created by the fundamental language,...
The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras...
This book represents a complete course in abstract algebra, providing instructors with flexibility in the selection of topics to be taught in individual classes. All the topics presented are discussed in a direct and detailed manner. Throughout the text, complete proofs have been given for all theorems without glossing over significant details or leaving important theorems as exercises. The book contains many examples fully worked out and a variety of problems for practice and challenge. Solutions to the odd-numbered problems are provided at the end of the book. This new edition contains...
Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of this acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme and demonstrates their importance in a variety of applications. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of...
The theory of Schur–Weyl duality has had a profound influence over many areas of algebra and combinatorics. This text is original in two respects: it discusses affine q-Schur algebras and presents an algebraic, as opposed to geometric, approach to affine quantum Schur–Weyl theory. To begin, various algebraic structures are discussed, including double Ringel–Hall algebras of cyclic quivers and their quantum loop algebra interpretation. The rest of the book investigates the affine quantum Schur–Weyl duality on three levels. This includes the affine quantum Schur–Weyl reciprocity, the... |
Essentials of Trigonometry -Text - 4th edition
Summary: Intended for the freshman market, this book is known for its student-friendly approach. It starts with the right angle definition, and applications involving the solution of right triangles, to help students investigate and understand the trigonometric functions, their graphs, their relationships to one another, and ways in which they can be used in a variety of real-world applications. The book is not dependent upon a graphing calculator |
The approach is developmental.
Although it covers the requisite
material by proving things,
it does not assume that
students are already able at abstract work.
Instead, it proceeds with a great deal of
motivation, many computational examples,
and exercises that range from
routine verifications to (a few) challenges.
The goal is,
in the context of developing the usual material of an
undergraduate linear algebra course, to help raise
each student's level of
mathematical maturity.
Extensive exercise sets, with worked answers to all exercises.
Sometimes material described on the web as a
book is really someone's lecture notes.
That's fine but from notes to a book is a long way.
That way includes things like an index and diagrams
but the most important difference is exercises.
In this text subsections have perhaps two dozen exercises,
spanning a range of difficulty.
with worked answers, including proofs.
Popular.
This book has been downloadable since 1996 and has been used
in hundreds of classes at many schools, as well as by many individuals
for independent study.
Applications.
Each chapter finishes with four or five short
supplemental topics.
These are good for reading or projects, or for
small group work.
Extras.
You can get beamer slides to use in class, and a lab manual
that supplements the material in the text.
Prerequisite:
typically one semester of calculus
(some examples use the derivative operation).
Here Is Linear Algebra
Click to download
Linear Algebra
and the
answers to exercises.
If you save the two files in the same directory
then
clicking on an exercise sends you to its answer and
clicking on an answer sends you to the exercise,
if you use a PDF reader that supports this.
(The cover artwork may not display correctly in some
PDF renderers but it is right in Adobe Reader.)
If you prefer a paper copy then you can buy an official
one from a number of internet
sources, including
Amazon (the price there is $20).
You may also be interested in beamer slides
for classroom presentation which draw from the text source,
and a lab manual that supplements the text material
using Sage.
Material of less general interest.
If you are into LaTeX then you can
clone the source repository.
You could also get the book's
printed version, which is slightly different
than the version above
(and slightly less suitable as an e-book),
along with its cover.
License
To bookstores: I appreciate your being concerned about my rights.
I give instructors permission
to make copies of this material, either electronic or paper,
and give or sell those copies to students.
(Instructors may like to make an extra copy and prorate the price of student
copies so that their copy is free.)
Many schools use this text in this way.
If you have further questions, don't hesitate to contact me.
If you want to modify the text:
please, feel free.
If you can share back your modifications
then I'd be glad to see them, but if not that is fine.
However, as a favor, I ask that you make clear which material
is yours and which is from the main version of the text.
I often get questions or bug reports
and working out what is going on gets frustrating all
around unless authorship is clear.
In particular, one approach that would help is
changing the cover to include a statement about your modifications.
Something like this would be great: " \fbox{The material in the second appendix on induction is not from the main version of the text but has been added by Professor Jones of UBU. For this material contact \url{sjones@example.com}.}"
Can You Help With Linear Algebra?
Feel free to write me with any comments.
I enjoy hearing about people's experiences and I find suggestions helpful,
especially bug reports.
I save these and periodically revise.
If you are an instructor who
has some material that you are able to share back,
such as the additional material above,
then I'd be delighted to see it.
Of course, I reserve the ability to choose whether to use it.
I gratefully acknowledge all the contributions that I use,
or I can keep you anonymous.
In particular, I would welcome exams or problem sets.
Some instructors have reservations about using a text
where the answers to the exercises are downloadable.
(I can't resist noting here that this objection is misguided:
anyone college student knows how to use the Internet to
get copies of the answers to
all widely-available texts.
I'll also say that I have tried witholding the answers and
asking class instructors to email me for copies but
that left me trying to determine identity via email,
which just is not practical.)
Thus additional sets of exercises without answers would help some
instructors.
If you could contribute
your TeX or LaTeX source that'd be great because then
instructors could cut and paste.
I would also welcome contrbutions related to the emerging electronic
tools.
For instance, if you have sets of questions that are suitable for
Moodle quizzes and that you could
share with other users of this book then
write me and we can see about making them available.
The same holds for WeBWorK problem
sets. |
took several discrete mathematics courses in college. I have a few students who I've helped improve their understanding of sets, relations, functions, algorithms, mathematical induction, finance and difference equations. Usually when meeting discrete math students I bring Discrete Mathematics by Richard Johnsonbaugh. |
Almost every student has to study some sort of mathematical proofs, whether it be in geometry, trigonometry, or with higher-level topics. In addition, mathematical theorems have become an interesting course for many students outside of the mathematical arena, purely for the reasoning and logic that is needed to complete them. Therefore, it is not uncommon to have philosophy and law students grappling with proofs. This book is the perfect resource for demystifying the techniques and principles that govern the mathematical proof area, and is done with the standard "Demystified" level, questions and answers, and accessibility. |
College Algebra: A Graphing Approach
Part of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, College Algebra: A Graphing Approach, 5/e, is an ideal student ...Show synopsisPart of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, College Algebra: A Graphing Approach, 5/e, is an ideal student and instructor resource for courses that require the use of a graphing calculator. The quality and quantity of the exercises, combined with interesting applications and innovative resources, make teaching easier and help students succeed. Continuing the series' emphasis on student support, the Fifth Edition introduces Prerequisite Skills Review. For selected examples throughout the text, the Prerequisite Skills Review directs students to previous sections in the text to review concepts and skills needed to master the material at hand. In addition, prerequisite skills review exercises in Eduspace (see below for description) are referenced in every exercise set. The Larson team achieves accessibility through careful writing and design, including examples with detailed solutions that begin and end on the same page, which maximizes the readability of the text. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles |
Mathematics
Courses
MATH 001: Basic Mathematics (3) F, S
Basic math course, including place value, reading and writing numbers; arithmetic
operations on whole numbers; fraction concepts and operations on fractions; decimal
concepts an operations on decimals; ratio and proportion; percentage; U.S. and metric
systems of measurement; numerical geometry; graph reading; operations on signed numbers.
Application of arithmetic to everyday life (word problems) is emphasized throughout
the course.
Introduction to algebra: signed numbers, exponents, roots, evaluation of algebraic
expressions, simplification of algebraic expressions, translation from English to
algebra, solution of linear equations.
MATH 074: Pre-algebra Refresher (1) F,S, Summer
Short course intended for those students who have assessed into Math 4 and wish to
improve their assessment level; those who have successfully completed Math 4 but need
more review; or students who unsuccessfully attempted Math 100 and need review of
pre-algebra skills. Features a computer program to refresh those concepts identified
as needed for each student, plus weekly contact with the instructor. Successful completion
of this course may serve as a petition to challenge Math 4. Course does not replace
a failing grade in Math 4.
MATH 080: Elementary Algebra Refresher (1) F, S, Summer
Short course intended for those students who have assessed into Math 100 and wish
to improve their assessment level; those who have successfully completed Math 100
but need more review; or students who unsuccessfully attempted Math 107 and need review
of elementary algebra skills. Features a computer program to refresh those concepts
identified as needed for each student, plus weekly contact with the instructor. Successful
completion of this course may serve as a petition to challenge Math 100. Course does
not replace a failing grade in Math 100.
MATH 087: Intermediate Algebra Refresher (1) F,S, Summer
Short course intended for those students who have assessed into Math 107 and wish
to improve their assessment level; those who have successfully completed Math 107
but need more review; or students who unsuccessfully attempted Math 120 and need review
of intermediate algebra skills. Features a computer program to refresh those concepts
identified as needed for each student, plus weekly contact with the instructor. Successful
completion of this course may serve as a petition to challenge Math 107. Course does
not replace a failing grade in Math 107.
MATH 090: Orientation To Mathematics Assessment (0.2) F, S
Orientation to math assessment at SBCC. Four testing levels are reviewed. Discussion
of test-taking strategies. Supervised practice testing in a test-like situation.
Beginning algebra, similar to a standard first-year high school algebra course, including
a review of signed numbers and their properties, equations and inequalities in one
variable, graphing linear equations, systems in two variables, integer exponents,
rational and polynomial expressions, quadratic equations, the quadratic formula, graphing
parabolas.
Designed for Nursing and Allied Health professionals, focuses on math skills necessary
to be successful in an Allied Health occupational area. After reviewing basic math
skills and algebra, students learn metric system conversions, conversion among and
between the metric, apothecary and household units of measure, and computational methods
used in the preparation of medications.
This short course is intended for students who wish to review trigonometry topics
before or while taking Calculus or higher courses. A computer program is used to refresh
concepts identified as needed for each student, plus weekly contact with the instructor.
This course is in no way intended to replace Math 138.
Pre-requisites: MATH 138 or MATH 150 with grade of "C" or better or qualifying score
on SBCC placement exam
A supplementary problem-solving course designed for students currently enrolled in
Math 150 or 160.
Introductory course in the theory and applications of ordinary and partial differential
equations. Topics include constant coefficient equations, series techniques, introduction
to LaPlace transforms, separable partial differential equations, and introduction
to stability in nonlinear systems. The necessary linear algebra is developed for a
study of systems of linear differential equations. Topics from linear algebra include
linear transformations and their matrices relative to a given basis and eigenvalues
and eigen-vectors. (CAN MATH 24 or CAN MATH 26 [with MATH 250])
Limitation on Enrollment: Completion of two courses in the Mathematics Department
at SBCC prior to enrolling in an internship course. Five to 10 hours weekly on-the-job
experience.
Structured internship program in which students gain experience in community organizations
related to the discipline. |
Here you'll find everything from ANCIENT MATHEMATICS (with links to materials at the Vatican and in the U.S. Library of Congress) to WOMEN MATHEMATICIANS, (biographies written by students at Agnes Scott College).
There are so many good sites, however, that it can take a long time to find just what you're looking for.
We recommend that for general history, chronologies, and biographies of mathematicians, you begin with the searchable MACTUTOR HISTORY OF MATHEMATICS ARCHIVE from St. Andrews, Scotland - don't miss the articles in their History Topics Index:
As math resources on the Internet continue to expand, the Math Forum has added new categories for classifying sites. The full list of MATHEMATICS ON THE INTERNET BY TOPIC - from Arithmetic, Algebra, and Geometry to Ordinary and Partial Differential Equations or Fourier Analysis and Wavelets - with links to exhaustive lists of resources for each topic, can be found at:
Choose OTHER ADVANCED MATH to find collections assembled by AMS and SIAM or to try the Forum search tools.
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THIRD INTERNATIONAL MATHEMATICS AND SCIENCE STUDY (TIMSS)
The goal of this study is to assess school achievement in mathematics and science in nearly 50 countries. Designed to provide a comparative assessment of international educational achievement to date, it studies student outcomes, instructional practices, curricula, and cultural context.
Don't miss The TIMSS Videotape Classroom Study, a video survey of eighth-grade mathematics lessons in Germany, Japan, and the United States:
TIMSS is a project of the National Center for Education Statistics (NCES), the primary federal entity for collecting and analyzing data related to education in the United States and other nations. For comprehensive information consult: |
Finite Mathematics - 07 edition
ISBN13:978-0618332939 ISBN10: 0618332936 This edition has also been released as: ISBN13: 978-0618732661 ISBN10: 0618732667
Summary: Geared toward business and social science majors in a single semester finite mathematics course, this text equips students with the analytical tools and technological skills they will need in the workplace. Plain language and an easy-to-read style help stress conceptual understanding and reinforce key terms and concepts. At the same time, the incorporation of real-life applications, examples, and data help engage students--even those who have never enjoyed mathematic...show mores. Pedagogy throughout the text helps students analyze data from a variety of approaches, including numeric, algebraic, graphical, literal, and technological. A robust supplement package and exciting new technology program provide students with extensive learning and support, so that instructors can spend more time teaching.
Make It Real projects ask students to collect and analyze data that often mirrors their own experiences. Students are able to better process this information and connect math to their own lives.
Technology Tips are scattered throughout the text and help guide students through new techniques on the graphing calculator such as graphing a function, solving an equation, or finding the value of a function.
Examples, Exercises, and Applications focus on the business or social science student and are based on engaging real-life data.
Eduspace, powered by Blackboard, Houghton Mifflin's online learning tool brings your students quality online homework, tutorials, multimedia, and testing that correspond to the Finite Mathematics text. This content is paired with the recognized course management tools of Blackboard. Use Eduspace to teach all or part of a course online.
Streamlined content allows instructors to cover the text in its entirety 0618332920 +$3.99 s/h
LikeNew
Barnes And Nooyen Books tx Spring, TX
2006-01-05 Hardcover Like New New condition except name on first page.
$4.23 +$3.99 s/h
LikeNew
BookCellar-NH Nashua, NH
0618332995 +$3.99 s/h
Good
omgtextbooks Pueblo West, CO
2006 Hard cover New ed. Good. Sewn binding. Cloth over boards. 608 p.
$18.74 +$3.99 s/h
Good
Big Planet Books Burbank, CA
2006-01-05 |
p>This Saxon Math Homeschool 7/6 Tests and Worksheets book is part of the Saxon Math 7/6 curriculum for 6thA testing schedule and five have been using Saxon Math for 3 years and we will continue to use it until our son has graduated from HS. the curriculum is very easy to follow, the lessons are clearly explained. Before Saxon Math my son did not like to do math now it's all he wants to do! thank you Saxon Math!
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Review 2 for Math 76, Fourth Edition, Tests & Worksheets
Overall Rating:
5out of5
Saxon Math 7/6 Wonderful program
Date:November 4, 2011
Homeschooling Mom
Location:Niles, Michigan
Age:35-44
Gender:female
Quality:
4out of5
Value:
5out of5
Meets Expectations:
5out of5
I was skeptical about this math program at first but I decided to try it. It has turned out to be perfect for my son. I wish I would have used it sooner. It moves along at a nice, steady pace...not too fast, and I love how much review there is. Great program no doubt about it!
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Review 3 for Math 76, Fourth Edition, Tests & Worksheets
Overall Rating:
5out of5
Very useful for the homeschooling parent
Date:June 3, 2011
Stephanie
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
I think this is a great tool. The worksheets are great for daily practice and reinforcement. My child has gone from struggling in math to excelling with Saxon. It is a wonderful program that I highly recommend. My only complaint would be that the test problems are all on one side of a sheet which can become visually overwhelming. It would have been better to use an extra page and spread the material out so the student would not only have more space to work, but also be able to focus on each problem.
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Review 4 for Math 76, Fourth Edition, Tests & Worksheets
Overall Rating:
5out of5
Saxon is the best math ever
Date:March 25, 2011
famcole5
Location:Stone Mountain, GA
Age:35-44
Gender:female
Quality:
5out of5
Value:
4out of5
Meets Expectations:
5out of5
My kids all have used Saxon Math. This purchase of 76 was for my 3rd and final homeschooled child. Not only is it great for them, it also has beefed up my own math skills over the years, making it easier to teach math to each successive child. I would recommend Saxon to any homeschool family.
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+1point
1of1voted this as helpful.
Review 5 for Math 76, Fourth Edition, Tests & Worksheets
Overall Rating:
5out of5
Date:May 8, 2008
Lori A Snyder
We really enjoy SAXON Math. The sequential learning style enables the student to really learn the concepts and use them, rather than just learning for the test and forgetting it. Our children, 5th and 8th graders, have both benefitted greatly from this program. The placement tests at SAXON's website were extremely helpful. Definitely worth 5 stars! |
Students who need additional practice solving linear systems can use this short worksheet that provides immediate feedback as it is self-checking. The answers correspond to letters that decode a... More > secret message, increasing fun and satisfaction for students. Students will also interpret results graphically, indicating that the lines are parallel, the same, or intersecting at a single point. Similar PUZZLE MATH worksheet and books are available including, "PUZZLE MATH: Mixed Derivatives", and "PUZZLE MATH: Trigonometry and Logarithms".< Less
Linear Algebra I is a book for university students of any university branch of science. You will find summaries of theory and exercises solved, of the following topics: Matrices, Resolution of Linear... More > Systems Equations, Vector Spaces, Linear Transformations, Diagonalization of endomorphism, and Bilinear and Quadratic Forms.
I have 20 years of experience teaching mathematics at the university level. And, as a teacher of Algebra, Calculus, Statistics, etc., of university students, and, as a result of the needs that I have seen in my students, I have written this book.
This book is characterized by being practical and didactic. It is also useful as a guide for the student.
I hope it will be useful to you, above allPUZZLE MATH: Solve Equations Using Four Operations is a self-checking decoder-style worksheet. Students will solve 20 equations using addition, subtraction, multiplication, and division. The... More > numbers are positive integers, so performing operations with signed numbers is not a necessary prerequisite skill. If you like this worksheet, you'll enjoy the many other books and worksheets available at Less
This free self-checking decoder-style worksheet has twelve equations. Eight require performing one or two inverse operations to both sides of the equation, and four require performing three inverse... More > operations. Students square root and have to recall to put 'plus or minus' on their answers. This worksheet provides students with instant feedback and fun, and it reduces grading overhead for teachers. If you enjoy this worksheet, you'll love the other PUZZLE MATH publications at Less |
Maths Frameworking - SATs Revision Guide Levels 5-7
Maths Frameworking is the Number 1 Framework mathematics scheme in schools. The Maths Frameworking Revision Guides provide focussed study support and ...Show synopsisMaths Frameworking is the Number 1 Framework mathematics scheme in schools. The Maths Frameworking Revision Guides provide focussed study support and test practice at the right level for every student. Written by highly experienced teachers, these books help offer the best possible preparation for KS3 National Tests. Features include: / Complete summary and practice support in preparation for SATs at level 5-7 / Full-colour, clear layout / Easy-to-navigate layout to help students revise more efficiently / Sample answers with advice on how to achieve maximum marks / Hints and tips on how to raise grades / Links to the number one KS3 school maths scheme Maths Frameworking.. Illustrations |
MathHomeworkAnswers.org is a free math help site for student, teachers and math enthusiasts. Ask and answer math questions in algebra I, algebra II, geometry, trigonometry, calculus, statistics, word problems and more. Register for free and earn points for questions, answers and posts. Math help is always 100% free.
Note: This site is intended to help students and non-students understand and practice math. we do not condone cheating. Users attempting to abuse the system will be permanently blocked from further use. |
Euclidean plane geometry is one of the oldest and most beautiful topics in mathematics. Instead of carefully building geometries from axiom sets, this book uses a wealth of methods to solve problems in Euclidean geometry. Many of these methods arose where existing techniques proved inadequate. In several cases, the new ideas used in solving specific problems later developed into independent areas of mathematics. This book is primarily a geometry textbook, but studying geometry in this way will also develop students' appreciation of the subject and of mathematics as a whole. For instance, despite the fact that the analytic method has been part of mathematics for four centuries, it is rarely a tool a student considers using when faced with a geometry problem. Methods for Euclidean Geometry explores the application of a broad range of mathematical topics to the solution of Euclidean problems.
A unique and refreshing approach to teaching Euclidean geometry which will also serve to enhance a student's understanding of mathematics as a whole
Over a third of the book is given over to detailed problems of varying difficulty, and their solutions
Some of the same exercises are repeated in different chapters so that the student may see how the same problem may be tackled by a number of different methods |
05214235plex Algebraic Curves (London Mathematical Society Student Texts)
This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex |
For introductory courses in Differential Equations. This best-selling text by these well-known authors blends the traditional algebra problem ...Show synopsisFor introductory courses in Differential Equations. This best-selling text by these well-known authors blends the traditional algebra problem solving skills with the conceptual development and geometric visualization of a modern differential equations course that is essential to science and engineering students. It reflects the new qualitative approach that is altering the learning of elementary differential equations, including the wide availability of scientific computing environments like Maple, Mathematica, and MATLAB. Its focus balances the traditional manual methods with the new computer-based methods that illuminate qualitative phenomena and make accessible a wider range of more realistic applications. Seldom-used topics have been trimmed and new topics added: it starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the text1561076-5 Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780131561076-4Acceptable. Book has Damage to SPINE. Book has Damage to COVER....Acceptable. Book has Damage to SPINE. Book has Damage to COVER. Moderate Amount of Wrinkled PagesI like the cover graphic and the cover notes that provide inspiration/motivation for slogging through this often frustrating material. The book is well organized and includes excellent graphics. Lots of homework problems with answers in back. Not quite on par with James Stewart's calculus books, but not far from them in terms of quality |
Sharpen your skills and prepare for your precalculus exam with a wealth of essential facts in a quick-and-easy Q&A format!
Get the question-and-answer practice you need with McGraw-Hill's 500 College Precalculus Questions. Organized for easy reference and intensive practice, the questions cover all essential precalculus topics and include detailed answer explanations.
The 500 practice questions are similar to course exam questions so you will know what to expect on test day. Each question includes a fully detailed answer that puts the subject in context. This additional practice helps you build your knowledge, strengthen test-taking skills, and build confidence. From ethical theory to epistemology, this book covers the key topics in precalculus. |
Math Skills Self-Assessment and Math Preparation
Your Core course faculty require that you be proficient in certain math skills as you begin the program. The Math Self-Assessment will help you to assess your skills in this area. The Math Preparation Course is designed for you to learn or refresh your skills in order to help you be as prepared as possible for your first year's Core classes.
Math Skills Requirement
For economics, statistics and finance Core courses, you will need to be proficient in:
Familiarity with basic calculus, at the level of taking derivatives of polynomial expressions and using first-order conditions to determine maxima and minima
Math Skills Self-Study
We strongly advise you to study and/or review the Math Skills Self-Study Materials before you start your Core classes at Stern. Your professors will assume that you have this knowledge, and many of the concepts that you will be learning will rely on these skills.
Math Skills Self-Assessment
Your Core course faculty have created a 20-question Math Skills Self-Assessment to help you assess your skills in this area. It should take you between 20-45 minutes to complete. We ask that you take the self-assessment before arriving at Stern. Use the Answer Key to score your exam and learn your results. Your results will be a good indicator as to whether you need to take the Math Preparation Course.
Process
Study and/or review the Math Skills Self-Study Materials
Take the Math Skills Self-Assessment (20 questions)
Score your assessment exam using the Answer Key
15 or fewer questions correct – Enroll in the Math Preparation Course
All others: If you'd like a refresher on your math skills, you may also enroll.
Math Preparation Course
This preparation course contains three classroom sessions, designed to help you refresh or learn the required math skills so that you are best prepared for your first-year Core classes.
The following topics will be covered in three sessions, totaling about nine hours.
Methodologies are illustrated with business examples.
Session Number
Topic Description
Session 1
1. Algebra Review
a. Laws of Exponents
b. Adding and Subtracting Rational Expressions
c. Quadratic Formula and Factoring
d. Summation Notation
e. Solving Systems of Equations
f. Solving Systems of Inequalities
2. Functions and Linear Models
a. Rate of Change
b. Average Rate of Change
Session 1/
Session 2
3. The Mathematics of Finance and Nonlinear Models
a. Simple Interest
b. Compound Interest
c. Exponential functions and logarithmic functions
d. Continuously Compounded Interest
e. Effective Rate
f. Logarithmic Rate of Return
Session 2/
Session 3
4. Introduction to the Derivative
a. Introduction to Limits
b. Definition of the Derivative
5. The Shortcuts of Differentiation
a. Power Rule
b. Product Rule
c. Quotient Rule
d. Chain Rule
e. The derivative of exponential functions
f. The derivative of logarithmic functions
Session 3
6. Applications of the Derivative: Optimization
a. Critical Points
b. First Derivative Test
c. The Second Derivative
d. The Second Derivative Test
e. Convexity and Concavity
f. Inflection Points
g. Applications to Problems in Economics and Business
You may register for the Math Preparation Course when you complete the Summer/Fall registration forms. A fee of $350 will be charged to your Bursar account for the preparation course. |
Elementary Algebra
Give your students the text that makes algebra accessible and engaging - McKeague's "Elementary Algebra, 9th edition, International Edition". Pat ...Show synopsisGive your students the text that makes algebra accessible and engaging - McKeague's "Elementary Algebra, 9th edition, International Edition". Pat McKeague's passion for teaching mathematics is apparent on every page, and this Ninth Edition continues to provide students with a thorough grounding in the concepts central to their success in mathematics. Attention to detail, an exceptionally clear writing style, and continuous review and reinforcement are McKeague hallmarks that constitute the solid foundation of the text, while new pedagogy help students 'bridge the concepts.' These 'bridges' guide students and help them make successful connections from concept to concept-and from this course to the next. "Elementary Algebra, 9th edition, International Edition" is one of the most current and reliable texts you will find for the course, and is ideally structured and organized for a lecture-format. Each section can be discussed in a 45- to 50-minute class session, allowing you to easily construct your course to fit your64219Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780840064219-4 0840064217 -used book-book appears to be recovered-has...Fair. 0840064217 |
Descriptive analysis and presentation of either single-variable data or bivariate data, probability, probability distributions, normal probability distributions, sample variability, statistical inferences involving one and two populations, analysis of variance, linear correlation and regression analysis. Statistical computer software will be extensively integrated as a tool in the description and analysis of data.
Math 9 is the first step in preparing students for the study of calculus by providing important skills in algebraic manipulation and interpretation at the college level. Topics will include a review of basic algebraic concepts; lines; polynomial and rational functions; exponential and logarithmic functions; trigonometric functions, identities, inverse functions and equations; applications of trigonometry; systems of equations and matrices; conic sections; sequences, series and combinatorics. Hand-held graphing calculators will be used extensively to highlight their strengths and their limitations as a problem-solving tool. Real world applications will be numerous.
Prerequisite: Mathematics 9 with a grade of 'C' or better and proficiency with the TI-83 graphing calculator as gained from, for instance, Math 209.
Transferable: CSU; UC; CSU-GE: B4; IGETC: 2A; GAV-GE: B4
Math 10 prepares students for the study of calculus by providing important critical thinking and problem solving skills. The central theme of the course is the analysis of mathematical functions as models of change. Families of functions - linear, exponential, logarithmic, power, periodic, polynomial, rational - will be introduced, compared and contrasted. Course content will include an introduction to functions and functional notation; transformation of functions; composite, inverse and combinations of functions; vectors and polar coordinates; series; parametric equations; complex numbers. Hand-held graphing calculators will be used extensively to highlight their strengths and their limitations as a problem-solving tool. Real world applications will be numerous.
Sect#
Type
Room
Instructor
Units
Days
Time Start-End
Footnotes
8074
LEC
BU117
HINDERER K
3.0
DAILY
0830A - 1015A
Class meets 06/14/04 - 07/23/04
MATH 205:
Elementary Algebra
Beginning Fall 2005: MATH 402 with a grade of 'C' or better or assessment test recommendation.
Advisory: Mathematics 402 or assessment test recommendation.
Transferable: GAV-GE: B4
This course is a standard beginning algebra course, including algebraic expressions, linear equations and inequalities in one variable, graphing, equations and inequalities in two variables, integer exponents, polynomials, rational expressions and equations, radicals and rational exponents, and quadratic equations. Mathematics 205, 205A and 205B, and 206 have similar course content. This course may not be taken by students who have completed Mathematics 205B or 206 with a grade of "C" or better. This course may be taken for Mathematics 205B credit (2.5 units) by those students who have successfully completed Mathematics 205A with a grade of "C" or better.
A survey of practical geometry with an emphasis on applications to other disciplines and everyday life. Parallel lines, triangles, circles, polygons, three dimensional figures, vectors, and right triangle trigonometry will be covered. There will be a weekly lab.
Beginning Fall 2005: Completion of Math 400 with a 'C' or better, or assessment test recommendation.
Advisory: Mathematics 400.
Transferable: No
This course covers operations with integers, fractions and decimals and associated applications, percentages, ratio, and geometry and measurement, critical thinking and applications. Elementary algebra topics such as variables, expressions, and solving equations are introduced.
Sect#
Type
Room
Instructor
Units
Days
Time Start-End
Footnotes
8079
L/L
PH103
WANG C
3.0
DAILY
0800A - 1020A
Class meets 06/14/04 - 07/23/04
Address of this page:
Please feel free to send us your comments and questions.
Send e-mail messages to webmaster@gavilan.edu
Page generated on Monday, June 14, 2004 at 11:51 PM |
Description: UCI Math 176 covers the following topics: reviewing of tools from probability, statistics, and elementary differential and partial differential equations, concepts such as hedging, arbitrage, Puts, Calls, the design of portfolios, the derivation and solution of the Blac-Scholes, and other equations. |
Calculus, Early Vectors - 99 edition
Summary: Once again keeping a keen ear to the needs of the evolving calculus community, Stewart created this text at the suggestion and with the collaboration of professors in the mathematics department at Texas A&M University. With an early introduction to vectors and vector functions, the approach is ideal for engineering students who use vectors early in their curriculum. Stewart begins by introducing vectors in Chapter 1, along with their basic operations, such as add...show moreition, scalar multiplication, and dot product. The definition of vector functions and parametric curves is given at the end of Chapter 1 using a two-dimensional trajectory of a projectile as motivation. Limits, derivatives, and integrals of vector functions are interwoven throughout the subsequent chapters.As with the other texts in his Calculus series, in Early Vectors Stewart makes us of heuristic examples to reveal calculus to students. His examples stand out because they are not just models for problem solving or a means of demonstrating techniques - they also encourage students to develop an analytic view of the subject. This heuristic or discovery approach in the examples give students an intuitive feeling for analysis.In the Preliminary Edition, Stewart incorporates a focus on problem solving; meticulously attends to accuracy; patiently explains the concepts and examples; and includes the same carefully graded problems that make his other texts work so well for a wide range of students. --This text refers to an out of print or unavailable edition of this title63.56 |
Complex Analysis
9780387950693
ISBN:
0387950699
Pub Date: 2001 Publisher: Springer Verlag
Summary: The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of to...pics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis.
Gamelin, Theodore W. is the author of Complex Analysis, published 2001 under ISBN 9780387950693 and 0387950699. Six hundred thirty seven Complex Analysis textbooks are available for sale on ValoreBooks.com, one hundred twelve used from the cheapest price of $35.74, or buy new starting at $55.23.[read more]
Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). This Book has Notes in Pencil. May have minimal notes/highlighting, minimal wear/tear. Please con... [more]RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). This Book has Notes in Pencil An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the fi [more]
This item is printed on demand. An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The r.[ |
Elementary Number Theory
9780073051888
ISBN:
0073051888
Edition: 6 Pub Date: 2005 Publisher: McGraw-Hill College
Summary: Elementary Number Theory, Sixth Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton's engaging style, Elemen...tary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.
Burton, David M. is the author of Elementary Number Theory, published 2005 under ISBN 9780073051888 and 0073051888. Eighteen Elementary Number Theory textbooks are available for sale on ValoreBooks.com, thirteen used from the cheapest price of $26.99, or buy new starting at $60.53.[read more]
Highlighting, underlining, marks & notes in pen & pencil, covers scuffed/scratched, minimal shelf wear, stickers on back cover & spine |
Let's Review : Math A - 2nd edition
Summary: This major revision prepares students to succeed on the New York State Math A Regents Exam as it is now given. The book places increased emphasis on mathematical modeling and the use of the graphing calculator. In line with New York State Regents core curriculum, it shows how given problems can be solved in several different ways. The author also includes new Regents question types dealing, for instance, with motion problems and mathematical systems defined by tables...show more. New contextualized word problems further enhance the presentation. The totally rewritten chapter on problem-solving offers students a core set of strategies that apply to a variety of curriculum-related exercises. In addition to subject review, demonstraton examples, and practice exercises with answers, the book includes several complete recent Math A Regents exams with answers |
Contemporary Mathematics in Real-world Phenomena
Students apply algebraic and geometric principles to environments and phenomena in society, nature, architecture and art. Through an elementary study of game theory, fractals, symmetry, patterns, etc., students investigate how humans play, interact and employ mathematics to understand and optimize real-world events. |
The most helpful favorable review
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214 of 220 people found the following review helpful
5.0 out of 5 starsManhattan GMAT's Review and Analysis of the 13th Ed of The Official Guide for GMAT Review
The OG13 has flaws, as did its predecessors. Why five stars, then? Because, hands down, this latest incarnation of the OG is still the one book to get, if you can only get one book to prepare for the GMAT with. There's simply no better all-in-one source of retired GMAT problems.
Could the explanations be better? Sure. Could the GMAT folks have replaced...
1.0 out of 5 starsWARNING
Received everything ok and on time. Warning: This is not the book to study for GMAT.UNLESS you already have a math AND english degree. The way they explain how they got to the conclusion of each question is missing data that leaves the reader clueless on how the math equation or reading question was reached. I still stuggled through the entire book anyway and I do not...
There are no signifficant diferences between this and the 12th edition. You could study from previous editions and complement with another book for the new section. if you don't have a previous edition then it is ok.
I am a math tutor. I wrote a GMAT test-pre book on amazon and very familiar with the test. This official guide is a good resource for tutors and test-takers. Some of the math solution explanations are unclear and hard to understand by my students, even myself. The GMAT math covers only the basic math areas with arithmetic, algebra, geometry, and word problems. It is not necessary to emphasize the ranking of difficulty on each math question, just need practice, practice, and practice.
The people who sold and shipped this book to me are dishonest. They broke the seal of the "Integrated Reasoning Online Access Code" and stole what should be my access code for their own use. DON'T BUY BOOKS FROM THIS DISHONEST RETAILER!!!
This is the book written by the test makers so it is essential to have it in order to get familiar with the test format and standard question types. However, you may need some other books for your preparation |
A Guide to MATLAB, 2e: for Beginners and Experienced Users
Written for novice and experienced MATLAB users, this book contains explanations of essential MATLAB commands, as well as instructions for using the programming features, graphical capabilities, and desktop interface of MATLAB. Chapters contain examples of how to use MATLAB applications to solve problems in mathematics, engineering, economics, and physics.
MATLAB Courseware
Trials Available
About This Book
Kevin R. Coombes Brian R. Hunt, University of Maryland at College Park Ronald L. Lipsman, University of Maryland at College Park John E. Osborn, University of Maryland at College Park Garrett J. Stuck Jonathan M. Rosenberg, University of Maryland at College Park |
More About
This Textbook
Overview
R is the amazing, free, open-access software package for scientific graphs and calculations used by scientists worldwide. The R Student Companion is a student-oriented manual describing how to use R in high school and college science and mathematics courses. Written for beginners in scientific computation, the book assumes the reader has just some high school algebra and has no computer programming background.
The author presents applications drawn from all sciences and social sciences and includes the most often used features of R in an appendix. In addition, each chapter provides a set of computational challenges: exercises in R calculations that are designed to be performed alone or in groups.
Several of the chapters explore algebra concepts that are highly useful in scientific applications, such as quadratic equations, systems of linear equations, trigonometric functions, and exponential functions. Each chapter provides an instructional review of the algebra concept, followed by a hands-on guide to performing calculations and graphing in R.
R is intuitive, even fun. Fantastic, publication-quality graphs of data, equations, or both can be produced with little effort. By integrating mathematical computation and scientific illustration early in a student's development, R use can enhance one's understanding of even the most difficult scientific concepts. While R has gained a strong reputation as a package for statistical analysis, The R Student Companion approaches R more completely as a comprehensive tool for scientific computing and graphing.
Editorial Reviews
From the Publisher
"This book requires no prior knowledge of calculus, programming, or statistics. … the commands and real-world examples are explained very thoroughly. This should make the book suitable for self-study and hold interest for the target group (high school and college-level students) …"
—Joonas Kauppinen, International Statistical Review (2013), 81, 2
"An R book for high schoolers! This is an excellent idea, and the quality of the product is equally excellent. It may be suitable for non-calculus-based introductory courses at the college level as well. … Dennis does a good job dispelling the 'steep learning curve' myth concerning R … . The writing style is clear and lively, and the examples should appeal to high school students. It is high time that introductory statistics be taught in an engaging manner that reflects our own enthusiasm for the subject, with meaningful data sets, attractive graphics, and so on. Dennis' book is a fine contribution toward that goal."
—Norman Matloff, Journal of Statistical Software, Volume 52, February 2013
Product Details
Meet the Author
Brian Dennis is a professor with a joint appointment in the Department of Fish and Wildlife Sciences and the Department of Statistical Sciences at the University of Idaho. He received a master's degree in statistics and a Ph.D. in ecology from The Pennsylvania State University. He has authored over 70 scientific articles on applications of statistics and mathematical modeling in ecology and natural resource management. He has been enthusiastically using R in his scientific work and teaching R in his courses |
Work out what the normal vector is to a line of the form ax + by = c, and make up a simple example that helps you remember how this works.
Work out what the normal vector is to a plane of the form ax + by + cz = d, and make up a simple example that helps you remember how this works.
(Note an earlier version mistakenly had "line" instead of "plane." My thanks to alert readers.)
Sketch the vector u = <cos A, sin A>, where A is some angle measured in degrees counterclockwise from the x-axis, and also sketch the vector v = <1,0>.
Explain in this simple example why the dot product of u and v is equal to the product of the length of u, the length of v, and the cosine of the angle between u and v.
What happens to this equality when you stretch u by a factor of 2? (Draw a picture. Compute the dot product of this new u and v. Compute the product of the lengths of this new u and v and the cosine of the angle between them.
Does the equality still hold?) Try this again when you stretch v by a factor of 3. What happens to the sides of the equality when you do both: scale u by a factor of 2 and scale v by a factor of 3?
Now consider again the original u and v. Sketch the vectors obtained by rotating both u and v by B degrees counterclockwise (say, B = 90 degrees).
Explain in your own words why the equality described above still holds. (For fun: What happens to the equality if you combine rotations and stretches of u and/or v? What about reflections?)
Read through the sections of the text listed above to acquaint yourself with the author's style and the layout.
Visit the website listed above ("MIT OPEN COURSE LINEAR ALGEBRA WEBSITE"). Watch some parts (or all) of a few of the lectures there to get a feel for what this course is about, and what the lectures look like.
Look at some of the assigned problems (on the MIT site) along with the worked solutions.
Come with questions!
STUDENTS WITH DISABILITIES
Students with disabilities who require special accommodations for access and
participation in this course must be registered with the
Office of Disability Services (ODS). Students who need exam accommodations
must contact ODS in the first week of the term to arrange a
meeting with a Disability Specialist. |
Web Site Dave's Short Trig Course Check out the short trigonometry course and learn the new way of learning trig. This short course breaks into sections and allows user to learn at his/her o... Curriculum: Mathematics Grades: 9, 10, 11, 12
33.
Web Site S.O.S. Mathematics - Calculus Check out a good list of calculus problems with solutions. This is a free resource for math review material from Algebra to Differential Equations!
Web Site Order of Operations When a numerical expression involves two or more operations, there is a specific order in which these operations must be performed. The phrase PEMDA (Parenth... Curriculum: Mathematics Grades: 5, 6, 7, 8
38.
Web Site Intermediate Algebra Tutorials 42 Tutorials that math teachers can use with student or students can work on their own to reinforce skills, as homework, or review during class. Tutorials in... Curriculum: Mathematics Grades: 6, 7, 8, 9, 10, 11, 12, Junior/Community College, University
39.
Web Site Variables This site covers symbol variables and substitution of symbols to discover unknown values. In simple terms it shows you how a box is waiting for a value. (Key... Curriculum: Mathematics Grades: 6, 7, 8
40.
Web Site Introduction to Algebra Think Algebra is hard? Think again - this site explains the history along with simple equations. Each paragraph scaffolds skills until you get it. Than at th... Curriculum: Mathematics Grades: 3, 4, 5, 6 |
Should College Classes Ditch the Calculator?
Should College Classes Ditch the Calculator?
According to Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research and Development Center, using calculators in college math classes may be doing more harm than good. In a limited study conducted with undergraduate engineering students and published in the British Journal of Educational Technology, King has determined that our use of calculators may be serving as an alternative to an actual, deep understanding of mathematical material.
"We really can't assume that calculators are helping students," says King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
King, along with co-author and director of the Mathematics Education Centre at Loughborough University, Carol Robinson, conducted the study by interviewing 10 second-year undergraduate students who were enrolled in a competitive engineering program. The students were given a number of mathematical questions dealing with sine waves, which are mathematical curves that describe a smooth repetitive oscillation. To help solve the problems, the students were given the option of using a calculator instead of completing the work entirely by hand. Over half of the students questioned opted to utilize their calculators in order to solve the problems and plot the sine waves.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," says King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the work, King and Robinson interviewed the students about how they approached the material. One student who used the calculator stated that she had trouble remembering the rules for how sine waves operate, and found it generally easier to use a calculator instead. In contrast, however, a student who opted to complete the work without a calculator stated that they couldn't see why anyone would have trouble completing the question, but did admit that it would likely be easier with a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," says King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
Given the small sample size used in the study, it is entirely possible that King's findings are largely anecdotal in how our usage of calculators and understanding of mathematical concepts may positively or negatively correlate. However, King does stress that while all the evidence may not be in, his study does raise important questions regarding how, when and why students choose to use calculators, and in doing so, we may develop a more holistic approach to math instruction |
Algebra Syllabus
Department of Mathematics
University of Colorado
Group Theory. Basic definitions and examples, lattice of subgroups/normal subgroups, quotient
groups, isomorphism theorems, the characterization of products, Lagrange's Theorem, Cauchy's The-
orem, Cayley's Theorem, the structure of finitely generated abelian groups, group actions, the class
o
equation, Sylow's Theorems, the Jordan H¨lder Theorem, simple groups, solvable groups, semidirect
products, free groups, presentations of groups.
Ring Theory. Basic definitions and examples, lattice of subrings/ideals, quotient rings, chain condi-
tions, rings of fractions, Chinese Remainder Theorem, Euclidean domains, PID's, UFD's, polynomial
rings, irreducibility criteria for polynomials.
Modules and Linear Algebra. Basic definitions and examples, lattice of submodules, quotient
modules, tensor products of modules, the matrix of a linear transformation, minimal polynomial of
a transformation, Cayley-Hamilton Theorem, trace and determinant, dual spaces, modules over a
PID, rational canonical form, Jordan canonical form.
Field Theory. Basic definitions and examples, field extensions, simple extensions, algebraic exten-
sions, transcendental extensions, separable and inseparable extensions, cyclotomic extensions, solu-
tion of the Greek construction problems, splitting fields and normality, algebraic closure, the Galois
correspondence, Galois groups of extensions/polynomials, sovable and radical extensions, the insolv-
ability of the quintic, Fundamental Theorem of Algebra, Casus Irreducibilis, finite fields, Frobenius
endomorphism.
Miscellaneous. Axiom of choice and Zorn's lemma, universal constructions (products, coproducts),
lattices.
References.
M. Artin, Algebra
D. Dummit and R. Foote, Abstract Algebra
T. Hungerford, Algebra
N. Jacobson, Basic Algebra I
S. Lang, Algebra
ALGEBRA PRELIM JANUARY 2008
1. Let G be a nonabelian finite simple group, and let p be a prime divisor of its order |G|. Show
that if the number of Sylow p-subgroups of G is n, then |G| divides n!.
2. Let G be a finite solvable group. Show that
(a) G has a nontrivial abelian normal subgroup of prime power order, and
(b) Every maximal proper subgroup of G has prime power index in G.
3. Let R be a UFD such that any ideal generated by two elements of R is principal. Prove that R is
a PID. [Hint: If a ∈ I is to generate the ideal I, consider what the factorization of a must look like.]
4. Let A be an n × n matrix over C such that tr(Ak ) = 0 for all k > 0. Show that An = 0. (The
trace tr(M ) of a matrix M is the sum of its diagonal entries.)
5. Find the splitting field of x4 + x3 + 1 over the 32-element field.
6. True or false? Justify your answer.
(a) Every field extension of degree 2 is Galois.
(b) Every algebraically closed field is infinite.
√ √
(c) If α = 5 2 + i + 5 2 − i, then Gal(Q[α]/Q) ∼ S5 .
=
ALGEBRA PRELIM AUGUST 2007
1. Prove that in a group of order 12, any two elements of order 6 must commute.
2. Show that any group of order 105 has an element of order 35.
3. Let R be an integral domain in which every nonzero element factors into a product of finitely
many irreducible elements up to a unit. For any a, b ∈ R − {0}, define the ideal
Ia,b = {x ∈ R : ax ∈ (b)},
where (b) is the ideal of R generated by the element b. Then show that R is a UFD ⇐⇒ Ia,b is
principal for any a, b ∈ R − {0}.
4. Let R be an associative ring with 1 = 0 and let N ⊆ M be left R-modules. Suppose that N and
M/N are Noetherian. Then show that M is Noetherian.
5. Let ◦ be a binary operation on the field R of real numbers. Show that R has a countable subfield
F with the following properties:
(i) Every positive element of F has a square root.
(ii) Every polynomial of odd degree over F has a root.
(iii) F is closed under ◦.
6. Determine the splitting field of the polynomial x5 + 2x4 + 5x2 + x + 4 over F11 and its Galois
group.
ALGEBRA PRELIM JANUARY 2007
1. There exists an injective group homomorphism σ : S4 → A7 given by
σ((12)) = (12)(56),
σ((23)) = (23)(56),
σ((34)) = (34)(56).
List the elements in one Sylow 2-subgroup of S4 and hence, or otherwise, write down a Sylow
2-subgroup of A7 . Deduce that A7 contains precisely 315 Sylow 2-subgroups, each of which is self-
normalizing. [Hint: each Sylow 2-subgroup of A7 contains precisely two elements of cycle type
(4,2,1).]
2. Classify up to isomorphism all groups of order 8. (Your argument should contain full proofs,
although you may use general theorems without proof if you state them clearly.)
3. Let S be the subring of the field of fractions of R[x] consisting of those fractions whose de-
nominators are relatively prime to x2 + 1, i.e., of the form p(x)/q(x) with q(x) relatively prime to
x2 + 1.
(a) What are the units of S?
(b) Identify the ideals of S.
(c) Is S a unique factorization domain? Explain.
(d) If R is replaced by C and the set of rational functions corresponding to S constructed, would
it have a unique maximal ideal? Explain.
4. Let R be a ring with identity 1, and let f ∈ R be an idempotent (i.e., f 2 = f ).
(a) Show that Rf = {rf : r ∈ R} is a projective left R-module under the action r1 ·(rf ) = (r1 r)f .
(b) Now let R = M2 (C), and let M be the left R-module
a
M= : a, b ∈ C
b
with the usual action. Prove that M is projective.
5. Let α be a zero of the polynomial p(x) = x3 − x − 1 over Z3 in some splitting field.
(a) Express the multiplicative inverse of α as a polynomial of minimum degree in α.
(b) Express the other zeros of p(x) as polynomials of minimum degree in α.
(c) What is the minimal polynomial q(x) for α2 ?
(d) Express the other zeros of q(x) as polynomials of minimum degree in α.
6. Let f (x) = x3 − 5 ∈ Q[x].
(a) Find a splitting field for f over Q.
(b) Find the Galois group for f .
(c) Find all proper, nontrivial subgroups of this Galois group and the fields to which they corre-
spond according to the fundamental theorem of Galois theory.
ALGEBRA PRELIM AUGUST 2006
1. Show that if G is a simple group of order 4pq, where p and q are distinct odd primes, then
|G| = 60.
2. Let G be a non-trivial finite group. Show that if M ∩ N = {1} whenever M and N are distinct
maximal subgroups of G, then some maximal subgroup of G is normal. (Recall that a subgroup
of a group is called maximal if it is a proper subgroup not properly contained in any other proper
subgroup.)
3. Let F be a field of order 1024, and let G = GL(6, F), the group of invertible 6 × 6 matrices with
entries in F.
(a) How many conjugacy classes of G contain an element of order 3?
(b) How many conjugacy classes of G contain an element of order 4?
4. Let I be a nonzero ideal in Z[i]. Show that I is a prime ideal if and only if it is a maximal ideal.
5. Let E ≤ K be a field extension of degree n. Show that if there are more than 2n−1 intermediate
subfields E ≤ F ≤ K, then there are infinitely many intermediate subfields. (Hint: At some point
consider minimal polynomials.)
6. Find the Galois group of x6 − 3 over Q.
ALGEBRA PRELIM JANUARY 2006
1. Let G be the alternating group A6 .
(a) How many Sylow 2-subgroups does G have?
(b) To what well-known group is a Sylow 2-subgroup of G isomorphic?
2. Let G be a group each of whose elements is its own inverse.
(a) Prove that G is abelian.
(b) If G is finite, what are the only possibilities for its order?
(c) Prove that if |G| > 2 and is finite, then its automorphism group Aut(G) is not abelian.
3. Let R be a commutative and associative ring with multiplicative identity 1 = 0 and let I be an
ideal of R. Suppose that I is not finitely generated and that the only ideal of R not finitely generated
and containing I is I itself. Then show that I is a prime ideal. [Hint: You may want to make use of
Ja := {r ∈ R : ra ∈ I} for a ∈ R.]
4. For any vector spaces V and W over a field k, let Homk (V, W ) be the set of k-linear maps (=
k-linear transformations) from V to W and let V ∗ = Homk (V, k).
Now let V and W be finite-dimensional vector spaces over a field k. Then:
(a) Show that Homk (V, W ) is a vector space over k under the natural operations of addition and
k-scalar multiplication;
(b) Calculate dimk Homk (V, W );
(c) Calculate dimk (V ∗ ⊗k W ); and
(d) Construct an explicit isomorphism to show that Homk (V, W ) and V ∗ ⊗k W are isomorphic as
vector spaces over k.
5. Let K be a field of characteristic p = 0, and let f = xp − x − a ∈ K[x]. Show that either f splits
(completely) in K[x] or f is irreducible over K.
6. Find a splitting field L/Q and the Galois group G = Gal(L/Q) for f = x5 − 3 ∈ Q[x]. Find 3
nontrivial, proper subgroups of G and the intermediate fields to which they correspond according to
the fundamental theorem of Galois theory.
ALGEBRA PRELIM AUGUST 2005
1. If P is a Sylow p-subgroup of a finite group G, where p is a prime factor of |G|, show that
(a) For any subgroup H of G containing NG (P ), we have NG (H) = H,
(b) NG (NG (P )) = NG (P ).
2. Let G be a finite group for which x2 = 1 for all x ∈ G.
(a) Prove that G is abelian of order 2n for some n.
(b) Prove that the product of all elements of G is equal to the identity if the order of G is
sufficiently large. (Your answer should make it clear what "sufficiently large" means.)
3. (a) Let n ∈ Z, n ≥ 1, and let I be the ideal generated by n and x in Z[x]. Show that I is a
maximal ideal if and only if n is prime.
(b) Show that Z[x] is not isomorphic, as a ring, to Z.
Recall that if G is a group, the group ring ZG is the free Z-module on G with associative
multiplication inherited from the multiplication in G, so that every element in ZG is uniquely
represented by a sum
ng1 g1
g1 ∈G
with ng1 ∈ Z, and
n g1 g 1 ng2 g2 = ng g,
g1 ∈G g2 ∈G g∈G
where ng = g1 g2 =g ng1 ng2 .
(c) Show that if G is any nontrivial group, the group ring ZG has at least four units. Deduce
that Z[x] is not isomorphic to any group ring ZG.
4. Let S be a commutative ring. We say that S is a graded ring if we can decompose S into the direct
sum of additive subgroups S = n≥0 Sn , such that for all integers k, l ≥ 0 we have Sk Sl ⊆ Sk+l .
(For example, if R is a commutative ring, then S = R[x1 , . . . , xm ] is a graded ring, where Sn consists
of the elements of total degree n.)
(a) If S is a graded ring, verify that S0 is a subring, and that for every n, Sn is an S0 -module.
(b) Show that if S is a graded ring, then S+ = n>0 Sn is an ideal of S, and that it is a prime
ideal if and only if S0 is an integral domain.
5. (a) Let p be an odd prime. By considering the action of the Frobenius automorphism, show that
xp − x − 1 is irreducible over Fp , the field with p elements.
(b) Show that the Galois group of x5 − 6x − 1 over Q is S5 .
6. Let p1 , . . . , pn be distinct odd prime numbers, m = n pi , and ζ a primitive mth root of unity.
i=1
Let K = Q(ζ). Determine with proof the number of subfields E, Q ⊆ E ⊆ K, with [E : Q] = 2.
ALGEBRA PRELIM JANUARY 2005
1. Show that Q under addition does not have any proper subgroup of finite index.
2. Show that if G is a group, |G| = 315, and G has a normal subgroup of order 9, then G is abelian.
You may assume that if p < q are primes such that p does not divide q − 1, then a group of order pq
is cyclic, and if Z is the center of G and G/Z is cyclic, then G is abelian.
3. (a) (i) Prove that the integral domain Z[i] (the Gaussian integers) is a Euclidean domain.
(ii) What are its units?
(iii) Give an example of a maximal ideal of Z[i].
(b) (i) Prove that the integral domain Z[x] is not a Euclidean domain.
(ii) What are its units?
(iii) Give an example of a maximal ideal of Z[x].
√
(c) (i) Prove that the integral domain Z[ −5] is not a Euclidean domain.
(ii) What are its units?
4. Let R be a ring and M a left R-module. For N any submodule of M , define A(N ) = {a ∈ R :
aN = 0}. For J any ideal of R, define N (J) = {n ∈ M : Jn = 0}.
(a) Prove that A(N ) is an ideal of R.
(b) Prove that RN is a submodule of M .
(c) Prove that N (J) is a submodule of M .
(d) Prove: If N and L are submodules of M and N ⊆ L, then A(L) ⊆ A(N ).
(e) Prove: If N1 and N2 are submodules of M , then A(N1 + N2 ) = A(N1 ) ∩ A(N2 ).
In (f) and (g) assume that R is nilpotent, i.e., there exists a positive integer n such that the product
of n elements of R is 0.
(f) Prove: If N = 0, then RN = N .
(g) Prove: If RM = 0, then M is not the direct sum of RM and N (R).
5. Suppose that L : K is a field extension, γ ∈ L with γ transcendental over K. Suppose that
f ∈ K[x], deg f ≥ 1.
(a) Show f (γ) is transcendental over K.
(b) Suppose that β ∈ L with f (β) = γ. Show β is transcendental over K.
/
(c) Suppose that α ∈ L, α ∈ K, with α algebraic over K. Show K(α, γ) is not a simple extension
of K.
(d) Suppose that α is a root of f , f ∈ K[x] irreducible of degree n. Prove that K[α] : K = n
by displaying a basis for K[α] over K; prove this is indeed a basis. Then prove K[α] is a
field.
6. Find a splitting field L and the Galois group G for x4 − 2 ∈ Q[x]. Determine the degree of L : Q.
Find at least 3 subgroups and the intermediate fields to which they correspond according to the
Fundamental Theorem of Galois Theory.
ALGEBRA PRELIM AUGUST 2004
1. Show there is no simple group of order 90.
2. Let p and q be distinct prime numbers with p ≡ 1 mod q, and q ≡ 1 mod p. Show that every
group of order pq is cyclic.
√
3. Let d ≥ 1 be an integer. Let Rd = {a + b −d : a, b ∈ Z} ⊂ C, which is a subring of C. Recall
that in a ring with multiplicative identity, an element is called a unit if it has a 2-sided multiplicative
inverse. Recall also that in an integral domain, an element which is nonzero and not a unit is called
irreducible if whenever it is written as a product of two elements, one of these elements is a unit.
(a) Show that complex conjugation restricts to an automorphism of Rd .
(b) Show that ±1 are the only units of Rd if d > 1.
√ √
(c) Show that 2 + −5, 2 − −5, and 3 are irreducible elements of R5 .
√ √
(d) From the equation 3 · 3 = (2 + −5)(2 − −5), show that R5 is not a principle ideal domain.
4. Let (R, +, ·) be a ring that contains a field F as a subring. Then R has the structure of an
F -vector space, where addition is given by + and scalar multiplication is performed via ·. Suppose
that R is a finite-dimensional F -vector space. Show that if R is an integral domain, then R is a field.
5. Find the Galois group of x3 + 10x + 20 over Q.
6. Let p be an odd prime, and φp = (xp − 1)/(x − 1) = xp−1 + · · · + 1 ∈ Z[x]. Let z be a root of φp
in a splitting field over Q, and let K = Q(z). Show there is precisely one subfield L of K such that
[K : L] = 2. In addition, show that this L is Q(z + 1/z).
ALGEBRA PRELIM JANUARY 2004
1. Let p be a prime number. Show that
(a) The center of any p-group is a p-group (that is, the center cannot be trivial),
(b) Any group of order p2 must be abelian.
2. Let G be a nonabelian group of order pq, with p, q prime and p < q.
(a) Prove that p divides q − 1.
(b) Prove that the center of G is trivial.
(c) How many distinct conjugacy classes are there in G?
3. The 2 × 2 trace-zero Hermitian matrices form a real vector space H of dimension 3. Let SU (2) =
{g = (gij )2×2 : gij ∈ C, t gg = g t g = I2 , det g = 1}; it is the special unitary group. An element
g ∈ SU (2) acts on H by ρ(g) : x ∈ H → gx t g ∈ H.
(a) Show that there is a (positive-definite) inner product on H that is invariant under the SU (2)
action. (Hint: You may want to consider the determinant of the matrices in H.)
Consequently, for any g ∈ SU (2) we have ρ(g) ∈ SO(3), where SO(3) is the special orthogonal
group defined by SO(3) = {q = (qij )3×3 : qij ∈ R, t q q = q t q = I3 , det q = 1}.
(b) Show that ρ : SU (2) → SO(3) is a homomorphism.
(c) Find the kernel of ρ : SU (2) → SO(3).
(d) Show that ρ : SU (2) → SO(3) is surjective.
4. Prove that if R is a domain and a = 0 is not a unit in R, then A = a, x is not a principle ideal
in R[x]. Explain why Q[x] is a Euclidean domain, but Q[x, y] is not.
5. Let R be a ring with identity 1 and let M be a left R-module on which 1 acts as the identity.
(a) Show that if e ∈ R is in the center of R and satisfies e2 = e, then we have M = M1 ⊕ M2
as modules, where M1 = eM and M2 = (1 − e)M . Prove that EndR (M ) ∼ EndR (M1 ) ⊕
=
EndR (M2 ) as rings.
(b) Now suppose 1 = e1 + · · · + en , where ei (1 ≤ i ≤ n) are elements in the center of R and they
are orthogonal idempotents, that is, they satisfy e2 = ei (for all 1 ≤ i ≤ n) and ei ej = 0 (for
i
all 1 ≤ i = j ≤ n). State and prove a generalization of the above result.
(c) Let R = C[Z5 ] be the group algebra1 of Z5 . Find a decomposition of the unit element 1
into five nonzero orthogonal idempotents. Let M = R, with the R-action given by the left
multiplication. Show that M is isomorphic to a direct sum of five one-dimensional submodules
that are pairwise nonisomorphic.
6. Let ζ be a primitive complex ninth root of unity.
(a) What is its minimal polynomial over Q?
(b) What is the degree of Q(ζ) over Q?
(c) Find primitive elements for each field intermediate between Q and Q(ζ). Express them as
polynomials in ζ.
1The group algebra of a finite group G is the set C[G] of formal sums ag g(ag ∈ C) with the obvious
g∈G
multiplication
ALGEBRA PRELIM AUGUST 2003
1. Let G be a group, GL the group of left translates aL (a ∈ G) of G, and Aut(G) the group of
automorphisms of G. The set GL Aut(G) = {στ : σ ∈ GL , τ ∈ Aut(G)} is called the holomorph of
G and is denoted Hol G.
(a) Show that Hol G is a group under composition and that if G is finite, then |Hol G| = |G| ×
|Aut(G)|.
(b) Prove that Hol (Z2 × Z2 ) is isomorphic to S4 .
2. Let G be a group of order pqr where p < q < r are prime. Show that G has a normal Sylow
subgroup.
3. Let R be a commutative ring with identity, I1 and I2 ideals in R, and φ : R → R/I1 × R/I2 the
canonical mapping.
(a) Describe ker φ and show that if I1 + I2 = R then ker φ = I1 I2 .
(b) Prove that when I1 + I2 = R the mapping φ is surjective.
(c) Show that (Z100 )× is isomorphic to (Z4 )× × (Z25 )× .
4. Let V be a finite-dimensional vector space and let T : V → V be a linear transformation from V
to itself. Define a mapping T ∗ : V ∗ → V ∗ by T ∗ (f ) = f ◦ T .
(a) Show that T ∗ is a linear transformation.
(b) Let B = {e1 , . . . , en } be a basis for V and let B ∗ = {e∗ , . . . , e∗ } be a basis for V ∗ . Show that
1 n
the matrix for T ∗ relative to B ∗ is the transpose of the matrix for T relative to B.
5. Suppose that F is a finite field and that x3 + ax + b ∈ F[x] is irreducible. Explain why −4a3 − 27b2
must be a square in F.
6. Let g(x) = xp − x − a ∈ Zp [x], where p is a prime and assume a is nonzero.
(a) Show that g(x) has no repeated roots in a splitting field extension.
(b) Show that g(x) has no roots in Zp .
(c) Show that if α is a root of g(x) in a splitting field extension then so is α + b for any b ∈ Zp .
Conclude that {α + b : b ∈ Zp } is a complete set of roots of g(x).
(d) Show that g(x) is irreducible in Zp [x].
(e) Construct a splitting field L for g(x) and determine |Gal(L/Zp )|.
ALGEBRA PRELIM JANUARY 2003
1. Let G be a finite simple group of order n. Determine the number of normal subgroups of G × G.
2. (a) State the Feit-Thompson theorem.
(b) Without using the Feit-Thompson theorem, show that there is no simple group of order
6545 = 5 · 7 · 11 · 17.
3. (a) Let R be a ring with ideals I, J such that I ⊆ J. Prove that
(R/I)/(J/I) R/J.
(b) Give an example of an unique factorization domain that is not a principle ideal domain (PID).
Prove that this ring is not a PID.
(c) Suppose R is a PID. Say a, b, c ∈ R such that gcd(a, b) = 1 = gcd(a, c). Show that gcd(a, bc) =
1.
4. (a) Let F be a field, V and W finite-dimensional vector spaces over F , and T : V → W a
linear transformation. Let {w1 , w2 , . . . , wr } be a basis for T (V ), and take v1 , . . . , vr ∈ V
such that T (vj ) = wj (1 ≤ j ≤ r). Show that v1 , . . . , vr are linearly independent. Then,
let U be the space spanned by v1 , . . . , vr , and K = ker T . Prove the theorem that states
rank(T ) + nullity(T ) = dim(V ) by showing V can be realized as a direct sum of U and K.
(b) Let V be as above. Show that any linearly independent subset {v1 , . . . , vm } of V can be
extended to a basis {v1 , . . . , vn } of V .
5. Suppose that K[α] : K is an extension, that α is algebraic over K, but not in K, and that β is
transcendental over K. Show that K(α, β) is not a simple extension of K.
6. Let h(x) = x4 + 1 ∈ Q(x).
√
(a) Show that the four complex numbers ± 22 (1 ± i) are the four roots of h(x) in C.
(b) Find an α ∈ C such that L = Q(α) is a splitting field extension for h(x) over Q.
(c) Describe Gal(L/Q) as a group of permutations of the roots of h(x), and as a group of
automorphisms of L. (The latter means: write an arbitrary a ∈ L out in terms of a basis for
L over Q, and then describe what σ(a) looks like in terms of this basis, for each σ ∈ Gal(L/Q).
(d) Find all intermediate fields M between L and Q; for each such field M find a subgroup H of
Gal(L/Q) such that M = Fix(H) and H = Gal(L/M ). Which of the extensions M : Q are
normal?
1
ALGEBRA PRELIM AUGUST 2002
1. (a) Suppose that G is a finite group and that there is a group homomorphism
h : G −→ S,
where S is the multiplicative group of roots of unity in the complex numbers, and which
satisfies
3
h(g) = 1
for every element g ∈ G, but for which not every h(g) has the value 1. Prove that G contains
an element of order 3.
(b) Let F7 be the finite field of 7 elements, and GL(2, F7 ) the group of nonsingular 2 × 2 matrices
A with entries in F7 , and multiplication of matrices as group law. Use the determinant
function to construct a homomorphism
t : GL(2, F7 ) −→ S
which satisfies
3
t(A) =1
for all A ∈ GL(2, F7 ), but for which not every t(A) has the value 1.
2. (a) For which prime divisors p of n! are all the elements of the Sylow p-subgroups of the symmetric
group Sn even permutations?
(b) In the symmetric group Sn the conjugacy class of a particular element a (i.e., the set of
elements conjugate to a) consists of all elements with the same cycle structure as a (i.e.,
whose decomposition as a product of disjoint cycles agrees with that of a in having the same
number of cycles and of the same lengths). For what even permutations a is this also the
case for the conjugacy class of a in the alternating group An (n > 1)?
3. Let A be a commutative ring with identity 1, and let M be an A-module. If there exists a chain
of submodules
M = M0 ⊃ M1 ⊃ M2 ⊃ · · · ⊃ Mr = {0}
such that for i = 1, . . . , r, Mi−1 /Mi A/Pi for some maximal ideal Pi , then r is called the length of
M and is denoted by LA (M ), and M is said to have finite length.
(a) Prove that LA (M ) is well-defined.
(b) If
0 −→ M −→ M −→ M −→ 0
is an exact sequence of A-modules and two of the modules have finite length, then the third
module also has finite length. Furthermore,
LA (M ) = LA (M ) + LA (M ).
(c) If
0 −→ Mn −→ Mn−1 −→ · · · −→ M0 −→ 0
is an exact sequence of modules of finite length, then
n
(−1)i LA (Mi ) = 0.
i=1
please turn over
2
ALGEBRA PRELIM AUGUST 2002
4. An ideal a in a commutative ring R is called primary iff a, b ∈ R and ab ∈ a implies that either
a ∈ a or there is an n ∈ N such that bn ∈ a.
(a) Provide an example of a prime ideal in C[x, y].
(b) Let a be the ideal in C[x, y] generated by xy and x2 . Prove that a is not primary.
√
(c) Prove that the radical of a, a, is a prime ideal.
√
(d) Is a maximal?
5. (a) Prove that the polynomial x4 − 27 is irreducible over Q.
(b) Determine a (minimal) splitting field for the polynomial x4 − 27 over Q. Determine the order
of its Galois group (over Q) and prove that it is not commutative.
6. (a) Let Q denote the field of rational numbers, and let K be a (minimal) splitting field for x2 − 2
over Q. For what other monic irreducible polynomial in Q[x] is K a splitting field?
(b) Let L be a (minimal) splitting field for x3 + x + 1 over F2 , the field of 2 elements. Find all
other irreducible polynomials in F2 [x] for which L is a splitting field over F2 .
ALGEBRA PRELIM JANUARY 2002
1. Let G be a finite group and N a normal subgroup. Show that
(a) The intersection with N of a Sylow p-subgroup of G is a Sylow p-subgroup of N and every
Sylow p-subgroup of N is obtained in this way.
(b) The image in G/N of a Sylow p-subgroup of G is a Sylow p-subgroup of G/N and every
Sylow p-subgroup of G/N is obtained in this way.
2. Let G and H be groups and θ : H → Aut(G) a homomorphism. Let G ×θ H be the set G × H
with the following binary operation: (g, h)(g , h ) = g[θ(h)(g )], hh .
(a) Show that G×θ H is a group with the identity element (e, e ) and (g, h)−1 = θ(h−1 )(g −1 ), h−1 .
(You may assume without proving it that the operation is associative.)
(b) Use the construction of (a), with G a cyclic group of order 7, to show that there is a group
K with 105 elements generated by elements a, b, c such that a5 = e, b3 = e, c7 = e, ab = ba,
bc = cb, ac = c4 a.
(c) In the group described in (b), determine the number of Sylow subgroups.
3. (a) Suppose 0 → A → A → A → 0 is a short exact sequence of abelian groups. Show that
rank A is finite if and only if rank A and rank A are finite. If so, show that rank A =
rank A + rank A .
dn dn−1 d2 d1
(b) Suppose 0 / Cn / Cn−1 / ··· / C2 / C1 / C0 / 0 is a chain of abelian
groups, i.e., Ci is an abelian group and di : Ci −→ Ci−1 is a homomorphism such that
ker di
di−1 ◦ di = 0, for each i. Let Hi = (i = 0, 1, . . . , n). Assume that rank Ci is finite,
Im di+1
for all i. Define two polynomials
n n
m(t) = rank Ci ti , p(t) = rank Hi ti .
i=0 i=1
Show that there is a polynomial q(t) with nonnegative coefficients such that m(t) = p(t) +
(1 + t)q(t).
4. Let R be a commutative ring and M be a module over R. A submodule N is a characteristic
submodule if ϕ(N ) ⊂ N for any R-endomorphism ϕ of M . Show that
(a) ∀ r ∈ R, rM and Ann(r) = {m ∈ M : rm = 0} are characteristic submodules of M .
(b) If N is a characteristic submodule of M , and P , Q are complementary submodules of M ,
i.e., P ⊕ Q = M , then N ∩ P , N ∩ Q are complementary submodules of N .
5. (a) Suppose H is a subgroup of Sn (n ≥ 2) which contains both an n-cycle and a transposition.
Show that H = Sn .
(b) Show that the roots of the polynomial P (x) = x5 − 6x + 3 cannot be expressed by radicals.
6. Let K be a field of characteristic 0, and let K(x) be a simple transcendental extension. Let G be
the subgroup of the group of K-automorphisms of K(x) generated by an automorphism that takes
x to x + 1. Show that K is the fixed field of G.
ALGEBRA PRELIM AUGUST 2000
1. Determine the Galois groups of the following polynomials in Q[x]:
(a) x4 − 7x + 10.
(b) x3 − 2.
(c) x5 − 9x + 3.
2. (a) If G is a group of order 53 · 7 · 17 show that G has normal subgroups of sizes 53 , 53 · 7, and
53 · 17.
(b) Show that there is a nonabelian nilpotent group of order 53 · 7 · 17. [Hint: To construct a
nonabelian group of order 53 , work in S25 to find nonidentity elements a, b such that a is of
order 25, b is of order 5, and b−1 ab = a6 . A finite group is nilpotent if it is the direct product
of its Sylow subgroups.]
3. Let R be a ring with 1. An element x in R is called nilpotent if xm = 0 for some positive integer
m.
(a) Show that if n = ak b for some integers a and b then the coset ab is a nilpotent element of
Z/nZ.
(b) If a ∈ Z is an integer, show that the element a ∈ Z/nZ is nilpotent if and only if every prime
divisor of n is also a divisor of a. In particular, determine the nilpotent elements of Z/36Z
explicitly.
(c) If R is any commutative ring with 1 and x is a nilpotent element, show that 1 + x is a unit
for R (i.e., is invertible). [Hint: As motivation, think of the sum of the geometric series.]
4. Let R be a ring with 1 and M a left unitary R-module. An element m in M is called a
torsion element if rm = 0 for some nonzero element r ∈ R. The set of torsion elements is denoted
Tor(M ) = {m ∈ M : rm = 0 for some nonzero r ∈ R}.
(a) Prove that if R is an integral domain then Tor(M ) is a submodule of M (called the torsion
submodule of M ).
(b) Give an example of a ring R and an R-module M such that Tor(M ) is not a submodule.
[Hint: Consider letting R be itself a left R-module where R is some ring which is not an
integral domain.]
(c) Show that if R has zero divisors then every nonzero R-module has nonzero torsion elements.
5. Give a representative element of each conjugacy class of the elements of the alternating group A5 ,
and determine the number of elements in its class.
6. (a) Prove that f (x) = x4 + x3 + x2 + x + 1 is irreducible over Z2 .
(b) What are the other irreducible quartic polynomials over Z2 ?
(c) If θ is one of the roots of f (x), what are the others (expressed as polynomials in θ of least
possible degree)?
(d) Give a method for finding an element ϕ (expressed as a polynomial in θ) of the splitting field
Z2 (θ) such that [Z2 (ϕ) : Z2 ] = 2.
ALGEBRA PRELIM JANUARY 1999
1. Let G be a finite group, and C be the center of G.
(a) Show that the index [G : C] is not a prime number.
(b) Give an example where [G : C] = 4.
2. Let G be a finite group that acts transitively on a set S. Recall that G is said to act doubly
transitively if for every pair (a, b), (c, d) there is a g ∈ G such that g(a) = c and g(b) = d.
In (a) and (b) below, assume that G is a finite group that acts transitively on a set S. Let s be in
S, and let
H = {g ∈ G : g(s) = s}
be its isotropy group. Note then H acts on the complement S − {s}.
(a) Show that G acts doubly transitively on S if and only if H acts transitively on S − {s}.
(b) Suppose there is a subgroup T of G of order two, T not contained in H, such that G = HT H.
Show that G acts doubly transitively on S.
3. Let R be a commutative ring with identity. Suppose that for some a, b ∈ R, the ideal Ra + Rb is
principal. Prove that the ideal Ra ∩ Rb is principal.
4. Let S be a commutative ring with identity, R = S[x1 , . . . , xn ]. Let I be the ideal of R generated
by the quadratic monomials {xi xj : 1 ≤ i, j ≤ n}, and φ the natural projection
φ : R → R/I.
(a) Show that R/I is a free S-module and find its rank.
(b) For f ∈ R define f ∈ R/I by f = φ(f ) − φ(f (0, . . . , 0)). Show that
(f g) = φ(f )g + φ(g)f .
(c) Show that for all positive integers n, (f n ) = nφ(f )n−1 f .
5. Determine the Galois group (using generators and relations if you would like) over K of x5 − 3
when:
(a) K = Q.
(b) K = F11 , the finite field with 11 elements.
6. We call a six degree polynomial symmetric if x6 f (1/x) = f (x). Let f be a symmetric six degree
polynomial in Q[x].
(a) Suppose r is a root of f in a splitting field of f . Show that [Q(r + 1/r) : Q] ≤ 3.
(b) Deduce from (a) that the Galois group of f is solvable. [Hint: All groups of order less than
60 are solvable.]
1
ALGEBRA PRELIM JANUARY 1998
1. (a) Show that there is no simple nonabelian group of order 76.
(b) Show that there is no simple nonabelian group of order 80.
2. Let p be an odd prime. Show that a group of order 2p is either cyclic, or is isomorphic to the
dihedral group D2p . (Recall that the dihedral group Dn is the group of symmetries of a regular n-gon
in a plane.)
√ √ √
3. Let R = Z[ −3] = {a + b −3 : a, b ∈ Z}, where −3 is a root of x2 + 3 in some splitting field.
Let
√
1 + −3
S=Z
2
√
1 + −3
= a+b : a, b ∈ Z .
2
(a) Show that S is a Euclidean domain with respect to the norm
√
1 + −3
δ a+b = a2 + ab + b2 .
2
(b) Show that R is not a Euclidean domain with respect to the norm
√
δ(a + b −3) = a2 + 3b2 .
[Hint: Is R a unique factorization domain?]
4. Let F be a field and let t be transcendental over F . Recall that if P (t) and Q(t) are nonzero
relatively prime polynomials in F [t], which are not both constant, then
[F (t) : F (P (t)/Q(t))] = max{deg P, deg Q},
a fact you may use, if needed.
(a) Prove that Aut(F (t)/F ) ∼ GL2 (F )/{λI : λ ∈ F × }, where
=
a b 1 0
GL2 (F ) = : a, b, c, d ∈ F and ad − bc = 0 and I = .
c d 0 1
(b) Let F2 be the field with two elements. Show that Aut(F2 (t)/F2 ) ∼ S3 .
=
(c) Find the subfields of F2 (t) which are the fixed fields of the subgroups of Aut(F2 (t)/F2 ).
5. Show that f (x) = 2x5 − 10x + 5 is not solvable by radicals over the rational numbers.
please turn over
2
ALGEBRA PRELIM JANUARY 1998
6. An ultrafilter on N = {0, 1, 2, . . . } is a collection U of subsets of N such that the following
conditions hold:
(i) N ∈ U.
(ii) /
∅ ∈ U.
(iii) If x ∈ U and x ⊆ y ⊆ N, then y ∈ U .
(iv) If x, y ∈ U , then x ∩ y ∈ U .
(v) For any x ⊆ N, x ∈ U or N − x ∈ U . (N − x is the complement of x in N.)
Suppose that Fi : i ∈ N is a system of fields, and U is an ultrafilter on N. Consider the full
direct product i∈N Fi , which is a commutative ring with identity, consisting of all functions a
with domain N, with ai = a(i) ∈ Fi for all i, the ring operations being coordinate-wise. Let
I = {a ∈ i∈N Fi : {i ∈ N : ai = 0} ∈ U }.
(a) Show that I is a maximal ideal of i∈N Fi .
(b) Suppose that for each i ∈ N, every polynomial in Fi [x] of positive degree at most i has a root
in Fi . Suppose that N − F ∈ U for every finite subset F of N. Show that i∈N Fi /I is an
algebraically closed field.
ALGEBRA PRELIM AUGUST 1997
1. Let G be a group of order 429 = 3 · 11 · 13.
(a) Show that every subgroup of order 13 in G is normal in G. (Use the Sylow theorems.)
(b) Show that every subgroup of order 11 in G is normal in G.
(c) Classify (up to isomorphism) all groups of order 429.
√ √
2. Let Q denote the field of rational numbers and let K = Q( 5, 7).
(a) Find the Galois group of K over Q and show that K is a Galois extension of Q. Express all
of the elements of the Galois group as permutations of the roots of (x2 − 5)(x2 − 7).
(b) Find all the subfields of K and match them up with the subgroups of the Galois group as is
indicated by the Fundamental Theorem of Galois Theory.
3. Let K = GF (pm ) be the finite field with q = pm elements (p is a rational prime number). Let V
be an n-dimensional vector space over K. Give explicit formulas for the following numbers:
(a) The number of elements of V .
(b) The number of distinct bases of V . Give it for both ordered and unordered bases.
(c) The order of the general linear group GLn (K).
(d) Let K = GF (3) be the field with 3 elements. Verify that there are 48 nonsingular 2 × 2
matrices over K. Also show that the only nonsingular 2 × 2 matrix A over K that satisfies
2 0 2 0
the equation A5 = is the matrix itself.
0 2 0 2
4. Let V be an n-dimensional vector space over an arbitrary field K and let f : V → V be a linear
transformation. Show that there exists a basis for V such that the matrix representation for f with
respect to that basis is diagonal if and only if the minimal polynomial for f is a product of distinct
linear factors.
5. Let Zn denote the cyclic group of order n. Let G = Z81 ⊕ Z30 ⊕ Z16 ⊕ Z45 .
(a) What is the largest cyclic subgroup of G? Give a generator for this group in terms of the
generators for the cyclic components of G. Please denote the generators for the groups Z81 ,
Z30 , Z16 , and Z45 by a, b, c and d, respectively.
(b) How many elements of order three does G have?
(c) How many elements of order nine does G have?
6. Recall that a Euclidean domain is an integral domain R together with a natural number valued
function N defined on the nonzero elements of R which has the property that, given a and b in R
with b nonzero, we can find q and r in R such that a = bq + r and either r = 0 or N (r) < N (b).
√ √
Now let R = Z[ −2] = {m + n −2 : m, n ∈ Z}, where Z is the ring of rational integers. Let
√
N (m + n −2) = m2 + 2n2 .
(a) Show that R is a Euclidean domain.
√
(b) Decide whether x3 +2 −2x+4 is irreducible in Q(x), where Q is the field of rational numbers.
√
√
1+ −7 1+ −7 (2m + n)2 + 7n2
7. Let R = Z 2 and let N m + n = , where Z is the ring of
2 4
rational integers and m, n ∈ Z. Show that R is a Euclidean domain. (Your proof should also work
√
1 + −11 (2m + n)2 + 11n2
if −7 is replaced by −11 and N m + n = .
2 4
ALGEBRA PRELIM JANUARY 1997
1. Suppose the group G has a nontrivial subgroup H which is contained in every nontrivial subgroup
of G. Prove that H is contained in the center of G.
2. Let n be an odd positive integer, and denote by Sn the group of all permutations of {1, 2, 3, . . . , n}.
Suppose that G is a subgroup of Sn of 2-power order. Prove that there exists i ∈ {1, 2, 3, . . . , n} such
that for all σ ∈ G one has σ(i) = i.
3. Let p be an odd prime and Fp the field of p elements. How many elements of Fp have square roots
in Fp ? How many have cube roots in Fp ? Explain your answers.
4. Suppose that W ⊆ V are vector spaces over a field with finite dimensions m and n (respectively).
Let T : V → V be a linear transformation with T (V ) ⊆ W . Denote the restriction of T to W by
TW . Identifying T and TW with matrices, prove that det(In − xT ) = det(Im − xTW ) where x is an
indeterminate and Im , In denote the m × m, n × n identity matrices.
θ
5. Let Q be the field of rational numbers. For θ a real number, let Fθ = Q(sin θ) and Eθ = Q(sin 3 ).
Show that Eθ is an extension field of Fθ , and determine all possibilities for dimFθ Eθ .
6 (Hint: Study h(x + 1) by
first writing h(x) = g(x)/(x − 1). Use Eisenstein's criterion to show h(x + 1) is irreducible.)
(b) Show that G = Gal(K/Q) is cyclic of order 6, and has as a generator the map that takes
ω → ω 3 for any root ω Q(x1 ) and Q(x2 ) are the only fields M with Q ⊂ M ⊂ Q(ω). (Here ⊂ denotes
proper containment.)
ALGEBRA PRELIM AUGUST 1996
1. Suppose p > q are prime numbers and that q does not divide p − 1. Show that every group G of
order pq is cyclic.
2. Let R be a ring with multiplicative identity 1. An element r ∈ R is called nilpotent if rn = 0 for
some positive integer n > 0. Let N denote the set of nilpotents in R.
(a) Show that if R is commutative then N is an ideal. Give an example of a noncommutative R
for which N is not an ideal.
(b) An ideal I in a commutative ring is called primary if for every xy ∈ I, either x ∈ I or y m ∈ I
for some positive integer m. Suppose that R is commutative and that I is an ideal in R.
Show that I is primary if and only if every zero divisor in R/I is nilpotent.
√
1+ −15 √
3. Consider the set of numbers R = a + b 2 : a, b ∈ Z ⊂ Q( −15).
√ √ √
(a) Show that R is a ring, and that the automorphism −15 → − −15 of Q( −15) induces an
automorphism of R. √
(b) What is the norm of a + b 1+ 2−15 for integers a, b?
(c) Find all the units in R.
(d) Find all factorizations of 4 into irreducibles in R.
(e) Give an example in R of an irreducible which isn't prime.
4. Let ζ be a primitive 12th root of unity.
(a) Find the Galois group of Q(ζ) over Q.
(b) Let Φn (x) denote the nth cyclotomic polynomial over Q. What is the degree of Φ24 (x) over
Q?
(c) When Φ24 (x) is factored over Q(ζ), how many factors are there, and what are their degrees?
5. Let q be a power of a prime, and r a positive integer. Let Fq and Fqr denote, respectively, the
fields with q and q r elements. Let G denote the Galois group of Fqr over Fq , and let N denote the
norm map, N (α) = σ∈G σ(α) from Fqr to Fq . Show that
N : F× → F×
qr q
is a surjective homomorphism.
6. Let G be a finite group of order n, and suppose for each prime p dividing n there is a unique
Sylow p-subgroup. Show that G is solvable. (Be sure to carefully state any theorems about solvable
groups that you use.)
1
ALGEBRA PRELIM JANUARY 1996
1. Let G be a group, GL the group of left translates aL (a ∈ G) of G (that is, for a ∈ G, aL : G → G
is defined by aL (g) = ag).
(a) Show that GL Aut(G) (that is, the set {xy : x ∈ GL , y ∈ Aut(G)}) is a group of transfor-
mations of G. GL Aut(G) is called the holomorph of G and is denoted Hol (G).
(b) Show GR ⊂ Hol (G), where GR is the group of right translates of G.
(c) Show that if G is finite, then |Hol (G)| = |G||Aut(G)|.
2. Let h(x) = x4 + 1 ∈ Q[x], and let L ⊂ C be a splitting field for h(x) over Q.
(a) Find the four roots of h(x) in C.
(b3. Let R be a ring with identity and M an R-module.
(a) Show that, if m ∈ M , then {x ∈ R : xm = 0} is a left ideal of R.
(b) Let A be a left ideal of R, and m ∈ M . Show that {xm : x ∈ A} is a submodule of M .
(c) Suppose that M is irreducible, which means that M has no submodules other than (0) and
M . Let m0 ∈ M , m0 = 0. Show that A = {x ∈ R : xm0 = 0} is a maximal left ideal in R.
4. Let g(x) = xp − x − a ∈ Z/pZ[x], where p ∈ Z is a prime, and a a nonzero element of Z/pZ.
(a) Show that g(x) has no repeated roots in a splitting field extension.
(b) Show that g(x) has no roots in Z/pZ.
(c) Show that, if c is a root of g(x) in a splitting field extension, then so is c + i for any i ∈ Z/pZ.
Conclude that {c + i : i ∈ Z/pZ} is a complete set of roots of g(x).
(d) Show that g(x) is irreducible in Z/pZ[x].
(e) Consturct a splitting field extension L for g(x) over Z/pZ.
(f) Find the Galois group Gal(L/(Z/pZ)). Describe this group as a group of permutations of
the roots of g(x).
5. Let N be a positive integer, and let LN denote the set of functions f : Z → C such that
f (t) = f (t + N ) for all t ∈ Z. Define the convolution f ∗ g of functions f, g ∈ LN by
1
f ∗ g(t) = f (t − y)g(y) (t ∈ Z).
N
0≤y≤N −1
(a) Show that, under the usual addition of functions and the above convolution of functions, LN
is a commutative ring, which identity δN given by
N if N |x,
δN (x) =
0 if not.
You may assume (that is, you needn't prove) that LN is an abelian group under addition.
please turn over
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ALGEBRA PRELIM JANUARY 1996
5. (b) Suppose M |N for some positive integer M , and define
M if M |x,
δM (x) =
0 if not.
Show that δM is an idempotent element of LN : that is, δM ∗ δM = δM .
(c) Let M, N be as above, and let f, g ∈ LM , so that also f, g ∈ LN . Suppose, for clarity, we
denote the convolution in LM by ∗ . Show that f ∗ g = f ∗ g, where ∗ again denotes the
convolution in LN .
(d) Show that, for M and N as above, the map f → f ∗ δM is a ring homomorphism of
(LN , +, ∗, 0, δN ) onto (LM , +, ∗ , 0, δM ).
6. Let H and K be subgroups of a group G.
(a) Show that the set of maps {x → hxk : h ∈ H, k ∈ K} is a group of transformations of the
group G.
(b) Let HxK denote the orbit of x relative to the above group of transformations of G. Show
that if G is finite then |HxK| = |H|[K : x−1 Hx ∩ K] = |K|[H : x−1 Kx ∩ H].
ALGEBRA PRELIM AUGUST 1995
1. Let G be a group of order 3 × 11 × 17 = 561. Let H be a group of order 11 × 17 = 187.
(a) Prove that H is abelian and cyclic. [Hint: Use Sylow theorems.]
(b) Prove that the Sylow 11- and Sylow 17-subgroups of G are both normal in G.
(c) Is G necessarily abelian? If "yes," prove it; if "no," give an example of a nonabelian group
of order 561. Are all abelian groups of order 561 cyclic?
2. Let Zn denote the cyclic group of order n. Let G = Z9 ⊕ Z27 ⊕ Z25 ⊕ Z5 ⊕ Z35 ; let Z9 = a ;
Z27 = b ; Z25 = c ; Z5 = d ; Z35 = e ;, i.e., a, b, c, d, e are generators for the summands of G.
(a) What is the largest cyclic subgroup of G? Give a generator of that subgroup in terms of
a, b, c, d, e. You do not need to justify your answer.
(b) How many elements of order 5 does G have? Justify your answer.
(c) How many elements of order 25 does G have? Justify your answer.
3. (a) Let F be a field and F [x] the ring of polynomials in one indeterminate over F . Note that
F [x] is an integral domain.
(i) Is F [x] a Euclidean domain?
(ii) Is F [x] a principal ideal domain?
(iii) Is F [x] a unique factorization domain?
(iv) Are all its nonzero prime ideals maximal?
(Explain your answers. You may quote relevant theorems. In some cases, counterexam-
ples may be appropriate.)
(b) Answer the same questions ((i)-(iv)) for the integral domain F [x, y], the ring of polynomials
in two indeterminates over F . Again, explain your answers.
4. Let R be a commutative ring. An R-module M is said to be cyclic if it is generated by one of its
elements.
(a) Show that every nonzero cyclic R-module M is isomorphic to R/J, where J is an ideal of R.
(b) Show that if R is a principal ideal domain, then every submodule of a cyclic R-module is
again cyclic.
5. Let p be a prime number. Let Fp denote the field Z/pZ.
(a) Suppose that K is an extension of Fp of degree n. Show that K is the splitting field for
n
f (x) = xp − x.
(b) Prove that the Galois group of K (in part (a)) over Fp is cyclic.
(c) Let Fpm denote a field with pm elements. Show that Fpm contains a subfield Fpn of pn elements
if and only if n divides m.
6. (a) Suppose that a and b are complex numbers and that K is a subfield of the complex numbers
such that [K(a) : K] = 2 and [K(b) : K] = 3. Suppose that K(b) is a normal extension of K.
Prove that K(a, b) is a normal extension of K and that K(a + b) = K(a, b).
(b) Suppose that K and b are as in (a), except that K(b) is not a normal extension of K (but
still [K(b) : K] = 3). Let L be an extension of K(b) which is a splitting field for the minimal
polynomial of b over K. Show that there exists an element a in L such that [K(a) : K] = 2.
Show that L = K(a, b). Let b be another zero of the minimal polynomial for b over K. Show
that K(b + b ) = L, but that K(b − b ) = L.
1
ALGEBRA PRELIM JANUARY 1995
1. Let G be a finite group. A character on G is a homomorphism χ : G → C∗ taking its values in
the multiplicative group of the complex numbers. Let G denote the set of all characters on G. Show:
(a) If χ1 and χ2 are in G, then the definition χ1 χ2 (g) = χ1 (g)χ2 (g) for all g in G makes G into
a group.
(b) If χ and g are in G and G, respectively, then χ(g) is a root of unity.
(c) For any x in G, G/ ker(χ) is cyclic.
(d) If χ is in G, then χ(g), where the sum is taken over the elements of G, is either n = [G : 1]
or 0 depending on whether χ is the identity element of G or not.
2. Let p be a rational prime and let k be a field with q = pn elements. Let M2 (k) denote the
ring of 2 × 2 matrices over k, and GL2 (k) the subset of M2 (k) consisting of matrices with nonzero
determinant.
(a) Show GL2 (k) is a group under matrix multiplication.
(b) Show that order of GL2 (k) is r = (q 2 − q)(q 2 − 1).
(c) Show that for any matrix A ∈ M2 (k),
Ar+2 = A2 .
[Hint: Part (c) can be done using part (b) or by using the Theory of Canonical Forms.]
n
3. A field K is called formally real if the conditions xi ∈ K and x2 = 0 for some n > 0 imply
i
i=1
that each xi vanishes. It is called real, closed if it is formally real and no proper algebraic extension
is formally real.
(a) Show that K is formally real if and only if −1 cannot be expressed as a sum of squares in K.
(b) Show that if K is real, closed then every sum of squares in K is a square in K.
(c) Let K be real, closed and let P = {all nonzero finite sums of squares in K}. Show that P
satisfies the following properties: (i) If a and b are in P , then so are ab and a + b. (ii) For
any a in K, exactly one of the following holds: a = 0, a is in P or −a is in P .
4. Let ω1 and ω2 be a pair of complex numbers which are linearly independent over the reals. Let
L = Zω1 ⊕ Zω2 be the (necessarily free) abelian group generated by these complex numbers. Clearly
nL ⊆ L for any integer n. Let R = {z ∈ C : zL ≤ L} and suppose R contains a non-integer z.
Show:
(a) τ = ω1 /ω2 generates a quadratic extension of the rational numbers.
(b) If the minimal polynomial for τ is of the form τ 2 − rτ − s for suitable integers r and s, then
R = Z[τ ].
please turn over
2
ALGEBRA PRELIM JANUARY 1995 their
degrees are when Φ20 is factored into irreducible factors over Q(ω).
6. Let C2 be the set of Sylow 2-subgroups of the symmetric group S5 , and let C3 be the set of Sylow
3-subgroups.
(a) What are the cardinalities of C2 and C3 ?
(b) Let G2 ∈ C2 and G3 ∈ C3 . Describe G2 and G3 in terms of a faithful action on a set of 5
symbols. (In the case of the Sylow 2-groups, look at the symmetries of a labelled square.)
ALGEBRA PRELIM AUGUST 1994
1. Let G be a simple group of order 60. Determine how many elements of order 3 G must have. (Do
not assume that you already know that G A5 ).
2. Let the vertices of a regular n-sided polygon (n ≥ 3) be labelled consecutively from 1 to n, i.e.,
with vertices i, i + 1 endpoints of one side of the polygon. The only symmetries are rotations ϕj ∈ Sn
(j = 1, . . . , n) where ϕj (i) = j + i and reflections ψj ∈ Sn (j = 1, . . . , n) where ψj (i) = j − i. (The
addition and subtraction in these definitions are modulo n.) Let Γ be the subgroup of Sn generated
by ϕj , ψj (j = 1, . . . , n).
(a) Show that {ϕ1 , ψ1 } generate Γ.
(b) Show that Γ is dihedral, i.e., isomorphic to Dn , the group generated by a, b subject to the
relations an = b2 = e, bab−1 = a−1 .
(c) (i) For which n are all ϕj 's and ψj 's even permutations of {1, . . . , n}? (ii) For which n are
all ϕj 's even permutations and all ψj 's odd ones? (iii) For the remaining n, which ϕj 's and
ψj 's are even?
3. Consider the polynomial ring R = Z[x]. Consider the ideals
I = (x), J = (5, x), K = (2x, x2 + 1).
Which of these are prime ideals? Which are maximal ideals? Give explanations!
4. Let R be a principal ideal domain and M an R-module that is annihilated by the nonzero proper
ideal (a). Let a = pα1 · · · pαk be the (unique) factorization of a into distinct prime powers in R.
1 k
Let Mi = {m ∈ M : pαi m = 0}. Show that M1 + · · · + Mk is in fact a direct sum and that
i
M = M1 ⊕ · · · ⊕ Mk .
5. Let K1 , K2 , K3 , K4 denote splitting fields for x3 − 2 over Q, (x3 − 2)(x2 + 3) over Q, x9 − 1 over Q
and x64 − x over Z2 , respectively. Consider the Galois groups Gal(K1 /Q), Gal(K2 /Q), Gal(K3 /Q)
and Gal(K4 /Z2 ).
(a) Which of these groups are isomorphic?
(b) If ζ is a primitive 9th root of unity (over Q), find an element α ∈ Q expressed as a polynomial
/
in ζ such that the field Q(α) = K3 .
6. (a) Let f (x) = ax3 + bx2 + cx + d ∈ Z[x] be of degree 3 and irreducible over Q. In its splitting
field K, f (x) = a(x − α1 )(x − α2 )(x − α3 ). Show that [K : Q] = 3 or [K : Q] = 6 when
(α1 − α2 )(α2 − α3 )(α3 − α1 ) does or does not lie in Q, respectively.
(b) Determine the degree [L : F3 ] where F3 is the field with 3 elements and L is the splitting field
(over F3 ) of x3 − x + 1.
ALGEBRA PRELIM AUGUST 1993
1. Let G be a group. If U and V are subgroups of G, we let U ∨ V denote the smallest subgroup of
G containing U ∩ V .
(a) Suppose that H, K, and L are normal subgroups of G. Show: If H ⊆ L, then H ∨ (K ∩ L) =
(H ∨ K) ∩ L.
(b) Give an example showing that part (a) does not work for arbitrary subgroups (i.e., if the
subgroups are not assumed to be normal). [Hint: Look at subgroups of A4 .]
2. (a) Let G be a group, Z its center. Prove that if the factor group G/Z is cyclic, then G is abelian.
(b) Let p be a prime and let P be a nonabelian group of order p3 . Prove that the center Z of
P is a cyclic group of order p, and the factor group P/Z is the direct product of two cyclic
groups of order p. [Note: You may use without proof standard results about finite p-groups.]
3. Let R be a ring with unit element 1. Using its elements we define a ring R by defining
c ⊕ d = c + d + 1 and
c ∗ d = cd + c + d for all elements c, d in R (where the addition and multiplication of the right
hand side of these relations are those of R).
(a) Prove that R is a ring under the operations ⊕ and ∗.
(b) Which element is the zero element of R ?
(c) Which element is the unit element of R ?
(d) Prove that R is isomorphic to R .
4. Let A be a commutative ring satisfying the ascending chain condition for ideals. Let φ : A → A
be a ring homomorphism of A onto itself. Prove that φ is an automorphism. [Hint: Consider the
powers φn .]
5. (a) Let K be the splitting field of x4 − 2 over the field of rationals Q. Find two subfields E1 and
E2 of K such that [K : E1 ] = [K : E2 ] = 2 but E1 and E2 are not isomorphic.
(b) Let K be the splitting field of x7 − 3x3 − 6x2 + 3 over Q and let E1 and E2 be any subfields
of K such that [K : E1 ] = [K : E2 ] = 7. Prove that E1 and E2 are isomorphic.
[Hint: Use the Fundamental Theorem of Galois Theory.]
6. (a) Determine the splitting field K of the polynomial x12 − 1 over the field of rational numbers
Q. Give generators for K over Q and find the degree [K : Q].
(b) Prove that for all positive integers n, cos(2π/n) is an algebraic number.
1
ALGEBRA PRELIM JANUARY 1993
do all three questions in part a. do any three of the four questions in part b.
part a
1. Let A be an associative ring with identity 1. Suppose that 1 = e1 + · · · + en where ei is in
A and ei ej = δij for all i and j. (δij equals 1 or 0 depending upon whether i = j or not.) Let
Ai = Aei = {all aei where a is in A}. Prove:
(a) Ai is a left ideal of A for each i.
(b) If a is any member of A then a is uniquely expressible in the form a = ai where ai ∈ Ai .
2. Let f (x) = x4 + x + 1 be a polynomial over F = GF (2), the field with two members.
(a) Show that f (x) is irreducible in F[x].
(b) Let K be the splitting field of f (x) over F. How many members does K have?
(c) Describe an automorphism of K over F having the maximum possible order in the Galois
group of K over F.
(d) Find a subfield of K distinct from F and K. List its elements as polynomials in α over F
where α is a root of f (x) in K.
3. Let G be a group of order 7 · 13 = 91 and let H be a group of order 5 · 7 · 13 = 455.
(a) Prove that G is abelian. [Hint: Use the Sylow theorems.]
(b) Prove that the Sylow 7- and Sylow 13-subgroups of H are both normal in H.
(c) Is H abelian? If "yes," prove it. If "no," give an example of a nonabelian group of order 455.
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ALGEBRA PRELIM JANUARY 1993
part b
1. Let V be the set of all rational numbers expressible in the form a/b where a and b are integers
and b is odd. Show:
(a) V is a subring of the rational numbers.
(b) The field of quotients of V is the field of rational numbers.
(c) Exhibit all the units of V .
(d) Exhibit all the ideals of V and determine which are prime ideals and which are maximal
ideals.
(e) Prove V /M , where M is a maximal ideal of V , is isomorphic to Zn (= Z/nZ) for some n.
Which n?
2. Let Q be the field of rational numbers. An absolute value on Q is a real-valued function |a| having
the following properties:
(1) |a| ≥ 0 and |a| = 0 if and only if a = 0.
(2) |ab| = |a||b|.
(3) |a + b| ≤ |a| + |b|.
Suppose that |n| ≤ 1 for all natural numbers n. Show:
(a) |a + b| ≤ max{|a|, |b|} for all a and b.
(b) Either |a| = 1 for all nonzero a or there is a prime number p such that, if a = pr m/n, with
m and n relatively prime to p and to each other while r is an integer, then |a| = |p|r .
3. Recall that an ordered field is a field K together with a distinguished subset P (the "positive"
elements) with the properties:
(1) For all a in K exactly one of the following holds: a is in P , −a is in P , or a = 0.
(2) If a and b are in P , then so are a + b and ab.
Show:
(a) Any ordered field has characteristic zero.
(b) The rational numbers can be ordered in exactly one way.
(c) Any subfield of an ordered field can be ordered by an order induced by the larger field.
4. Describe, up to isomorphism, all groups of order 27. Describe the two nonabelian groups in terms
of generators and defining relations.
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ALGEBRA PRELIM AUGUST 1992
1. Prove that every group of order 1645 = 5 · 7 · 47 is abelian and cyclic.
2. Let B = {v1 , . . . , vm } be a basis over Q for the m-dimensional vector space V . Using B, we
identify the vectors in V with m × 1 column vectors over Q:
α1
.
v = α1 v1 + · · · + αm vm ←→ vB = . .
.
αm
Let A be a symmetric m × m rational matrix. Then with respect to the basis B, A defines a
"symmetric bilinear form" ·, · from V × V to Q by
u, v = t uB AvB for all u, v ∈ V.
(Here t uB denotes the transpose of the matrix uB ). Note that u, v = v, u . For a subspace W of
V , let W ⊥ denote the subspace of V given by
W ⊥ = {v ∈ V : v, w = 0 for all w ∈ W }.
(a) Show that V = V ⊥ ⊕ W for some subspace W which satisfies W ⊥ ∩ W = {0}.
(b) Suppose W is a subspace of V such that W ⊥ ∩ W = {0}. Show that there is some x ∈ W
such that x, x = 0. [Hint: Argue that for w ∈ W , we can find some w ∈ W such that
w, w = 0; now expand w + w , w + w .]
(c) Suppose still that W is a subspace of V such that W ⊥ ∩ W = {0}. Let x ∈ W satisfy
x, x = 0. Show that W = Qx ⊕ W where x, w = 0 for all w ∈ W .
(d) FACT: Induction on r = dim W and (c) may be used to show that V = V ⊥ ⊕Qx1 ⊕· · ·⊕Qxr ,
where xi , xi = 0 and xi , xj = 0 whenever i = j. Use this fact to show that for some
nonsingular matrix S, t SAS = D where D is a diagonal matrix of rank r = dim W .
3. Let G be a finite group of permutations of order N acting on s symbols. Let GP denote the
subgroup of G consisting of all elements fixing a given letter P .
(a) Let m be the number of elements in the transitivity class (orbit) containing P . Show that
|GP |m = N where |GP | denotes the order of the subgroup GP .
(b) Suppose that P and Q are in the same transitivity class. Show that GP and GQ are isomorphic
groups.
(c) Let σ(g) stand for the number of symbols left fixed by an element g in the group and let t
be the number of transitivity classes under G. Show that
σ(g) = tN.
g∈G
(d) Let G be the symmetry group of the rectangle shown below whose vertices are labelled 1, 2,
3, 4 and the midpoints of whose sides are labelled 5, 6, 7, 8. G is a Klein 4-group. Use its
realization as a permutation group on the 8 points {1, 2, . . . , 8} to illustrate the theorem in
part (c).
1 5 2
8 6
4 7 3
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ALGEBRA PRELIM AUGUST 1992
√ √
4. Consider the ring Z[ −3] = {a + b −3 : a, b ∈ Z}. In answering the following questions, you
√
may want to use the "norm": N (a + b −3) = a2 + 3b2 .
√
(a) What are the units in Z[ −3]?
(b) Find all factorizations into irreducible elements of the number 4 in this ring, showing that
√
Z[ −3] is not a Unique Factorization Domain. Is it a Euclidean domain? Explain.
√
(c) Test 5 and 7 to see if they are irreducible in Z[ −3].
√
(d) Give an example of an element of Z[ −3] which is irreducible but not prime.
5 [Hint: Study h(x + 1) by
first writing h(x) = g(x)/(x − 1). Use Eisenstein's criterion to show h(x + 1) is irreducible.]
(b) Show that G = Gal(K/Q) is cyclic of order 6, and has as generator the map that takes r → r3
for any root r, besides Q, Q(ω), Q(x1 ) and Q(x2 ), there are no fields M with Q ⊂ M ⊂ Q(ω).
(Here ⊂ denotes proper containment.)
6. Let h(x) = x4 + 1 ∈ Q[x], and let L be a splitting field for h(x) over Q.
√
(a) Show that the four complex numbers ± 22 (1 ± i) are the four roots of h(x) in C.
(a1
ALGEBRA PRELIM JANUARY 1992
1. Recall that a finite group G is called solvable if there is a sequence of groups
G = G0 ⊇ G1 ⊇ · · · ⊇ Gn = 1
such that for i = 1, . . . , n, Gi is normal in Gi−1 and Gi−1 /Gi is abelian.
(a) Let p be a prime number, and a a positive integer. Show that any group of order pa is
solvable.
(b) Let p and q be 2 distinct prime numbers, and a and b be any 2 positive integers. Show that
any group of order pa q b is solvable.
2. Let R3 denote the 3-fold Cartesian product of the real numbers with itself. We will consider all
its elements as column vectors.
(a) Recall that the standard inner product of 2 vectors v and w in R3 is given by v · w = t vw,
where t denotes the transpose. Prove that if w1 and w2 are vectors in R3 , and for all v in
R3 , v · w1 = v · w2 , then w1 = w2 .
(b) Let E denote the symmetric matrix
1 0 0
0 1 0 ,
0 0 −1
and define φ(·, ·) to be the symmetric bilinear form on R3 given by
φ(v, w) = v · Ew = w · Ev
(where the second equality follows from the symmetry of E).
Prove that for an arbitrary 3 × 3 matrix A, the following conditions are equivalent:
(i) φ(Av, Aw) = φ(v, w) for all v, w ∈ R3 .
(ii) det A = ±1, and A satisfies: If φ(v, v) = 0, then φ(Av, Av) = 0.
(iii) The columns ci of A satisfy: φ(c1 , c1 ) = φ(c2 , c2 ) = −φ(c3 , c3 ) = 1 and φ(ci , cj ) = 0 for
i = j.
(iv) t AEA = E.
(c) Let us call a matrix A hyperbolic if it satisfies any (hence all) of the conditions in part (b).
Prove that the hyperbolic 3 × 3 real matrices form a group.
3. Recall that a regular dodecahedron is a convex polygon whose faces comprise twelve regular
pentagons, symmetrically arranged. In this problem, we will let G stand for the group of orientation-
preserving rigid motions of the regular dodecahedron. In a coordinate system centered at the center
of the dodecahedron, each element of G is a rotation centered at the origin. In particular, elementary
geometric considerations show that G consists of elements of 4 types: (i) The identity; (ii) A two-fold
rotation that fixes the midpoints of 2 antipodal edges; (iii) A three-fold rotation about each of its
twenty vertices; and (iv) Five-fold rotations that cyclically permute each of its twelve pentagons.
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ALGEBRA PRELIM JANUARY 1992
To make this clearer, below is the Schlegel diagram of the dodecahedron (a combinatorially correct,
but metrically inaccurate, representation).
1
3
4 5
5 4 2
2 1 2
3 5 3 1
4
1 2 5
4 3
Note that the vertices of the Schlegel diagram have been numbered from 1 to 5. You may assume the
slightly-painful-to-verify fact that every element of G permutes the labels in the Schlegel diagram
above in a consistent manner. That is to say, if vertices A and B have the same label, and if g ∈ G,
then g(A) and g(B) also have the same labels. In other words, G acts on the set of labels. The
purpose of this exercise is to show that G is isomorphic to the alternating group A5 on five letters.
(a) Make a table showing the number of elements in G of order 1, of order 2, of order 3, and 5.
(b) Do the same thing for A5 .
(c) Prove that G is isomorphic to A5 .
4. Let A be a commutative ring. An element a ∈ A is called nilpotent if an = 0 for some positive
integer n. Let
N = {a ∈ A : a is nilpotent}.
(a) Show that N is an ideal.
(b) Let p be a prime ideal in A. Show that N ⊆ p.
(c) Show that A/N contains no nonzero nilpotent elements.
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ALGEBRA PRELIM JANUARY 1992
5. Let R be a ring. Recall that a sequence of R-modules A, B, and C with R-module homomorphisms
f and g
f g
A / B / C
is called exact if the image of f is equal to the kernel of g. Also recall that a diagram of R-modules
A, B, C, and D, with R-module homomorphisms f, g, h, and i
f
A /B
h g
i
C /D
is called commutative if g ◦ f = i ◦ h.
Consider the following diagram of modules and homomorphisms, where each square is commutative,
and each sequence in the top and bottom rows is exact:
g1 g2 g3
A1 / A2 / A3 / A4
f1 f2 f3 f4
h1 h2 h3
B1 / B2 / B3 / B4
Prove that if f1 is surjective, and f2 and f4 are injective, then f3 is injective.
6. Let K = Fq be the finite field with q elements, and let K(x) denote the field of rational functions
over K in the variable x. Let G = GL2 (K) denote the group of 2 × 2 invertible matrices with entries
a b
in K. If g = is in G, then we associate to g an automorphism φ(g) of K(x) which leaves
c d
K pointwise fixed, and
ax + b
φ(g)(x) = .
cx + d
You may assume the well-known fact that the map φ from G into the group of automorphisms of
K(x) is a homomorphism.
(a) Show that the order of G is q 4 − q 3 − q 2 + q.
(b) Let H = φ(G). Show that the order of H is q 3 − q.
2
(xq − x)
(c) Use field theory to show that f = is relatively prime to xq − x.
(xq − x)
(d) Prove that the fixed field of H is the field K(y), where
2
(xq − x)q+1 f q+1
y= q 2 +1 = .
(x − x)q (xq − x)q2 −q
[Hint: Use the fact that G is generated by the set of matrices of the form
a 0 1 a 0 1
, , and ,
0 1 0 1 1 0
as a varies over all nonzero elements of K.]
ALGEBRA PRELIM AUGUST 1991
do any 5 of the following 7 problems
1. Let C denote the unit circle, x2 + y 2 = 1, in the real plane. Let E be the point (1, 0) =
(cos(0), sin(0)) on C and let A = (cos(α), sin(α)) and B = (cos(β), sin(β)) be an arbitrary pair of
points on C. Define a binary operation on C as follows: A ∗ B = U is the second point of intersection
on C and the straight line through E parallel to the straight line through A and B. (When A = B,
the line through A and B is the line tangent to C at A.)
Prove: This operation induces the structure of an abelian group on C with E as the identity element.
[Hint: All of the group laws except for associativity are easy to verify. In order to show associativity
you must show that C is isomorphic to the circle group R/2πZ and, thus, note that associativity of
the binary operation is inherited]
2. Using the result of Problem 1, show that if C is the ellipse 3x2 + 5y 2 = 1 in the real plane then
the geometric operation described above makes C into an abelian group. Specifically, let O be an
arbitrary point on C - this will be the identity element of the group. If A and B are two points on
C, then A + B is the second point of intersection of the line through O that is parallel to the line
through A and B.
3. Let R be an associative ring with identity element such that a2 = a for all a in R. Show that R
is necessarily commutative and that a = −a for all a.
4. (a) Give an example of a homomorphism of rings f : R → S with multiplicative identities 1R
and 1S such that f (1R ) does not equal 1S .
(b) If f : R → S is an epimorphism of rings with identity, show that f (1R ) = 1S .
(c) Now assume only that f : R → S is a homomorphism of rings both of which have multi-
plicative identities and that there is a unit u of R such that f (u) is a unit in S. Prove that
f (1R ) = 1S and that f (u−1 ) = f (u)−1 .
5. Let G be a group of order 10, 000 having a normal subgroup K of order 100. Show that G has a
normal subgroup of order 2500.
6. Let p(x) = xn − 1 and suppose that p(x) splits in the field K. Let G be the set of all roots of p(x)
in K.
(a) Show that any finite subgroup of K ∗ (the multiplicative group of K) is cyclic.
(b) Show that G is a cyclic group under multiplication.
(c) What is the order of G?
Note: Characteristic 0 and p must be handled separately.
7. Using the fact (obtained in Problem 6) that G is cyclic:
(a) Show that the field Q(G) is an abelian extension of the field Q of rational numbers.
(b) Show that the Galois group of Q(G) over Q in the case of p(x) = x8 − 1 is the Klein 4-group.
Note: Q(G) is the field extension of Q obtained by adjoining the elements of G.
1
ALGEBRA PRELIM JANUARY 1991
do 6 of the following 7 problems
1. Let G be a group and Aut(G) the group of automorphisms of G. Let C be a characteristic
subgroup of G, i.e., C is a subgroup of G such that α(C) = C for all α ∈ Aut(G). Now let
B = {β ∈ Aut(G) : β(c) = c for all c ∈ C}.
(a) Show that B is a normal subgroup of Aut(G).
(b) Suppose that p is a prime integer and G is a group such that G Z/pZ × Z/pZ. Show that
Aut(G) GL2 (Z/pZ) = {invertible 2 × 2 matrices with entries from Z/pZ}.
2. Prove that every group of order 45 is abelian.
3. Let G = {a1 , . . . , an } be a finite abelian group of order n and with identity e.
(a) Prove that ( n ai )2 = e.
i=1
(b) Prove that if G has no elements of order 2 or if G has more than one element of order 2 then
n
ai = e.
i=1
[Hint: Consider the subgroup {x ∈ G : x2 = e}.]
(c) Prove that if G has exactly one element x of order 2, then
n
ai = x.
i=1
(d) Prove Wilson's Theorem, which states that if p is a prime integer then
(p − 1)! ≡ −1 mod p.
4. (a) Let Z[x] be the ring of polynomials in the indeterminate x with integer coefficients. Find an
ideal in Z[x] which is not principal. Justify your result.
(b) Find a nonzero prime ideal in Z[x] which is not maximal. Justify your result.
(c) Let R be a principal ideal domain. Prove that a proper nonzero ideal in R is a maximal ideal
if and only if it is prime.
5. Let F = Z/pZ where p is a prime; take a ∈ F, a = 0, and let x be an indeterminate.
(a) Show that if α is a root of xp − x − a then so is α + 1.
(b) Show that xp − x − a is irreducible in F[x].
(c) Let K be a splitting field over F for xp − x − a. Compute the Galois group of K over F.
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ALGEBRA PRELIM JANUARY 1991
6. Let F be a finite field with q elements and characteristic p > 2.
(a) Show that exactly half the nonzero elments of F are squares in F. [Hint: Consider the
mapping a → a2 .]
(b) Show that for a ∈ F× = {x ∈ F : x = 0}, we have a = b2 for some b ∈ F if and only if a is
q−1
the root of the polynomial X 2 − 1.
(c) Show that −1 is a square in the field Z/pZ if and only if p ≡ 1 mod 4. (Recall that p ∈ Z+
is an odd prime.)
7. Assumptions: Let V be a vector space over a field F and W a subspace of V ; suppose that
dimF V = n < ∞ and dimF W = m < n (where dimF V denotes the dimension of V as a vector
space over F ). Thus each element α ∈ F acts on V ; notice that this action is a linear transformation
on V . Let R be a commutative ring of linear transformations on V such that (i) F ⊆ R, and (ii) for
T ∈ R we have T (W ) ⊆ W . Let S = {T ∈ R : T (V ) ⊆ W }. Note that S is an ideal of R.
Prove:
(a) Show that α → α + S gives an embedding of the field F into the ring R/S; using the fact
that a ring containing a field is a vector space over that field, show that R/S is a vector
space over F .
(b) Show that V /W is a module over R/S.
(c) Suppose S is a maximal ideal of R. Show that
dimF R/S · dimR/S V /W = dimF V /W.
1
ALGEBRA PRELIM AUGUST 1990
answer 2 of the questions of part i and 4 of the questions of part ii
part i
1. Show that the center of a nonabelian group of order p3 has order p.
2. Let G be the infinite dihedral group = v, t with generators v and t where t has infinite order,
v is of order two, and vt = t−1 v. Let H be a subgroup of index 2 in G and let T = t2 , a normal
subgroup.
(a) Show that T ⊆ H. (You may use the fact that H must be normal because it has index 2.)
(b) Describe the quotient group G/T .
(c) List the subgroups of index 2 in G/T .
(d) Use an appropriate correspondence theorem to find all subgroups of index 2 in G (note any
subgroup may be described by listing its generators).
3. Inside the symmetric group S4 , let
H = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)},
where e is the identity.
(a) Show that H is a normal subgroup of S4 , and that A4 /H is cyclic.
(b) Show that H is its own centralizer in S4 .
(c) By considering the homomorphism
ϕ : S4 → Aut(H) defined by ϕ(s) = {h → shs−1 },
show that
S4
Aut(H) S3 .
H
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ALGEBRA PRELIM AUGUST 1990
part ii
4. Let Z2 denote the field with two elements.
(a) If f is an irreducible polynomial of degree 17 over Z2 , how many elements are there in the
splitting field of f ?
(b) How many irreducible polynomials of degree 17 are there over Z2 ?
5. Let R = Z[x, y].
(a) Prove or disprove: The ideal I = (x, y) is a prime ideal in R.
(b) Find all maximal ideals in R which contain I.
(c) Prove or disprove: The ideal J = (y 2 − x3 ) is a prime ideal in R.
6. (a) Let φ : V → V be a linear transformation of a vector space V over a field K. Let E be
a collection of eigenvectors of φ whose eigenvalues are distinct. Prove that E is a linearly
independent set.
(b) Suppose that φ : V → V is a linear transformation of a vector space V where dim V = n.
Suppose that φ has n distinct eigenvalues. Prove that there is a basis B of V such that the
matrix representation of φ with respect to B is a diagonal matrix.
7. Let f (x) = x5 − 5 be a polynomial defined over the rational field Q. Let E be the splitting field
of f .
(a) Find the degree of E over Q.
(b) Give the structure of the Galois group of E over Q presenting generators and relations.
8. Let F be a field. Let V be the subgroup of the multiplicative group of the reals given by
V = {2n : n ∈ Z}.
F is called isosceles if there exists a surjective map f : F → V ∪ {0} with the following properties:
(i) f (0) = 0 and f (α) > 0 if α = 0.
(ii) f (αβ) = f (α)f (β) for all α, β ∈ F .
(iii) f (α + β) ≤ max{f (α), f (β)}.
(a) Suppose F is an isosceles field. Let R = {α ∈ F : f (α) ≤ 1}. Show that R is a ring with
identity.
(b) Let P = {α ∈ F : f (α) < 1}. Prove that P is a prime ideal of R.
(c) Show that P is a principal ideal.
1
ALGEBRA PRELIM JANUARY 1990
do two of the first three problems and four of the last five
1. Let H be a proper subgroup of a finite group G. Show that G is not the union of all the conjugates
of H. [Hint: How many conjugates does H have?]
2. Classify (up to isomorphism) all groups of order 286 = 2 × 11 × 13.
3. If G is a group, and x ∈ G, then the inner automorphism of G determined by x is the automorphism
αx : G → G, αx (g) = x−1 gx for g ∈ G. If an automorphism is not an inner automorphism, then it
is called an outer automorphism.
(a) Does the set of all inner automorphisms of G form a group? Justify your answer, i.e., either
prove the set is a group, or give a counterexample.
(b) Does the set of all outer automorphisms along with the identity automorphism form a group?
Justify your answer.
(c) Show that every finite abelian group, with the exception of one abelian group, has an outer
automorphism. What is the exceptional abelian group?
4. Let R be a commutative ring with unity. We call an element e ∈ R an idempotent if e2 = e.
Suppose M is an R-module, and e is an idempotent of R. Set
M1 = {em : m ∈ M }, M2 = {(1 − e)m : m ∈ M }.
(a) Show that M1 and M2 are submodules of M .
(b) Show that M = M1 ⊕ M2 .
5. (a) Give an example of a non-principal ideal I in a Noetherian integral domain A.
(b) Give an example of a not finitely generated ideal I in an integral domain A.
(c) Give an example of a Unique Factorization Domain which is not a Principal Ideal Domain.
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ALGEBRA PRELIM JANUARY 1990
6. Let f (x) = x4 + ax2 + b be an irreducible polynomial over Q, with roots ±α, ±β and splitting
field K. Show that Gal(K/Q) is isomorphic to a subgroup of the dihedral group D4 (this is the
noncommutative group of order 8 which is not isomorphic to the quaternion group) and therefore is
isomorphic to Z/4Z, or Z/2Z × Z/2Z, or D4 .
7. Let p be an odd prime, and let ζ = e2πi/p . Recall that the map
a → (σa : ζ → ζ a )
gives an isomorphism between (Z/pZ)× and the Galois group of Q(ζ) over Q.
Recall also that (Z/pZ)× is a cyclic group, so has a unique subgroup S of index 2 consisting of the
elements which are squares. Let χ be the composite homomorphism
(Z/pZ)× / (Z/pZ)× /S {±1}
7
χ
In other words,
1 if t is a square in (Z/pZ)×
χ(t) =
−1 if t is not a square in (Z/pZ)× .
Let g = χ(t)ζ t .
t∈(Z/pZ)×
(a) Show that σa (g) = χ(a)−1 g = χ(a)g, for any a ∈ (Z/pZ)× .
(b) Show that g 2 ∈ Q, but g ∈ Q.
/
(c) Show that Q(g) is the unique degree 2 extension of Q contained in Q(ζ).
√ √ √
8. Let p1 , p2 , . . . , pn be distinct primes. Show that Q( p1 , p2 , . . . , pn ) is of degree 2n .
ALGEBRA PRELIM AUGUST 1989
1. Let k be an arbitrary field and let k(z) be the field of rational functions in one variable over k.
Let a1 , . . . , an be distinct elements of k. Let e1 , . . . , en be natural numbers (i.e., ej > 0). Let L be
the set of all elements R(z) = f (z)/g(z) of k(z) satisfying (i) gcd(f, g) = 1, (ii) the roots of g(z) in
the algebraic closure of k are among the points a1 , . . . , an and have multiplicities at most e1 , . . . , en ,
respectively, and (iii) deg f (z) ≤ deg g(z).
(a) Show L is a vector space over k.
(b) Find an explicit basis for L over k.
(c) Prove dimk L = e1 + · · · en + 1.
(d) What can be said if deg f (z) ≥ deg g(z)?
2. Let C be the hyperbola xy = 1 in the real plane. Let (a, b) and (c, d) be points on C (i.e.,
ab = cd = 1). Let L be the line through (a, b) and (c, d). When (a, b) = (c, d) then L is assumed to
be the line tangent to C at that point. Next let M be the line through (1, 1) parallel to L. Let (x, y)
be the other point on C where M intersects C. When M is tangent to C then (x, y) is set equal to
(1, 1). Define a binary operation on C by setting (a, b) · (c, d) = (x, y). Show:
(a) C is an abelian group under this binary operation with (1, 1) as identity element.
(b) C is isomorphic to R× (the multiplicative group of nonzero real numbers).
[Hint: Set up the isomorphism first and use it to show that all of the group properties on C are
inherited from R× .]
3. (a) Let G be a group and let H be a normal subgroup. Suppose that every element of G/H has
finite order and every element of H also has finite order. Show that every element of G has
finite order.
(b) Show that no group of order 56 is simple.
4. Let Zn = Z/nZ be the cyclic group of order n. Let G = Z45 ⊕ Z54 ⊕ Z36 ⊕ Z10 .
(a) What is the order of the largest cyclic subgroup of G?
(b) How many elements of order 3 are there in G?
5. Let f (x) = (x2 − 3)(x3 − 5) and g(x) = (x2 + 3)(x3 − 5). Let K and L be the splitting fields,
respectively, of f (x) and g(x) over the rational numbers, Q.
(a) Find generators for K and L over Q.
(b) Find the degrees [K : Q] and [L : Q].
ALGEBRA PRELIM JANUARY 1989
1. Let G be a finite group and let H be a subgroup of G of index n. Show: there exists a normal
subgroup H ∗ of G with the properties that H ∗ ≤ H and [G : H ∗ ] ≤ n!. [Hint: Construct a
homomorphism of G into the symmetric group Sn whose kernel is contained in H.]
2. Let p be a rational prime and let q = pt . Let GF (p) and GF (q) be the fields with p and q
elements, respectively. Let T : GF (q) → GF (p) be the trace map (i.e., T (x) equals the sum of the
conjugates of x). Let Φ : GF (q) → GF (q) be defined by Φ(x) = xp − x. Show:
(a) Φ is a homomorphism of the additive group GF (q).
(b) The kernel of Φ is GF (p).
(c) The image of Φ equals the kernel of T .
(d) T maps GF (q) onto GF (p).
3. Let fm (x) = (x − µ1 ) · · · (x − µr ) where the roots range over the (complex) primitive mth roots of
unity. Show:
(a) The coefficients of fm are rational integers.
(b) If p is a rational prime not dividing m then fm (xp ) = fm (x)fpm (x).
(c) fm is irreducible over Q (the rational numbers).
4. A field is said to be formally real if the equation x2 + · · · + x2 = −1 is not solvable in the field. A
1 n
field is said to be real closed if it is formally real and does not possess a proper, algebraic, formally
real extension. Show:
(a) A formally real field has characteristic zero.
(b) If K is formally real, with algebraic closure M then there exists a real closed field L such
that K ≤ L ≤ M .
(c) A real closed field K is ordered in exactly one way (i.e., there is a subset P of K such that
(i) for all a in K exactly one of the following holds: a = 0, a is in P , −a is in P and (ii) if
x and y are in P then so are x + y and xy). Moreover, the set of positive elements of K is
precisely the set of nonzero squares. [Hint: Show that if a nonzero element a of the field is
not a square then −a is a square.]
(d) An example of a formally real subfield K of C (the complex numbers) which is algebraic over
Q (the rational numbers) but which is not itself a subfield of R (the real numbers).
ALGEBRA PRELIM AUGUST 1988
1. Determine the number of (isomorphism classes of) groups of order 21. (Justify your answer.)
2. Let R be an associative ring with multiplicative identity 1.
(a) Let x ∈ R be arbitrary. Show that (x) = {z ∈ R : zx = 0} is a left ideal in R.
(b) Let x be any element of R that has a mutiplicative left inverse y in R.
(i) Prove that y is unique iff (x) = 0.
(ii) Prove that y is unique iff x is a unit in R.
3. Let L be a free Z-module of rank 2 contained in C. Let {w1 , w2 } be a Z-basis for L and assume
that u = w1 /w2 is not real. Assume also that there exists a complex number z, not in Z, such that
zL ⊆ L. Show the following facts:
(a) The minimal polynomial for u over Q is quadratic (hence u is algebraic).
(b) If w is any complex number for which wL ⊆ L, then w ∈ Q(u) and satisfies a quadratic
equation of the form w2 + rw + s = 0 where r and s are in Z.
(c) If R denotes the set of all complex numbers w for which wL ⊆ L, then R is a subring of Q(u)
and L is isomorphic to an ideal of R.
4. Let F be a field and let R = F x be the ring of formal power series in one variable with coefficients
in F .
(a) Describe the units in R. (Justify your answer.)
(b) Show that every nonzero ideal in R is of the form xk R, k ∈ Z, k ≥ 0.
(c) Show that x is the unique prime element in R, up to associates.
ALGEBRA PRELIM JANUARY 1988
1. Determine up to isomorphism all groups of order 8.
2. Let G be a finite group and f : G → G be an automorphism of G. If f (x) = x implies x = e, and
f 2 = f ◦ f equals the identity map, show that G is abelian. [Hints: Prove that every element in G
has the form x−1 f (x) and that if φ(x) = x−1 for all x ∈ G is an isomorphism, then G is abelian.]
3. (a) Let R be a ring with 1. State the axioms for a unitary left R-module M .
(b) Let M be a cyclic unitary left R-module with generator m, i.e., M = R m . Let J = {r ∈
R : rm = 0}. Show that J is a left ideal of R.
(c) Regarding both R and J as left R-modules, show that R/J ∼ M , i.e., R/J and M are
=
isomorphic as left R-modules.
4. Find the Galois group of x5 − 3 over Q. Give the order of the group, find the splitting field and
give a set of generators for the Galois group by describing their effect on the roots of the polynomial.
5. (a) Let x be an indeterminate (transcendental) over the complex numbers C and suppose that
r(x) = p(x) where p(x) and q(x) are elements of C[x] and are relatively prime. Define
q(x)
deg r(x) = max{deg p(x), deg q(x)}. Show that if deg r(x) ≥ 1, then r(x) is transcendental
over C and that C(x) is an algebraic extension of C(r(x)) of degree equal to deg r(x).
(b) Show that there is an automorphism φ of C(x) fixing C defined by φ(x) = r(x) precisely if
r(x) = ax+b where ad − bc = 0 (i.e., the automorphisms of C(x) fixing C correspond to the
cx+d
set of linear fractional transformations).
6. Prove that the integral domain Γ of Gaussian integers (i.e., complex numbers of the form a + bi,
with a and b integers) is a unique factorization domain.
ALGEBRA PRELIM JANUARY 1987
1. Show that every group of order 77 is cyclic.
2. Let G be the direct sum of cyclic groups of order m and n where m | n. Let G be written
additively.
(a) Determine the order of the subgroup G(m) consisting of elements x with mx = 0.
(b) Determine the order of the group of endomorphisms of G, i.e., the homomorphisms of G into
itself.
3. Let R be a ring with identity and M a unitary R-module.
(a) If m ∈ M ,i.e., M is
irreducible). Let m0 ∈ M , m0 = 0. Show that A = {x : xm0 = 0} is a maximal left ideal of
R (that is, if A is contained properly in a left ideal B, then B = R).
4. An ideal I in a commutative ring R with identity is primary if for any a, b in R with a · b ∈ I, if
a ∈ I, then bn ∈ I for some n ≥ 1.
/
(a) Show that I is primary iff every zero divisor in R/I is nilpotent.
√ √
(b) Let I = {x ∈ R : xn ∈ I for some n}. Prove that if I is a primary ideal, then I is a
prime ideal.
(c) When R is a principal ideal ring, show that I is primary iff I = P e for some prime ideal P
with e ≥ 0. are
their degrees when Φ20 is factored into irreducible factors over Q(ω).
ALGEBRA PRELIM JANUARY 1986
do five of the six problems
1. Let α be an element of the alternating group An . Prove that the number of conjugates of α in
An (i.e., under conjugacy by the elements of An ) is either the same as or only half as large as the
number of conjugates of α in the symmetric group Sn that contains An .
2. Let G be a group of order 780 = 22 · 3 · 5 · 13 which is not solvable. What are the orders of its
composition factors? Explain your reasoning. (You may assume without proof that all groups of
order less than 60 are solvable.)
3. (a) Prove that prime elements in an integral domain are irreducible.
(b) Let D be a principal ideal domain. Prove that if P is a nonzero prime ideal in D, then P is
a maximal ideal.
(c) Let R[x] be the ring of polynomials in one indeterminate over an integral domain R. Prove
that if R[x] is a principal ideal domain, then R is a field.
4. Let R be a commutative ring (not necessarily with multiplicative identity). Prove that if the only
ideals in R are (0) and R, then either:
(a) R is the zero ring: R = {0},
(b) R contains a prime number p of elements, and a · b = 0 for all a, b ∈ R, or
(c) R is a field.
5. (a) Let Fp denote a finite field with p elements, where p is an arbitrary prime, x be transcendental
over Fp , K = Fp (x), and f (z) = z p − x ∈ K[z], where K[z] is the ring of polynomials in a
transcendental element z over the field K. Prove:
(i) f (z) is irreducible in K[z].
(ii) If θ is a root of f (z) in its splitting field over K, then K(θ) is an inseparable (algebraic)
extension of K.
(b) Prove: If F is a subfield of a field E such that [E : F ] = n = (degree of E over F ) < ∞, x
is transcendental over F , f (x) ∈ F [x] is irreducible of degree d ≥ 1 in F [x] and (d, n) = 1,
then f (x) is irreducible in E[x].
6. (a) Let Q denote the field of rational numbers. Determine the subfield K of the complex field C
that is the splitting field over Q of the polynomial f (x) = x4 − x2 − 6.
(b) Determine the Galois group Gal(K/Q) = Gal(f /Q) and all of its subgroups.
ALGEBRA PRELIM AUGUST 1984
1. G is a finite group of order 2p, where p is a positive odd prime number. You are given that x, y
are elements of G of order 2, p, respectively.
(a) Prove from first principles (i.e., using only the notion of a group) that xyx−1 = y m for some
integer m.
(b) Prove that one may take m = 1 or p − 1.
2. There exists a simple group G or order 168. Prove that G is isomorphic to a subgroup of S8 , the
symmetric group on eight letters. [Hint: Consider Sylow subgroups of G.]
3. Let R = Z[x, y], where Z denotes the ring of rational integers and x, y are algebraically independent
over Z. For each of the ideals I in R as defined below:
(i) Briefly describe the quotient ring R/I. (If you wish, you may describe an isomorphic image.)
(ii) Determine whether or not I is prime. (Justify your answer.)
(iii) Determine whether or not I is maximal. (Justify your answer.)
(a) I = (y), the principal ideal generated by y in R.
(b) I = (y, 5, x2 + 1), the ideal generated by the three elements y, 5, x2 + 1 in R.
(c) I = (y, 3, x2 + 1), the ideal generated by the three elements y, 3, x2 + 1 in R.
4. Let M = {0} be an arbitrary left R-module of an arbitrary ring R = {0}. M is called a simple
left R-module if and only if its only proper left R-submodule is {0}. Prove:
(a) If M is simple, then either
(i) RM = {0} and M is finite of prime order, or
(ii) RM = {0} and M is a unitary cyclic left R-module generated by each of its nonzero
elements.
(b) If either (i) or (ii) above holds, then M is simple.
5. Let F = GF (2), the field with two elements. Let K be a splitting field for f (x) = x4 + x + 1 over
F. Let α be an element of K such that f (α) = 0. Find all elements β ∈ K such that K = F (β).
(Express each β as a polynomial in α over F of least possible degree.) Prove that your list is complete.
6. Determine the Galois group G of x6 − 3 over Q (the rational number field).
ALGEBRA PRELIM JANUARY 1984
4. Let w(x, y) = xm1 y n1 · · · xmr y nr , mi and nj are any integers (of any sign), different from 0 and
r ≥ 1. Find two permutations p and q of a finite set such that
w(p, q) = pm1 q n1 · · · pmr q nr
is a permutation different from the identity.
5. (a) Consider the matrix
15 12 −16
1
A= −20 9 −12
25 0 20 15
as a linear mapping from R3 into itself. You may assume without proof that this mapping is
a rotation around a certain axis through an angle θ. Find the axis and find θ.
(b) Find two different square roots of A, one a rotation and one not. For full credit, include a
numerical solution; up to 6 out of 8 points will be awarded for a geometric description and a
√
description of how one would proceed in calculating A, in lieu of the calculation itself.
6. In the following problem, you may assume the following fact, which holds for cubic polynomials
over any field:
if x3 + px + q = (x − α)(x − β)(x − γ), then [(γ − α)(γ − β)(β − α)]2 = −4p3 − 27q 2 .
You may assume the fundamental facts of Galois theory, but apart from these assumptions, please
base your proofs on fundamentals of field theory.
(a) Prove that f (x) = x3 − 3x + 1 is irreducible over the field Q of rational numbers.
(b) Prove that f (x) has three distinct real roots (alias "zeros"). Let us call them α, β, γ with
α < β < γ.
ALGEBRA PRELIM AUGUST 1983
1. Let Zn denote the (additive) cyclic group of order n. Let G = Z15 ⊕ Z9 ⊕ Z54 ⊕ Z50 ⊕ Z6 .
(a) What is the order of the largest cyclic subgroup in G?
(b) How many elements are there of order 5?
(c) How many elements are there of order 25?
(d) How many subgroups are there of order 25?
2. Let K = GF (pn ) be a finite field of characteristic p which has degree n over its prime field GF (p).
(a) Prove that K has pn elements.
(b) Prove that K is a Galois extension of GF (p) and describe its Galois group.
(c) Prove: GF (pm ) is (isomorphic to) a subfield of GF (pn ) if and only if m divides n. Show that
in this case GF (pn ) has exactly one subfield with pm elements.
3. Let P3 be the vector space of all polynomials over the real field R of degree ≤ 3. Define a mapping
φ : P3 → R by φ(a0 + a1 x + a2 x2 + a3 x3 ) = a0 + a1 + a2 + a3 for every a0 + a1 x + a2 x2 + a3 x3 ∈ P3 .
∗
(a) Prove: φ ∈ P3 , the dual space of P3 (by definition, the dual space of a real vector space is
the space of all linear functions from the space to R).
∗
(b) Let φ0 , φ1 , φ2 , φ3 be the basis of P3 which is dual to the basis {1, x, x2 , x3 }, i.e., φj (xi ) = 0
if i = j and φi (xi ) = 1, for i = 0, 1, 2, 3. Express the linear function φ of part (a) in terms of
this dual basis.
ALGEBRA PRELIM JANUARY 1983
1. (a) What is meant by the statement that a field is a normal extension of the rational field Q?
(b) Let K = Q(21/2 , 21/3 ). Determine the relative degree [K : Q].
(b) Prove that K is not a normal extension of Q.
2. (a) State any one of Sylow's Theorems on finite groups. Consider the set of nonsingular matrices
α β
with elements α, β in the field with 3 elements.
0 α
(b) Prove that they form a group under multiplication.
(c) Determine the structure of this group, in particular whether it is abelian.
3. Let K be an arbitrary field and K(x) the field of rational functions in one variable over K. Let
u be an element of K(x) not in K. Show:
(a) u is not algebraic over K.
(b) If u = f (x)/g(x) where f (x) and g(x) are relatively prime polynomials in K[x] then [K(x) :
K(u)] = m where m = max{deg f (x), deg g(x)}.
4. Let Q be the rational field and let α be a root of x4 + 1. Show:
(a) [Q(α) : Q] = 4.
(b) Q(α) is a Galois extension of Q.
(c) The Galois group of Q(α) over Q is the Klein 4-group.
ALGEBRA PRELIM JANUARY 1982
1. (a) Consider a group G of order 2n which contains exactly n elements of order 2. Show that n
must be odd.
(b) Let A = {a1 , . . . , an } be the set of those elements of G which are of order 2. Prove that
ai aj = aj ai for all i = j.
(c) Give an example of a group of the type given in part (a).
2. (a) Let G be a finite group, H a normal subgroup, p a prime, p [G : H]. Show that H contains
every Sylow p-subgroup of G.
(b) Show that a group of order 992 (= 31 · 32) is not simple.
3. Let R be a commutative ring with 1, and let I, J be ideals in R with I + J = R.
(a) Let a, b ∈ R. Prove that there exists c ∈ R such that c ≡ a mod I and c ≡ b mod J.
(b) Deduce from the above that R/I ∩ J is isomorphic to the direct product (R/I) × (R/J).
[Note: You may do part (b) for partial credit, assuming the result for part (a), even if you haven't
done part (a).]
4. Let R be a commutative ring with 1. Let f1 , f2 , . . . , fr be r elements of R and let (f1 , . . . , fr ) be
the ideal generated by this set. Suppose that g and h are elements of R and that a certain positive
power of g belongs to (f1 , . . . , fr , h) while a positive power of gh belongs to (f1 , . . . , fr ). Show that
there is a positive power of g which belongs to (f1 , . . . , fr ).
ALGEBRA PRELIM JANUARY 1981
solving completely any 4 of the problems secures the maximum score of 100 points
1. Let ω = e2πi/5 .
(a) If possible, find a field F ⊂ Q(ω) such that [F (ω) : F ] = 2.
(b) If possible, find a field F ⊂ Q(ω) such that [F (ω) : F ] = 3.
2. If p is a prime, let Fp denote the finite field with p elements. Find the Galois group of x4 − 3 over
each of the following fields:
(a) F7 .
(b) F13 .
3. Let G be a finite group with nm elements and K a subset with m elements. Define a "coset" of
K to be Kg = {kg : k ∈ K} where g is an element chosen from G.
Suppose that there exist exactly n distinct cosets of K in G. Prove that one of these "cosets" is a
subgroup H and that the other "cosets" are then really the right cosets of the subgroup H in G.
4. (a) Show that a group of order 12 is not simple.
(b) Show that a group of order p2 q is not simple where p and q are distinct odd primes.
5. (a) Let B be a nontrivial Boolean ring (so B = {0} and for all b ∈ B, b2 = b). Prove:
(i) B is commutative.
(ii) If P is any prime ideal in B, then P is maximal.
(b) Let R be a noncommutative ring with multiplicative identity 1.
(i) Let x ∈ R be arbitrary. If r(x) = {y ∈ R : xy = 0}, prove that r(x) is a right ideal in
R.
ALGEBRA PRELIM AUGUST 1980
1. Let K be a field and K x the ring of all formal power series with coefficients in K. Prove:
∞
(a) an xn is a unit in K x if and only if a0 = 0.
n=0
(b) K x has only one maximal ideal.
2. Let R be a commutative ring with only one maximal ideal P . Let M be a finitely generated
R-module for which P M = M . Prove that M = 0.
3. Prove: There is no simple group of order 36.
n−1
4. Prove: The order of GLn (Fq ) is (q n − q j ).
j=0
5. Let k be a field and k(x) the field of rational functions in one variable over k. Prove: GL2 (k)/k ∗ =
P GL2 (k) is the Galois group of k(x) over k.
2
(xq − x)q+1
6. Let K be the fixed field of P GL2 (Fq ) acting on Fq (x). Prove K = Fq (y) where y = q .
(x − x)q2 +1
ALGEBRA PRELIM JANUARY 1980
1. Let Fq be a finite field with q elements. What is the number of quadratic (of exact degree 2)
irreducible polynomials in Fq [x]?
2. (a) Prove that the polynomial x4 − 3 is irreducible over the field Q of rational numbers.
(b) What is the degree of a splitting field K of x4 − 3 over Q? Give a set of field generators for
K over Q. (Take K to be a subfield of the complex numbers.)
(c) Prove x4 − 3 is irreducible over Q(i).
(d) Determine the Galois group of x4 − 3 over Q(i) as an abstract group.
3. Let V be a vector space over a field K, R a subring of the ring Hom(V ) of linear transformations
from V to V , and HomR (V ) the ring {S ∈ Hom(V ) : ST = T S for all T ∈ R}.
Prove: If R is 1-transitive, i.e., for all x, y ∈ V with x = 0 there is a T ∈ R with T (x) = y, then
HomR (V ) is a division ring.
4. Let G be a finite group of order n. Assume that, for each prime dividing n, G has a unique Sylow
p-subgroup P , and that P is cyclic. Prove that G is cyclic.
5. Let p, q, r be distinct primes.
√ √
(a) Show that [Q( p, q) : Q] = 4.
√ √ √
(b) Show that [Q( p, q, r) : Q] = 8.
6. (a) Determine for which pairs k, n with 1 ≤ k ≤ n there is a k × k matrix A over the rationals
Q such that An = 2I.
(b) Give, with proof, an example, for each n, of a linear transformation T : Qn → Qn such that
the only T -invariant subspaces of Qn are {0} and Qn .
ALGEBRA PRELIM JANUARY 1979
1. Let G be an arbitrary group whose center is trivial. Prove: The center of the automorphism
group of G is also trivial.
2. Let L be a separable extension of degree n of the field K. Assume L is contained in a given algebraic
(j)
closure K of K. Let {v1 , . . . , vn } be a vector space basis for L over K. Let vi , j = 1, . . . , n be the
conjugates of vi in K. Prove
.
.
.
.
.
.
. . . . . . . . v (j) . . . . . . . . = 0.
det i
.
.
.
.
.
.
3. Determine all maximal ideals in the polynomial ring Z[x]. (Z is the ring of rational integers.)
10 −1
4. How many irreducible factors does the polynomial x2 − 1 have over GF (2)?
5. Let ω1 , ω2 be two complex numbers whose ratio ω = ω1 /ω2 is not real. Let Λ = {mω1 + nω2 :
m, n ∈ Z} be the abelian group generated by ω1 and ω2 . Let R = {λ ∈ C : λΛ ⊆ Λ} (C is the field
of complex numbers).
(a) Prove: R is a ring.
(b) Prove: If λ is a unit in R then λ is a root of unity. (λn = 1 for some n > 0.)
(c) If λ is an nth root of unity in R then n is a divisor of ??.
6. Let R be a ring (associative with identity) for which b2 = b for every b in R. Let p be a prime
ideal of R.
(a) Prove: R is commutative.
(b) Prove: R/p is a field.
ALGEBRA PRELIM AUGUST 1978
1. An element r in a ring is called nilpotent if rn = 0 for some positive integer n.
(a) Show that in a commutative ring R the set of nilpotent elements forms an ideal N .
(b) In the notation of (a) show that R/N has no nilpotent elements.
(c) Show by example that part (a) need not be true for noncommutative rings.
2. Let F = Q(θ) where Q denotes the rational number field and θ the fifth root of unity e2πi/5 .
Discuss the Galois group of the polynomial x5 − 7 over Q(θ), including a determination of the degree
of the root field (justify this), a description of the Galois group in purely group-theoretic language,
and a representation of each automorphism as a permutation.
3. Either: Let G be a finite group of order 2p, p and odd prime.
Let a be an element of order 2, b an element of order p.
Let H be the subgroup of G which is generated by b.
(i) Prove H is a normal subgroup of G.
2
(ii) Prove that aba = br for some integer r, and hence that br = b.
Deduce that one of the relations aba = b; aba = b−1 must hold.
Or: State some theorem involving Sylow subgroups and use it to show that a group of order 30
cannot be simple.
4. fj (x), j = 1, . . . , k (k ≥ 2) are polynomials in x with complex coefficients. Assume that they have
no common root; thus for any x
ALGEBRA PRELIM JANUARY 1978
1. (a) Let G be a cyclic group of order n. Let d ∈ Z+ , and let ν = gcd(d, n). Show that n/ν of the
elements of G are dth powers (i.e., are of the form y d for some y ∈ G).
(b) Let d, s ∈ Z+ , and let p ∈ Z+ be an odd prime (so the group of units U of the ring Z/ps Z is
cyclic). When is the dth power mapping (y → y d ) on U surjective?
2. Let G be the abelian, non-cyclic group of order 25. Let the field K be a Galois (finite, separable,
normal) extension of the field F , with Galois group G.
(a) Find [K : F ], the degree of the field extension.
(b) How many intermediate fields Σ are there between F and K? (F ≤ Σ ≤ K)
(c) Which of the above fields Σ are normal extensions of F ?
3. Let ω1 , ω2 be a pair of complex numbers that are linearly independent over the reals. Let Λ be the
free abelian group generated by ω1 , ω2 . That is, Λ = Zω1 + Zω2 . Now let R = {λ ∈ C : λΛ ≤ Λ}.
Show:
(a) R is a commutative ring containing Z as a subring.
(b) Z R ⇐⇒ ω = ω1 /ω2 generates a quadratic extension of Q.
(c) Suppose that [Q(ω) : Q] = 2 and ω 2 + rω + s = 0 with suitable r, s ∈ Q. Let r = r1 /r2 ,
s = s1 /s2 where r1 , r2 , s1 , s2 are integers and gcd(r1 , r2 ) = gcd(s1 , s2 ) = 1. Finally let
c = lcm(r2 , s2 ). Prove R = Z[cω].
1
ALGEBRA PRELIM JANUARY 1977
1. Let K be a field of degree n over the rational numbers, Q. Moreover, let {w1 , . . . , wn } be a basis
for K as a vector space over Q. Next, when α ∈ K let pα (x) be its minimal polynomial over Q and
n
nα = [Q(α) : Q]. Since αK ⊂ K we can write αwi = aji wj , i = 1, . . . , n. Let
j=1
.
.
.
Aα = · · ·
aji · · · .
.
.
.
We define Φ : K → Mn (Q) by Φ(α) = Aα .
(a) Show that Φ is a monomorphism of the field K into the ring Mn (Q).
(b) For a given α ∈ K what are the minimal and characteristic polynomials of Aα ? (Give these
explicitly.)
√
(c) Compute the minimal and characteristic polynomials in the case: K = Q(i, 2), α = i,
√ √
w1 = 1, w2 = i, w3 = 2, w4 = i 2. Also find Φ(α) in this case.
√ √
2. Let R = Z 1+ 2−11 be the ring of all complex numbers of the form m + n 1+ 2−11 where m
and n are ordinary integers. When a ∈ R we let |a| be its length as a complex number.
(a) Show that R is a Euclidean ring. That is, show that for all a, b = 0 in R there exist q, r in R
such that a = bq + r and |r| < |b|.
(b) Since Euclidean rings are unique factorization domains, factor 37 into prime factors in R.
3. Let H be a subgroup of a group G. Let NG (H), CG (H) be, respectively, the normalizer and the
centralizer of H, i.e., NG (H) = {x ∈ G : x−1 Hx = H}, CG (H) = {x ∈ G : xg = gx for all g ∈ G}.
(a) Prove that CG (H) is a normal subgroup of NG (H), and that NG (H)/CG (H) is isomorphic
to a subgroup of the automorphism group of H.
(b) A celebrated theorem (credited to Burnside) is: "Let the order of a finite group G be pα m,
where p is a prime, and (p, m) = 1. Let P be a Sylow p-subgroup of G. Suppose NG (P ) =
CG (P ). Then G has a normal subgroup of order m."
Use this theorem to prove the following: Let G be a finite group of order pα m, where p is
the smallest prime dividing the order of G, and (m, p) = 1. Suppose P is cyclic, where P is
a Sylow p-subgroup of G. Then G has a normal subgroup of order m.
4. Let K be a splitting field of x12 − 1 over Q, where Q is the field of rational numbers.
(a) Describe the Galois group of K over Q (what are its elements and what is the group struc-
ture?).
(b) How many subfields does K have and what are their degrees over Q?
(c) Let θ be a primitive 12th root of unity in an extension field of Q (i.e., θ12 = 1 and θm = 1 if
0 < m < 12). Find the irreducible polynomial for θ over Q.
please turn over
2
ALGEBRA PRELIM JANUARY 1977
5. Let R be a ring with identity and M a unitary R-module.
(a) If m ∈ Mone says that
M is irreducible). Let m0 ∈ M , m0 = 0. Show that A = {x : xm0 = 0} is a maximal left
ideal of R (that is, if A is contained properly in a left ideal B, then B = R).
6. Let ω1 , ω2 be a pair of complex numbers such that ω = ω1 /ω2 lies in the upper half plane (i.e.,
Im (ω) > 0). Let Λ = {mω1 + nω2 ∈ C : m, n ∈ Z}. Let E(Λ) = {α ∈ C : αΛ ≤ Λ}. (Note:
Z ≤ E(Λ).)
(a) Show: if Z E(Λ) then [Q(ω) : Q] = 2.
(b) Show: If [Q(ω) : Q] = 2 then Z E(Λ) and every α ∈ E(Λ) satisfies an integral equation
(i.e., α2 + aα + b = 0 for some a, b in Z).
√
(c) Compute E(Λ) explicitly in the case ω = −1. (Be careful of this one!)
ALGEBRA PRELIM AUGUST 1976
1. Suppose R is a Boolean ring, i.e., a ring such that x2 = x for all x ∈ R.
(a) Prove that R is commutative and of characteristic 2.
From now on assume there is a unit element 1 ∈ R. For a ∈ R, we let (a) denote the principal
ideal generated by a.
(b) Prove that for a ∈ R, (a) is itself a Boolean ring with unit element a.
(c) Prove that, for any a ∈ R, the ring R is the direct sum of the ideals (a), (1 + a):
R = (a) ⊕ (1 + a).
(d) Prove that any finite Boolean ring is isomorphic to a direct power of the two-element ring Z2
(a direct sum of several copies of Z2 ).
2. (a) A group G is decomposable if it is isomorphic to a direct product of two proper subgroups.
Otherwise G is indecomposable.
Prove that a finite abelian group G is indecomposable if and only if G is cyclic of prime power
order.
(b) Determine all positive integers n for which it is true that the only abelian groups of order n
are the cyclic ones. Justify your answer.
t
3. Let S be the set of all 2 × 2 Hermitian matrices of trace 0, i.e., {A : A and tr(A) = 0} (B t =
transpose of B).
(a) Prove that the mapping
x y + iz
(x, y, z) →
y − iz −x
is an isomorphism of R3 onto S.
t t
Let G be the set of all unitary 2 × 2 complex matrices, i.e., {A : A · A = A · A = I}. For
each matrix A ∈ G define ϕA (B) = ABA−1 for any 2 × 2 complex matrix B.
(b) Prove that ϕA maps S → S, and is a linear transformation of S into itself.
(c) Making use of the isomorphism in part (a), prove that the mapping A → ϕA is a group
homomorphism of G onto a group of distance-preserving linear transformations of R3 .
4. (a) List, without proof, the standard results you know on finite fields (including their Galois
theory).
For any prime p, let F = Zp , the field with p elements. Let K be an algebraic closure of F,
and let G be the group of automorphisms of K.
You may use, in the following, any result quoted in part (a).
(b) Prove that for any positive integer n, K contains one and only one subfield with q = pn
elements.
(c) Let E be any finite subfield of K, and let σ ∈ G. Prove σ(E) = E.
(d) Prove that G is an abelian group.
ALGEBRA PRELIM JANUARY 1975
1. (a) Determine the splitting field K for the polynomial x4 −5 over Q (the field of rational numbers)
and give the degree [K : Q].
(b) Find a set of automorphisms of K which generate the Galois group of K over Q (but do not
list all the elements of the Galois group).
(c) What is the order of the Galois group G of K over Q?
(d) Give an example of intermediate fields F1 , F2 : Q F1 K, Q F2 K such that F1 is
normal over Q and F2 is not normal over Q.
(e) Find the subgroups H1 and H2 of G which correspond to F1 and F2 , respectively, under the
Galois correspondence.
2. If a matrix A has a minimal polynomial (x − 3)3 (x − 5)2 (x − 2) and characteristic polynomial
(x − 3)5 (x − 5)5 (x − 2), give the possible Jordan canonical forms that might correspond to A.
3. Let R be a noncommutative ring with multiplicative identity 1.
(a) Let x ∈ R. If r(x) = {y ∈ R : xy = 0} prove that r(x) is a right ideal of R.
(b) Let x be an element of R which has a right multiplicative inverse z in R. Prove that z is also
a left inverse of x if and only if r(x) = 0.
(c) Prove that if an element x of R has more than one right inverse then it has infinitely many.
[Hint: Note if xz = 1 and a ∈ r(x) then x(z + a) = 1.
4. Prove the theorem: If G is a nonabelian group then G/Z(G) is not cyclic (where Z(G) denotes
the center of the group G).
5. Let G be a group of order p2 q where p and q are distinct odd primes. Prove that G contains a
normal Sylow subgroup.
ALGEBRA PRELIM JANUARY 1974
1. Prove that all groups of order 45 are abelian, and determine how many nonisomorphic groups of
order 45 there are.
2. Let p be an odd prime. For any positive integer n, call an integer, a, a quadratic residue mod pn
if (a, p) = 1 and the equation x2 = a is solvable mod pn . Prove that for any n, the quadratic residues
mod pn are precisely the quadratic residues mod p. [Hint: Use the fact that the group of units of
the ring Zpn form a cyclic group of order pn−1 (p − 1).
3. Prove that the multiplicative group of an infinite field is never cyclic.
4. Let K be the splitting field of (x3 −2)(x2 −2) over the rational numbers Q. Determine all subfields
of K which are of degree four over Q. Explain how you know you have found them all.
5. Let A be a 4 × 4 matrix over the field F . Suppose that
(i) A = I,
(ii) A − I is nilpotent, i.e., there exists a positive integer n such that (A − I)n = 0, and
(iii) A has finite multiplicative order, i.e., there exists a positive integer m such that Am = I.
For what fields F does such a matrix A exist? Clearly indicate your reasoning.
6. Let T be a linear transformation of V into V , where V is a finite-dimensional vector space over
the complex numbers. Let p be any polynomial with complex coefficients. Show p(T ) has exactly
the eigenvalues p(λ1 ), . . . , p(λn ) if λ1 , . . . , λn are the eigenvalues of T .
ALGEBRA PRELIM AUGUST 1973
q n−1
1. Let N : GF (q n )∗ → GF (q)∗ by N (a) = a1+a+···+a . Prove: N is onto.
2. Let V be an n-dimensional vector space over the field k. Let S be a set of pairwise commuting
linear transformations of V into V . Prove: If each f in S can be represented by a diagonal matrix
with respect to some basis of V (depending on f ), then there is a basis of V with respect to which
all of the endomorphisms in S are diagonal.
3. Prove: The group of units of the ring Z/pn Z is a cyclic group of order (p − 1)pn−1 when p is an
odd rational prime. [Hint: Use induction on n.]
4. Let K be the splitting field of x7 − 3x3 − 6x2 + 3 over Q. Let E1 , E2 be subfields of K such that
[K : E1 ] = [K : E2 ] = 7. Prove E1 ∼ E2 .
=
5. Find the Galois group of the splitting field K of x4 − 2 over Q. Find two subfields E1 , E2 of K
such that [K : E1 ] = [K : E2 ] = 2 but E1 and E2 are not isomorphic.
6. Let K be a field in which −1 cannot be represented as a sum of squares and such that in every
proper algebraic extension −1 can be represented as a sum of squares. Prove: If a ∈ K is not a
square in K then a is not a sum of squares in K.
ALGEBRA PRELIM JANUARY 1973
please do 5 out of 6 problems
1. Show that no real 3 × 3 matrix satisfies x2 + 1 = 0. Show that there are complex 3 × 3 matrices
which do. Show that there are real 2 × 2 matrices that satisfy the equation.
2. Prove: Let G be a finite group, let H be a subgroup of G. Let i(H) be the index of H. Let o(G)
be the order of G. Suppose o(G) does not divide i(H). Then H must contain a nontrivial normal
subgroup of G. In particular, G cannot be simple.
[Hint: Let S be the set of right cosets of H. Let a, g ∈ G. Let θa : Hg → Hga. θa is a one-to-one
mapping of S. Consider the collection of {θa : a ∈ G}.]
Use this theorem to show that a group of order 75 cannot be simple. You may use Sylow's theorem.
3. Let G be a group of order pn , where p is a fixed prime and n is a positive integer. Prove:
(a) The center of G is nontrivial, i.e., there is a g ∈ G, g = 1, g ∈ center of G. The center of a
group is {x ∈ G : xg = gx for all g ∈ G}.
(b) For every m, m < n, G has a subgroup of order pm .
(c) Every subgroup of order pn−1 is normal.
4. Let R be a unique factorization domain, and let K be its field of quotients. In the following we
fix a prime element p in R.
(a) Let Rp = { a ∈ K : a ∈ R, b ∈ R and p does not divide b in R}. Prove Rp is a subring of K.
b
(b) Find the units of Rp . What are the primes of Rp ? Prove Rp is a unique factorization domain.
(c) Show that Rp has a unique maximal ideal.
(d) Prove that Rp is a maximal subring of K, i.e., if S is a subring of K which contains Rp then
S = Rp or S = K.
5. Let R be a ring with more than one element and with the property that for each element a = 0
in R there exists a unique element b in R such that aba = a. Prove:
(a) R has no nonzero divisors of zero.
(b) bab = b.
(c) R has a unity.
(d) R is a division ring.
Note: If you can't do one part of this problem assume the result and go on to the next part.
6. Consider p(x) = x8 + 1 as a polynomial over the rationals Q. Let K be the splitting field of p(x)
over Q. Find the Galois group G of p(x), i.e., the group of automorphisms of K relative to Q. Is
this group abelian? If so, express it as a direct sum of cyclic groups. List all the subgroups of G |
UA preparatory math goes virtual
Apr 26, 2011 By La Monica Everett-Haynes
Is this what flashes across your mental screen when you think about math? The UA's mathematics department is piloting a new course, Math 100, which is designed to help students who struggle with university-level math. The course provides personalized instruction with a heavy emphasis on tutoring, peer support and the use of technology.
The University of Arizona's math department is experimenting with a novel approach to early math instruction – one with a heavy emphasis on technology and peer-to-peer tutoring.
Arguably, few other required college-level courses elicit the same frustration or the intimidation factor as mathematics.
Some commonly talk about holding a hatred for math, believe they are no good at it or think up strategies to avoid it all together.
But one University of Arizona team is working to unravel the enigmatic nature of math for the very students who struggle the most with it – those who do not test into college-level math.
Math 100, now in the second semester of its pilot phase, has a heavy emphasis on both self-paced progress and peer-to-peer support while being offered through Elluminate, a web-conferencing system.
"Students are so used to being online. We thought that if we put the course online we could interact more," said Michelle Woodward, who coordinates the pilot course being offered by the UA mathematics department.
The number of section offerings will be expanded during the fall to accommodate more UA students who do not test into algebra-level mathematics.
Woodward said the course is being emphasized and expanded because it is especially important for new students to grasp college math, especially algebra – a curricular core – early.
Algebraic skills have long been associated with giving students the ability to think in more complex ways. A student's ability to comprehend algebra has long been upheld as an indication of college-readiness, particularly for study in science and engineering-related disciplines.
"It's the foundational material they need to be prepared for college algebra," Woodward said.
"My whole goal in this is to make an online environment that is as close to what students would do in person. I want the environment to be as interactive as possible," Woodward said, adding that another program, the ALEKS Learning Module, provides both structure and flexibility while also offering the course content.
"I have done a lot of work with students who needed individualized plans. ALEKS does that for me," she said. "I could not do that for 300 students, it doesn't replace me – it frees me up to work with students individually, the kind of work I didn't have time to do before."
Over the course of the semester, the 300 students currently enrolled in one dozen Math 100 sections meet three hours weekly, receiving self-paced instruction mediated by Elluminate. Students complete assignments, learning to master algebraic expressions and graphing techniques and, all the while, ALEKS tracks their progress.
"We are able to personalize the lessons much better than we have. It's been wonderful," said Cheryl Ekstrom, a mathematics lecturer who initiated the idea to incorporate Elluminate. "You aren't stuck listening to a lecture on things you already know or breezing by things you don't understand."
This is in direct contrast to more established and traditional ways of teaching math.
"In a traditional class, it doesn't matter if it's hard for you," said Shailendra Simkhada, an electrical engineering senior also studying math.
"Each day in a regular class, you might get a new chapter or deadline to meet but, here, they can work at their own pace," he said.
"It's not that they do less work, but if you don't understand something you get more information and one-on-one help so that they stay on track," he added.
If fact, students designate their goals at the start of the class, deciding what sections they want to master and what math class they hope to test into at the end of the term.
Students also engage in weekly virtual classroom meetings, sharing their computer screens and conversing online with student leads and support staff – UA students who are advanced in math and receive more than 15 hours of training.
Kirandeed Banga, a UA sophomore studying biology, is a member of the student lead and support staff.
Each week, Banga joins the other leads and support staff members in a classroom in the Math Building where they each log online to tutor and monitor student work.
"With it being completely online, it's hard to get their trust. But we try to talk to them as much as possible," said Banga who, like others on the team, also offer office hours.
"And we put them into virtual groups, so they are also able to help one another," she added. "They obviously are used to the technology, so they can adapt to it."
Also built into the design of the course is extensive support to the UA students facilitating the class.
Ivvette Rios, a UA math and French major, observes the virtual sessions and conducts weekly meetings with all of the students offering tutoring and support. Her role is to ensure that the leads and support staff have everything they need to appropriately help the hundreds of students enrolled.
Rios said the time for self-evaluation and self-reflection is critical for those involved, and helps to ensure that the structure is working well for all involved.
"We are always thinking of ways we can do this better; to make it more and more like our everyday experience," Rios said. "It's work out way better than we thought it would."
Leo Shmuylovich knows a lot about how tutoring can take a student from confused to confident. The Washington University graduate student has worked as a tutor for several test preparatory companies over the years, helping ...
(PhysOrg.com) -- New research from the University of Notre Dame suggests that even though adults tend to think in more advanced ways than children do, those advanced ways of thinking don't always override old, incorrect"Considering how many fools can calculate, it is surprising that it should be thought either a difficult or tedious task for any other fool to learn how to master the same tricks. Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the text-books of advanced mathematics-and they are mostly clever fools-seldom take the trouble to show you how easy the calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way." Calculus Made Easy, Silvanus P. Thompson, Prologue, 1910. |
Essentials of College Algebra with Modeling and Visualization - 4th edition
Gary Rockswold teaches algebra in context, answering the question, &''Why am I learning this?&'' By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswold&'s focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses. This streamlined text covers linear, qu...show moreadratic, nonlinear, exponential, and logarithmic functions and systems of equations and inequalities, which gets to the heart of what students need from this course. A more comprehensive college algebra text isalso52 +$3.99 s/h
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The Solutions Manual provides answers for all problems in the lesson (including Warm-up, lesson practice, and mixed practice exercises), as well as solutions for the supplemental practice found in the back of the student text. It also includes answers for the facts practice tests, activity sheets, and tests in the separate tests & worksheets book.
This Saxon Math Homeschool 6/5 Solutions Manual provides answers for all problems in the textbook lesson (including warm-up, lesson practice, and mixed practice exercises), as well as solutions for the investigations and supplemental practice found in the back of the student text. It also includes answers for the facts practice tests, activity sheets, and tests in the tests & worksheets book. Answers are line-listed, and are organized by type (lessons & investigations, facts practice tests, tests, etc.).
313 perforated pages, softcover. 3rd Edition.
Customer Reviews for Math 65, Third Edition, Solutions Manual
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Interval Methods for Systems of Equations
An interval is a natural way of specifying a number that is specified only within certain tolerances.An interval is a natural way of specifying a number that is specified only within certain tolerances.Hide synopsis
Description:New. This item is printed on demand. An interval is a natural...New. This item is printed on demand. An interval is a natural way of specifying a number that is specified only within certain tolerances.
Description:NEAR FINE. Paperback, 272pp., This listing is a new book, a...NEAR FINE. Paperback, 272 |
1001 Basic Math & Pre-Algebra Practice Problems For Dummies, with free access to online practice problems, takes you beyond the instruction and guidance offered in Basic Math & Pre-Algebra For Dummies, giving you 1,001 opportunities to practice solving problems from the major topics in your math From the book, go online and find:
One year free subscription to all 1001 practice problems
On-the-go access any way you want it—from your computer, smart phone, or tablet
Multiple choice questions on all you math course topics
Personalized reports that track your progress and help show you where you need to study the most
Customized practice sets for self-directed study
Practice problems categorized as easy, medium, or hard
The practice problems in 1001 Basic Math & Pre-Algebra Practice Problems For Dummies give you a chance to practice and reinforce the skills you learn in class and help you refine your understanding of basic math & pre-algebra.
Note to readers:1,001 Basic Math & Pre-Algebra Practice Problems For Dummies, which only includes problems to solve, is a great companion to Basic Math & Pre-Algebra I For Dummies, which offers complete instruction on all topics in a typical Basic Math & Pre-Algebra course.
This handy guide, with free access to online practice problems, gives you 1,001 opportunities to practice solving problems that you'll encounter in your basic math and pre-algebraSolving problems online, you'll track your progress, see where you need more help, and create customized problem sets to get you where you need to be.
On-the-go access any way you want it — from your computer, smartphone, or tablet
Multiple choice questions on all your math course topics
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Practice problems categorized as easy, medium, or hard
About the Author
Mark Zegarelli is a math and test prep tutor and instructor in SanFrancisco and New Jersey. He is the author of Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, ACT Math For Dummies, Logic For Dummies, and Calculus II For Dummies .
More About the Author
Mark Zegarelli is the author of Logic For Dummies. He holds degrees in both English and math from Rutgers University. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. Along the way, he's also paid a few bills doing housecleaning, decorative painting, and (for ten hours) retail sales. He likes writing best, though.
Most Helpful Customer Reviews
I am a math teacher and when I saw this book I thought I would try is a resource. The book starts with the basic four operations and works its way up. It starts easy and works its way up. It does not start with 1+1 but is does start with 47+21. It covers a huge arrange of topics. Each chapter starts with what I would almost call a helpful hints page. Covers what you will work on and what to watch out for. This book is not going to teach you how to do this but it is a great practice. This book covers all basic math and pre-algebra. Some of the topics I personally would classify as algebra but that would depend on the curriculum being used. The book as word problems, graphing, statistics. The answers at the back of the book are step by step and also with a small description of what is being done. Again this is useful as a refresher but not as a lesson. Many of the problems are good practice for not using a calculator. The only issue I found annoying is I found finding the answers at the back a pain. They have grouped the answers together in hundreds but finding just one answer can take a minute or two. Make sure you mark you answers for the section you a working on otherwise you maybe flipping pages for a long time. Since the questions are grouped together in sections of around 10 I wish in the little description they had just added question 135-140 answers 177-179.
Overall I think this has a good selection of problems and is a nice supplement to a class. I will probably use this problems as warm ups and bonus questions within my classes. If you have had a summer off and you are about to start algebra this is a great book to work through to refresh those math skills.
I wanted this book to brush up on my math skills. When I was a kid I never did do well in these subjects. Good to see there is a reference that you can drill and practice with math situations. The Dummies books are mis-named as they are not really for "dummies" but for people who want to learn more about a field than the layman does. Recommended.
This book is good for the initial learning of basic math and pre-algebra. They even review how to add and subtract on a more efficient basis, although it's obvious that it starts much more advanced level than kindergarten and first grade. It's terrific for people like me, who have studied a lot of these items in the past but have gotten rusty on a few of the details.
However, this book I am reviewing is a WORKBOOK and would best be used in tandem with BASIC MATH AND PRE-ALGEBRA FOR DUMMIES. If you have a basic knowledge of basic math and remember a little bit about algebra, this book could be used on its own or perhaps with other books and/or web searches. however, in my observation the FOR DUMMIES book on this subject would probably be the best bet.
Having said that, the answers to the problems are still given with good, solid instructions on how to set up the steps. I can see this workbook being valuable for practice, review, or learning for the first time. I'd strongly recommend it for homeschoolers.
When I was somewhere between about 18-22 years old, someone told me I was "mentally deficient" in math, and it took me only about 20 years to get over it. Math didn't come easy to me, but with helps like this workbook and the book that goes with it, I feel like I am much more competent with math than I was. I doubt that I'll ever become a CPA, but at the same time, I can do much more than I thought I could, and it has become easier for me to write out answers and do more figuring in my head.
Recently back in college the one class I fear is math. It's been a long time! This is helping get back to speed without crushing my brain. Starting with basic math and moving into pre-algebra is just enough for me. It has on-line access also though being a bit old school at the moment I prefer to just use the book. This is the kind of text you skim and skip through and it is designed to help you plateau and move on. I've been attacking a page or two every night. I can't say I enjoy it but I can says its very helpful.
I am terrible at math. Always been. So, not too long ago I decided to get some books to keep my skills in a somehow working level. I used the Basic Math and pre-algebra book and workbook in the dummies series and i truly enjoyed it.
This book added to the "sharpening." One exercise after another; it's like doing Sudoku, you can't just do one. I found myself going through the pages with great pleasure.
Also, there is an on-line subscription that added value to the book. Whether you want to keep your math skills going or you are helping someone, this book is a great tool.
This book is a very different type of format from what you may expect from the "For Dummies" series. As with the other books in this series, Basic Math and Pre-Algebra is a wonderful way of brushing up on the basics. This book is exactly what it says it is: a huge book of problems to work through. And it is wonderful that at the end of the book not only are answers given, but the entire problem is explained and worked out for you. Just a wonderful aid for someone who needs to brush up on their math. |
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Publication Date: 1 July 1984--This text refers to an out of print or unavailable edition of this title.Product Description
Book Description
As in the previous edition, the purpose of this text is to introduce mathematical techniques to economics students. Through a complete integration of mathematics and economics along with a very patient exposition, the author attempts to maintain the emphasis on economics. Economic topics of equilibrium analysis, comparative-static analysis, economic dynamics and optimization are covered using mathematical techniques such as matrix algebra, differential equations, convex sets and mathematical programming in their solution.
--This text refers to an out of print or unavailable edition of this title.
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--This text refers to an out of print or unavailable edition of this title.
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First Sentence
Mathematical economics is not a distinct branch of economics in the sense that public finance or international trade is. Read the first page
I purchased this book while I was doing the literature review for my PhD; it had been a few years since I had taken a calculus class, and I was coming across quite a few mathematical models in my review. This book was invaluable in refreshing my memory about calculus. The book was written very clearly and in a logical manner.
This is a really well written book which can be of value to students with a wide range of mathematical abilities and backgrounds. The basic mathematical tools required to study economics at a post graduate level are explained with the minimum of fuss, yet in a way which enables the reader to understand the intuition behind the manipulations. There are many useful diagrams and plenty of well thought out exercises. If you are planning to study economics at a post graduate level and you don't have a first degree in mathematics then take the time to read this book before you start!
This book is an excellent introduction to the mathematics needed by economists. It covers all the background needed to begin in a typical economics Ph.D. program, and presents it understandably. I found that it is rigorous enough that I can apply what I have learned from it confidently. It uses total differentials where more advanced books might use vector calculus. This is a more intuitive approach, and better suited for an introduction. It includes some very good illustrations which clarify the concepts. |
Algebra for College Students
9780495105107
ISBN:
0495105104
Edition: 8 Pub Date: 2006 Publisher: Thomson Learning
Summary: Kau fundam...ental problem solving skills necessary for future mathematics courses in an easy-to-read format. The new Eighth Edition of ALGEBRA FOR COLLEGE STUDENTS includes new and updated problems, revised content based on reviewer feedback and a new function in iLrn. This enhanced iLrn homework functionality was designed specifically for Kaufmann/Schwitters' users. Textbook-specific practice problems have been added to iLrn to provide additional, algorithmically-generated practice problems, along with useful support and assistance to solve the problems for students.
Kaufmann Schwitters Staff is the author of Algebra for College Students, published 2006 under ISBN 9780495105107 and 0495105104. Seventy one Algebra for College Students textbooks are available for sale on ValoreBooks.com, twenty four used from the cheapest price of $0.20, or buy new starting at $340495105104 Brand new book. Hardcover US edition. Ship from multiple locations, including USA, UK, ASIA. 3-5 business days Express Delivery to USA/UK/Europe/Asia/Worldwide. Tracking number will be provided. Satisfaction guaranteed. ISBN: 0495105104 |
9780471129578ometric Modeling
A comprehensive, up-to-date presentation of the indispensable core concepts of geometric modeling
Now completely updated to include the most recent developments in the field, Geometric Modeling, Second Edition presents a comprehensive discussion of the core concepts of this subject. It describes and compares all the important mathematical methods for modeling curves, surfaces, and solids, and shows how to transform and assemble these elements into complex models. Written in a style free of the jargon of special applications, this unique book focuses on the essence of geometric modeling and treats it as a discipline in its own right.
It integrates the three important functions of geometric modeling: to represent elementary forms (i.e., curves, surfaces, and solids), to shape and assemble these into more complex forms, and to determine concomitant derivative geometric elements (i.e., intersections, offsets, and fillets).
With more than 300 illustrations, Geometric Modeling, Second Edition appeals to the reader's visual and intuitive skills in a way that makes it easier to understand its more abstract concepts. An extensive bibliography lists many supporting works, directing the reader to more specialized treatments of this subject. |
have helped students of all ages backgrounds and abilities make studying more efficient. Discrete math is a catch-all term encompassing many diverse areas of mathematics. There is no universal agreement as to what constitutes discrete math. |
analysis
analysis, branch of mathematics that utilizes the concepts and methods of the calculus. It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit are subjected to rigorous examination; differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations; complex variable analysis, in which the variables are of the form z = x + iy, where i is the imaginary unit; vector analysis and tensor analysis; differential geometry; and many |
MIT OpenCourseWare: New Translated Courses (Korean)New Translated courses (Korean) in all departments from MIT OpenCourseWare, provider of free and open MIT course materials.
2014-07-29T08:07:3103 Differential Equations (MIT)Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.
Miller, HaynesMattuck, Arthur2011-03-16T14:26:50+05:0018.03en-USOrdinary Differential EquationsODEmodeling physical systemsfirst-order ODE'sLinear ODE'ssecond order ODE'ssecond order ODE's with constant coefficientsUndetermined coefficientsvariation of parametersSinusoidal signalsexponential signalsoscillationsdampingresonanceComplex numbers and exponentialsFourier seriesperiodic solutionsDelta functionsconvolutionLaplace transform methodsMatrix systemsfirst order linear systemseigenvalues and eigenvectorsNon-linear autonomous systemscritical point analysisphase plane diagrams Solid State Chemistry (MIT)Introduction to Solid State Chemistry is a first-year single-semester college course on the principles of chemistry. This unique and popular course satisfies MIT's general chemistry degree requirement, with an emphasis on solid-state materials and their application to engineering systems.
Sadoway, Donald2011-01-11T17:49:10+05:003.091SCen-USsolid state chemistryatomic structureatomic bondingcrystal structurecrystalline solidperiodic tableelectron shellx-ray spectroscopyamorphous solidreaction kineticsaqueous solutionsolid solutionbiomaterialpolymersemiconductorphase diagrammaterial processing Highlights of Calculus (MIT)Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject. The series is divided into three sections: Introduction Why Professor Strang created these videos How to use the materials Highlights of Calculus Five videos reviewing the key topics and ideas of calculus Applications to real-life situations and problems Additional summary slides and practice problems Derivatives Twelve videos focused on differential calculus More applications to real-life situations and problems Additional summary slides and practice problems About the Instructor Professor Gilbert Strang is a renowned mathematics professor who has taught at MIT since 1962. Read more about Prof. Strang Acknowledgements Special thanks to Professor J.C. Nave for his help and advice on the development and recording of this program. The video editing was funded by the Lord Foundation of Massachusetts.
Strang, Gilbert2010-04-30T12:11:42+05:00en Computer Science and Programming (MIT)ThisGrimson, EricGuttag, John2009-09-10T12:22:44+05:006.00en-UScomputer sciencecomputationproblem solvingPython programmingrecursionbinary searchclassesinheritancelibrariesalgorithmsoptimization problemsmodulessimulationbig O notationcontrol flowexceptionsbuilding computational modelssoftware engineering Algorithms (SMA 5503) (MIT)This course teaches techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics covered include: sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; amortized analysis; graph algorithms; shortest paths; network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5503 (Analysis and Design of Algorithms).
Leiserson, CharlesDemaine, Erik2006-04-24T14:03:24+05:006.046J18.410Jen-USalgorithmsefficient algorithmssortingsearch treesheapshashingdivide-and-conquerdynamic programmingamortized analysisgraph algorithmsshortest pathsnetwork flowcomputational geometrynumber-theoretic algorithmspolynomial and matrix calculationscachingparallel computing Chemical Science (MIT)5.112 is an introductory chemistry course for students with an unusually strong background in chemistry. Knowledge of calculus equivalent to MIT course 18.01 is recommended. Emphasis is on basic principles of atomic and molecular electronic structure, thermodynamics, acid-base and redox equilibria, chemical kinetics, and catalysis. The course also covers applications of basic principles to problems in metal coordination chemistry, organic chemistry, and biological chemistry.
Cummins, ChristopherCeyer, Sylvia2006-04-21T13:12:22+05:005.112en-USintroductory chemistryatomic structuremolecular electronic structurethermodynamicsacid-base equilibriumredoxchemical kineticscatalysislewis structuresVSEPR theory Introductory014 focuses on the application of these fundamental principles, toward an understanding of microorganisms as geochemical agents responsible for the evolution and renewal of the biosphere and of their role in human health and disease.AcknowledgementsThe study materials, problem sets, and quiz materials used during Spring 2005 for 7.014 include contributions from past instructors, teaching assistants, and other members of the MIT Biology Department affiliated with course 7.014. Since the following works have evolved over a period of many years, no single source can be attributed.
Walker, GrahamKhodor, JuliaMischke, MichelleChisholm, Penny2005-10-26T13:23:14+05:007.014en-USmicroorganismsgeochemistrygeochemical agentsbiospherebacterial geneticscarbon metabolismenergy metabolismproductivitybiogeochemical cyclesmolecular evolutionpopulation geneticsevolutionpopulation growthecologycommunities012 focuses on the exploration of current research in cell biology, immunology, neurobiology, genomics, and molecular medicine.AcknowledgmentsThe study materials, problem sets, and quiz materials used during Fall 2004 for 7.012 include contributions from past instructors, teaching assistants, and other members of the MIT Biology Department affiliated with course #7.012. Since the following works have evolved over a period of many years, no single source can be attributed.
Gardel, ClaudetteLander, EricWeinberg, RobertChess, Andrew2005-04-27T15:15:23+05:007.012en-US Also by Walter Lewin Courses: Electricity and Magnetism (8.02) - with a complete set of 36 video lectures from the Spring of 2002
Lewin, Walter2004-03-31T23:48:33+05:008.01en-USunits of measurementpowers of tendimensional analysismeasurement uncertaintyscaling argumentsvelocityspeedaccelerationacceleration of gravityvectorsmotionvector productscalar productprojectilesprojectile trajectorycircular motioncentripetal motionartifical gravityforceNewton's Three Lawseightweightlessnesstensionfrictionfrictionless forcesstatic frictiondot productscross productskinematicsspringspendulummechanical energykinetic energyuniversal gravitationresistive forcedrag forceair dragviscous terminal velocitypotential energyheat; energy consumptionheatenergy consumptioncollisionscenter of massmomentumNewton's Cradleimpulse and impactrocket thrustrocket velocityflywheelsinertiatorquespinning rodelliptical orbitsKepler's LawsDoppler shiftstellar dynamicssound waveselectromagnetsbinary starblack holesrope tensionelasticityspeed of soundpressure in fluidPascal's Principlehydrostatic pressurebarometric pressuresubmarinesbuoyant forceBernoulli's EquationsArchimede's Principlefloatingbaloonsresonancewind instrumentsthermal expansionshrink fittingparticles and wavesdiffraction Electricity and Magnetism (MIT)In addition to the basic concepts of Electromagnetism, a vast variety of interesting topics are covered in this course: Lightning, Pacemakers, Electric Shock Treatment, Electrocardiograms, Metal Detectors, Musical Instruments, Magnetic Levitation, Bullet Trains, Electric Motors, Radios, TV, Car Coils, Superconductivity, Aurora Borealis, Rainbows, Radio Telescopes, Interferometers, Particle Accelerators (a.k.a. Atom Smashers or Colliders), Mass Spectrometers, Red Sunsets, Blue Skies, Haloes around Sun and Moon, Color Perception, Doppler Effect, Big-Bang Cosmology. OpenCourseWare presents another version of 8.02T: Electricity and Magnetism. Also by Walter Lewin Courses: Classical Mechanics (8.01)- with a complete set of 35 video lectures from the Fall of 1999
Lewin, Walter2003-09-28T09:02:30+05:008.02en-USIntroduction to electromagnetism and electrostaticselectric chargeCoulomb's lawelectric structure of matterconductorsdielectricsConcepts of electrostatic field and potential, electrostatic energyElectric currentsmagnetic fieldsAmpere's lawMagnetic materialsTime-varying fieldsFaraday's law of inductionBasic electric circuitsElectromagnetic wavesMaxwell's equationslightningpacemakerselectric shock treatmentelectrocardiogramsmetal detectorsmusical instrumentsmagnetic levitationbullet trainselectric motorsradiosTVcar coilssuperconductivityaurora borealisrainbowsradio telescopesinterferometersparticle accelerators (a.k.a. atom smashers or colliders)mass spectrometersred sunsetsblue skieshaloes around sun and mooncolor perceptionDoppler effectsuper-novaebinary starsneutron starsblack holes Media, Education, and the Marketplace (MIT)
How can we harness the emerging forms of interactive media to enhance the learning process? Professor Miyagawa and prominent guest speakers will explore a broad range of issues on new media and learning - technical, social, and business. Concrete examples of use of media will be presented as case studies. One major theme, though not the only one, is that today's youth, influenced by video games and other emerging interactive media forms, are acquiring a fundamentally different attitude towards media. Media is, for them, not something to be consumed, but also to be created. This has broad consequences for how we design media, how the young are taught in schools, and how mass media markets will need to adjust.
Miyagawa, Shigeru2003-06-12T07:20:53+05:00CMS.93021F.034en-USeducational technologymedia designCMS.93021F.034 |
Intermediate Algebra : Text and Workbook - 9th edition
Summary: For the modern student like you--Pat McKeague's INTERMEDIATE ALGEBRA, 9E--offers concise writing, continuous review, and contemporary applications to show you how mathematics connects to your modern world. The new edition continues to reflect the author's passion for teaching mathematics by offering guided practice, review, and reinforcement to help you build skills through hundreds of new examples and applications. Use the examples, practice exercises, tutorials, videos, and e-Book ...show moresections in Enhanced WebAssign to practice your skills and demonstrate your knowledge. ...show less
2012 Paperback Fair CONTAINS SLIGHT WATER DAMAGE/STAIN, STILL VERY READABLE This item may not include any CDs, Infotracs, Access cards or other supplementary material.
$57.43 +$3.99 s/h
New
textbook_rebellion2 Troy, MI
113310364269.36 +$3.99 s/h
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Campus_Bookstore Fayetteville, AR
Used - Acceptable moderate-heavy water damage. still classroom useable. some spine and binding wear. 9th Edition Not perfect, but still usable for class. Ships same or next day. Expedited shipping tak...show morees 2-3 business days; standard shipping takes 4-14 business days. ...show less
PAPERBACK Very Good 1133103642 USED-9th(US)Edition paperback textbook only-clean pages-crease on the cover. We ship same day, if order received by 1 pm CST (excluding weekends) orders received afte...show morer 1 pm ship next business day. ...show less
$12122 +$3.99 s/h
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Textbookcenter.com Columbia, MO
Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book
$177 |
Morrow, GA SAT Math...ACT Math is a collection of pre-algebra, elementary algebra, intermediate algebra, geometry, and trigonometry; basically all the courses that should have been taking by the end of the eleventh grade year. Elementary math is basic computation skills such as adding, subtracting, multiplying and di... |
This textbook offers an accessible, modern introduction at undergraduate level to an area known variously as general topology, point-set topology or analytic topology with a particular focus on helping students to build theory for themselves. |
hands on, multi-media package provides a motivating introduction to fundamental concepts, specifically discrete-time systems, for beginning engineering readers. This class-tested learning package can also be used as a self-teaching tool for anyone eager to discover more about DSP applications, multi-media signals, and MATLAB. Presents basic DSP concepts in a clear and intuitive style. Integrated laboratory projects related to music, sound and image processing and new MATLAB functions for basic DSP operations are also included. Appropriate for readers interested in mastering fundamental concepts in today's electrical and computer engineering curriculum. |
Math-o-mir
This product is basically an Equation Editor. However it is not centered over one single equation but you can write your mathematical text over several pages. Inside your mathematical document, you can copy equations and expressions easily by mouse click. You can also make simple drawings or sketches. Added function plotter and symbolic calculator can assist you... Engineers can use it to make quick informal calculations; students can use it as a real-time math note-taking tool; math teachers can prepare electronic exams to be solved by their students |
Matrix Calculator is a site containing an interactive applet that let a user to input a square matrix and then with a press...
see more
Matrix Calculator is a site containing an interactive applet that let a user to input a square matrix and then with a press of a button compute a power of this matrix, determinant, inverse, characteristic polynomials and other useful matrix characteristics.
This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the...
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This. It's definitely worth checking out. Quote from site: "A parabola is actually a locus of a point and a line. The point is called the focus and the line the directrix. That means that all points on a parabola are equidistant from the focus and the directrix. To change the equation and the graph of the interactive parabola below just click and drag either the point A, which is the focus, or point B, which controls the directrix." This is an interactive site that allows people to change the graph to understand why directrix and focus dictate parabolic graphs. |
There is a newer edition of this item:
-- More than 44,000 copies sold of second edition-- More than 230,000 students are enrolled in trigonometry courses-- Required study for all mathematics majors-- Hundreds of practical problems solved step by step-- Complements most popular textbooks
Editorial Reviews
From the Back Cover
Get a firm grasp of trigonometry with this simple-to-use guide! It can help you pump up your problem-solving skills, ace your exams, and reduce the time you need to spend studying. Students love Schaum's Outlines! Each and every year, students purchase hundreds of thousands of the best study guides available anywhere. Students know that Schaum's delivers the goodsin faster learning curves, better test scores, and higher grades!
If you don't have a lot of time but want to excel in class, this book helps you:
Brush up before tests
Find answers fast
Study quickly and more effectively
Get the big picture without spending hours poring over lengthy texts
Schaum's Outlines give you the information teachers expect you to know in a handy and succinct formatwithout overwhelming you with unnecessary details. You get a complete overview of the subjectand no distracting minutiae. Plus, you get plenty of practice exercises to test your skill. Compatible with any classroom text, Schaum's lets you study at your own pace and reminds you of all the important facts you need to rememberfast! And Schaum's is so complete it's the perfect tool for preparing for graduate or professional exams!
Inside, you will find:
Hundreds of detailed problems, including step-by-step solutions
Hundreds of additional practice problems, with answers supplied
Clear explanations of trigonometry and the underlying algebra
Understandable coverage of all relevant topics
If you want top grades and excellent understanding of trigonometry, this powerful study tool is the best tutor you can have!
--This text refers to an out of print or unavailable edition of this title.
About the Author
McGraw-Hill authors represent the leading experts in their fields and are dedicated to improving the lives, careers, and interests of readers worldwide
--This text refers to an out of print or unavailable edition of this title.
Most Helpful Customer Reviews
I found the book very useful for brushing up one's knowledge of trigonometry.It gets down to the point right away without wasting any time on unnecessary theory.After all, its your problem solving ability that counts and not how well you know the theory.:) What I like most about this book is that it treats the subject matter from a student's point of view (like it teaches you how to get the answer using a calculator STEP by STEP.....it actually tells you which buttons to push).Also, throughout the book ,the application of trigonometry in surveying, construction,astronomy and air-navigation is emphasized. I recommend this book for anyone who's taking computer science/engineering/technology courses in college.
Trigonometry was never a good subject for me--I never "got" it. But when I was taking advanced math and science courses, I needed trig. This book helped me to "get" it, finally, and be able to solve trigonometric problems. It's very clear, up-to-date, and well-written.
I am observing that my test scores on tests on tests involving trigonometery are increasing, thanks to this thin aid. It is thin, yet it is good. That is almost impossible. This book is one of the best created. Trigonometry is hard, and is mentioned and applied almost everywhere. This book is comphrehensive and is both easy and advanced. Everyone should have this. You can have knowledge of math, science, and computers with it.
I used Schaum's for a Summer School Trigonometry course. It doesn't replace a textbook but it covered all the necessary topics effectively and provided a good alternative to derivations and problems found in the text. I relied on it as a backup and a good check on the work being covered in class.
I had to have this book for a trigonometry class, and it sucks. There were so many miss prints in the book, that it was actually a disgrace, and I personally think this book is unfit for learning. There are way too many miss prints in this book, and it is piss poor at explaining concepts and how to work problems. When I went to study for a test on graphing sines, cosines, and tangents, I actually had to look the material up online because I could not understand the book, and almost everyone that's in my class says that they cannot understand the book all that great, so I know it's not just me, and my professor even went so far as to say that the book had a lot of miss prints. If you ever have to buy this for a class, BEWARE! My best suggestion would be to buy an alternate text that's a lot better if this is a requirement for a class.
The preface claims that the book can be used by students who are studying trig for the first time. What a load of bunk. As a "first time" trig student, I couldn't get through the first chapter without getting confused. I moved on to the second chapter, and the information was just as muddled. This book may be more useful as a study guide to be used together with a real trig textbook, but I don't believe it is useful for the beginner. Kahn Academy on Youtube gave more clearer explanations and examples on the subject. Beginners should stay away from this book!
However, I used this book from beginning to end and now I am able to pass a trig test. I would like to suggest a better write-up on explaining trigonometric graphs such as "Y=1/2 cosine 3x-16". This book truly helped me make sense of something that is abstract to me. |
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learning is underway.
Completely Common Core aligned, these grade-specific probes eliminate the guesswork and will help you
Editorial Reviews
Lyneille Meza
"High school mathematics can be one of the most difficult subjects for students to learn. This book provides the teacher with tools to help the student get to the root of their thinking in order to correct the underlying misconceptions that students amass during their school years. "
Randy Wormald
"Not only does this book make strong connections to the CCSS, it provides a framework for teachers to improve their craft. Identifying student misconceptions is an extremely important aspect of assessment and this book provides a solid format to aide teachers. "
Related Subjects
Meet the Author
Cheryl Rose Tobey is a Senior Mathematics Associate at EDC. She is the implementation director for the Pathways to Mathematics Achievement Study and a mathematics specialist for the NSF-funded Formative Assessment in the Mathematics Classroom: Engaging Teachers and Students (FACETS) and Differentiated Professional Development: Building Mathematics Knowledge for Teaching Struggling Students (DPD) projects. She also serves as a project director for an Institute for Educational Science (IES) project, Eliciting Mathematics Misconceptions (EM2). Her work is primarily in the areas of formative assessment and professional development.
Prior to joining EDC, Tobey was the senior program director for mathematics at the Maine Mathematics and Science Alliance (MMSA), where she served as the co-principal investigator of the mathematics section of the NSF-funded Curriculum Topic Study, and principal investigator and project director of two Title IIa State Mathematics and Science
Partnership projects. Prior to working on these projects, Tobey was the co-principal investigator and project director for MMSA's NSF-funded Local Systemic Change Initiative, Broadening Educational Access to Mathematics in Maine (BEAMM), and she was a fellow in Cohort 4 of the National Academy for Science and Mathematics Education Leadership. She is the coauthor of four published Corwin books, including three books in the Uncovering Student Thinking Series and Mathematics Curriculum Topic Study: Bridging the Gap Between Standards and Practice. Before joining MMSA
in 2001 to begin working with teachers, Tobey was a high school and middle school mathematics educator for 10 years. She received her BS in secondary mathematics education from the University of Maine at Farmington and her MEd from City University in Seattle.
Carolyn B. Arline is a secondary mathematics educator, currently teaching high school students in Maine. Carolyn also works as a teacher leader in the areas of mathematics professional development, learning communities, assessment, systematic school reform, standards-based teaching, learning and grading, student-centered classrooms, and technology. She has previously worked as a mathematics specialist at the Maine Mathematics and Science Alliance (MMSA) and continues her work with them as a consultant. Carolyn is a fellow of the second cohort group of the Governor's Academy for Science and Mathematics Educators and serves as a mentor teacher with the current cohort. She participated as a mathematics mentor in the NSF-funded Northern New England Co-Mentoring Network (NNECN) and continues her role as a mentor teacher. She serves as a board member of the Association of Teachers of Mathematics in Maine (ATOMIM) and on local curriculum committees. Carolyn received her B.S. in secondary mathematics education from the University |
Translator
The Google translation of this page's content may not be completely accurate. Please contact the school directly for clarification of official information.
Resources
The "math lab" is a place where students can get help on homework and/or get answers to any general math questions they have. The lab can also be used as a place to just sit and do your work, with someone nearby in case you needed help. Please have a task in mind when you come to the lab. Friends of math lab students are not allowed to come in and "linger" while waiting for their friend to finish their math.
2013-14 Math Lab:
Currently the math lab is open from 2:50-3:50 all days except Wednesdays. (In the library)
Looking for a tutor???
The University of Portland has many education majors that are willing to do tutoring on the side! Many parents in the community are able to find great tutors this way!
What to do (info from UofP):
If parents are interested in hiring a University of Portland student as a tutor, please have parents send an email requesting a tutor to Kathleen Staten, the office manager. Be sure to include hours, days, subject, etc. She will then forward the request on to all of the education majors.
Kathleen's email is: staten@up.edu
Other available resources to help out...
The book Math on Call is a great general mathematics resource guide (ISBN# 0669508195). It is basically a math dictionary that provides a short description on any topic/term and also an example. It's a good resource just to have around the house.
The website Khan Academy ( is a great website! Go to the site, fine the topic that you're working on or struggling with and then watch a virtual lesson on the topic! It is narrated by the person doing the work and you get to watch the problem being done, with step by step explanations, right before your eyes! A very valuable resource when in need of extra help!
For Algebra Connections students - you can get online help by following these steps:
1. Go to
2. On the left-hand side of the page, under Student Support, click on "Homework Help"
3. On the next page, find the picture of your math book (it should be in the second to the last row of books)
4. From there, select the appropriate chapter and then enter the number of the Review/Preview problem you need help on |
Looking for Pythagoras is filled with investigations that develop a fundamentally important relationship connecting geometry to algebra: the Pythagorean Theorem. Students are not merely introduced to a meaningless formula. To encourage deep understanding of what the theorem means, students explore squares created with various lengths, their areas and how these relate to the side lengths of right triangles. Students also explore square roots and strategies for estimating square roots. Irrational numbers are introduced and students are expected to be able to estimate where these occur on a number line.
For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Looking for Pythagoras
Complete policy statement |
Ck-12 Trigonometry
Ck-12 Trigonometry
This is a free, online textbook offered by CK-12 Foundation. Although designed for high school students, it could also be used by college freshmen. Chpater topics include the following: 1. Trigonometry and Right Angles2. Circular Functions3. Trigonometric Identities4. Inverse Functions and Trigonometric Equations5. Triangles and Vectors6. Polar Equations and Complex Numbers
This is a free, online textbook offered by CK-12 Foundation. Although designed for high school students, it could also be used by college freshmen. Chpater topics include the following: |
books.google.com - Classical harmonic analysis is an important part of modern physics and mathematics, comparable in its significance with calculus. Created in the 18th and 19th centuries as a distinct mathematical discipline it continued to develop, conquering new unexpected areas and producing impressive applications... Harmonic Analysis II |
this book sold more than 100,000 copies—and this new edition will show you why! Schaum's Outline of Discrete Mathematics shows you step by step how to solve the kind of problems you're going to find on your exams. And this new edition features all the latest applications of discrete mathematics to computer science! This guide can be used as a supplement, to reinforce and strengthen the work you do with your class text. (It works well with virtually any discrete mathematics textbook.) But it is so comprehensive that it can even be used alone as a text in discrete mathematics or as independent study tool! |
MATH-182 Project-Based Calculus II
Tue, 02/04/2014 - 11:10am -- gpltwc
This is the second in a two-course sequence intended for students majoring in mathematics, science or engineering. It emphasizes the understanding of concepts, and using them to solve physical problems. The course covers techniques of integration including integration by parts, partial fractions, improper integrals, applications of integration, representing functions by infinite series, convergence and divergence of series, parametric curves and polar coordinates. (C or better in MATH-181 Project-Based Calculus I) Class 4, Workshop 2, Credit 4 (F, S, Su) |
Managing the Mean Math Blues - 2nd edition
Summary: Is it possible for math-phobic students to learn math? Yes! Not only can they learn math, they can excel by learning the truly unique techniques in Managing the Mean Math Blues. Written by an experienced math teacher and psychotherapist, this book helps students overcome their negative perceptions about math using positive psychology, brain-based learning, and study skills techniques. Students turn failure into success as they practice these new skills on basic m...show moreath content. With clear psychological models for concentration and focus (called flow) into math, students learn how to match their skills with math challenges, set short-term goals and seek feedback in order to learn math successfully |
Algebra and Trigonometry and Useful Mathematical and Physical Formulae | 22 Mb Beecher, Penna, and Bittinger's Algebra and Trigonometry is known for enabling students to "see the math" through its focus on visualization and early introduction to functions. With the Fourth Edition, the authors continue to innovate by incorporating more ongoing review to help students develop their understanding and study effectively.
A compact volume of mathematical and physical formulae presented in a concise manner for general use.
Updating the Artech House bestseller, Fundamentals and Applications of Microfluidics, this newly revised second edition provides electrical and mechanical engineers with complete and current coverage of microfluidics – an emerging field involving fluid flow and devices in microscale and nanoscale. The second edition offers a greatly expanded treatment of nanotechnology, electrokinetics and flow theory |
Casio Takes New Approach to Graphing Calculator for Students
Casio's Prizm fx-CG10 plots graphs over full-color images to help students visualize concepts.
Casio Education has introduced the Prizm fx-CG10, a new concept in educational graphing calculators that aims to impart mathematical concepts in addition to providing standard graphing functions. Using a new tool known as Picture Plot, the Prizm enables users to plot graphs over full-color photographic images, such as an Egyptian pyramid or the jets of an outdoor fountain, as way of relating complex mathematical functions to real-world concepts such as design and engineering.
Casio also offers teachers online training using streaming video and downloadable supplemental activities, as well as a loaner program, which enables interested educators to try the Prizm for 30 days. An application for the program |
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Overview
A state-of-the-art introduction to the powerful mathematical and statistical tools used in the field of finance
The use of mathematical models and numerical techniques is a practice employed by a growing number of applied mathematicians working on applications in finance. Reflecting this development, Numerical Methods in Finance and Economics: A MATLAB®-Based Introduction, Second Edition bridges the gap between financial theory and computational practice while showing readers how to utilize MATLAB®—the powerful numerical computing environment—for financial applications.
The author provides an essential foundation in finance and numerical analysis in addition to background material for students from both engineering and economics perspectives. A wide range of topics is covered, including standard numerical analysis methods, Monte Carlo methods to simulate systems affected by significant uncertainty, and optimization methods to find an optimal set of decisions.
Among this book's most outstanding features is the integration of MATLAB®, which helps students and practitioners solve relevant problems in finance, such as portfolio management and derivatives pricing. This tutorial is useful in connecting theory with practice in the application of classical numerical methods and advanced methods, while illustrating underlying algorithmic concepts in concrete terms.
What People Are SayingEditorial Reviews |
Online Distance Learning
Practical math applications are featured in this course. You will develop
skill in whole numbers, fractions, decimals, percents, ratio and proportion,
linear measurement, basic geometry and volume. Both the imperial and
metric systems are utilized |
Numerical Solution of Ordinary Differential Equations
A concise introduction to numerical methodsand the mathematical framework neededto understand their performance "Numerical Solution of Ordinary ...Show synopsisA concise introduction to numerical methodsand the mathematical framework neededto understand their performance "Numerical Solution of Ordinary Differential Equations" presents a complete and easy-to-follow introduction to classical topics in the numerical solution of ordinary differential equations. The book's approach not only explains the presented mathematics, but also helps readers understand how these numerical methods are used to solve real-world problems. Unifying perspectives are provided throughout the text, bringing together and categorizing different types of problems in order to help readers comprehend the applications of ordinary differential equations. In addition, the authors' collective academic experience ensures a coherent and accessible discussion of key topics, including: Euler's method Taylor and Runge-Kutta methods General error analysis for multi-step methods Stiff differential equations Differential algebraic equations Two-point boundary value problems Volterra integral equations Each chapter features problem sets that enable readers to test and build their knowledge of the presented methods, and a related Web site features MATLAB(R) programs that facilitate the exploration of numerical methods in greater depth. Detailed references outline additional literature on both analytical and numerical aspects of ordinary differential equations for further exploration of individual topics. "Numerical Solution of Ordinary Differential Equations" is an excellent textbook for courses on the numerical solution of differential equations at the upper-undergraduate and beginning graduate levels. It also serves as a valuable reference for researchers in the fields of mathematics and engineering47004294X Brand-New, Unread Copy in Perfect Condition. To...New. 047004294 |
Applied Linear Algebra - 06 edition
Summary: For in-depth Linear Algebra courses that focus on applications.
This text aims to teach basic methods and algorithms used in modern, real problems that are likely to be encountered by engineering and science students--and to foster understanding of why mathematical techniques work and how they can be derived from first principles. No text goes as far (and wide) in applications. The authors present applications hand in hand with theory, leading students thro...show moreugh the reasoning that leads to the important results, and provide theorems and proofs where needed. Because no previous exposure to linear algebra is assumed, the text can be used for a motivated entry-level class as well as advanced undergraduate and beginning graduate engineering/applied math |
When Maple's ease-of-use features combined with Maple's point-and-click tools, Clickable Calculus was born. Clickable Calculus is a syntax-free way of accessing the power of Maple and flattening the learning curve, making it possible to use Maple to teach mathematics without first having to teach the tool.
Join Dr. Robert Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology, in a hands-on workshop where the magic of Clickable Calculus will be revealed. Bring your laptop. We'll provide the software. In this hands-on workshop, Dr. Lopez will lead participants through a selection of problems drawn from precalculus, calculus, differential equations, linear algebra, and vector calculus. Participants will learn to solve these problems in a syntax-free way, using the new interface features in Maple.
Please register in advance. There are only a limited number of spaces available. If you don't have a working copy of Maple 14 on your laptop, a temporary download copy of Maple 14 will be emailed to you. At the workshop, the files needed for the exercises will be provided. Register Now
P.S. While you're at the 2011 Joint Mathematics Meetings, stop by the Maplesoft booth #109.
Bring math to life in your classroom! Download the course content for FREE directly from the Teacher Resource Center. Each topic includes a demo video, testing content, and Maple worksheets. Click here to explore the Maplesoft Teacher Resource Center and the new course content.
This webinar presents techniques and applications for rapidly developing computationally optimized models for HIL simulation of high-fidelity physical plant models. Using symbolic computation techniques, multibody models can be effectively preprocessed to select optimal coordinate frames, eliminate redundant calculations, simplify algebraic constraints, and generate computationally minimal code for real-time deployment. Furthermore, novel mathematical techniques can be deployed for efficient parameter optimization and other advanced analysis. Applications in robotics, including space and industrial robotics will be presented. The symbolic computation system Maple and the related modeling system MapleSim will be used to illustrate examples. This webinar consists of two parts:
Part I:
General survey of techniques and applications of symbolic computation
techniques in robotics.
Part II:
Applications in space and industrial robotics.
Presenters:
Dr. Amir Khajepour, President AEMK Systems, Professor, Mechanical Engineering, University of Waterloo, and Canada Research Chair in Mechatronic Vehicle Systems: Dr. Khajepour is an authority in mechatronics and robotics. His research includes modeling and development of advanced systems for the automotive, space, and manufacturing industries.
Dr. Tom Lee, Vice President Applications Engineering and Chief Evangelist, Maplesoft: Dr. Lee is the principal technology spokesperson for Maplesoft, an emerging leader in advanced engineering modeling and simulation. As Vice President of Applications Engineering, Dr. Lee directs the principal engineering engagements for Maplesoft, including custom model development, consulting, and training, with key customers and partners. Dr. Lee holds a Ph.D. in automation and control and has been a leading influence in the development of new modeling tools for physical modeling and symbolic
Find out what's happening at the heart of Maplesoft, and provide your comments and feedback to the Maplesoft experts.
Science Fiction Double Feature Dr. Tom Lee
Like many in the technology industry, I am a big fan of science fiction films and I've written in the past about how exciting it is for me to have a job where science fiction and reality literally meet. Over the past few months, several key projects from various Maple and MapleSim users caught my attention for various reasons and once again, I was forced to giggle publicly as the shear cool factor of these applications overcame my normal mature demeanor. Read More>>
Green Cars Dr. Tom Lee
It's ironic that my life in North America started with a green car and has come full circle with green cars all over the place as far as my job is concerned. But of course, today's green car is really about highly fuel efficient cars or cars powered by entirely difference energy sources than with the old Skylark. Read More>>
As output, Maple can display the partial derivative ∂/∂x f(x,y) as fx; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.
With the holiday's right around the corner, there is a festive vibe going around the office. Decorations have been hung, lights have been strewn, and we've just had our first snow fall of the season. Our application engineering group has also got in the spirit and modeled a flight path of Santa on his sleigh. Using MapleSim, they've created a simple flight path for Santa and have modeled the effects his body would experience as he is flying through the night sky. You can watch a video of it here. Granted this doesn't have all the features, like the weight of his bag of toys, or the aerodynamic effects 8 reindeer would have on his flight, but it's a good starting point.
For a number of years, just before Christmas, an article explaining the physics involved in Santa's Christmas Eve trip has circulated the internet. It is re-created here, this time with up-to-date values, and of course, using Maple.
Extract: "The speed of all most computations has been increased by more efficient algorithms. The new Maple Cloud Document Exchange facilitates the creation, distribution and exchange of documents with colleagues. Of course all of our old friends such as the TA's, Tutor's, Help Database and extensive mathematical algorithms are still there, as are the very useful palettes. The many application briefs and add-ons available on-line complete a very nice package for the student, engineer or industrial mathematician."
Extract: "Maplesoft is releasing Maplesim 4.5, the latest version of its high-performance physical modelling and simulation tool. Based on advanced symbolic computation technology, Maplesim is an engineering software tool for design, modelling, and high-performance simulation, including real-time and hardware-in-the-loop applications. Unlike traditional, numeric-based modelling tools, the symbolic approach of Maplesim exposes and provides access to the model equations."
In addition to these articles, check out the Media Center for all the latest coverage on Maplesoft.
You are receiving this newsletter in an effort to keep you up-to-date on the latest developments at Maplesoft. To manage subscriptions or to opt out of all commercial email communications from Maplesoft, please click here. To view our privacy policy, click here. |
Viewpoints: Mathematical Perspective and Fractal Geometry in Art for an Amazon Gift Card of up to £3.10, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book DescriptionMore About the Author
Product Description
Review
"The book goes a long way trying to convey to its audience--through both theory and practice--professional techniques that could not fail but empower students to make accurate, sophisticated drawings. The book presents an elegant fusion of mathematical ideas and practical aspects of fine art."--Cut the Knot
"[T]his is an excellent text that I will certainly consider using for a future class. The material on perspective is accessible, thorough and well-written, and the text is designed for a hands-on pedagogy that is well-suited to the intended audience. And as an elementary, but thorough, discussion of both the mathematics and practice of perspective drawing, it deserves a place in any collection of books on mathematics and the arts."--Blake Mellor, Journal of Mathematics and the Arts
"The writing is extremely clear, the material is fresh, and the many excellent diagrams clarify the ideas under discussion. The authors use relevant artwork to illustrate the mathematical principles. . . . The exercises are original and promote active learning. . . . This is an excellent work for academic curricula and an outstanding resource for self-study in mathematical perspective, fractals, and the relationship between art and mathematics."--Choice
"This is not a book to read passively and, indeed, you will want to read this book with a pencil in hand. The text is designed to be experienced first hand, sketching out examples whilst following the text, as well as doing the exercises at the end of each chapter that develop the material well. . . . Prerequisites for this book are a minimum, effectively being an understanding of basic coordinate geometry. I would recommend this book to anyone who is interested in the interplay between mathematics and art."--George Matthews, Mathematics Today
From the Inside Flap
"This practical, hands-on, and significant book makes clear the connections between mathematics and art, and demonstrates why artists need to know mathematics. Viewpoints appeals to students' visual intuition and engages their imaginations in a fresh way."--Barbara E. Reynolds, SDS, coauthor of College Geometry: Using the Geometer's Sketchpad
"This entire book is a thing of beauty: the mathematics, the visual art, the writing, the exercises, and the organization. The authors' passion and excitement for their subject matter is apparent on every page. I am in awe."--Robert Bosch, Oberlin College
"The book's emphasis on a workshop approach is good and the authors offer rich insights and teaching tips. The inclusion of work by contemporary artists--and the discussion of the mathematics related to their work--is excellent. This will be a useful addition to the sparse literature on mathematics and art that is currently available for classroom use."--Doris Schattschneider, author of M. C. Escher: Visions of Symmetry
"Concentrating on perspective and fractal geometry's relationship to art, this well-organized book is distinct from others on the market. The mathematics is not sold to art students as an academic exercise, but as a practical solution to problems they encounter in their own artistic projects. I have no doubt there will be strong interest in this book."--Richard Taylor, University of Oregon
I have to admit, this book is pretty challenging. There is a ton of geometry math that is used to describe the mathematical aspects of perspective. But it is a face, it increased my understanding of perspective. It is not an easy book to get through, and frankly I probably only understood about 20% of it, but that 20% was useful, and some day I will probably go back and actually try to do the exercises.
3 of 3 people found the following review helpful
5.0 out of 5 starsExcellent textbook on the mathematics of Perspective5 Sep 2011
By Ed Pegg - Published on Amazon.com
Format:Hardcover
At the start of the book, students are looking at normal hallways, rooms, and buildings through sheets of plexiglas, and tracing the outlines of what they see with drafting tape. From there, it's easy to see the concept of vanishing points.
A few pages later, an image from Jurassic Park with a velociraptor walking towards Sam Neill is shown. As an exercise, the students must compare the position of a clawtip to the bottom of a doorframe. I've messed up this issue of image placement many times, so this simple exercise brought home a lesson for me.
The core part of the book is 1, 2, and 3 point perspective, but with the idea that you'll be using a modern program of some sort. Then they introduce fractal geometry in a way I didn't expect, by taking a picture of a patch of grass and a small rock, and photopasting in a toy gnu and a climber. Small rocks look like big rocks look like mountains. I knew that, but I hadn't been tricked by it before, so I got the lesson better this time.
Recommended.
2 of 2 people found the following review helpful
4.0 out of 5 starsGood information20 July 2013
By Genita Cartwright - Published on Amazon.com
Format:Hardcover|Verified Purchase
This is a very specialized book, and after looking at it I have created a wonderful sculpture that we sold to a local hospital. Very fine information. |
Solitons are explicit solutions to nonlinear partial differential equations. They are waves that behave in many respects like particles. The founding story of soliton theory, repeated so often it is now almost indistinguishable from myth, tells of John Scott Russell and his observation in 1834 of a peculiar solitary wave in a canal near Edinburgh. He followed this wave on horseback as it kept its speed and shape for a mile or two until he lost it. The response to this amazing discovery was … not much, mostly scoffing, because everyone thought that such a wave would disperse and distort and could not propagate. In 1895 Korteweg and de Vries modeled water waves in a canal, derived the KdV equation named after them and found a number of wave-like solutions that travel and maintain their shape. But not even they seemed particularly interested in what they found. It wasn't until the twentieth century and computational work by Fermi-Pasta-Ulam and later Kruskal-Zabusky that the soliton got a name and some respect.
It's an odd thing. We can write down many explicit exact solutions of the nonlinear KdV equation. Why is this possible when we usually can't find even one explicit solution for most nonlinear partial differential equations? Moreover, an n-soliton solution to the KdV equation (with n peaks) bears an unusually close relationship to n individual one-soliton solutions: it looks almost — but not quite — like a linear combination of the others. Is there a geometric structure analogous to the vector spaces we see with solutions of ordinary differential equations?
This book explores the ramifications of these questions for advanced undergraduates who have had basic calculus and linear algebra. It's very challenging material for undergraduates, but it presents an exciting opportunity too. As a capstone course, or for independent study, soliton theory ties together several important applications to science and engineering with an extraordinary range of mathematical topics from PDEs to elliptic curves, differential algebras and Grassmanians. The author doesn't expect to bring students to the research frontiers with his book; his aim is rather to provide a "glimpse" that intrigues and engages. Partial differential equations and algebraic geometry meet in a most remarkable and unexpected way.
After an introductory review of differential equations that emphasizes the differences between linear and nonlinear equations, the author tells the story of solitons. He begins with James Scott Russell and continues to twentieth century developments and applications. These include examples in telecommunications (where solitons travel down optical fibers) and biology (where solitons play a role in DNA transcription and energy transfer).
The book's real work begins with an examination of Korteweg and de Vries' solution to the KdV equation. We see that the general solution they find can be written in terms of a Weierstrass ℘-function. The connection to elliptic curves and algebraic geometry begins here.
The author makes extensive use of Mathematica throughout the book; in particular, that program is used to introduce the Weierstrass ℘-function without requiring the background in complex analysis that would otherwise be necessary. (It is also used throughout for a variety of straightforward, if messy, calculations and for animations of wave dynamics.) This innovative use of Mathematica works well here where the object is to offer glimpses of a broad and subtle theory. It does, however, tie the book to the software — it would be unsatisfying to read the book without having access to Mathematica, and difficult to derive full benefit from it.
After dipping into algebraic geometry, the author goes on to discuss the n-soliton version of the KdV equation and its solutions that look asymptotically like linear combinations of solutions to one-soliton equations. The next few chapters try to explain the special nature of the KdV equations, and along the way discuss the algebra of differential operators, isospectral matrices, and the Lax form for KdV and other soliton equations. It's only with an additional spatial dimension and analysis of the corresponding generalization of the KdV equation (the KP equation) that the picture gets a little clearer. We then finally get a glimmer of the geometry of the solution space, and a way to describe it using the Grassman cone.
This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar.
Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics. |
Geometry (Cliffsstudysolver) - Study Notes
by David Alan Herzog Publisher Comments
The learn-by-doing way to master Geometry Why CliffsStudySolver™ Guides? Go with the name you know and trust Get the information you need–fast! Written by teachers and educational specialists Inside you'll get the practice you need to... (read more)
Algebra & Trigonometry (Rea's Problem Solvers)
by Rea Publisher Comments
REAs Algebra and Trigonometry Problem Solver Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. Answers to all of your questions can be found in one convenient source from one of... (read more)
Sharp Math: Building Better Math Skills (Sharp)
by Kaplan Publisher Comments
Features: A 10-question diagnostic quiz in every chapter to show readers where they need the most help.Math from basic arithmetic to Algebra 2, broken down by subject and then building up from chapter to chapter so readers can group concepts together for... (read more)
Algebra II
by Edward Kohn Publisher Comments... (read more)
Precalculus With Unit Circle Trigono 3RD Edition
by David Cohen Publisher Comments
David Cohen's PRECALCULUS, WITH UNIT-CIRCLE TRIGONOMETRY, Fourth Edition, focuses on teaching mathematics, using a graphical perspective throughout to provide a visual understanding of college algebra and trigonometry. The author is known for his clear... (read more)
Differential Geometry
by Heinri Guggenheimer Publisher Comments
This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix... (read more)
Overcoming Math Anxiety
by Sheila Tobias Publisher Comments
The new edition retains the author's pungent analysis of what makes math "hard" for otherwise successful people and how women, more than men, become victims of a gendered view of math. It has been substantially updated to incorporate new research on what... (read more)
Trigonometry
by David A Kay Publisher Comments
Cliffs Quick Reviews are Produced by the People Who Know Student Needs and Respond to Them. This logically presented, easy-to-grasp review gives you the reference you want to effectively organize your introductory-level course work. Each Review Gives... (read more)
Math Essentials (Math Essentials)
by Learningexpress Publisher Comments
"Simple" math like fractions, decimals, and percentages can prove maddeningly difficult. This guide helps readers learn to solve these math problems and to meet the demands of both the workplace and the marketplace. Includes free access to online... (read more)
Pre-Calculus Problem Solver (Rea's Problem Solvers)
by M Fogiel Publisher Comments
REAs Pre-Calculus Problem Solver Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. Answers to all of your questions can be found in one convenient source from one of the most... (read more)
Algebra Workbook for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
From signed numbers to story problems — calculate equations with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear — this hands-on-guide focuses on... (read more)
Pre-Calculus Demystified
by Rhonda Huettenmueller Publisher Comments
HERE'S AN ABSOLUTE VALUE FOR ANYONE WISHING TO MASTER PRE-CALCULUS! Calculus is a cinch with pre-calculus under your belt -- and calculus is a must for any science, math, or computer science major. Pre-calculus by itself deepens your understanding of... (read more)
Doodle Yourself Smart . . . Geometry (Doodle Books)
by Sonya Newland Synopsis
Do you remember the Pythagorean theorum? How about quadratic equations? If its been years since your last geometry class, these terms may sound like theyre from another language. But theres an easy way to get back up to speed. All you have to do is... (read more)
A New Look at Geometry (Dover Books on Mathematics)
by Irving Adler Publisher Comments
This richly detailed overview surveys the development and evolution of geometrical ideas and concepts from ancient times to the present. In addition to the relationship between physical and mathematical spaces, it examines the interactions of geometry |
with the most basic notions, Universal Algebra: Fundamentals and Selected Topics introduces all the key elements needed to read and understand current research in this field. Based on the author's two-semester course, the text prepares students for research work by providing a solid grounding in the fundamental constructions and concepts of universal algebra and by introducing a variety of recent research topics.
The first part of the book focuses on core components, including subalgebras, congruences, lattices, direct and subdirect products, isomorphism theorems, a clone of operations, terms, free algebras, Birkhoff's theorem, and standard Maltsev conditions. The second part covers topics that demonstrate the power and breadth of the subject. The author discusses the consequences of Jónsson's lemma, finitely and nonfinitely based algebras, definable principal congruences, and the work of Foster and Pixley on primal and quasiprimal algebras. He also includes a proof of Murskiĭ's theorem on primal algebras and presents McKenzie's characterization of directly representable varieties, which clearly shows the power of the universal algebraic toolbox. The last chapter covers the rudiments of tame congruence theory.
Throughout the text, a series of examples illustrates concepts as they are introduced and helps students understand how universal algebra sheds light on topics they have already studied, such as Abelian groups and commutative rings. Suitable for newcomers to the field, the book also includes carefully selected exercises that reinforce the concepts and push students to a deeper understanding of the theorems and techniques. |
97898102158Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry Theorems (Series on Applied Mathematics)
This book begins by introducing the area method, and recent results in automating the area method. It can either be used as a geometry text for students and geometers, or be regarded as a monograph on machine proofs in geometry. By automating the area method, this book presents a systematic way of proving geometry theorems using traditional methods. The authors aim to make learning and teaching geometry easier through |
More About
This Textbook
Overview
Dr Smithies' analysis of the process whereby Cauchy created the basic structure of complex analysis, begins by describing the 18th century background. He then proceeds to examine the stages of Cauchy's own work, culminating in the proof of the residue theorem. Controversies associated with the the birth of the subject are also considered in detail. Throughout, new light is thrown on Cauchy's thinking during this watershed period. This authoritative book is the first to make use of the whole spectrum of available original |
Synopses & Reviews
Publisher Comments:
This graduate-level textbook is a detailed exposition of key mathematical tools in analysis aimed at students, researchers, and practitioners across science and engineering. Every topic covered has been specifically chosen because it plays a key role outside the field of pure mathematics. Although the treatment of each topic is mathematical in nature, and concrete applications are not delineated, the principles and tools presented are fundamental to exploring the computational aspects of physics and engineering. A central theme of the book is the structure of various vector spaces--most importantly, Hilbert spaces--and expansions of elements in these spaces in terms of bases. Key topics and features include: * More than 150 exercises * Abstract and normed vector spaces * Approximation in normed vector spaces * Hilbert and Banach spaces * General bases and orthonormal bases * Linear operators on various normed spaces * The Fourier transform, including the discrete Fourier transform * Wavelets and multiresolution analysis * B-splines * Sturm-Liouville problems As a textbook that provides a deep understanding of central issues in mathematical analysis, Functions, Spaces, and Expansions is intended for graduate students, researchers, and practitioners in applied mathematics, physics, and engineering. Readers are expected to have a solid understanding of linear algebra, in Rn and in general vector spaces. Familiarity with the basic concepts of calculus and real analysis, including Riemann integrals and infinite series of real or complex numbers, is also required. Functions, Spaces, and Expansions is the main textbook for the e-course Mathematics 4: Real Analysis currently being taught at the Technical University of Denmark. Please click the "Course Materials" link on the right to access videos of the lectures, problem sheets, and solutions to selected exercises.
Synopsis:
This text presents a detailed exposition of key mathematical tools in analysis. Each topic covered was chosen because it plays a role beyond the field of pure mathematics, so the text is useful in exploring the computational areas of physics and engineering.
"Synopsis"
by Springer,
This text presents a detailed exposition of key mathematical tools in analysis. Each topic covered was chosen because it plays a role beyond the field of pure mathematics, so the text is useful in exploring the computational areas of physics and engineering |
The 2014 International Conference on Future Communication, Information and Computer Science (FCICS 2014) was held May 22-23, 2014 in Beijing, China. The objective of FCICS 2014 was to provide a platform for researchers, engineers and academics as well as industrial professionals from all over the world to present their research results and development... more...
A True Textbook for an Introductory Course, System Administration Course, or a Combination Course
Linux with Operating System Concepts merges conceptual operating system (OS) and Unix/Linux topics into one cohesive textbook for undergraduate students. The book can be used for a one- or two-semester course on Linux or Unix. It is complete with... more... validation), this text concentrates on numericalThe intriguing tale of why the United States has never adopted the metric system, and what that says about us. The American standard system of measurement is a unique and odd thing to behold with its esoteric, inconsistent standards: twelve inches in a foot, three feet in a yard, sixteen ounces in a pound, one hundred pennies to the dollar. For... more...
Give Your Students the Proper Groundwork for Future Studies in Optimization
A First Course in Optimization is designed for a one-semester course in optimization taken by advanced undergraduate and beginning graduate students in the mathematical sciences and engineering. It teaches students the basics of continuous optimization and helps them... more...
Originally published in 1963, this book was one of the first to explore group process and working with groups. The introductory chapter tells us that working with groups requires three skills: and understanding of theory, a knowledge of its application, and trained experience in its use. It goes on to discuss these points, helping the reader towards... more...
This bookis aboutthe subject of higher smoothness in separable real Banach spaces.It brings together several angles of view on polynomials, both in finite and infinite setting.Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves around two main broad questions: What is the best smoothness... more... |
Basic College Mathematics - 4th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics, Fourth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for20.26 +$3.99 s/h
VeryGood
BookCellar-NH Nashua, NH
03216494002023.00 +$3.99 s/h
VeryGood
Mini City Media Nc Raleigh, NC
2010 Paperback Very good ***TOO LARGE for INTERNATIONAL SHIPPING! ! *** Light wear. No CD or access code included. All items shipped to US include delivery confirmation. Thanks for looking!
$24.5524.8425.00 +$3.99 s/h
Good
Chamblin Bookmine Jacksonville, FL
N/a Boston, MA 2011 Soft Cover 4th Edition Good 4to. No writing, highlighting, or creases. Cover has edgewear, an upcurl, & a yellow stain on the fore-edge & bottom corner. First few pages also have...show more a faint stain on the bottom corner. [#54328] |
Math 220
Matrices
Course Description
Many problems we have to solve in day-to-day business, engineering, and science practice require the simultaneous study of several different but interrelated factors. Although problems of this form have been studied throughout the long history of mathematics, only in the early 20th century did the systematic approach we now refer to as linear algebra based on matrices emerge. Matrices and linear algebra are now recognized as the fundamental tool for foundational methods in statistics, optimization, quantum mechanics, and many other fields, and are an essential component of most subfields of mathematics. Linear algebra provides students their first introduction to the concept of dimension in an abstract setting where things with 4, 5, or even more dimensions are often encountered.
MATH 220 is a 2 credit course that teaches the core concepts of matrix arithmetic and linear algebra. It is a required course for many students majoring in engineering, science, or secondary education. In past coursework, students should have gained practice solving pairs of equations like 3 x + 4 y = 10, x - y = 1. This is a system of two linear equations with two unknowns and as a unique solution students can find by isolating and substituting. In linear algebra, this system is represented as A x = b, where x is a vector of unknowns, A is a matrix, and b is a vector of constants. Linear algebra is the field of mathematics that grew out of a need to solve systems like these and related problems with many unknown variables.
Topics covered in MATH 220 include matrix algebra, vectors, linear transformations, solution to systems of linear equations, determinants, matrix inverses, concepts of rank and dimension, eigenvalues, eigenvectors, and others as time permits. Course prerequisites can be filled by one semester of calculus. Students may take MATH 220 concurrently with MATH 141, MATH 230, or MATH 250. Students seeking a linear algebra course without a calculus prerequisite may consider MATH 018 as an alternative. After completing MATH 220, students can enroll in MATH 441 or MATH 484. MATH 441 provides more in-depth perspective on linear algebra. MATH 484 studies widely used applications of linear algebra to optimization problems.
Class size, frequency of offering, and evaluation methods will vary by location and instructor. For these details check the specific course syllabus. |
This course addresses a range of
Euclidean and Non-Euclidean geometries, as summarized in the table below:
Course Topics
Related Standards
Euclidean and Non-Euclidean geometries, systems of
postulates in a comparison of Euclidean and Non-Euclidean geometries.
Axiomatic systems and their role in problem-solving in mathematics
development of Geometry as an axiomatic system
G1; G6; G7;
Role of geometry in understanding the world and of
how this understanding can be developed in their own classrooms
G2; G5; G8; G9; G10
Conceptions of geometry beyond plane figures and
their properties
G13; G14
Development of geometric topics addressed in high
school and discussion of current SOLs and Career Readiness Expectations
G1; G2; G3; G4; G5; G6; G7; G8; G9; G10; G11; G12
Other topics may include: structures of
transformational, fractal, projective geometry with a brief history of the
development of axiomatic systems of geometry, trigonometry.
G3; G4; G11; G12
Applied Statistics for Teachers (RU Stat 644)
Applied Statistics for Teachers This course explores ways to
collect, organize, display, and analyze data and make reasoned decisions based
on it. Students use statistical methods based on data, develop and evaluate
inferences and predictions about data, and apply probability and distribution
theory concepts. The course helps prepare teachers to teach statistical
concepts and AP statistics and to critically examine and comprehend data
analysis in education literature. Graphing calculators and computer software
are incorporated.
Students use matrices and determinants to solve systems of linear
equations. Applications of matrices and matrix inverses are used in solving
real world problems, and graphing calculators are used extensively in solving
large scale problems using matrix techniques.
Course Topics
Related
Standards
Linear Equations and Matrices
A.4(e), MA.14, AII.7(g), AII.7,(h)
Applications of Linear Equations
MA.14, A.4(e), AII.7(g), AII.7,(h)
Determinants
MA.2, AII.7(g)
Vectors in
AFDA.2
Real Vector Spaces
A.4(e), MA.2
Eigenvectors and Eigenvalues
MA.4
Linear Transformations and Matrices
AFDA.2, AII.7(g), MA.2, AII.7, (h)
Mathematical Modeling (RU Math 641)
This course examines mathematical models of real life phenomena and
develops solution strategies for open-ended problems. Models include extensive
algebraic reasoning, calculus and linear algebra. They may include discrete and
continuous population growth models, exponential and logistic growth,
predator-prey models, diffusion processes such as heat conduction or pollution
models, and optimization with several variables. Software such as Excel and
Maple, or similar programs, is utilized.
Course Topics
Related
Standards
Linear and Quadratic Equations
A.4, MA.14, AII.7
Applications of Linear Equations
MA.14, A.4, AII.7(g), AII.7,(h)
Exponential and Logistic Growth
A.2, AII.6, MA.9
Solving and Graphing Equations
A.5, A.6, A.7, AII.1, AII.5
Least Squares Data Approximation
A.9, A.11
Optimization with Several Variables
AII.8, AII.9
Mathematical Modeling (RU Math 641)
Educational Technology (VCU Math 554):This course emphasizes
educational technologies appropriate for use in Algebra I, Algebra II, AFDA,
and Geometry. The course strengthens teachers' understandings of algebra, data
analysis, and geometry by integrating instructional technologies in these areas.
Content is aligned with Virginia SOLs and national standards for technology and
for Algebra I, Algebra II, AFDA, and NCTM Communication Standard for grades
9-12. The course emphasizes research, practice, and policy involving current
technologies in education and uses mathematics software such as Fathom,
GeoGebra, Excel, and Mathematica. Students learn mathematics applications of
word processors, databases, spreadsheets, fundamentals of Internet tools, and
rudimentary hypermedia tools to create multimedia projects. They discuss what
it means to be a responsible and effective technology user in classrooms and
how to appropriately assess student learning using technology. Students analyze
and synthesize their learning through presentations, group work, reflection
papers, computer projects and discussions. |
...
Show More of rules and procedures, to an arena of inquiry. The volume provides plentiful exercises that give users the opportunity to participate and investigate algebraic and geometric ideas which are interesting, important, and worth thinking about. The volume addresses algebraic themes, basic theory of groups and products of groups, symmetries of polyhedra, actions of groups, rings, field extensions, and solvability and isometry groups. For those interested in a concrete presentation of abstract algebra |
201401_Skills Set
MATH 1010 Essential Mathematics Testable Skills Spring 2014 Sect.Num Skill Related Problems 1C.1 Use set notation (braces) to write the members of sets. Recognize when a set has no members. 1C(29-36) 1C.2 Draw a Venn diagram with two circles showing the relationship between the two sets. 1C(37-44) 1C.3 Given a categorical proposition, state the subject and predicate sets and draw a Venn diagram for the proposition. Label all regions of the diagram clearly. 1C(45-52) 1C.4 Draw a Venn diagram with three overlapping circles for the three given sets. Label the contents of each region. Identify empty regions, if any. 1C(53-58) 1C.5 Use a given Venn diagram to answer questions in an applied problem. 1C(59-62) 1C.6 Draw a Venn diagram to represent the given situation in an applied problem and answer questions. 1C(63-66) 1C.7 Draw a Venn diagram from a Two-Way Table. 1C(67-72) 1C.8 Complete a Two-Way Table. 1C(77-80) 1C.9 Use Venn diagrams to organize propositions and information. 1C(81a-86) 1D.1 Identify if an argument is inductive or deductive. 1D(15-22) 1D.2 Evaluate an inductive argument. (Determine the truth of the premises, the strength of the argument and the truth of the conclusion.) 1D(23-28) 1D.3 Evaluate a deductive argument. (Draw a Venn diagram to determine whether the argument is valid. Determine the truth of the premises and state whether the argument is sound.) 1D(29-44) 2A.1 Identify the units for the given quantity. 2A(19-26) 2A.2 Carry out unit conversions. 2A(27-36) 2A.3 Perform area and volume calculations. 2A(37-38) 2A.4 Convert units with raised powers. 2A(39-46) 2A.5 Compute currency conversion. 2A(47-54) 2A.6 Use unit conversions to solve applied problems. 2A(55-68, 73-75) 2B.1 Review: Evaluate expressions using powers of ten. 2B(19-22) 2B.2 Perform conversions using the US Customary System. 2B(23-30) 2B.3 Perform US Customary System-Metric conversions. 2B(37-46) 2B.4 Perform Celsius-Fahrenheit and Celsius-Kelvin conversions. 2B(47-50) 2B.5 Calculate and compare densities. 2B(55-56) 2B.6 Use standardized units to solve applied problems. 2B(70-71, 85)... View Full Document
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Summary: This tried-and-true text from the pioneer of the basic technical mathematics course now has Addison-Wesley's amazing math technologies MyMathLab and MathXL helping students to develop and maintain the math skills they will need in their technical careers.
Technical mathematics is a course pioneered by Allyn Washington, and the eighth edition of this text preserves the author's highly regarded approach to technical math, while enhancing the integration of te...show morechnology in the text. The primary strength of the text is the heavy integration of technical applications, which aids the student in pursuit of a technical career by showing the importance of a strong foundation in algebraic and trigonometric math.
Allyn Washington defined the technical math market when he wrote the first edition of Basic Technical Mathematics over forty years ago. His continued vision is to provide highly accurate mathematical concepts based on technical applications. The course is designed to allow the student to be simultaneously enrolled in allied technical areas, such as physics or electronics. The material in the text can be easily rearranged to fit the needs of both instructor and students. Above all, the author's vision of this book is to continue to enlighten today's students that an understanding of elementary math is critical in many aspects of life.
Special Caution and Note indicators identify and aid students on difficult topics throughout the text.
Flexibility of Material Coverage. An important and critical feature to the Washington approach to technical math is the flexibility of the table of contents. The chapters of the text are easily adapted to the specific needs of the students as well as the instructor. Certain sections or chapters may be omitted without loss of continuity, and chapters may be reorganized for a customized syllabus. Notes and suggestions on how to reorganize the material are contained in the Answer Book.
Graphing Calculator.The graphing calculator is integrated and emphasized throughout the text, though it is still not required for the course. This integration includes over 160 graphing calculator screens pictured in the text.
Design.The open design is an important aspect that continues in the eighth edition. The spacious layout allows for additional graphing calculator screen graphics in the margin, which helps students visualize the graphing technology.
Word Problems and Solutions.Throughout the text, approximately 120 examples present complete solutions to word problems. These examples are clearly indicated in the margin by the phrase ''solving a word problem.'' In addition, the text includes over 900 word problems within the exercise sets.
Exercises and Figures.The eighth edition includes over 1500 new exercises. Over 200 new figures have been added to help students visualize applications and concepts.
Writing Exercises. The number of writing exercises has been increased. An icon highlights the writing exercises in the text. These exercises reinforce student understanding, as they require students to verbalize their answers.
Hardcover Very Good 0321306899 A very nice used copy. No folds, tears, creasing or writing. Tightly bound and ready to go63.00 +$3.99 s/h
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Biblio Books Mississauga, ON
Canada 2005 Hardcover Fine Book Edges are sharp and fine. No tears or creases. No stains, writing or reminder marks. The binding is straight and tight. Only very mild touch of edge rubbing. Book has...show more appearance of only minimal use. The book itself is very nice. Clean, tight, sound |
Pre-Calculus for Dummies
9780470169841
ISBN:
0470169842
Pub Date: 2008 Publisher: Wiley, John & Sons, Incorporated
Summary: Brush up on algebra and trig concepts and get a glimpse of calculusUnderstand the principles and problems of pre-calculusGetting ready for calculus, but feel confused? Have no fear! This unintimidating, hands-on guide walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. You'll understand the concepts -... not just the number crunching - and see how to perform all tasks, from graphing to tackling proofs.Apply the major theorems and formulasGraph trig functions like a proFind trig values on the unit circleTackle analytic geometryIdentify function limits and continuity[ |
Accessible to students and flexible for AccessAdditional Product Information
Features/Benefits
Visualize the Solution When appropriate, both algebraic and graphical solutions are provided to help you visualize the mathematics of the example and to create a link between the algebraic and visual components of a solution.
Exploring Concepts with Technology The optional Exploring Concepts with Technology feature appears after the last section in each chapter and provides you the opportunity to use calculators or computers to solve computationally difficult problems. In addition, you are challenged to think about pitfalls that can be produced when using technology to solve mathematical problems.
Integrating Technology Integrating Technology boxes show how technology can be used to illustrate concepts and solve many mathematical problems. Examples and exercises that require a calculator or a computer to find a solution are identified by the graphing calculator icon.
Modeling sections and exercises rely on the use of a graphing calculator or a computer. These optional sections and exercises introduce the idea of a mathematical model and help you see the relevance of mathematical concepts.
Chapter Review Exercise Sets and Chapter Tests The Chapter Review Exercise Sets and the Chapter Tests, at the end of each chapter, are designed to provide you with another opportunity to assess your understanding of the concepts presented in a chapter. The answers for all exercises in the Chapter Review Exercise Sets and the Chapter Tests are provided in the Answers to Selected Exercises appendix along with a reference to the section in which the concept was presented.
What's New
Mid-Chapter Quizzes New to this edition, these quizzes help you assess your understanding of the concepts studied earlier in the chapter. The answers for all exercises in the Mid-Chapter Quizzes are provided in the Answers to Selected Exercises appendix, page XX, along with a reference to the section in which a particular concept was presented.
Chapter Test Preps The Chapter Test Preps summarize the major concepts discussed in each chapter. These Test Preps help you prepare for a chapter test. For each concept there is a reference to a worked example illustrating the concept and at least one exercise in the Chapter Review Exercise Set relating to that concept.
Table of Contents
P. PRELIMINARY CONCEPTS. The Real Number System. Integer and Rational Number Exponents. Polynomials. Factoring. Rational Expressions. Complex Numbers. 1. EQUATIONS AND INEQUALITIES. Linear and Absolute Value Equations. Formulas and Applications. Quadratic Equations. Other Types of Equations. Inequalities. Variation and Applications. 2. FUNCTIONS AND GRAPHS. A Two-Dimensional Coordinate System and Graphs. Introduction to Functions. Linear Functions. Quadratic Functions. Properties of Graphs. The Algebra of Functions. Modeling Data Using Regression 3. POLYNOMIAL AND RATIONAL FUNCTIONS. The Remainder of Theorem and the Factor Theorem. Polynomial Functions of Higher Degree. Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. Graphs of Rational Functions and Their Applications. 4. EXPONENTIAL AND LOGARITHMIC FUNCTIONS. Inverse Functions. Exponential Functions and Their Applications. Logarithmic Functions and Their Applications. Properties of Logarithms and Logarithmic Scales. Exponential and Logarithmic Equations. Exponential Growth and Decay. Modeling Data with Exponential and Logarithmic Functions. 5. TOPICS IN ANALYTIC GEOMETRY. Parabolas. Ellipses. Hyperbolas. 6. SYSTEMS OF EQUATIONS AND INEQUALITIES. Systems of Linear Equations in Two Variables. Systems of Linear Equations in More Than Two Variables. Nonlinear Systems of Equations. Partial Fractions. Inequalities in Two Variables and Systems of Inequalities. Linear Programming. 7. MATRICES. Gaussian Elimination Method. The Algebra of Matrices. The Inverse of a Matrix. Determinants. Cramer's Rule. 8. SEQUENCES, SERIES, AND PROBABILITY. Infinite Sequences and Summation Notation. Arithmetic Sequences and Series. Geometric Sequences and Series. Mathematical Induction. The Binomial Theorem. Permutations and Combinations. Introduction to ProbabilityThe Complete Solutions Manual provides worked-out solutions to all of the problems in the text.
Solution Builder
(ISBN-10: 0538797878 | ISBN-13: 9780538797870)
This is an electronic version of the complete solutions manual available via the PowerLecture and Instructor's Companion Website. It provides instructors with an efficient method for creating solution sets to homework or exams that can then be printed or posted.
The CD-ROM provides the instructor with dynamic media tools for teaching college algebra. PowerPoint® lecture slides and art slides of the figures from the text, together with electronic files for the test bank and a link to the Solution Builder, are available. The algorithmic ExamView allows you to create, deliver, and customize tests (both print and online) in minutes with this easy-to-use assessment system. Enhance how your students interact with you, your lecture, and each other your students need to retake the course and you are using the same book AND edition then they will not need to buy a new code.
This instant access code will be delivered via email when purchased. If you are not certain this is the correct access code for your course, please contact your Cengage Learning Consultant.
Cengage Learning's Enhanced WebAssign allows you to create online homework assignments that draw from thousands of end of chapter questions that match your adopted textbook. Problems are enhanced with rich tutorial content like Watch It Videos and Animations, Practice It and Master it Tutorials and access to relevant textbook sections. Flexible assignment options allow you to choose how feedback and tutorial content is released to students as well as the ability to conditionally release assignments based on your student's prerequisite assignment scores. Increase student engagement, improve course outcomes and experience the superior service offered through CourseCare. Visit us at to learn moreReinforces student understanding and aids in test preparation with detailed explanations, worked-out examples, and practice problems. Lists key ideas to master and builds problem-solving skills. Includes worked solutions to the odd-numbered problems in the text.Reinforces student understanding and aids in test preparation with detailed explanations, worked-out examples, and practice problems. Lists key ideas to master and builds problem-solving skills. Includes worked solutions to the odd-numbered problems in the text you need to retake the course and your instructor is using the same book AND edition then you will not need to buy a new code.
This instant access code will be delivered via email when purchased. If you are not certain this is the correct access code for your course, please contact your instructor.
AccessMeet the Author
About the Author
Richard N. AufmannVernon C. Barker
Vernon Barker has retired from Palomar College where he was Professor of Mathematics. He is a co-author on the majority of Aufmann texts, including the best-selling developmental paperback series.
Vernon Barker has retired from Palomar College where he was Professor of Mathematics. He is a co-author on the majority of Aufmann texts, including the best-selling developmental paperback series.
Richard D. Nation
Richard Nation is Professor of Mathematics at Palomar College. He is the co-author of several Aufmann titles.
Richard Nation is Professor of Mathematics at Palomar College. He is the co-author of several Aufmann titles |
Numbers offers an intuitive interface and elegant organization features we've come to expect from Apple, allowing you to focus more time on analysis and presentation. David Rivers helps you learn all that you can accomplish with Numbers. First, take a quick tour of the interface and the Template Chooser, and then see how to quickly create a basic spreadsheet. He also walks through more complex features, like tables, styles, formulas and functions, charts, and formatting options, as well as the array of objects you can add to make a Numbers spreadsheet even more effective and eye-catching. Finally, learn how to share your spreadsheets with the world as printouts and emails, or through iCloudllDescription CreateThe pioneering work of French mathematician Pierre de Fermat has attracted the attention of mathematicians for over 350 years. This book was written in honor of the 400th anniversary of his birth, providing readers with an overview of the many properties of Fermat numbers and demonstrating their applications in areas such as number theory, probability theory, geometry, and signal processing.
Program Numbers will help you realize the dream and achieve the goal, and gorgeous Apple-designed templates help you get started. Tables are already made ??up. Formulas are derived. Fonts are installed. Everything is ready. Simply select one of 30 templates for home, school or work, and adjust it as you like for you. Spreadsheets are made ??on a flexible canvas freeform Easily filter through large tables. Automatically format cells based on numbers, text, dates, and durations with new conditional highlighting. And with an all-new calculation engine, Numbers is faster than ever before. |
Geometry - Hardcover
Author:
Unknown
ISBN-13:
9780395977286
ISBN:
0395977282
Publisher: Holt McDougal
Summary: The theorems and principles of basic geometry are clearly presented in this workbook, along with examples and exercises for practice. All 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 encourag...e students to enjoy working the pages while gaining valuable practice in geometry |
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This article provides a convenient entry point into the content of the PlanetMathsite. It is organized around the principal subject areas of mathematics. Each of the articles pointed to herein should provide an overview of the content of that area of mathematics. |
Internet courses/tutorials to help students do homework, prepare for a test, or get ready for a class. The material presented here reviews the most important results, techniques and formulas in college and pre-college mathematics. The learning units are presented in worksheet format and require the student's active participation. Subjects covered include Algebra, Trigonometry, Calculus, Differential Equations, Complex Variables, and Matrix Algebra. |
Algebra and Trigonometry - 01 edition
ISBN13:978-0534434120 ISBN10: 0534434126 This edition has also been released as: ISBN13: 978-0534380298 ISBN10: 0534380298
Summary: James Stewart, the author of the worldwide best-selling calculus texts, along with two of his former Ph.D. students, Lothar Redlin and Saleem Watson, collaborated in writing this book to address a problem they frequently saw in their calculus courses. Many students were not prepared to "think mathematically" but attempted to memorize facts and mimic examples. Algebra and Trigonometry was designed specifically to help readers learn to think mathematically an...show mored to develop true problem-solving skills. Patient, clear, and accurate, the text consistently illustrates how useful and applicable mathematics is to real life. The new book follows the successful approach taken in the authors' previous books, College Algebra, Third Edition, and Precalculus, Third Edition95 +$3.99 s/h
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elson Principles of Mathematics 9 Student Success Workbook is specially designed to help struggling students be successful. It provides accessible, on-grade math to support students in the Grade 9 Academic Math course MPM1D (revised 2005).
Features & Benefits:
? All lessons written to meet the same goals as equivalent lessons in each textbook ? Clear instructions provided for all lessons with exercises scaffolded in manageable steps ? Written at a level appropriate for struggling readers ? Predictable layout assists students with weak organizational skills ? Provides extra support and differentiated instruction opportunities |
Info for
Punch Line Algebra Book A 7 11
What is the answer to page 8 . 2 in punchline algebra book, What is the answer to page 8.2 in punchline algebra book a did you hear about the antelope who was getting dressed when he was trampled by a herd of buffalo?.
Classzone - algebra 1, Welcome to algebra 1. this course will make math come alive with its many intriguing examples of algebra in the world around you, from baseball to theater lighting to.
Algebra - wikipedia, the free encyclopedia, The start of algebra as an area of mathematics may be dated to the end of 16th century, with françois viète's work. until the 19th century, algebra consisted.
Ok are book a little messed up. we are learning about, You can put this solution on your website! let's find the slope of the line through the points a(5,8) and b(7,11) note: is the first point . so this means that and ..
Phschool.com - prentice hall bridge page, Take a closer look at some of the leading instructional materials for secondary school classrooms..
Cpm educational program, Cpm educational program strives to make middle school and high school mathematics accessible to all students. it does so by collaborating with classroom teachers to.
Cpm educational program, Cpm educational program strives to make middle school and high school mathematics accessible to all students. it does so by collaborating with classroom teachers to.
Untitled document [ The punchline algebra set consists of two binders, each containing 192 pages. book a includes topics often taught in the first semester of an algebra 1 course, while.
Algebra homework help, algebra solvers, free math tutors, Pre-algebra, algebra i, algebra ii, geometry: homework help by free math tutors, solvers, lessons. each section has solvers (calculators), lessons, and a place where.
Abstract algebra: the basic graduate year, Abstract algebra: the basic graduate year (revised 11/02) click below to read/download chapters in pdf format. pdf files can be viewed with the free program adobe.
Cpm homework help page for students - cpm educational program, Select the appropriate chapter from the drop down menu above each book icon.click on the problem number for most current devices. click on "j" for java enabled computers..
ClassZone - Algebra 1
Welcome to Algebra 1. This course will make math come alive with its many intriguing examples of algebra in the world around you, from baseball to theater lighting to ... Last update Sun, 20 Jul 2014 05:18:00 GMT Read More |
Precalculus - With 2 CDS - 4th edition
Summary: Bob Blitzer's background in mathematics and behavioral sciences, along with his commitment to teaching, inspired him to develop a precalculus series that gets students engaged and keeps them engaged. Presenting the full scope of mathematics is just the first step. Blitzer draws students in with applications that use math to solve real-life problems.0321559843 Good Condition used book34 +$3.99 s/h
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Nivea Books Lynnwood, WA
Hardcover Fine 0321559843 Like New copy, without any marks or highlights. 2 of 2 CDs included. Might have minor shelf wear on covers. This is Student US Edition. Same day shipping with free trackin...show moreg number |
0135709 Algebra-Preliminary Edition
This text is based upon NCTM and AMATYC Standards as well as the authors' five main goals for beginning algebra students: (1) to develop conceptual thinking (2) to develop algebraic skills (3) to integrate topics so that students experience a "holistic" math curriculum (4) to provide practical application problems within all topics (5) to provide students with the experience of working collaboratively on exploratory |
Arlington algebra project answers unit 3 l7. is not in the code or scrambled answer list, the student knows it is incorrect. and concepts required for each puzzle are limited so that students with different .
. The expectation for this course is that all students will become better problem solvers by interweaving skills How does knowledge of mathematical skills correlate to real-life experiences? Alignment. Find area, perimeter and circumference.
. understanding of algebraic equations by graphing the solution set to a linear algebraic A.7 Simplify expressions of the first degree by combining like terms, and. Based Learning packet is located in the Supplemental Materials section.
. The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry. Placement Test Sample Questions. 1. A. 6. B. 24. C. 27. D. 30. A total of 50 juniors and seniors were given a mathematics test. The 35.
. Probably the most important lab in our book is the lab entitled Math Alphabet Soup become evident when you ask them to estimate the percent of each color from duced to the concept of the average or heard of the average temperature for .
. school mathematics teachers. Students were placed in pre-algebra, algebra and geometry classes based on prior P.A.G.E. instructors utilized the item analysis to guide instruction, providing direct. Q3: More confidence in math ability . |
Course Descriptions
This course is a review of basic arithmetic operations
with an algebraic approach (including whole numbers,
fractions, decimals, percents, and ratios). It provides
preparation/review for Intro to Algebra, Business Math,
Math for Food Service Records, and Concepts in Mathematics.
F,S |
Maths Quest General Mathematics Preliminary Course
2nd edition
Maths Quest General Mathematics Preliminary Course by Rowland
Book Description
Maths Quest General Mathematics Preliminary Course Second edition is specifically designed for the General Mathematics Stage 6 Syllabus. This text provides comprehensive coverage of the five areas of study: Financial mathematics, Data analysis, Measurement, Probability and Algebraic modelling. This student textbook offers these new features: * graphics calculator tips throughout the text * a quick and easy way for students to identify formulae that will appear on the HSC examination formula sheet * A CD-ROM that contains the entire student textbook with links to: * interactive technology files; * SkillSHEETS, which assist students to revise and consolidate essential skills and concepts; *2 WorkSHEETS for each chapter, which assist students to further consolidate their understanding * Test Yourself multiple-choice questions.
The following award winning features continue to be offered in this edition: * full colour with photographs and graphics to support real-life applications * carefully graded exercises with many skill and application problems, including multiple-choice questions * cross-references to relevant worked examples matched to questions throughout the exercises * comprehensive chapter summaries and chapter review exercises with practice examination questions * a glossary of mathematical terms, simply defined * investigations, spreadsheet applications and more. The teacher edition contains everything in the student edition package and more: * answers printed in red next to most questions in each exercise * annotated syllabus information * detailed work programs The teacher edition CD-ROM contains 2 tests per chapter, complete with fully worked solutions, WorkSHEETS and their solutions, and syllabus advice - all in editable World format.
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A comprehensive guide to concepts and problem-solving techniques taught in typical high school Intermediate Algebra or Algebra 2 class. It introduces the foundational concepts in each topic area - from linear equations to polynomials, and radical functions - and provides a wealth of tips, step-by-step examples, practice problems, and solutions.
Help fourth grade students master Common Core skills such as determining a story's theme, using prepositional phrases, understanding fractions, and more with Common Core Language Arts and Math for grade 4.
This book helps young learners master fundamental skills necessary for success in school. This title provides more than 90 full-color activities that focus on skills aligned to grade-level curricula. Plus, this book includes access to online timed tests and activities, adding interactive practice of a related skill.
Leo teaches his cat Pallas all about circles by applying his knowledge of geometry to their Stone Age world. Engaging illustrations and stories provide a fun introduction to math concepts, including tangents, ellipses, and more. Information boxes accompany each story to explore real applications of circles in the natural and designed world.
Books By Author Rowland
Further FP3 is a new title in Oxford A Level Mathematics for Edexcel, a new series that covers the latest curriculum changes and takes a completely fresh look at presenting the challenges of A Level. The author, Mark Rowland, is an experienced teacher who also wrote the other two Further Pure books in this series, FP1 and FP2 |
books.google.com - Ren.... Algebra and Its Applications
Linear Algebra and Its Applications
Ren. This book is written in an informal and personal style and teaches real mathematics. The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics.
User ratings
Review: Linear Algebra and Its Applications
Good one, but confusing concepts on Gaussian Elimination. Good reference book, not for learners!Read full review
Good for Undergrads and those who try to start from basics
User Review - Dinesh Dileep - Flipkart
The book is really nice if you are not exposed to linear algebra before. It patiently takes the reader through each concepts. Good if you are not familiar with linear algebra. If you do have a mathematical background, I would suggest Carl D Meyer's Matrix Analysis.Read full review |
book presents 49 space-related math problems published weekly on the SpaceMath@NASA site during the 2011-2012 academic year. The problems utilize information, imagery, and data from various NASA spacecraft missions that span a variety of math...(View More) skills in pre-algebra and algebra |
Find an Oakland Park, FL will cover relations, functions, graphs, trigonometry, polar coordinates, complex numbers, limits, and derivatives. Will analyze and graph mathematical functions. There will be an emphasis on verification of trigonometric identities using all of the basic trigonometric identitiesAs the student becomes more familiar with the subject, advanced analysis is taught to deal with differential equations. Differential calculus is just what it sounds like: dealing with differences. The student learns to develop skills in problem solving dealing with rates of change and develops |
Geometry was derived from real world measurements of lines, planes and
solids. These developed into concepts that were idealized and defined. A systematic
logical approach was then made with the relations these idealized figures have with
themselves. Therefore, it is always useful to start with tangible figures and intuitively
develop definitions and agreed upon propositions as a basis for study in geometry.
1. Along with numbers, geometry uniquely connects mathematics with the
physical world.
2. Geometry uniquely enables ideas from other areas of mathematics to
be pictured.
3. Geometry non-uniquely provides an example of a mathematical system.
The direct connection that geometry makes with the physical world takes
the form of shapes for buildings, city layouts, and construction of many types. It answers
questions like, "How far?", "How big?" or "How long?".
Areas, perimeters, volumes and the Pythagorean relations are examples of the usefulness of
geometry. Also analysis and classifications of shapes and relationships between figures
using congruence or similarity are useful ideas explored in geometry. Geometry also can be
used to picture algebraic ideas. Using coordinate geometry, graphs of lines and curves can
be generated. Sine, cosine and tangent curves can be pictured. The derivative of a
function as a tangent to a curve at a point on that curve, and statistics using bar and
circle graphs and curve fitting are uses of geometry. Despite these unique and direct
aspects for studying geometry, the non-unique aspect of geometry as a mathematical system,
historically has been the most influential focus contained in the content of the geometry
course. The emphasis on proofs must be there but not to the exclusion of the other aspects
of geometry. In order to do that:
1. Treat obvious statements informally and not as rigorous proofs.
2. Shorten the prolonged periods for proofs using the same two column
format.
3. Include topics from coordinate geometry, and transformational
geometry and simple uses of statistics.
Geometry has traditionally been taught between Algebra I and Algebra
II. Flexibility should be kept so that geometry could be taught after Algebra II.
Critical Components
In order to satisfy the graduation requirement, geometry should deal
significantly with:
Geometry as a logical System
Problem Solving
Lines and Angles
Triangles
Geometric Constructions
Polygons
Circles, Arcs
Coordinate Geometry Transformations
Solid Figures
I. Geometry as a logical System
A. Goal:
Students will understand and approach geometric problems using measurement.
Objectives: Students will
1. Use measurement to derive definitions and assumptions.
2. Use the deductive system to prove basic (important) theorems.
3. Use other than formal methods of demonstration for non-essential
problems and theorems.
II. Problem Solving
A. Goal:
Students will understand problem solving and select strategies generally used in
geometry.
Objectives: Students will
1. Learn and apply the four step problem solving procedure of
a. Identifying and analyzing the problem.
b. Formulating a plan to solve the problem.
c. Solving the problem.
d. looking back for patterns that can be useful for solving other problems.
2. Learn and use the following problem solving strategies
a. Drawing a picture or diagram.
b. Solving part of the problem.
c. Looking for a pattern.
d. Work backwards from conclusion to condition.
III. Lines and Angles
A. Goal:
Students will understand and demonstrate proficiency in the geometry of lines.
a. Congruence of triangles.
b. Calculation and application of area of triangles.
c. Similarity and proportionality of triangles.
d. The Pythagorean theorem and apply this principle to meaningful problems.
e. Elementary right triangle trigonometry functions (sine, cosine, tangent) and apply to
meaningful problems.
V. Geometric Constructions
A. Goal
Students will understand how to construct certain geometric figures.
Objectives:
Students will
1.Explain methods
and construct
a.line
segments, angles, triangles
b. The bisection of segments and angles, subdivision of a line into "N" equal
segments.
c. Perpendiculars, parallels and simple polygons.
2. Use the concept of locus of points to define a curve.
VI. Polygons
A. Goal:
Students will understand parts and uses of polygons.
Objectives: Students will
1. Name the components of polygons and their properties.
2. Compute areas of polygons.
3. Measure interior and exterior angles of polygons.
VII. Circles, Arcs
A.Goal:
Students will understand the properties of the circle and arc of a circle. |
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