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058223185ation Maths (Essential Maths for Students)
"Foundation Maths" is designed to pave the way into higher education for those students who have not specialized in mathematics at A level. It is intended for non-specialists who need some, but not a great deal of mathematics as they embark on careers in higher education. It takes students from around the lower levels of GCSE to a standard which will enable them to participate fully in a degree or diploma course. It is suitable for foundation and access courses in mathematics and for those who wish to enter a wide range of courses such as marketing, business studies, management, science, engineering, social science, geography, combined studies and design. The style of the book also makes it suitable for self-study or distance learning. Objectives are clearly stated at the beginning of each chapter, and key points and formulas are highlighted throughout the book. Self-assessment questions are provided at the end of most sections. These test understanding of important features in the section and answers are provided. These are followed by exercises for which answers are also available. A further set of test and assignment exercises is given in each chapter, but solutions are not provided to these | 677.169 | 1 |
Elementary Linear Algebra (2nd Edition)
9780131871410
ISBN:
0131871412
Edition: 2 Pub Date: 2007 Publisher: Pearson
Summary: "Elementary Linear Algebra, 2/e" -- Lawrence Spence, Arnold Insel, and Stephen Friedberg Embracing the recommendations of the "Linear Algebra Curriculum Study Group, the authors have written a text that" students will find both accessible and enlightening. Written for a matrix-oriented course, students from a variety of disciplines can expect a greater understanding of the concepts of linear algebra. Starting with ma...trices, vectors, and systems of linear equations, the authors move towards more advanced material, including linear independence, subspaces, and bases. The authors also encourage the use of technology, either computer software (MATLAB) or super-calculators, freeing students from tedious computations so they are better able to focus on the conceptual understanding of linear algebra. Lastly, students will find a variety of applications to engage their interest, demonstrated via economics, traffic flow, anthropology, Google searches, computer graphics, or music to name a few. By leveraging technology and incorporating engaging examples and numerous practice problems and exercises, this text best serves the needs of students attempting to master linear algebra.
Lawrence E. Spence is the author of Elementary Linear Algebra (2nd Edition), published 2007 under ISBN 9780131871410 and 0131871412. Two hundred forty four Elementary Linear Algebra (2nd Edition) textbooks are available for sale on ValoreBooks.com, sixty seven used from the cheapest price of $90.95, or buy new starting at $102.699780131 [more]
0131871410 shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less] | 677.169 | 1 |
Math for Strategists
Abstract
Great strategists rely heavily on numbers as they go about their work. Offers an overview of the high- and low-brow quantitative tools that students encounter during the Strategy course. The class explores high-brow tools in detail; the focus here is on low-brow calculations. Such calculations come up often in class but because they seem so simple, they get little airtime or explanation. From past class experience, roughly 20% of the students in each section come into the course with the intuition and experience to do these simple calculations themselves. The other 80% understand the calculations after they see them and grasp their value, but don't spot the opportunities to do the math themselves, before class | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Biological systems are often best explored and explained using the power of maths - from the rate at which enzymes catalyse essential life processes, to the way populations ebb and flow as predators and prey interact. Mathematical tools lie at the heart of understanding biological systems - and mathematical skills are essential for success as a bioscientist.
Core Maths for the Biosciences introduces the range of mathematical concepts that bioscience students may encounter - and need to master - during the course of their studies. Starting from fundamental concepts, the book blends clear explanations and biological examples throughout as it takes the reader towards some of the most sophisticated yet elegant mathematical tools in use by biologists today.
Reflecting the use of maths in the exploration of biology today, the book shows how computational approaches are applied to probe biological questions, and makes extensive use of computational support to help the reader develop mathematical skills for themselves - both through online graph-plotting software, and interactive ExcelRG workbooks.
Core Maths for the Biosciences is the ideal course companion as you master the mathematical skills you need to complete your undergraduate studies.
Online Resource Centre
The Online Resource Centre to accompany Core Maths for the Biosciences features
For registered adopters of the book:
DT Figures from the book in electronic format
DT Solutions to all end of chapter exercises
DT A test bank of questions for both formative and summative assessment
For students:
DT Solutions to around half of the end of chapter exercises
DT Access to FNGraph, the graph-plotting software featured in the book
An extensive range of interactive ExcelR workbooks, to help the reader master some of the concepts presented in the book through hands-on learning
About the Author
Dr. Martin Reed is Director of Teaching in the Department of Mathematical Sciences at the University of Bath, and has taught maths in universities since 1973, both in the UK and overseas (Swaziland, Papua New Guinea, Tanzania); the latter experience has developed his ability to explain subtle concepts in simple, clear language. Prior to his position at Bath, Martin was a member of the Biosciences Department at Brunel University, where he taught core skills to all first year students. | 677.169 | 1 |
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This book provides comprehensive answers to all the exercises in French for Common Entrance 13+ Exam Practice Questions, which are modelled on the style and level required of ISEB exam papers.- Endorsed by ISEB- Provides comprehensive practice and answers for Common Entrance French at 13+ - Enables pupils to practise exam-style questions before they enter the exam room, so that they can build …Lire la suite
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French for Common Entrance 13+ Exam Practice Exercises provides comprehensive practice for the Reading and Writing sections of the 13+ Common Entrance exam. All questions from a varied range of topics are modelled on the style of ISEB exam papers. This is an ideal book for pupils needing extra practice and, together with the answer book, it provides a comprehensive tool to understand …Lire la suite
A Story of Units, Grade 3
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Eureka Math is a comprehensive, content-rich PreK-12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. The companion Study Guides to Eureka Math gather the key components of the curriculum for each grade into a single location, unpacking the …Lire la suite | 677.169 | 1 |
Mathematics (MTH) Courses
Descriptions of courses offered in the Mathematics Department
MTH 1113 College Algebra
Three hours
Topics include solving equations and systems of equations, functions and graphing, inequalities, logarithms, exponentials, sequences, and series. An emphasis is placed on applied problems in physical, life, and social sciences. Offered in online format.
MTH 2003 Introduction to Statistics
Three hours
A course to give students an understanding of the concepts of statistics and tools to become critical readers of current issues involving quantitative data. Applications of the use of data from a wide variety of professions, public policy, and everyday life are made. The course focuses on methods of producing data, organizing data, and drawing conclusions from data. Topics include descriptive statistics, frequency distributions, correlation, regression, inference, and significance. Selected topics in research design and the consumer price index are also taught. Offered in online format. Prerequisite: MTH 1113. | 677.169 | 1 |
This learning object from Wisc-Online covers the fundamental laws of algebra. The interactive activity includes slides which cover the following topics: the commutative, associative and distributive laws as applied to addition and multiplication.
Syllabus, course notes, and various proofs, in PDF, PostScript and dvi format. The course assumes that students know about groups, rings, and vector spaces, and covers all the topics in algebra commonly used by analysts and topologists, such as commutative diagrams, the tensor product, functors, and Nakayama's Lemma. | 677.169 | 1 |
Product Description
Too many students end their study of mathematics before ever taking an algebra course. Others attempt to study algebra, but are unprepared and cannot keep up. "Key to Algebra" was developed with the belief that anyone can learn basic algebra if the subject is presented in a friendly, non-threatening manner and someone is available to help when needed. Some teachers find that their students benefit by working through these books before enrolling in a regular algebra course--thus greatly enhancing their chances of success. Others use "Key to Algebra" as the basic text for an individualized algebra course, while still others use it as a supplement to their regular hardbound text. Allow students to work at their own pace. The "Key to Algebra" books are informal and self-directing. The authors suggest that you allow the student to proceed at his or her own pace. Book 1 covers Operations on Integers.
We really like the Key to Mathematics series. We've used all of the topics except the geometry. These affordable books are just as effective as the pricier programs. The main difference is that each book covers ONE topic. There's no 'spiral' review built in, however, each topic builds on the last.
We love that the student can work independently, and that it encourages a true understanding of math with small steps and lots of simple line drawings. These books are black and white, and each individual book is short and not overwhelming.
I would recommend the algebra series:
-as an intro for advanced younger students,
-as a complete course for math-challenged kids who don't need a rigorous program for college, or
-as a review/explanation for kids who are stuck on a specific topic (ie trinomials). The individual books make it easy to get exactly what you need.
This is a great book especially for those who have trouble learning math. My son was having difficulty in Algebra I so I decided to take a change on the "Key to Algebra" books. I bought only the first 4, plus the answer book, and the test book. He is now at the end of the first book. This book has been a blessing. He is finding Algebra a lot easier now and I am finding out where his trouble spots are. The book is written in a relaxed manner that i think is inviting and not intimidating to children. I would highly recommend this book set.
I recently bought the Key to Algebra and Key to Geometry series for my 9and 11yr old home-schooling children. We use the books for self-directed maths sessions for about one hr each per week, in addition to their 'taught' general maths syllabus.I am completely dazzled by these books, they begin at the beginning and very methodically and gently take the child through a very detailed, comprehensive and indeed joyful course. With the odd exception, the child is able to totally self direct. I am surprised by how much is covered in these series and how quickly the child moves on, while still feeling they are 'having fun'! My 9yr old often works on the geometry series in his free time and is now able to discuss with correct terminology, bisecting angles, congruence etc; and I stress this is all self taught. Would say that on completion of these series - which may take us a couple of years - a child would be able to complete the geometry and algebra elements of the UK GCSE public examination - aimed at 16yr olds. I cannot recommend highly enough; I've found nothing like this anywhere else and order from the US - in fact I have ordered other 'Key to' series: metric measurement, decimals, percentages and fractions. The real key to success in maths it is to lay a solid foundation without fear; these books do this and much more. | 677.169 | 1 |
Algebra is an area of math that is notorious for causing distress among many students. Some algebra concepts are so complex that they may seem unnecessary; most people would never come across a situation where they would need to calculate the speed of a...
More »
When you're getting started in algebra, you'll want to understand what expressions are and how to write them . If you are past the basics, try these worksheets on using the distributive property and combining like terms. I know when I first began taking Algebra, it wa... More » | 677.169 | 1 |
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Mathematica
Get Mathematica and Wolfman|Alpha Pro
Mathematica is currently installed in the following locations:
Computer labs
Mathematics/Computer Science labs, Several open labs, dorm labs
Computer clusters
California University's Mathematica license allows for grid computing on both dedicated research clusters and in distributed, or ad-hoc, grid environments. For more information about setting this up, please contact Andy Dorsett at adorsett@wolfram.com.
To request Mathematica and Wolfram|Alpha Pro, follow the directions below.
Tutorials
Mathematica
The first two tutorials are excellent for new users, and can be assigned to students as homework to learnMathematica outside of class time.
Hands-on Start to MathematicaFollow along in Mathematica as you watch this multi-part screencast that teaches you the basics—how to create your first notebook, calculations, visualizations, interactive examples, and more.
What's New in Mathematica 10Provides examples to help you get started with new functionality in Mathematica 10, including machine learning, computational geometry, geographic computation, and device connectivity.
How To TopicsAccess step-by-step instructions ranging from how to create animations to basic syntax information
Learning CenterSearch Wolfram's large collection of materials for example calculations or tutorials in your field of interest.
Mathematica for Teaching and Education (Free video course) Learn how to make your classroom dynamic with interactive models, explore computation and visualization capabilities in Mathematica that make it useful for teaching practically any subject at any level, and get best-practice suggestions for course integration.
How To Create a Lecture Slideshow (Video tutorial) | 677.169 | 1 |
Math/Stat Moments
Do You Know the Answer?
This seemingly benign question was given to students in Austrailia sitting for the Victorian Certificate of Education (VCE) Further Mathematics exam. The VCE is the credential awarded to secondary school students who successfully complete high school level studies (Year 11 and 12 or equivalent) in the Australian state of Victoria. Apparently the students felt the question was too difficult and took to social media to complain about it. Of course, social media replied in its usual understanding manner with, "Quit your whining! It's not that hard!".
So, what do you think? Are you smarter than an Australian high school student?
Spotlight on AMS
So You Want to Minor in Math?
And really, who doesn't?
With most students, math is really a love-hate relationship -- they love to hate it. In actuality mathematics and statistics are viewed as a valuable skill regardless of your major. A strong mathematics background implies a logical approach to problems while a strong statistics background carries with it the ability to interpret the massive amounts of data that we are confronted with on a daily basis.
We currently offer a minor in Statistics and two minors in Mathematics. Our Statistics minor begins with Probability & Statistics and continues through courses covering Spatial Statistics and Survivor Analysis. Our minors in Mathematics begin with Differential Equations and, for the Mathematical Science minor, follows with your choice of 5 additional courses as varied as your interests allow. Our minor in Computational & Applied Mathematics is a bit more focused on those applied courses that may be helpful in your engineering pursuits.
For more information, check out our Undergraduate offerings! or stop by our main office in Chauvenet 141. We're happy to answer any questions you have.
Check out the opportunities inside AMS or Ask Us
if you have any questions. | 677.169 | 1 |
-Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text - Provides a quick and useful introduction to research spanning the fields of combinatorial and computational commutative algebra, with a special focus on monomial ideals - Only a basic knowledge... more...
How is that you can walk into a classroom and gain an overall sense of thequality of math instruction taking place there? What contributes to gettingthat sense? In Math Sense, Chris Moynihan explores some of the componentsthat comprise the look, sound, and feel of effective teaching and learning.Does the landscape of the classroom feature such items... more...
How can we solve the national debt crisis? Should you or your child take on a student loan? Is it safe to talk on a cell phone while driving? Are there viable energy alternatives to fossil fuels? What could you do with a billion dollars? Could simple policy changes reduce political polarization? These questions may all seem very different, but they... more...
In teaching an introduction to transport or systems dynamics modeling at the undergraduate level, it is possible to lose pedagogical traction in a sea of abstract mathematics. What the mathematical modeling of time-dependent system behavior offers is a venue in which students can be taught that physical analogies exist between what they likely perceive... more...
Mobile Learning and Mathematics provides an overview of current research on how mobile devices are supporting mathematics educators in classrooms across the globe. Through nine case studies, chapter authors investigate the use of mobile technologies over a range of grade levels and mathematical topics, while connecting chapters provide a strong... more...
The Hindu?Arabic numeral system (1, 2, 3,...) is one of mankind's
greatest achievements and one of its most commonly used
inventions. How did it originate? Those who have written about the
numeral system have hypothesized that it originated in India; however,
there is little evidence to support this claim.
This book provides considerable evidence... more... | 677.169 | 1 |
most scientists and engineers, the only analytic technique for solving linear partial differential equations is separation of variables. In Transform Methods for Solving Partial Differential Equations, the author uses the power of complex variables to demonstrate how Laplace and Fourier transforms can be harnessed to solve many practical, everyday problems experienced by scientists and engineers. Unlike many mathematics texts, this book provides a step-by-step analysis of problems taken from scientific and engineering literature. Detailed solutions are given in the back of the book. This essential text/reference draws from the latest literature on transform methods to provide in-depth discussions on the joint transform problem, the Cagniard-de Hoop method, and the Wiener-Hopf technique. Some 1,500 references are included as well. | 677.169 | 1 |
What is contemporary math?
A:
Quick Answer
Contemporary math is a math course designed for college freshman that develops critical thinking skills through mathematics with an emphasis on practical applications. Contemporary math provides students with an alternative to more traditional college algebra courses.
Keep Learning
Traditionally, most college freshman meet their mathematics requirement with a college algebra course that extends high school mathematics into advanced algebra concepts. The math concepts taught in college algebra are not typically used in careers outside of the areas of math, science and technology. Contemporary mathematics teaches students skills used in other career areas while developing critical thinking skills that are useful in any career. Topics covered in contemporary math courses include statistics, practical geometry and logic.
In mathematics, the word "evaluate" is a verb that refers to finding or calculating the value of something. This can pertain to financial mathematics or to finding the exact value of an algebraic expression or numerical calculation.
While mathematics is a universal language, superficial differences in notation exist between English and Spanish-speaking cultures. The way numerals are rendered differ slightly, with 7 usually having a crosshatch mark and the number 9 often resembling a lower-case g. | 677.169 | 1 |
Mathematical software
Mathematical software is software used to model, analyze or calculate numeric, symbolic or geometric dataSides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric. This abbreviation is seldom if ever used in print; it is very informal. More generally, these terms may apply to an inequation or inequality. In the inequality case, there is no symmetry | 677.169 | 1 |
Spring 2016 Semester Courses
804-107 College Mathematics
Review and develop fundamental concepts of mathematics pertinent to the areas of arithmetic and algebra; geometry and trigonometry; and probability and statistics. Special emphasis is placed on problem solving, critical thinking and logical reasoning, making connections and using calculators. Topics include performing arithmetic operations and simplifying algebraic expressions, solving linear equations and inequalities in one variable, solving proportions and incorporating percent applications, manipulating formulas, solving and graphing systems of linear equations and inequalities in two variables, finding areas and volumes of geometric figures, applying similar and congruent triangles, converting measurements within and between U.S. and metric systems, applying Pythagorean Theorem, solving right and oblique triangles, calculating probabilities, organizing data and interpreting charts, calculating central and spread measures, and summarizing and analyzing data. | 677.169 | 1 |
$49Jon Lee focuses on key mathematical ideas leading to useful models and algorithms, rather than on data structures and implementation details, in this introductory graduate-level text for students of operations research, mathematics, and computer science. The viewpoint is polyhedral, and Lee also uses matroids as a unifying idea. Topics include linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and network flows. Problems and exercises are included throughout as well as references for further study.
Self contained (includes all linear-programming preliminaries)
Aimed as a one-semester textbook (not a research monograph)
Problems and exercises interspersed in the exposition, making the text a 'workbook' for the student
Reviews & endorsements
"Lee strikes a perfect balance between the specific and the general, between the concrete and the abstract."
CHOICE
"The author, with his light but rigorous mathematical writing style, takes delight in revealing the stars of combinatorial optimization. This is an excellent teaching book; I recommend it highly."
International Statistical Institute
"The book is attractively laid out on the page and there are lots of good diagrams. Algorithms are separated visually in special boxes and are easy to track down. There are plenty of problems (short proofs) and exercises (calculations) and they are well integrated with the text."
MAA Reviews, Bill Satzer
"Jon Lee's A First Course in Combinatorial Optimization is a brilliant introduction to the topic." - Ryan B. Hayward, University of Alberta | 677.169 | 1 |
Product Description
Provide insight into the many real-life uses of a scientific calculator with these 45 lessons. Boost student proficiency through clear explanations, sample problems, and practice exercises. Grades 7-Adult.
Product Information
Format: Paperback Number of Pages: 100 Vendor: Walch Education
ISBN: 0825129044 UPC: 7005070090 | 677.169 | 1 |
>I think speed and certain types of memorization are still important. For >example if a student learns sin(2x) = 2sin(x)cos(x), he may forget it 6 >months later but 2 years later he will see sin(2x) and at least know it can >be expanded. The student who doesn't learn these IDs OFTEN won't even know >enough to look it up in a book. >
Can anyone give me a real world *application* that might involve knowing the aforementioned trig identity? I'm not implying that it's not worth knowing. I'm really just curious because, in the 10 years or so since I first learned it, I don't recall ever using it outside of my trig and calculus classes. How do people use these trig identities? | 677.169 | 1 |
An Introduction to Analysis, Second Edition
9781577662327
ISBN:
1577662326
Edition: 2nd Pub Date: 2002 Publisher: Waveland Pr Inc
Summary: An Introduction to Analysis, Second Edition provides a mathematically rigorous introduction to analysis of real-valued functions of one variable. The text is written to ease the transition from primarily computational to primarily theoretical mathematics. Numerous examples and exercises help students to understand mathematical proofs in an abstract setting, as well as to be able to formulate and write them. The mater...ial is as clear and intuitive as possible while still maintaining mathematical integrity. The author presents abstract mathematics in a way that makes the subject both understandable and exciting to students.
James R. Kirkwood is the author of An Introduction to Analysis, Second Edition, published 2002 under ISBN 9781577662327 and 1577662326. One hundred thirteen An Introduction to Analysis, Second Edition textbooks are available for sale on ValoreBooks.com, six used from the cheapest price of $39.37, or buy new starting at $55.92.[read more | 677.169 | 1 |
cornerstone of ELEMENTARY LINEAR ALGEBRA is the authors' clear, careful, and concise presentation of material—written so that users can fully understand how mathematics works. This program balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system. The Sixth Edition incorporates up-to-date coverage of Computer Algebra Systems (Maple/MATLAB/Mathematica); additional support is provided in a corresponding technology guide. Data and applications also reflect current statistics and examples to engage users and demonstrate the link between theory and practice. This Enhanced Edition includes instant access to WebAssign®, the most widely-used and reliable homework system. WebAssign® presents over 500 problems, as well as links to relevant book sections, that help users grasp the concepts needed to succeed in this course. As an added bonus, the Start Smart Guide has been bound into this book. This guide contains instructions to help users learn the basics of WebAssign quickly. | 677.169 | 1 |
This resource includes number facts plus estimation, probability, graphing, ordering, telling time, fractions and more! Watch children smile as they chug along on a train and embark on other learning ...
Stage 4 (Year 8 Highschool) Mathematics MATHS QUEST 8 for New south Wales with eBook plus syllabus not changed since 2013. I have many Year 11 and Year 12 textbooks and study guides for HSC available ...
Maths Plus 5 Outcomes Edition by Harry O'Brien Greg Purcell. For multiple purchases please wait for a combined invoice to be sent BEFORE paying - we will issue the combined invoice usually within 24 h...
Publisher: Jacaranda Plus, Unit 3/4 VCE Further maths. 4TH Edition. VCE year 12 Further Maths Textbook and TI-Nspire calculator companion. Calculator Companion has not been used and is still in brand ...
Maths Plus 2 by Harry O'Brien ISBN: 9780195519587. Education. Homework and Mentals Books revise and consolidate what they have learnt. Shipping Delivery. Sign up to our newsletter to hear about about ... | 677.169 | 1 |
Your INTEGRAL tool for mastering ADVANCED CALCULUS Interested in going further in calculus but don't where to begin? No problem! With Advanced Calculus Demystified , there's no limit to how much you will learn. Beginning with an overview of functions of multiple variables and their graphs, this book covers the fundamentals, without spending tooBrownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessible introduction to the technical literature. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best...
When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented... more... | 677.169 | 1 |
Share the Learning!
The new TI-34 MultiView™ calculator is designed specifically for middle grades math. The MultiView display allows users to view multiple calculations at the same time to compare results and explore patterns. Expressions appear in familiar textbook format, for example, scientific notation with the proper superscripted exponents, mixed fractions, and the radical bar extending over the entire argument. The Toggle Key quickly converts fractions, decimals and exact pi terms to alternative forms. Ideal for Middle Grades Math, Pre-Algebra, Algebra 1 and 2, Geometry, General Science and Trigonometry. | 677.169 | 1 |
Elementary Analysis: The Theory of Calculus
Browse related Subjects ...
Read More and statistics. It is also recommended for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied. Many abstract ideas, such as metric spaces and ordered systems, are avoided. The least upper bound property is taken as an axiom and the order properties of the real line are exploited throughout. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics. Optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals. Designed for students having no previous experience with rigorous proofs, this text can be used immediately after standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, as well as for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied, while many abstract ideas, such as metric spaces and ordered systems, are avoided completely. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics, and optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals | 677.169 | 1 |
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About the book:
" . . . that famous pedagogical method whereby one begins with the general and proceeds to the particular only after the student is too confused to understand even that anymore. " Michael Spivak This text was written as an antidote to topology courses such as Spivak It is meant to provide the student with an experience in geomet describes. ric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average stu dent would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an in tegral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed.
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Hpb-Ohio via United States
Hardcover, ISBN 0387941029 Publisher: Springer41029 Publisher: SpringerEARTH SCIENCES MATHEMATICS GEOMETRY)
Hardcover, ISBN 0387941029 Publisher: Springer, 1993 1993. Corr. 2nd Printing 1997 1993. Corr. 2nd Printing 1997 ed.
Hardcover, ISBN 0387941029 Publisher: Springer, 1997Hardcover, ISBN 0387941029 Publisher: Springer, 1997 0387941029 Publisher: Springer, 1997Hardcover, ISBN 0387941029 Publisher: Springer | 677.169 | 1 |
This general math site offers reference material on a host of math topics, plus a math message board and links to relevant material online. The tables cover a range of math skills, from basic fraction-decimal conversion to the more advanced calculus and discrete math. The information is presented in notation form, with diagrams, graphs, and tables. The site is available in English, Spanish, and French. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. | 677.169 | 1 |
New Mexico Mathematics Content Standards, Benchmarks, and Performance Standards
State Board of Education Goal: Establish clear and high standards in all academic and vocational subjects and ensure that assessments are aligned with content, benchmarks, and performance standards; require alignment of school curricula with performance standards and revise on a regular basis.
This document groups the Mathematics Content Standards, Benchmarks, and Performance Standards into five strands:
" Numbers and Operations
" Algebra
" Geometry
" Measurement
" Data Analysis and Probability
The Mathematics Content Standards, Benchmarks, and Performance Standards has been designed to:
Establish an articulated, coordinated, and comprehensive description of the content and skills students should learn at specific grade levels in the study of mathematics;
Help teachers create classroom instruction and authentic assessments that address a substantive mathematics curriculum that can be applied to learning across all disciplines;
Serve as the basis for a statewide assessment of student learning; and
Stimulate thoughtful conversations and policy development regarding the acquisition and application of essential mathematical skills and concepts.
New Mexico Mathematics Content Standards, Benchmarks, and Performance Standards identify what students should know and be able to do across all grade levels. They form a spiraling framework in the sense that many skills, once introduced, develop over time. While the Performance Standards are set forth at grade-specific levels, they do not exist as isolated skills; each exists in relation to the others.
Each Content Standard is elaborated into three grade-span Benchmarks (K-4, 5-8, and 9-12) that are further defined by specific grade level Performance Standards. They illustrate how learners at every level apply mathematical concepts with increasing sophistication, refinement, and independence. In the 9-12 grade band, the five thematic strands have been focused into three fields of mathematical study to reflect more accurately how students engage in mathematics study during those grades. While the 9-12 Performance Standards do describe essential learning for high school students, they do not indicate grade-specific requirements because students are not required to enroll in specific courses in any particular grade.
High school students interested in various career directions may need to pursue mathematics topics beyond what is specified in the Standards. While all students will learn the mathematics content defined in the Standards, additional course offerings may provide a comprehensive continuum of learning experiences to prepare students to achieve their academic and career goals. Guidance for further mathematics study, in the form of Topics for Further Study, accompanies the Standards for grades 9-12. These topics are not required of all students and are not part of the State assessment system. Their purpose is to extend the depth and sophistication of students' knowledge and skills.
Guiding Principles
The New Mexico Mathematics Content Standards, Benchmarks, and Performance Standards provide a guide for focused, aligned, and sustained efforts to ensure that all students have access to high-quality mathematics education. The New Mexico Standards are based upon the framework developed by the National Council of Teachers of Mathematics (NCTM) and presented in Principles and Standards for School Mathematics. The NCTM recommends the following Guiding Principles that influence the development and delivery of successful mathematics programs. These Guiding Principles, although not unique to mathematics, establish the foundation for developing students' capabilities to mathematically reason, and solve problems.
Equity
Excellence in mathematics requires equity, including high expectations and strong support for all students.
Mathematics can and must be learned by all students. Teachers and schools should encourage high expectations in their interactions with students. They determine students' opportunities to learn and succeed in mathematics.
Curriculum
A curriculum is more than a set of activities - it must be coherent, focused on important mathematics content, and clearly articulated across grades.
In a coherent curriculum, mathematical ideas are linked and build upon each another so that students' understanding and knowledge deepens, and their ability to apply mathematics expands. An effective curriculum focuses on important mathematics that will prepare students for continued study and for solving problems in a variety of school, home, college and work settings. An articulated curriculum challenges students to learn increasingly more sophisticated mathematical ideas as they progress.
Teaching
Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.
Students' understanding of mathematics, their ability to use it to solve problems, and their confidence and disposition toward mathematics, are all shaped by the learning opportunities they encounter in school. To be effective, teachers must know and understand the mathematics they are teaching and be able to draw on that knowledge with flexibility in their teaching tasks. They need to be committed to their students as learners and as human beings, and be skillful in choosing and using a variety of pedagogical and assessment strategies. Effective teaching requires a challenging and supportive classroom learning environment. Teachers establish and nurture an environment conducive to learning mathematics through the decisions they make, the conversations they orchestrate, and the physical setting they create. Teachers' actions encourage students to think, question, solve problems, and discuss their ideas, strategies, and solutions.
Learning
Students must learn mathematics with understanding, actively acquiring new knowledge from experience and prior knowledge.
Mathematical proficiency requires conceptual understanding. In addition to factual knowledge and procedural facility, conceptual understanding allows students to use their knowledge flexibly, knowing when and how to use what they know, making sense of mathematics, remembering what they have learned, and connecting new knowledge to existing knowledge in meaningful ways. Learning with understanding is essential to enable students to solve the new kinds of problems they will inevitably face in the future.
Assessment
Multiple and varied assessments should support the learning and furnish useful information to both teachers and students.
Assessment should enhance students' learning. Good assessment conveys messages to students about what kinds of mathematical knowledge and performances are important. These messages influence the decisions students make. Teachers need to move beyond a simple 'right or wrong' judgment and discern how students are thinking about the problems. When teachers use assessment techniques that include observation, conversations and interviews with students, and interactive journals, students are more likely to learn and remember by articulating their ideas and communicating their thinking. Assessment that is a routine part of ongoing classroom activity rather than an interruption, helps students in setting goals, assuming responsibility for their own learning, and becoming more independent learners. Assessment can also help teachers make decisions about the content or form of the instruction as well as the students' mastery of the content. Exemplary mathematics assessment should:
" Measure the essential mathematics that students should know and be able to do
" Enhance mathematics learning
" Promote equity
" Be an open process
" Promote valid inference, and
" Be a coherent process.
Technology
Technology is essential; it influences the mathematics that is taught and enhances students' learning.
Electronic technologies such as calculators and computers are essential tools for teaching, learning, and doing mathematics. They furnish visual images of mathematical ideas, facilitate organizing and analyzing data, and compute efficiently and accurately. They support investigation by students in every area of mathematics and allow students to focus on decision-making, reflection, reasoning, and problem solving. Technology also supports effective mathematics teaching and can dramatically increase the possibilities for engaging students with challenging content using visualization, simulation, graphing, and advanced computing. In this context, technology is not used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster greater understanding. Technology provides a means of viewing mathematical ideas from multiple perspectives by enriching the range and quality of investigations, by assisting with feedback, and by providing an opportunity for students to discuss with one another the mathematical representations they view on the screen and the various dynamic transformations.
Processes of Problem Solving
Problem solving is an integral part of all mathematics learning. To solve problems, students must draw upon their knowledge of the concepts and skills they have learned and apply them to a novel situation; through this process, students develop new mathematical understanding. Problem solving should not be an isolated part of the program. Rather, problem solving should involve all content areas, numbers and operations, algebra, geometry, measurement, and data analysis and probability.
Instructional programs from kindergarten through grade 12 should enable students to:
Build new mathematical knowledge through problem solving
Solve problems that arise in mathematics and other contexts
Apply and adapt a variety of appropriate strategies to solve problems, and
Monitor and reflect on the process of problem solving.
To build mathematical knowledge through problem solving, teachers must first be able to choose good problems that give students the chance to solidify and extend what they know and that stimulate mathematics learning. In selecting worthwhile problems that help develop important mathematical ideas, teachers need to understand how to analyze and adapt problems, be able to focus on mathematical ideas that can be illuminated by working on the problem, and learn to anticipate students' questions to decide if particular problems will further develop the mathematical goals for the class.
Good problem solvers tend naturally to analyze situations carefully in mathematical terms and to properly pose problems based on particular situations. They consider simple cases before trying something more complicated, and they look for and examine patterns and relationships. They develop a disposition to analyze more deeply that leads to a more complete understanding of the situation and the correct solution. Throughout the grades, teachers build this disposition by asking questions that help students find the mathematics in their experiences, and by encouraging students to persist with interesting but challenging problems.
Students who can successfully solve problems are able to apply and adapt a variety of appropriate strategies. These strategies must receive instructional attention to assist students to learn them, and opportunities to use these strategies must be embedded naturally in the curriculum across the content areas. In the earliest grades, students first learn to express, categorize, and compare their strategies. In the middle grades, students should be skilled in recognizing when various strategies are appropriate to use and should be capable of deciding how to use them. By high school, students should have access to a wide range of strategies, be able to decide which one to use, and be able to adapt and invent strategies.
Effective problem solvers constantly monitor and adjust what they are doing. They make sure they understand the problem and they plan frequently, periodically taking stock of their progress to see whether they seem to be on the right track. If they are not making progress, they stop to consider alternatives and do not hesitate to take a completely different approach. Poor problem solving is often due not to lack of knowledge but to the ineffective use of what they do know. Good problem solvers become aware of what they are doing and frequently use reflective skills. Teachers who create classroom environments that support the development of reflective habits provide opportunities where students are more likely to monitor their understanding and more likely to make productive adjustments necessary when solving problems.
Reasoning and Proof
Reasoning is essential to understanding mathematics. By developing ideas, exploring phenomena, justifying results, and using mathematical conjectures in all content areas at all grade levels, students will learn that mathematics makes sense. Reasoning and proof cannot simply be taught in a single unit on logic or by 'doing proofs' in geometry. Reasoning and proof should be a consistent part of students' mathematical experience in kindergarten through grade 12. Reasoning mathematically is a habit of mind, and is developed through consistent use in many contexts.
Systematic reasoning is a defining feature of mathematics. It is found in all content areas and, with different degrees of rigor, at all grade levels. Effective instructional programs should enable students to:
" Recognize reasoning and proof as fundamental aspects of mathematics,
" Make and investigate mathematical conjectures,
" Develop and evaluate mathematical arguments and proofs, and
" Select and use various types of reasoning and methods of proof.
Conjecture, which is informed guessing, is a major pathway to discovery. Students can learn to make, refine, and test conjectures in very early grades, and they can develop their abilities to investigate their conjectures using concrete materials, calculators and other tools, and increasingly through the grades, mathematical representations and symbols. Beginning in the elementary grades, students can learn to disprove conjectures by finding counterexamples. At all levels, students will reason inductively from specific cases to larger patterns. Increasingly, they should learn to make effective deductive arguments. Students also need to work with other students to formulate their conjectures and to listen to and understand conjectures and explanations offered by classmates.
Early elementary students tend to justify general claims using specific cases. By the upper elementary grades, justifications can become more generalized and can draw on other mathematical results. In high school, students should be expected to construct relatively complex chains of reasoning. As students move through the grades they compare their ideas with other's ideas, which may cause them to modify, consolidate, or strengthen their arguments or reasoning.
Students need to learn how to select and use various types of reasoning and methods of proof to solve problems in a wide array of contexts. Students begin with informal reasoning, compared to formal logical deduction, and gradually become adept at various types of reasoning - algebraic and geometric reasoning, proportional reasoning, probabilistic reasoning, statistical reasoning, and so forth. Students need to encounter and build proficiency in all these areas with increasing sophistication as they move through the curriculum.
Communication
Though communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process helps build meaning and permanence for ideas and makes them public. When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others orally or in writing, they learn to be clear and convincing. Listening to others' explanations gives students opportunities to develop their own understanding. Conversations in which mathematical ideas are explored from multiple perspectives help students sharpen their thinking and make connections. Such activity also helps students to develop a language for expressing mathematical ideas and an appreciation for the need for precision in that language.
Instructional programs from kindergarten through grade 12 should enable students to:
" Organize and consolidate their thinking through communication,
" Communicate their mathematical thinking coherently and clearly to peers, teachers, and others,
" Analyze and evaluate the mathematical thinking and strategies of others, and
" Use the language of mathematics to express mathematical ideas precisely.
Reflection and communication are intertwined processes in mathematical learning. Writing in mathematics can also help students consolidate their thinking because it requires them to reflect on their work and clarify their thoughts about the ideas developed. Examining and discussing both exemplary and problematic pieces of mathematical writing can be beneficial at all levels. Students also need to test their ideas on the basis of shared knowledge and see whether they can be understood and are convincing.
Learning what is acceptable as evidence in mathematics should be an instructional goal from kindergarten through grade 12. Students benefit from analyzing and evaluating the mathematical thinking and strategies of others. Since not all methods and ideas have equal merit, students must learn to examine the methods and ideas of others to determine their own strengths and limitations.
As students articulate their mathematical understanding, they begin by using everyday, familiar language. Building upon this base, teachers can help students see that some words are also used in mathematics with different or more-precise meanings. While it is important that students have experiences that help them appreciate the power and precision of mathematical language, it is important to avoid imposing formal mathematical language prematurely. Students must first be allowed to grapple with their ideas and develop their own informal means of expressing them. Technology affords other opportunities and challenges for the development and analysis of language. Spreadsheets, algebraic symbols, and geometric shapes all contribute to building the vocabulary and an understanding of the language of mathematics.
Connections
Understanding involves making connections. Students that can connect their mathematical ideas develop a deeper and more lasting understanding of the rich interplay among mathematical topics and about the utility of mathematics. Viewing mathematics as a whole highlights the need for studying and thinking about the connections within the discipline, as reflected within the grade-level curriculum and across grades. Teachers help students build a disposition to recognize and use connections among mathematical ideas by asking guiding questions and providing opportunities for students to integrate mathematics in many contexts. Students begin to see the connections between arithmetic operations, understanding, for example, how multiplication can be thought of as repeated addition. As they see how mathematical operations can be used in different contexts, they develop an appreciation for the abstraction of mathematics.
As students progress through their school mathematics experience, their ability to see the same mathematical structure in different settings should increase. Students in Kindergarten through grade 2 recognize instances of counting, number, and shape. Upper elementary school students look for instances of arithmetic operations, and middle-grade students look for examples of rational numbers, proportionality, and linear relationships. High school students look for connections among the many mathematical ideas they are encountering.
As students develop a view of mathematics as a connected and integrated whole, they will have less of a tendency to view mathematical skills and concepts separately. When conceptual understanding is linked to procedures, students will not perceive mathematics as an arbitrary set of rules. This integration of procedures and concepts should be central in school mathematics.
Instructional programs from kindergarten through grade 12 should enable students to:
" Recognize and use connections among mathematical ideas,
" Understand how mathematical ideas interconnect and build on one another to produce a coherent whole, and
" Recognize and apply mathematics in contexts outside of mathematics.
The opportunity for students to experience mathematics in a context is important. Mathematics is used in science, the social sciences, medicine, engineering, construction, business, government, arts and architecture, finance and commerce, and many other fields. These links are not only through content, but also, as in the case of science, through process. The processes and content of science can inspire an approach to solving problems that applies to mathematics, or even result in the creation of new mathematical fields. Equally, students who see the connection of mathematics to the world and to other disciplines are better able to apply knowledge from several different areas and are more likely to be successful problem solvers.
Representation
The term representation refers both to process and product, meaning the act of capturing a mathematical concept or relationship in some form or the form itself. Representing applies to externally observable processes and products as well as to those that occur 'internally' in the minds of students as they are doing mathematics. When students have access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically.
Although some forms of representation, such as diagrams, graphical displays, and symbolic expressions, have long been a part of school mathematics, they have often been taught and learned in isolation. Representations should be treated as essential elements in supporting students' understanding of mathematical concepts and relationships, in communicating mathematical approaches, arguments, in understanding one's self and others, in recognizing connections among related mathematical concepts, and in applying mathematics to realistic problem situations through modeling. It is important to encourage students to represent their ideas in ways that makes sense to them, first in ways that are not conventional and later, in conventional form.
Instructional programs from kindergarten through grade 12 should enable all students to:
Create and use representations to organize, record, and communicate mathematical ideas,
Select, apply, and translate among mathematical representations to solve problems, and
Use representations to model and interpret physical, social, and mathematical phenomena.
Teachers gain valuable insight into students' ways of interpreting and thinking about mathematics by looking at their representations. Teachers can then build bridges from students' personal representations to more conventional ones when appropriate. It is important that students have opportunities to not only learn conventional forms of representation, but also to construct, refine, and use their own representations as tools to support learning and doing mathematics.
Computers and calculators change what students can do with conventional representations and expand the set of representations with which they can work. A variety of technological tools allow students to manipulate, visualize, and simulate more complex data and therefore represent and investigate mathematical ideas and situations not otherwise possible. As students' representational repertoire expands, it is important for students to reflect on their use of representations to develop an understanding of the relative strengths and weaknesses of various representations for various purposes.
Different representations can illuminate different aspects of a complex concept or relationship. To become deeply knowledgeable about many aspects of mathematics, students need a variety of representations to support their understanding. As they move through grades, students' repertoires of representations should expand to include more complex pictures, tables, graphs, and words to model problems and situations. As students become more mathematically sophisticated, they develop an increasingly large array of mathematical representations as well as the knowledge of how to use them productively.
The term model has many meanings. It can refer to physical materials, that is, manipulative models. It also refers to providing example behavior, such as when a teacher demonstrates a problem-solving process, or model can be used synonymously with representation. The term mathematical model means a mathematical representation of elements and relationships in an idealized version of a complex phenomenon. Mathematical models can be used to clarify and interpret the phenomenon and to solve problems.
In the early grades, students model situations using physical objects and simple pictures. As middle grade students model and solve problems that arise in the real and the mathematical worlds, they learn to use variables to represent unknowns and also learn how to employ equations, graphs, and tables to represent and analyze situations. High school students create and interpret models of phenomena drawn from a wider range of contexts by identifying the essential elements of the context and by devising representations that capture mathematical relationships among those elements. With technology tools, students can explore and understand complex concepts. These tools now allow students to explore iterative models for situations that were once studied in much more advanced courses.
Adapted from the National Council of Teachers of Mathematics. Principals and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 2000. | 677.169 | 1 |
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Boston College Math Experience
Program Description
This engaging and creative program is designed to challenge high school students with a strong interest in and talent for mathematics.
Students in this program will participate in math courses that are focused on topics outside of what is typically covered in a high school setting but does not require a background in Calculus. Under the guidance and instruction of faculty from the Mathematics Department and supported by graduate students in the field, students will have the opportunity to learn and explore a variety of problem solving skills. Their classroom experience will be supplemented with a Math workshop with an emphasis on collaborative problem solving. Students will also have access to supported evening study sessions.
Candidates should be able to complete pre-calculus level work and the teacher recommendation from a Math department teacher at your school should mention if you have any experience with writing proofs. Students without experience in writing proofs are still eligible for the program.
Course Titles and Description
MATH 100501 Excursions in Advanced Mathematics
In this course, we will explore several areas of advanced mathematics through lectures, classroom discussions, and labs. These include topics such as topology, projective geometry, transformation geometry, computational geometry, chaos, and coding, which are not normally taught in the high school curriculum. Through studying these exciting developments in modern mathematics, students can further develop their critical thinking and problem-solving skills. A good background in high school algebra and geometry is recommended.
June 22-July 30, M W , 11:00 a.m.-12:30 p.m. and T TH 12:00-1:00 p.m.
Instructor Bio
Professor Chi-Keung Cheung received his PhD from University of California-Berkeley, and has been a math professor at BC since 1993. His research interests are in geometric analysis and complex analysis. In recent years, he has also actively participated in training high school and elementary school math teachers.
MATH 221001 Linear Algebra
This course is an introduction to the techniques of linear algebra in Euclidean space. Topics covered include matrices, determinants, systems of linear equations, vectors in n-dimensional space, complex numbers, and eigenvalues. The course is required of mathematics majors and minors, but is also suitable for students in the social sciences, natural sciences, and management. | 677.169 | 1 |
Hardcover. Instructor Edition: Same as student edition with additional notes or answers. New Condition. SKU: 97803217858481785121.
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Customer Reviews
Beginning and Intermediate Algebra
by K Elayn Martin-Gay
So happy
I'm happy with my purchase! Couldn't beat the price. Exactly what I need for the class.
H. Marshall W
Jun 6, 2011
Excellent math book
Excellent book to assist students in math recovery. .
hollis517
Apr 23, 2010
a big help
i'm teaching myself algebra for fun, and this textbook has helped me wade through it. i still have trouble solving the word problems, though, so i can't give it 5 stars based on inadequate instruction in word problems.
c5c5
Mar 7, 2009
Just what I needed
This book helped me to grasp the basic concepts that are needed to understand and complete algebra | 677.169 | 1 |
Aimed at undergraduate mathematics and computer science students, this book is an excellent introduction to a lot of problems of discrete mathematics. It discusses a number of selected results and methods, mostly from areas of combinatorics and graph theory, and it uses proofs and problem solving to help students understand the solutions to problems. Numerous examples, figures, and exercises are spread throughoutThe goal of this book is to use the introduction to discrete mathematics … . Consequently, the authors … take a lot of time to explain proof techniques and to motivate definitions and style. The language is very informal and easy to read. The level is always introductory which makes it possible to give a taste of a wide range of topics … . There are a lot of exercises … which makes it perfectly suitable for self-study." (T. Eisenkölbl, Monatshefte für Mathematik, Vol. 144 (2), 2005)
"TheZentralblatt für Didaktik der Mathematik, January, 2004)
"The title of this book is quite apposite … . The text is, in fact, based on introductory courses in discrete mathematics … . the emphasis throughout the book is on finding efficient and imaginative ways to tackle problems and to develop general results. … I would see it as a valuable resource of enrichment activities for students … . is eminently suited for self-study (there are plenty of exercises and solutions) and can be warmly recommended for the school library." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (512), 2004)
"This book is an excellent introduction to a lot of problems of discrete mathematics. … The authors discuss a number of selected results and methods, mostly from the areas of combinatorics and graph theory … . This book is appealed to a broad range of readers, including students and post-graduate students, teachers of mathematics, mathematical amateurs. The authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book." (M.I Yadrenko, Zentralblatt MATH, Issue 1017, 2003)
"ThisL' Enseignement Mathematique, Vol. 49 (1-2), 2003)
"The aim of this book is NOT to cover discrete mathematics in depth. Rather, it discusses a number of selected results and methods … . The authors develop most topics to the extent that they can describe the discrete mathematics behind an important application of mathematics … . Another feature that is not covered in other discrete mathematics books is the use of ESTIMATES … . There are questions posed in the text and problems at the end of each chapter with solutions … ." (The Bulletin of Mathematics Books, Issue 43, February, 2003)
From the Back Cover
Discrete mathematics is quickly becoming one of the most important areas of mathematical research, with applications to cryptography, linear programming, coding theory and the theory of computing. This, mostly from the areas of combinatorics and graph theory, with a little number theory, probability, and combinatorial geometry. Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book.
László Lovász is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize and the Gödel Prize for the top paper in Computer Science. József Pelikán is Professor of Mathematics in the Department of Algebra and Number Theory at Eötvös Loránd University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.
Most Helpful Customer Reviews
I am really surprised at my fellow reviewer's statements indicating that you need to be a genius to understand this book. In fact, it is really the opposite; the authors took an effort to make the material approachable to the mathematically minded and provide motivating context for each example. While at the authors, you should note that these people are some of the most well known researchers in this area and Dr. Lovasz is also an exceptional lecturer. I believe all possess Erdos number 1 :) It is surely not a textbook, in the sense of Rosen's "Discrete Mathematics and its Applications" nor it strives for completeness like Reinhard Diestel's "Graph Theory". Instead it is a selection of topics that give a good introduction into discrete mathematics with carefully selected insightful problems with solution hints! So, yes, I think it is great for self study and especially for those (as the introduction suggests) who have had a more analysis-biased introduction into Mathematics. Instead of being a collection of theorems and proofs, the problems in this book build on the absolute necessary basics (often just high-school math) and, yes, skip unnecessary notation and pseudo-rigor. I should also note that I am basing this review on the Hungarian edition, which also reads well but I have not actually seen the original English text.
For my purposes, this textbook has no competitors. But first, let me explain my situation: I teach a 100-level college discrete math course. By "100-level," I mean something about as advanced as high school trigonometry/pre-calculus, with high-school-level algebra as the only prerequisite. Unlike many discrete math courses, mine is not primarily aimed at computer science majors -- they generally make up only about a third of the enrollment. As a whole, what my students need is to get a sense of what mathematics is like outside of the calculus sequence, and also a good introduction to reading and writing proofs. With all of that in mind, this is by far the best individual textbook I could use, to my knowledge (and I have looked over an absurd number of other discrete math texts). To be honest, sometimes I suspect I could write a better introductory discrete math textbook than this one, but I must be wrong, since apparently no one else can.
The best qualities of this textbook are its very broadly accessible style and, at the same time, the fact that it doesn't treat mathematics like a mere sequence of rules to be memorized and procedures to be "mastered." Unfortunately, that cookbook kind of presentation, followed by a mechanical regurgitation of pointless "skills," is what most of today's students seem to crave in math (witness the popularity of Khan Academy, for example). This textbook is one of those rare gems that puts mathematics in its proper light, as a field of real human curiosity, in some ways resembling an expressive art as much as a science.
One major problem with textbooks in this subject is that there are about a half dozen different versions of a "discrete math" course, some bearing almost no resemblance to others.Read more ›
Before the reader grumbles at my 1 star rating, let me please explain exactly what it is I am in fact rating. I bought this book with high hopes. Initially there were two front running candidates I had to choose from, this book and the very well known book of Norman Biggs published by Oxford University Press. The latter was more expensive so I took book of Lovasz. I chose poorly, very poorly. When this book arrived and I quickly noted that it was like many so-called "hard cover" Springer books, that is, it's not a hard cover book at all. It's a paperback book with a shabby cardboard "hard cover" glued onto it, which for some strange reason incurs a greater fee. However this is not the worst of it. The print quality was appalling. It's like a photocopied book (and most likely is a photocopy) where the photocopier's toner ran out three weeks ago. The physical quality of the book is just rubbish. This is not an isolated event, certainly not in my experience or my colleagues. A few months back I bought a copy of Bondy and Murthy's classic Springer GTM on Graph Theory. This is a large book, 600+ pages and a "hard cover", according to Springer. The spine of the book cracked as soon as I opened it and for the same reason. It was not properly bound and is effectively a paperback book with a cardboard cover glued on it. I took this book to a German book binder who pulled it apart and bound it properly. Now it "functions" as a book should. That was $50AUD extra cost. My advice to anyone buying a Springer "hard cover" book is to factor in $50 for a rebind, assuming of course that there was toner in the printer. If not even a rebind is a waste of time.
This books is a decent introduction to discrete mathematics. Lovasz does a good job of making material easier by putting it into words. This unfortunately comes at a cost though. For example, in the first few chapters about combinatorics Lovasz does a good job of distinguishing permutations from combinations. However, when he tries to present proofs in every day language the lack of mathematical preciseness can get really confusing. This unfortunately only gets worse as more topics get introduced. The section about fast modular exponentiation is very dense and requires careful reading to follow the math. I feel like these topics could've been presented better if Lovasz simply wrote out the equations and the manipulations. The sections on graph theory and convex geometry go a bit too fast. They start off quite easy and then ramp up rapidly at the end of the chapter which leaves the reader with more questions than answers. The section on RSA was surprisingly good and really brought Fermat's theorem to life, but I do wish that this was done nine chapters earlier.
So, I've complained a lot, and you may wonder why I've given the book four stars. The reason is that the book fulfilled it's purpose very well; it gave me a brief introduction to the many fields of discrete math without totally burying me. The tone and style was easy enough for me to read in my leisure time while still introducing to me some solid mathematical concepts. Most of the basic theorems were very clear (though the more advanced ones were typically presented poorly like I said). Exercises were generally easy and reinforced the topics in the chapter. One of my favorite things about the book was the number of open problems Lovasz explained. More authors should present these in order to stimulate the reader.Read more › | 677.169 | 1 |
one of the leading authors and researchers in the field, this comprehensive modern text offers a strong focus on enumeration, a vitally important area in introductory combinatorics crucial for further study in the field. Miklós Bóna's text fills the gap between introductory textbooks in discrete mathematics and advanced graduate textbooks in enumerative combinatorics, and is one of the very first intermediate-level books to focus on enumerative combinatorics. The text can be used for an advanced undergraduate course by thoroughly covering the chapters in Part I on basic enumeration and by selecting a few special topics, or for an introductory graduate course by concentrating on the main areas of enumeration discussed in Part II. The special topics of Part III make the book suitable for a reading course.
This text is part of the Walter Rudin Student Series in Advanced Mathematics.
Most Helpful Customer Reviews
I'm teaching an undergraduate course from the book covering the first three chapters. I like the material chosen by the author and how the book is organized. There are relatively few errors - I found two - and there are worked solutions to the exercises. My major complaint is that for an expensive hardcover, the make quality is very bad. The pages started coming apart after one month of use.
Besides sticker shock from the absurd price, all readers can look forward to is a watered down version of one of the author's earlier books. This incarnation appears to have been hastily written and lightly edited (if at all) -- the inevitable result: plenty of typos and even a few howlers. | 677.169 | 1 |
"Math Pro" will take you through high-school Math and beyond. It is a powerful tool that is overflowing with the tutorials, examples, and solvers from the following applications: Algebra Pro, Geometry Pro, Probability Pro, Statistics Pro, PreCalculus Pro, and Calculus Pro | 677.169 | 1 |
Product Description
This teacher's guide accompanies BJU Press' Pre-Algebra Grade 8 Student Text, 2nd Edition. Reduced student pages are included, and have teacher lesson notes in the margins. Algebraic expressions and linear equations are applied throughout a thorough review of operations on integers, fractions, decimals, percents, and radicals. Students explore relations and functions using equations, tables, and graphs, while chapters on statistics and geometry extend foundational concepts in preparation for high school courses.
This resource is also known as Bob Jones Pre-Algebra Grade 8 Teacher's Edition, 2nd Edition | 677.169 | 1 |
This book encompasses a wide range of mathematical concepts relating to regularly repeating surface decoration from basic principles of symmetry to more complex issues of graph theory, group theory and topology. It presents a comprehensive means of classifying and constructing patterns and tilings. The classification of designs is investigated and... | 677.169 | 1 |
The New Mathematical Library (NML) was begun in 1961 by the School Mathematics Study Group to make available to high school students short expository books on various topics not usually covered in the high school syllabus. In a decade the NML matured into a steadily growing series of some twenty titles of interest not only t o the originally intended audience, but to college students and teachers at al1 levels. Previously published by Random House and L. W. Singer, the NML became a publication series of the Mathematical Association of America (MAA) in 1975. Under the awpices of the MAA the NML will continue to grow and will remain dedicated to its original and expanded purposes.
Conventions. Published in Washington by the Mathematical Association of America Library of Congress Catalog Card Number : 6 1-1 21 85 Manufactured in the United States of America
sections containing thoroughly familiar material may be read very quickly. New York. he will have to make an intellectual effort. even within a single book. some of which may require considerable thought. The reader is urged to acquire the habit of reading with paper and pencil in hand. they vary in difficulty. If the reader has so far encountered rnathematics only in classroorn work. Y. N. NEW YORKUN~VERSITY. in this way mathematics will become increasingly meaningful to him. 10012.cannot be read quickly. Most of the volumes in the New Mathematical úibrary cover topics not usually included in the high school curriculum. Street. New Mathematical Library. Editor. while the reader needs little technical knowledge to understand most of these books. The authors and editorial committee are interested in reactions to the books in this series and hope that readers will write to: Anneli Lax. On the other hand. often an argument will be clarified by a subse quent remark. THECOURANT INSTITUTE OF WTHENATICAL251 Mercer SCIENCES.Note to the Reader
his book is one of a series written by professional rnathematicians in order to malte some important mathematical ideas interesting and understandable to a large audience of high school students and laymen. The best way to learn mathematics is to do mathematics. some parts require a greater degree of concentration than others. and. Thus. Nor must he expect to understand al1 parts of the book on first reading. He should feel free to skip complicated parts and return to them later. and each book includes problems. he should keep in mind that a book on mathemati. The Editors
T
.
The plan in this book is to present an easygoing discussion of simple continued fractions that can be understood by anyone who has a minimum of mathematical training. and then. Continued fractions were studied by the great mathematicians of the seventeenth and eighteenth centuries and are a subject of active investigation tc?day. and this attitude is reflected in the pages that follow. This chapter should be easy to read. how rational fractions can be expanded into continued fractions. but these accounts are condensed and rather difficult for the beginner. particularly into the nature of numbers. called "continued
fractions". that fractions of this form. Nearly al1 books on the theory of numbers include a chapter on continued fractions. Mathematicians often think of their subject as a creative art rather than a s a science.
. Chapter 1 shows how continued fractions might be discovered accidentally.Preface
At first glance nothing seems simpler or Iess significant than writing a number. however. more detailed tha n necessary . if anything. In Chapter 2 these results are applied to the solution of linear Diophantine equations. for example 3. in the form
It turns out. by means of examples. Gradually more general nohtion is introduced and prelimimry theorems are stated and proved. it is. provide much insight into many mathematical problems.
I a m also grateful to my wife who typed the original manuscript. 1901. he should plan to return to it later and tackle it once again until it is mastered. It goes without saying that one should not "read" a mathematics book. I n addition he should test his grasp of the subject by working the problems a t the end of the sections. refers to item 2 listed in the references. I n the text "Crystal [2]". Anneli Lax. who prepared the final ty pescript. Numbers:
Rational and írrational. C. Vinally. California. and to Mrs. and Appendix 1 is a collcction of mis1 cellaneous expansions designed to show how the subjcct has developed. Ruth Murray.D. Particular thanks are due to Dr. and to the Editorial Panel for suggestions which have irnproved the book. A student of mathematics should wrestle with every step of a proof .1 has no solution in integers. These are mostly of an elementary nature. there is a short list of refercnces. It is better to get out pencil and paper and rewrite the book. Here the famous theorem of Hurwitz is discussed. and should not present any difficulties. Chapter 5 is designed to give the reader a look into the future. and other theorems closely related to it are mentioned. The first of the two appendices &ves a proof that x2 . Olds Los Altos. but also for her critica1 reading of the text. many of these expansions are dificult to obtain. 1 wish t express my thanks to the School Mathematics Study o Group for including this book in the New Mathematical Library series. for example.
. Here one sees how continued fractions can be used to give better and better rational approximations to irrational numbers. The reader will find this chapter more challenging than the others. if he does not understand it in the first round. so freely given. These and later results are closely connected with and supplement similar ideas developed in Niven's book.
The periodic properties of continued fractions are discussed in Chapter 4. Their answers appear a t the end of the book. and to suggest further study of the subject. closely related to the text. The main part of the chapter develops a proof of Lagrange's theorem that the continued fraction expansion of every quadratic irrational is periodic after a cei-tain stage. but the end results are rewarding.3y2 = . not only for technical advice.4
PREFACE
Chapter 3 deals with the expansion of irrational numbers into infinite continued fractions. and includes an introductory discussion of the idea of limits. this fact is then used as the key to the solution of Pell's equation.
CHAPTER ONE
Expansion of Rational Fractions
1. This gives
+
Repeating this replacement of x by 3 obtains the expression
+ l/x several more times he
. narnely 3 l/x.1 Introduction
Imagine that an algebra student atternpts to solve the quadratic equation
as follows: He first divides through by x and writes the equation in the forrn
The unknown quantity x is still found on the right-hand side of this equation and hence can be replaced by its equal.
obhined by stopping at consecutive stages. is in agreernent to three decimal places with the last result above.303. what is meant when we say that the infinito dec&l 0. Then will the expression on the right of (1. First. if we calculate more and more convergents (1. he does not seem to be getting any closer to the solution of the equation (1.2) the nontermina ting expression
+
where the three dots stand for the words "and so on" and indicate that the successive fractione are continued without end. suppose we consider the process used to get (1.333 * is
+
.1).2).6
CONTINUED FRACTIONS
Since x continues to appear on the right-hand side of this "multipledecked" fraction.3)) will we continue to get better and better approximations to x = f (3 di3)? Second. when rounded to 3. These preliminary calculations suggest some interesting questions. The quadratic formula ehows that this root ie actually equal to
-
which. We see that it contains a succession of fractions. give in turn the nubere
It then comes as a very pleasant surpriee t o discover that these numbers (or convergents as we shall cal1 them later) give better and better approximations to the positive root of the given quadratic equation (1.4) actually be equal to +(3 fi) ? This reminds us of an infinite decimal. For example. These numbers.2) as being continued indefinitely.1). so that we have in place of (1. But let us look more closely at the right side of equation (1. when converted into fractions and then into decimals.
A study of these fractions and their many properties and applicatians forms one of the most intriguing chapters in mathematics. In this monograph. until we come to Chapter 3.RATIONAL FRACTIONS
7
equal to ? These and many other questions will eventually be discussed and answered. are positive integers. The first of these is the introduction of basic definitions. * 9 bi. we shall restrict our discussion to simple continued fractions. we shall further restrict the discussion to Jinite simple continued fractions. and the number of terms may be finite or infinite.
1. bz.2 Desnitions and Notation
+
An expression of the form
is mlled a continued fraction. In general. Multiple-decked fractions like (1.) a4. ba. These have the form
.a.2) and (1.a3. We must start with simpler things. and where the terms a2.. however. may be any real or complex numbers. the numbers al.4) are called continued fractions. however. In fact. These have the form
where the first term al is usually a positive or negative integer (but could be zero).a2.
a. continued fraction. From now on.7) is
signs after the first one are lowered to remind us of the where the "stepdown" process in forming a continued fraction. For exarnple. a. It is also convenient to denote the continued fraction (1. so that [al. .3 Expansion of Rational Fractions
A rational number is a fraction of the form p/q where p and q are integers with q # O.
. . unless the contrary is stated.a .8
CONTINUED FRACTIONS
with only a finite number of terms al. is
How did we get this result? First we divided 67 by 29 to obtain the quotient 2 and the remainder 9. so that
9
Note that on the right we have replaced gg by the reciprocal of
9.az. can be expressed as a Jinite simple continued fraction..8) by the symbol . Such a fraction is called a terminating continued fraction. or rational number. We shall prove in the next section that everg rational fraction. A much more convenient way of writing (1.as. az.. un.]. the words continued fraction will imply that we are dealing with a Jinite simple continued jrmtion.
are called the partial quotients of the
1..
+
The terms al. the continued fraction for 8.
.
RATIONAL FRACTIONS
Next we divided 29 by 9 to obtain
Finally, we divided 9 by 2 to obtain
at which stage the process terminates. Now substitute (1.12) intr (1.11), and then substitute (1.11) into (1.10) to get
We should notice that in equation (1.10) the number 2 29 is the largest multiple of 29 that is less than 67, and consequently the remainder (in this case the nurnber 9) is necessarily a number 2 0 but definitely <29. t Next consider equation (1.11). Here 3 9 is the largest rnultiple of 9 that is less than 29. The remainder, 2, is necessarily a number 2 0 but <9. In (1.12) the number 4 2 is the largest multiple of 2 that is less than 9 and the remainder is 1, a nurnber 2 0 but <2. Finally, we cannot go beyond equation (1.12), for if we write
then 2 1 is the largest multiple of 1 that divides 2 and we simply end up with
so the calculation terminates. t If a number a is l a s than a number b we write a < b. If a is less than or
equal to b we write q 5 b. Likewise, if a is greater than b, or if a is greater than or eqiial to b, we write, respectiveIy, a > b, a b. For a detailed discumioii of inequaIities, see E. Beckenbach and R. Bellman (11.
>
10
CONTINUED FRACTIONS
The process for finding the continued fraction expansion for can be arranged as follows: 20)67 (2 = al
We observe, in this exrtmple, that in the successive divisions the remainders 0, 2, 1 are exactly determined non-negative numbers each smaller than the corresponding divisor. Thus the remainder 9 is less than the divisor 29, the remainder 2 is less than the divisor 9, and so on. The remainder in each division becomes the divisor in the next division, so that the successive remainders become smaller and smaller non-negative integers. Thus the remainder zero must be reached eventually, and the process must end. Each remainder obtained in this process is a unique non-negative number. For example, can you divide 67 by 29, obtain the largest quotient 2, and end up with a remainder other than 0? This means that, for the given fraction +$, our process yields exactly one sequence of remaindew. As a second example, let us find the continued fraction expansion for gig. We obtain
RATIONAL F R A C T I O N S
Hence
Notice that in this example al = O. To check our results, al1 we have t do is simplify the continued fraction o
A comparison of the expansion = [2,3,4, 2 with the expan1 sion of the reciprocal # = [O, 2, 3,4, 21 suggests the result that, if p is greater than q and
then
The reader is asked to state a similar result for p
< q.
The following examples will help to answer some questions which may have occurred to the attentive student. Pirst , is the expansion
f the only expansion of +&as a simple finite continued fraction? I we go back and study the method by which the expansion was obtained, the answer would seem to be "yee". And this would be true except that a slight change can always be made in the last term, or last partial quotient, a4. Since a = 2, we can write 4
-1= - 1 - . =
a 4 Hence it is also true tbrtt
2
1
1 l+i
6. 1. aa are positive.
O
Thus
.3. For example. but as. 6.3.p / q . proceed as follows: (Search for a negative quotient which. a4. 3. aa]. f or (1. and then expand the resulting fraction. The third question is this: I we multiply the numerator and f denominator of by sume number. FVe shall see in the more general discussion which follows that this is the only way we can get a "different" expansíon. when multiplied by 44 and subtracted from . say 3. g$+. 21. 21 44
=
[-1. leaves the srnallest posz'title remainder.-l.1. az. ar. 6 -
a ? .4.1] can be changed back to its original form [2.[-1. let us consider how t obtain the expansion of a negative o rational number . 1 = [al.12
CONTINUED FRACTIONB
Clearly the expansion [2.14) 87)2oi(2 174 27)87 (3 81 6) 27 (4 24 3)6(2 .)
Thus
a . t o find the continued fraction expansion of . This requires only a slight variation of the process already explained. as. 1
Notice that al is negative. Next. a4.37. will the continued fraction for ZgiF be the same as that for We shall see that the expansions are identical.4.g .3.
al.
-
1.
(b)
$9.
4. Show that. Find p / q if
3. 21.
++.unJ. . and conversely.54 = 100
.
5.e.a2. 1..4 Expansion of Rational Fractions (General Discussion)
So far we have introduced the terminology peculiar t o the study of continued fractions and have worked with particular examples. . 2. . We always obtain a rational fraction p / q in its lowest terms. a.7.14159
2. Find p / q if p / q = [O. Problem Set 1
1. not to S#. Convert p / q to a decimal and compare with the value of u. a2.. 23 (e) -23 = 100 355 354 (c) 3. Find p / q if p / q = [3.]. a.
(g) 3. a fraction for which p and q have no factors greater than 1 i n common. Working with symbols instesd of with actual numbers frees
. If we calculated
we would get back to $g.15. al. 4. Can you discover a t this stage a reason for this? Later an explanation will be given. az.. Convert each of the folIowing into finite simple continued fractions.a. if q / p = [O. then q / p = [O. 11. Find the simple continued fraction expansions of (a) compare these with the expansions in Problem 1 (a). i. But to make real progress in our study we must discuss more general resulte. if p > q and p / q = [al. . (b).
6.RATIONAL FRACTIONS
13
This illustrates an interesting property of continued fractions. a*.].] then p / q = [al. .
dividing q by rl to obtain
Notice now that q/rl is a positive fraction. be any rational fraction. If rl f O. Thus.14
CONTINUED FRACTIONS
the mind and allows us to think abstractly. We divide p by q to obtain
where al is the unique integer so chosen as to make the remainder rl greater than or equal to O and less t h n q. f the process krminaks and the continued fraction expansion for P / Q is [all. I r2 = 0 . liero. let p/q. with the exceptions to be noted below. first sentence in this theorem is quite clear from what The we have explained in our worked examples. I rl = 0 .
PROOF.15) to obtain
P 9
=
al
1 + .= [al. a number bet'ween O and rl.16) into (1. any rational number p/q can be represented as a jinite simple continued fraction. THEOREM Any jinite simple continued fraction represents a rational number. once this has been accomplished a host of other ideas quickly follows. Conversely. the process stops and we substitute f q/rl = a2 from (1. we write
and repeat the division process. for if any expansion terminates we can always "back track" and build the expansion into a rational fraction.
1. or positive.
. az] a2
as the continued fraction expansion for p / q . so a2 is the unique largest positive integer that b k e s the remainder r. q > O. while our first theorem merely expresses in general terms what we did in the worked examples. al can be negative.1. t h representation. or expansion. i s unique. As we saw in the worked examples. To prove the converse.
1s it possible never to arrive a t an rn which is zero.16) in the form
and repeat the division process using rl/r2. which is equal to zero. with the equation in which the reminder rn is equal to zero. Hence. we shall be in the ridiculous position of having discovered an infinite number of distinct positive integers al1 less than a finite positive integer q. after a certain finite number of divisions. From the first two equations in (1. for the remainders rl. r2. sequence of non-negative integers q > rl > r2 > r3 > . by successive divisions we obtain a sequence of equations:
terminating. we write (1.RATIONAL FRACTIONS
If
r2
# O. TB.. and unless we come eventually to a remainder r. so that the division process continues indefinitely? This is clearly form a decreasing impossible.18) we have
. We observe that the calculations stop when we come to a remainder rn = O. It is now easy to represent p / q as a finite simple continued fraction.
1)
+1 1
-
so that (1.16
CONTINUED FRACTIONS
Using the third equation in (1. a. . This statement must be accompanied. as we choose. so that the nurnber of terms in the expansion is either euen or odd.19) can be replaced by
On the other hand. by the remark that once the expansion has been obtained we can always'modify the tast terrn a.18) we replace r1/r2 by
and so on. however.
1 .2.]. To see this. +a. if a.19) becomes
Hence we have the following theorem:
THEOREM Any rational nunzber p / q can be expressed as a 1. then
so that (1. .
=
1.
It is interesting to notice that the equations (1.az.19)
Q
=al+z
1
1 +G
+
.
.
The uniqueness of the expansion (1. Jinite sinzple continued fraction in which the East termican be mod$ed so as to make the nulnber of terms in the expansion eather even or odd.18) are precisely the equations used in a procedure known as Euclisl's algorithm for
. is greater than 1 we can write a. notice that if a.
(a.= fal. a3)
. until finally we obtain the expansion
(1.19) follows from the manner in which the si's are calculated.
.
.) of any two integers p and q is the largest integer which divides both p and q. any integer d which divides both b and c will also divide a.. similarly for the other equations. then b = dbi. The number d is the g. In the theory of numbers the g. ( p . . Likewise. theh any integer d which divides both a and b must divide c. of p and q.d. . We need only one more observation: I a.d. Since a . since (a) d = 3 . of the integere p and q is denoted by the symbol ( p . .18) by multiplying both sides by the denorninator q.c...-1 is the g. b. 5 divides both p and q. we see that so that d divides c.d.13. and c are integers f such that a = b+c..RATIONAL FRACTIONS
17
finding the greakst common divisor of the integers p and q. we first state the two conditions that the g. We shall prove that the last nonvanishing remainder r.18) in the form:
. .c...
+
t The greatest commn divisor (g..
m . and if d divides b.. For example. .. let p = 3 5 11 and let q = 32 5 . p = alq rl..b = c.C.c... . J. In order to do this. and (b) the common divisors 3 and 5 of p and q divide d . however it is known to be of earlier origin. [This procedure occurs in the seventh book of Euclid's Elemnts (about 300 B.
The first equation. of two integers must satisfy. bl an integer.] To find the greatest common divisor of p and q by means of Euclid's algorithm.c. .. q). of p and q is d = 3 5. we write the equations (1.). is obtained from the first equation in (1. of two integers p and q if (a) d divides both integers p and q. For if d divides a. and (b) any common divisor c of p and q divides d. Then the g. ..d..d. . thus. q) = d means that d is the largest integral factor common to both p and q. then a = d& where al is an integer.c.d.c..
Next we must show that if c is any common divisor of both p and q. shows that rn-1 divides.22) shows that c divides rt. of p = 6381 and q = 5163. I c divides both p f and q. then c divides 1 .-~ divides both p and q. let us use Euclid's algorithm to determine the g.d.2 .s and rn.a. and hence divides rl. and we conclude that r. and condition (a) is satisfied. the first equation in (1. and finally.22). of P q. we find that rn-1 divides ra and r2.
shows that rn-1 divides rn-3. Hence.22) and work our way down. Dividing r2 and rl. it divides p.-1 is the g.c. the second equation in (1. and hence divides rn-l. Thus condition (b) is satisfied. This time we start with the first equation in (1.18
C O N T I N U E D FRACTIONS
We now return to the equations (1. We find that
. In the same way. it dividea q .c. or is a factor of.-a and r. since it divides both rn. we arrive at the next to in the last equation. namely The equation
. r.-2. Working up from the bottom in thii fashion. since it divides rn-1 and r n 4 . from the equation we see that rn-1 divides rn-r. dividing both rl and q. in which c divides r. As an example.22) shows that c divides rl. The last equation there.d. But if c divides both q and rl. Contin~iing this manner. directly above it.
where 709 is a prime number.c. of 6381 and 5163. third.d. in succesaion. but before we can put them to effective use we must study some of their properties in greater detail. and hence is the g.1449 (b) 1517. Expand the following rational fractions into finite simple continued fractions with an even number of terms and also with an odd number of terms:
2.2015
( e ) 2299.) Thus 3 is the o d y factor common to these two numbrs.3800
(d) 3528. and 5163 = 3 1721 where 1721 is alm a prime number.d. a. are positive integers. .c. From'these we can form the fractions
obfained.1 of s the following pairs of numbers: (a) 1380. by cutting off the expansion procem after the firat. az. Actually.6 Convergents and Their Properties
Continued fractions are of great service in solving many interesting problems. These fractions are called the
m
.RATIONAL FRACTIONS
19
hence 3 is the g. From now on we will cal1 the a2. 6381 = 32.c. and where .d. the partid quotients or quotients of the numbers al.7455
1. aa. a. steps. In Section 1.
Problem Set 2
1. U e Euclid's algorithm to find the greatest common divisor (g.709.4 we saw that any rational fraction p/q could be expanded into a finite simple contiilued fraction
where al is a positive or negative integer. aecond. (A prime number is a number with precisely two positive integral divisors: 1 and the number itself. or zero. continued fraction.
Next we write
where p2
=
aia2
+1
and
q2
=
a2. then
and so on.23). The nth convergent. atl. ql = 1. It is important to develop a systematic way of computing ihese convergents. convergents. We write
where pl
=
al.].
it equal to the continued fraction itself.a. We notice that
so that
Again.
cn
= al
+ -r + a
1
.20
C O N T I N U E D FRACTIONS
first. second. Now let us take a closer Iook at the convergent c3.-+a. of the continued fraction (1. respectively.
1
.
. third. that
so that
. by factoring.= [al. from c4 we observe.
.RATIONAL FRACTIONS
From (1. n. of course. .
where
That the equations (1.25) we might guess that if
then
and that in general. .] satisfy the equations
with the initial vaEues
PROOF. although convinced of their correctness. but it is a genuine example of inductive thinking.28) we get
+
. 5.24) and (1.27) are true for i = 3 .
.3. U. . This. . 4 .n. a2. We guess the formulas from the first few calculations. ith convergent ci of the continued jraction [ a l . have seen already that cl = pl/ql = a l / l and that We c2 = p2/q2 = (azul l)/a2.. would not give us proof that the equations (1. I we substitute i = 3 in equations f (1. .4. Thus we state and then prove by induction the followirig theorem : THEOREM The nurnerators pi and the denorninators qí of the 1. then. for i
=
3. 5 .26)are correct can be confirmed by a direct calculation. we must still supply a forma1 proof. .
3 necessarily holds for the next 1. In equation (1. pk-1. that is. up to some integer k.30) with j replaced by k.4. To see this. first replace j by k .5.30). that is. simply compare
+
with
This suggests that we should be able to calculate c k + l from the formula for c k obtained from (1. that for the integers 3. qk-1 did not change their values when we tamper with ak.3 is true.1.2. Let us assume that Theorem 1. On the basis of this assumption.30) to help us supply a integer k proof that
+
The next few steps will require concentration. from
This we could certainly do if we were sure that the numbers pk-2.22
CONTINUED FRACTIONS
again in agreement with the direct calculation of Ca. k.1. qk-2. we wish to prove that Theorem 1.4. We obtain in succession :
and
. let us look at the manner in which they are calcillated. Notice first that ck+l differs from c k only in having (ak l/ak+l) in place of ak. 9 k .To do this we use equations (1. and then by k . or has been verified by direct calculation. To see that they do not.
m
for j = 3.5.
6. that if the expression for the convergent Cj.1 quotients al. a2. 5.30). This. .In (1. then it also holds for the next convergent c k + ~ = pk+l/qk+l.4. given by (1. Le.30) holds for j = k = 3. as we have explained. qk-1 depend only upon the number ak-1 and the numbers pk-2.32) replace a by k (ak l/ak+l) to obtain. But we actualiy know by a direct calculation that (1. we obtain
and rearranging the terms.
. . We are now ready to calculate ck+~. holds for the values j = 3. Ic. and q's.. thrtt akpk-1 pk-2 = pk. then. ak-1 and hence are independent of ak. Thus the numbers pk-2. This means that they wili not change when ak is replaced by (ak l/ak+l). 7. respectively. Qk-1 depend only upon the first k . multiplying numerator and denominator by ak+l. pk-1. . qk-2.30) hold for j = Ic. qk-3.
+
Hence the terms in parentheses in the numerator and denominator of our last expression for ck+l can be replaced.RATIONAL FRACTIONS
23
We notice that the numbers pk-1.
+
In studying this proof. al1 of which in turn depend upon preceding a's. we obtain
We have proved.proves Theorem 1. Hence it is true for the next integer Ic 1 = 4. . t n.3. . qk-2. p's. we get
At this point we use the assumption that formulas (1. Thus. notice that nowhere h v e we used the fact that the quoticnts ai are integers.
+
+
Now. by p k and q k . pk-3. and likewise for Ic = 5. Although each ai is an integer.
28)) and using (1.28) we get the undeJined terms po. and the first two values.28) could also reproduce the first two convergents given by (1. n.33))we get
for i
=
2 we get
Hence.Setting i equal to 1 in (1. I we put i = 1 . the assigned values (1.28) will hold for i = 1 . . Nevertheless its substitution for ak in the proof causes no breakdown of the argument. 2 . However. The calculation of successive convergents can now be systematized. 2. The continued fraction expansion for '.33) enable us to dispense with equations (1. qo. n.24
CONTINUED FRACTIONS
the number ak l/ak need not be one. q-1. will reproduce equations ( 1 -29). . 2 . p-1.29).1 . if we asxign the values
+
to these undefined terms. i = 1 .n . But notice that p-l/q-l and po/qo are not convergents. . I t would be convenient if the equations (1. 2 in f (1. 3.Ts" is
We form the following table :
TABLE 1
.29) and to use instead equations (1.28)) with i = 1 . then equations (1. An example will make this clear.
The special values p-1 = O. Thus. p = 49. First. for example. Then we calculate the pi's. . 0. q-l = 1.28). from equations (1. Express each of the following continued frsctions in an equivslent form but with s n odd number of psrtisl quotients. Cr = H. under i Problem Set 3
=
4. o under i = . 2t-5 . i = 0. Thus. For i = 3. ci have been listed. under the values of i to which they correspond. qo = O are entered a t the left. (a) (b) % o (c) M (d)
2. using i = 1. p = 1. pi. 2.)
2 under i
=
1 in the third row. Espsnd the following rationsI numbers into simple continued frsctions snd calculate the successive convergents ci for esch number. qr = 20.
Note: Starred problems are more difficult snd could be omitted the first time over. respectively. For i = 2. and so on.'s we follow the same scheme. entering the values we obtain in the row labeled qi. 4 We form our table in this way: We write the values a in the seci ond row. under i = 4 we find a4 = 2. p3 a3p2 pi = 4 5 2 = 22.1. we
2 which is recorded under i = 2 in the same row. 1. qi. we get
i =
-
(Follow the first system of arrows We record pl obtain
=
O+-1
# '
.
. To calculate the q.RATIONAL FRACTIONS
25
Explanation of tabb: The entries In the first row of the table are the values of i: i 1. Under each value of i the corresponding values of ai.
+
+
so 20 is recorded in the fourth row. 1.
26
C O N T I N U E D FRACTIONS
3.)
+ + + + +
5.3.t.
+
.n
.~ q . 51. 1. hneralize Problem 4. .4.l+
p+2.
*8.~ . (See Problem 7.p.4771 (d) 0. take pn/qs = pa/q6.2. (&e Problem 8 below. 1. 3.
(a) 3.) 6. Do the same with qa/qr. and add the multing expressions. . For each continued fraction in Problem 2. 4. al]. Note that a.6] and show that p6 = 5p6 5p4 4pa 3p2 2pl 2. .a. for example. show that
.n].
hence
and so on.2. the last convergent. 4. Calculate the convergents of the continued fraction [1. let i be equal to 1.-lp. -.718
(T)
(e)
Pn .a. .2. For 13. We know that p..
and
Rints. Calculate the successive convergents to the following approximations to the numbers in parentheses. of [ l . . let n be the numbe* oi par tia1 quotients and calculate pnqne1 . In 2 (a).
-. p2/q2. calculate p5 and p4. If pllql.[as.
hence
We also know that
p-1
= u.3.. then calculate the corresponding quantity after these fractions have been expressed with an odd number of partial quotients. 4.5.a
+ p.
Pn-I
(c) 0. Then convert p6/p4 into a simple continued fraction and compare it with the original fraction. = n.-~. G-2. ps/qn
are the convergents
Hint: In the relation pi = ipi-1 pi-2.3010
(Ioglo 3) (10glo 2)
S
S
S
.14159 (b) 2..
34).poql
= al O
-1
1 = (-1)l.
+
+
.3. i = k i equations (1. we see that
But this is the statement of the theorem for i = Ic 1. so we have proved t h t the theorern holda for i = k 1 if z't holds for i = h.28)] we know that for i = i 1.3 [see holds for the next integer.
.pk-lqk) *
We assume t h a t the theorem holds for i = k. When i
when i = 1 .
+
+
hence we can write
pk+lqk (1.1)')
where i
> 0.
Plqo
= 0.
PROOF: Direct calculations show that the theorem is true for
i = 0.
THEOREM If pi = aipi-1 1.4. This is a corollary to the following fundamental theorem.pkqk+l
= =
+ pk-1)qk ~k(ak+lqk+ qk-1) ak+lpkqk + pk-lqk .
when i
=
2. then it also 1.6 DBerences of Convergents
Those who worked the preceding exercises will already have guessed that the convergents to a finite simple continued fraction are always in their lowest terms.34)
. &$ned as in Theorern 1.pkqk-1
(ak+lpk
"
= (-
1 ) (pkqk-1
. 1 .
We shall prove that i f the theorem holds for i = Ic.RATIONAL FRACTIONS
1. that
Substituting this result into the last line in (1.pi-iqi
= (. From Theorem 1.ak+lpkqk . 2 . that is. then
piqi-1
+ pi-2
and qi = aiqi-i
+ qi-a
are
.
.pi-2qi-1 is the same as piqi-1 .c. and therefore for i r= 1 1 = 2. we obtain the final result.he symbol d = (a.
. Hence this reduction. 1. p.pi-iqi but with i replaced by i 1.2.
2. PROOF.piq2.8 . can be repeated.d. p2q1 . and so on for al1 values of i = 0. 1 . 51 by calculating in turn poq-1 .pi-tqi-i = (-1) ( p i .pi-tqi-1).1. Since
+
piqi-i
. 2 . n .z q i . performed in succession. continued fraction i s in its lowest terrns. of a simple 1.1 are the only common divisors of pi and qi.1)" But the only divisors of ( .1.
The expression pi-lqi-r .pi-tqi-2) After i reductions of the sarne sort. p 2 / q t r rational fraction in its lowest terrns. q i ) = 1. of a and b. The earliest traces of the idea of a continued fraction are somewhat confused. we used t.l)i are 1 and . and qi have no cornrnon divisors other than 1 or .
+
+
Problem Set 4
1.* . i 2 1. but there was no systematic development of the subject. that i s . Notice that
=
(-1)
(pi-iqi-2
. etc. yielding pi-iqi-2 .
it follows that any number which divides both p i and q i must be a divisor of ( . ~ 6 / q 6 is Also verify that each convergent p l / q l .pi-iqi
= (-l)i.I ~ O . for many ancient arithmetical results are suggestive of thesg fractions. . I n oiir discussion of Euclid's algorithm. hence the numbers 1 and .4 using the continued fraction [3.b) to indicate that d was the g. plqo .4 using the following hints. 1. since 1 is the largest number that divides both pi and qi. we can now state that ( p i .5.p . hence i t holds for i = O+ 1 = 1. 2.
+
COROLLARY Everg convergent ci = p i / q i .p0q1. or "stepdown" from i to i 1. Give another proof of Theorern 1. Check Theorern 1.28
CONTXNUED FRACTIONS
We know the theorem holds for i =O.
-
-
We end this chapter with a few brief remarks concerning the history of the theory of continued fractions.
A referente to continued fractions is found in the works of the Indian mathematician Aryabhata. of 177 and 233. he expressed in the form
0
This he modified.c. Further traces of the general concept of a continued fraction are fouiid occasionally in Arab and Greek writings. essentially. In our modern symbolism he showed. +#. who died around 550 A. Bis work contains one of the earliest attempts a t the general solution of a linear indeterminate equation (see next chapter) by the use of continued fractions. This is perhaps the earliest (c. Germany. into the form
which is substantially the modern form
A third early writer who deserves mention is Daniel Schwenter (1585-1 636)) who was a t vario us times professor of Hebrew. of two numbers is essentially that of converting a fraction into a continued fraction.D. 1530)' a native of Bologna. also a native of Bologna. +. and mathematics a t the University of Altdorf. +.d. and +. for example. that
This indicates that he knew. Oriental languages. His treatise on algebra (1572) contains a chapter on square roots.c. and from these calculations he determined the convergents *. 300 B. In his book Geornetrica Practica he found approximations to 8 8 by finding the g. for convenience in printing.d.
. Most authorities agree that the modern theory of continued fractions began with the writings of Rafael Bombelli (born c.) important step in the development of the concept of a continued fraction.C. In a treatise on the theory of roots (1613). that
The next writer to consider these fract'ions was Pietro Antonio Cataldi (1548-1626).RATIONAL FRACTIONS
29
We have already seen that Euclid's method for finding the g.
hgrange (1736-1813). New York: McGraw-Hill Book Company. laid the foundation for the modern theory. Wdlis stated a good many of the elementary properties of the convergents to general continued fractions. continued fractions are used to give approximations to various complicated functions. 1956 (Chapter 9). From this beginning great mathematicians such as Euler (17071783). Hildebrsnd. This is described in his treatise Descriptio Autornati Planetarii.
t See F. They constitute a most important tool for new discoveries in the theory of numbers and in the field of Diophantine approximatioiis. an extensive area for present and future research. Euler's great memoir. and once coded for the electronic machines. Christiaan Huygens (1629-1695) used continued fractions for the purpose of approximating the correct design for the toothed wheels of a planetarium (1698).t. astronomer. In the computes field. De Fractionibus Continius (1737). published posthumously in 1698. Continued fractions play an important role in present d a y mathematics. The great Dutch mathematician. He also used for the first time the name "continued fraction". and physicist. In particular. Lambert (1728-1777).30
CONTINUED FRACTIONS
The next writer of prominence to use continued fractions was Lord Brouncker (1620-1684). Introductwn io Numerical Anula(t?is. into the continyed fraction
but made no further use of these fractions.
. He transformed the interesting infinite product
discovered by Che English mathematician John Wallis (1655). mechanician. give rapid numerica1 results valuable to scientists and to those working in applied mathematical fields. In the discussion of Brouncker's fraction in his book Arithrnetica Infinitorum. including the rule for their formation. and many others developed the theory as we kiiow it today. B. There is the important generalization of coiitinued fractions called the analytic theory of continued fractions. published in 1655. the first President of the Roya1 Society.
we can f give x any value. riddles. we restrict the values of x and y to be integers. and a number of pigs at $50 each. we have the equation
which is equivalent to
I nothing limits the values of x and y in equation (2. and trick questions lead to mathematical equations whose solutions must be integers. If. say x = 9. this sense.21. (2. How many cows and how many pigs did he buy? If x is the number of cows and y the number of pigs. as the farmer is likely to do (since he is probably not interested in half a cow). His bill was $810.CHAPTER TWO
Diophantine Equations
2. which means that we can always find some value of y corresponding to any value we choose for x. getting y = 9.2) is a n indeterminate equation. and then solve the resulting equation
In for y. Here is a typical example: A farmer bought a number of cows a t $80 each. however.1 Introduction
A great many puzzles. then our example belongs to an extensive class of problems
.
The second method will show how the theory of continued fractions can be applied to solve such equations. it should be noted.1) can be solved in many ways. . we solve the equation for y.82 a non-negative multiple of 5. 3. if we write equation (2. Indeterminate equations ta be solved in integers (and sometimes in rat'ional numbers) are ofterl cdled Diophantine equations in honor of Diophantus.32
CONTINUED FRACTIONS
requiring that we search for integral solutions x and y of indeterminate equations. take on the values 0. I n f m t there is no harrn in solving such equations b y trial and error or b y rnakiq intelligent guesses. who wrote a book about such equations. So the farmer could buy 2 cows and 13 pigs.D. The calculations are
hence the two solutions to our problem are (3. We shall give two additional methods. getting
t For additional examples. see 0.5). Equation (2. 13) and (x..2) in the form 81 8 2 = 5y. in turn. There are other ways of solving Diophantine equations. . . Our problem. [lo].
-
we need only search for positive integral values of x such that 81 . a Greek mathematician of about the third century A.2) and hence equation (2. Ore
. y) = (2. Icor example. we find that x = 2 and x = 7 are the only non-negative values which make 81 .2 The Method Used Extensively by Eulert
Let us consider again the equation Since y has the smaller coefficient.
2. 10.8x is a mitltiple of 5 . y) = (7. The first of these was used extensively by Euler in his popular text Algebra. 1. published in 1770. Letting x.2. or 7 cows and 5 pigs. has the further restridion that both x and y must not only be integers but must be positive.
DIOPHANTINE EQUATIONS
Both 81 and 8 contain multiples of 5, that is, 81=5.16+1 therefore, from (2.4), we have and 8=5*1+3;
where
Since x and y must be integers, we conclude from equation (2.5) that t must be a n integer. Our .task, therefore, is to find integers x and t satisfying equation (2.6). This is the essential idea in Euler's method, i.e., to show that integral solutions of the given equation are in turn connected with integral solutions of similar equations with smaller coefficients. We now reduce this last equation to a simpler one exactly as we reduced (2.3) to (2.6). Solving (2.6) for x, the term with the smaller coefficient, we get
where
Again, since x and t must be integers, u must also be an integer.
34
CONTINUED FRACTIONS
Conversely, if u is an integer, equation (2.8) shows that is an integer; x also is an iriteger since, from (2.7)) Substituting x 2
=
- 5u and
t = 3u
-
1 in (2.5) gives
so that y is an integer. This shoivs that the general integral solution of (2.3) is
A direct substitution into (2.3) shows indeed that
Consequently (2.3) has an infinite number of solutions, one for each integral value of u. A few solutions are listed below:
I the problern is such that we are limited to positive values of x f and y, then two inequalities must be solved. For example, if in (2.9) both x and y are to be positive, we must solve the two inequalities
2
- 5u > 0,
2 u<-? 5
13
+ 8u > O,
u >
for u. These inequalities require that u be an integer such that and
13 -8
7
and a glance at Figure I shows that the only two possible integral values of u are O and -1. Substituting, in turn, u = O and u = 1 in (2.9) gives (x,y) = (2,13) and (x,y) = (7,5), the original answers to the farmer's problem.
-
D I O P H A N T I N E EQUATIONS
35
Figure 1 Going back over the solution of equation (2.3) we can raise certain questions. For example, why should we solve for y, rather than for x, simply because y has the smaller coefficient? If we had solved first for x, could we have arrived a t a shorter solution? In the second line below equation (2.4) we replaced 8 by 5 I 3. Why not replace 8 by 5 - 2 - 2 ' In solving equation (2.3) the writer did not ! have in rnind the presentation of the shortest solution. We leave i t to the reader to experiment and try to obtain general solutions in the least number of steps.
+
Problem Set 5
1. Use Euler's method to solve the following linear Diophantine equations.
2. Does the indeterminate equation 62 15y = 17 have integral solutions? Note that the left side of the equation is divisible by 3. What about the right-hand side? What happens if we go ahead and use Euler's method anyway? 3. Return to equation (2.9)and fill out the following table for the values of u indicated.
+
On ordinary graph paper plot the points (x,y) and join them by a straight line. Use this graph to pick out the positive solutions of the equation 8x 5y = 81.
+
4. A man buys horses and cows for a total amount of $2370. If one horse costs $37 and one cow $22, how many horses and cows does he buy?
7. Find a number N which leaves a remainder 2 when divided by 20 and a remainder 12 when divided by 30.Hint: Find integers x and y so that the required number N = 20x 2 = 30y 12. Hence solve the equation 20x - 30y = 10.
+
+
2.3 The Indeteminate Equation ax
- by
=
f1
We are now ready to show how continued fractions can be used to by = c where a, b, solve the linear indeterminate equation ax and c are given integers, and where x and y are the unknown integers. Our approach to this will be a step-by-step process, through easy stages, culminating in the final mastery of the solution of any solvabk equation of the form ax by = c. We start with the restrictions that the coefficients of x and y are of different signs and that they ha,ve no common divisor but 1. Thus we first learn to solve the equation
+
+
where a and b are positive integers. [The equation - a x by = 1 , ( a , b) = 1, is of the same form with the roles of x and y interchanged.] The integers a and b can have no ditrisors greater than 1 in common; for, if an integer d divides both a and b, it also divides the integer 1 on the right-hand side of the equation and hence can have only the value d = 1. In other words, a and b must be relatively prime, or d = ( a , b) = 1. We shall now state and prove
+
THEOREM The equation ax - by = 1, where a and b are 2.1. relatively prime positive integers, has un injinite number of integral aolutions (x,y).
We first convert a / b into a finite simple continued fraction
a - = [al, a2, b
- , a,-1,
CI,
a,],
and calculate the convergents
c2,
,cn-1, cn.
The last two
D I O P H A N T I N E EQUATIONS
con vergent S,
are the key to the solution, for they satisfy the relation stated in Theorem 1.4, namely that and since p,
=
a, q, = b, this gives
If n is even, that is if we have a n even number of partial quotients al, a2, , a,, then (- l)n = 1 and (2.12) berornes
Comparing this with the given equation we see that a solution to this equation is
This, however, is a particular solution and ilot the general solution We indicate particular solutions by the ilotation (xa,ya). On the other hand, if n is odd so that (-1)" = -1, we can modify the continued fraction expai~sion(2.11) by replacing
1 un or by replacing
by
1
(a. - 1)
+
1 1
if a,
> 1,
Thus, if (2.11) has an odd humber of partial quotients, i t may be transformed iiito [al, a2, or into [a1,a2,...,a,-~+l], if a n = l ;
, a - 1, 1
if an > 1,
38
CONTINUED FRACTIONS
in both cases the number of partial quotients is even. Using these continued fractions, one case or the other, we re-calculate p,- Jqn-r and p,/q, = a/b, and equation (2.13) is satisfied once more. Once a particular solution (xo, yo) of equation (2.10) has been found, i t is an easy matter to find the general solution. To this end, let (x, y) be any other solution of (2.10). Then and and a subtraction @ves (2.14) a(z
- xo)
=
b(y
- yo).
This shows that b divides the left side of the equation. But b cannot divide a since a and b are relatively prime; hence b must divide x - xo, that is, x - xo is an integral multiple of b, and we may write x-xo=tb (taninteger),
x
=
xo
+ tb.
But if this is true, (2.14) shows that a(tb) so that y form
=
b(y - yo),
- yo = at.
= 1
It follo\vs that any other solution (x,y) of ax - by
has the
Conversely, if (xo, yo) is any particular solution of ax - by = 1, and if we set up the equations (2.15) with t any integer whatever, then the values (x, y) will mtisfy the given equation, because (axo - byo)
=
+ tab - tab
We cal1 the values of x and given by equations (2.15) the general solution of the indeterminate equation ax - by = 1.
Here n = 6 . 21 has an odd SOLUTION. We convert a/b into a finite simple continued fraction with a n odd number of convergents.8.
Here the integers 205 = 5 . 4 1 and 93 = 3 31 are relatively prime.by = 205x .DIOPHANTINE EQUATIONS
39
EXAMPLE Find integral solutions of the indeterminate equation 1.12) becomes
.
= [2. I n this case equation (2. q .4. As a general check we have
since the terms involving t cancel.93y = 1 is
As a check. but it can be replaced by
s5
the equivalent expansion with an even number of quotients.29109 = 1. let t = 1 . y = 313 and 205(142) .15)) the general solution of the equation ax . . 1. The convergents are then computed.10). so the equation has solutions.
andhence. The continued fraction number of partial quotients. p n _ ~ = p 6 = 1 0 8 = y o .93(313) = 29110 .93y
=
1. then x = 142. The method for solving the equation
is quite similar to t h a t used to solve (2.l = q 6 = 4 9 = x o i by (2.
205x .
numbers 205 and 93 are relatively prime. hence the given The equation has integral solutions. .
SOLUTION. f2. Comparing this equation with
ax
we see that
XO
. f l . The continued fraction espansion for is
and has an odd number of partial quotients. . The general solution. take t
=
. so ( .1. then
(x.40
CONTINUED FRACTIONS
since n is odd. therefore.y) = ( -49. as before.
EXAMPLE Find integral solutions of the equation 2.by
=
-1.1) =
-1 a s
Our calculations show that en-r = pn-i/qn-i = p4/44 %.* . x
=
$0
+ tb
y = yo
+ ta
t = O . heme a particular solution of the given equation is xo = q4 = 44 and yo = p4 = 97. -3.
=
Qn-1)
Y O 'p n .
.
and
. the general solution being.1
is a particular solution of the @ven equation.108). To find the convergents we set up the table
=
(.1)" required. is
As a check.
& 2 . 1 3 ) we know that aq.
=
(q.p.
which a e with the aolution given for Example 2. = (49. 1 6 ) will be
yl=a-yo=afor then axl
Pn-1.
--•
9
EXAMPLE Show that we can solve Example 2 if we have already solved 3.bpnWl= 1 .qn-1)
.1 will then be (2.
=
a ( b .1 .18). 108) is a particular solution of 205x . P . Hence the general solution.DIOPHANTINE EQUATIONS
41
It is interesting to notice t h a t once we have calculated the
yo) particular solution ($0. That is.93y = +l. of y the equation
The particular solution of ( 2 . f
is according to (2.18) x = xl+tb 2/=y1+h and this can be checked by a direct substitution. k 3 . l ) .b(a .93y = . ms
. Using equations (2.-1. knowing that (xo. The general solution of the equation ax . cal1 it (si. yo)
SOLUTION.-1 .
.~ ) of the equation
we can immediately obtain a particular solution. f l .17)we find that
is a particuIar solution o 205x . . Example 1.by.-1)
since from ( 2 .by = .93y = .
t
=
O.1 . solve the equation 205x .
For example.yo) is any particular solution of (2. then
.1 7 y = . (a) 132 .1 (c) 652 .1 (e) 56x
. Check f each answer. Problem Set 6 1. Since (xo. .93y = .) and the general solution becomes
Notice that equations (2.20) for t = 3. = (-49. Find the general integral solutions o the following equations.20) reproduce the same values of x and y but not for the same values of t.
SOLUTION.17y = 1 (b) 1 3 2 .19) gives (x.56y = 1 (d) 6 5 2 .1.93y = . we know that
+
If we multiply through by
.5 6 y = = .21).507).21)
m-by=
where a and b are two relatively prime positive integers.y) = (230.
(a. the same values obtained from (2.
= c.4 The General Solution o ax f
.1 we see that
hence (xi. = (49. provided we know a particular solution of Example 1.
EXAMPLE Give a third solution of the equation 2052 4.42
C O N T I N U E D FRACTIONS
There is still another way to solve Example 2.108) is a particular solution of the equayo) tion 205x .
.1.19)and (2.by
1. suppose that (xo.108) is a particuIar solution of 2052 . This is illustrated in the following example. For.65y = 1
=1
2. y.933 = 1. t = 2 in (2.b)
Once we have learned t o solve the indeterminate equation (2. i t is a simple matter to soIve the equation where c is any integer.
93(745) = 69290
.
Multiplying both sides by 5 we get so that (5x0. The general solution. yo) = (49. cyo) is a particular solution of (2. 108) is a particular solution of theequation 205x .93y = 1. -540)
is a particular solution of the given
. Example 1 of this section we recalled that
205(49) .93(108) = 1.22). we know that (xo.22). T h u s the general solution of equation (2. take t
=
1 .
In SOLUTXON.
205(49) .
EXAMPLE Solve the equation 2. then (x. we obtain
so that (cxo.. I 2 .22) w i l be
This can easily be verified by a direct substitution into (2. according to (2. that is. Section 2.
EXAMPLE Solve the equation 1.69285 = 5 .DIOPHANTINE EQUATIONS
and multiplying both sides b y c.
Ir.745) and
205(338) .3.
m
S
m
9
As a check. 540) is a particular solution of the given equation.
SOLUTION. From Example 1 . 5yo) = (245. y = 540
+ 205t
t = o. will be
.
Multiplying through by -5 we get
so that (xOryo) = (-245.93(108) = 1. y) = (338.23).
Still assuming t h a t a a n d b are positive integers. The general solution.56y
=
7
(c) 562 .-.z.93(-130)
-12095
+ 12090 = -5.44
CONTINUED FRACTIONS
equation.-. take t 205(-59)
=
2.17y = 5
(b) 652 .. to that of the equation ax . and q.6 The General Solution of ax
+ by = c. then (x.b) = 1.3 to obtain the general integral aolutions of the following equations.
.x) = b(y
+ cpn-1). -130).
Prohlem Set 7
1. we first find a particular solution of the equation (a. but (4 b) = 1. b) = 1
=
-3
2. T o do this. so b cannot divide a. ax + b y = 1. Then
as before. Use particular solutions obtained from the yroblems at the eild of
Section 2.6%
(a.23). Therefore b divides cqn-1 .26)
x
= cq.24) a(cqn-1
+ by = c
.
T h e discussion of this equation is similar. expand a/b as a simple continued fraction with a n even numher of partial quotients. and
. Check each answer. The trick now is to write the given equation ax i n the form
Rearrange terms to obtain (2.by = c.-1
. From the table of convergents read off p. except for some minor changes. is then
To check this.
This shows that b divides the left side of the equation. y)
=
=
(-59. (a) 132 . so t h a t there is a n integer t such that
(2.tb. according to equation (2.
and the second two can be replaced. = -300.13t.
where A and B are positive integers. For example. or
y
=
and equation (2.30) can be solved in
. To see this.
SOLUTION.
13%
+ 171.
Hence the general solution of the given equation is
2.30)
A x + By
= fC. if d does not divi& C. let d be the greatest common divisor of A and B.
It follows that 17 divides x
x = -1200 and replacing x + 1200 by 171 gives
900
.46
CONTINUED FRACTIONS
EXAMPLE Solve the indeterrninate equation 2. any equation of the form
f A x k By= C
can be reduced to one or the other of the forms (2.
Not al1 equations of the form (2.
+ 17t. respectively.By
= fC. neither of the equations (2. Then. of the four equations
the first two are already in the required form.7y
=
-10.
A x .29%)
+ 17y = -300(13
+ 1200.30) have solutions. second equation in the solution of Example 1 now The becomes
132 (2.6 The General Solution of Ax & By = 1 C
By multiplying through by -1.900).17 3).29) is replaced by
13(x
+ 1200) = -17(y .?
3s
+ 7y = -10
and
3s . by
.
4 .
then we can divide both sides of the eqiiations (2. On the other hand. namely where a and b are relatively prime.c. The general solution of 205s . divide A.186y we find that
The main results obtained from our study of the linear Diophantine equation can be summarized as follows: Summary. Since d = 2 divides 10. the g. and where c is a positive or negative integer. for the left side of each would be divisible by d while the right side is not.30) by d. EXAMPLE Solve the equation l.the equation can be solved. This is the equation solved in Example 1 of Section 2.4. Divide the given equation by 2 to obtain where now 205 and 93 are relatively prime. Since 410 = 2 5 41.31) will aufomatically @ve solutions of equations (2. Any equation of the form Ax f By = f C has integral solutions x.
SOLUTION. The next step is t expand a/b as a simple continued fraction with an even number n
. and of which we know the solutions. and C by d = ( A . Conversely. if d does divide C. reducing them respectively to equations of the form we have just discussed.93y = 5 found there was x = 245 93t. any solution of equations (2. reducing the given equation to either the form or the form
where in both equations a and b are relatively prime positive integem. 186 = 2 3 31.d. y. B.D I O P H A N T I N E EQUATIONS
47
integers x. In this case. of 410 and 186 is d = 2. B).y only if the greatest common divisor of A and B divides C.30).
-
+
tlnd substituting it into 410% .
He divided the remainder
. Express f i as the sum of two fractions whose denominators are 7 and 11.=x +u 77 7 11
-
3 The sum of two positive integers a and b is 100. A little later a second sailor awoke and had the same idea as the first.48
CONTINUED FRACTIONS
n of partial quotients.-~ . He then hid his share and went back to sleep. and the general solution of (i) is
Likewise the general solution of (ii) is
The solutions (iii) and (iv) represent. b = 9y 5 and use the fact that a b = 100.174y = 9 (c) 772 $ 6 3 ~= 40
+
+ +
m Hint: Find integers x and y such that . Find the general solution in integers of the others.
+
+
+
4. He divided the nuts into five equal piles and discovered that one nut was left over. Five sailors were cast away on an island. and Monkeys
The following problem is of considerable age and. and if b is divided by 9 the remainder is also 5.
(d) 342 . respectively. Problem Set 8
1. Find a and b. y) of 13s
+ 17y = 300. so he threw this extra one to the monkeys. During the night one of the sailors awoke and decided to take his share of the coconuts. To provide food.49y = 5 (e) 34x 49y = 5 (f) 562 208 = 11 2. (hey collected al1 the coconuts they could find. and from the table of convergents read off pn-i and qn-1. Coconuts. Find positive integral solutions (x. continues to appear from time to time.bpn-l = 1. If a is divided by 7 .
(a) 1832 174y = 9 (b) 183s . Then aq. the remainder is 5.7 Sailors. Two of these six equations do not have integral solutions.
2. Hint: Let a = 7% 5. in one form or another. for the cases (i) and (ii) the general solution of Ax f By = f C .
discovered also that one was left over. The problem is to Jind the smallest number of nuts in the original pile. and threw it to the monkeys. The next morning the sailors. we find that the third. divided the remaining nuts into five equal piles. We first seek a particular solution (xl. Then he hid his share. Now the number of nuts in the last pile must be a multiple of 5 since it was divided evenly into five piles with no nuts left over. no nuts being left over this time. al1 looking as innocent as possible.
nuts. The first sailor took +(x . aild fifth sailors left. Similarly the second sailor took
coconuts and left four times this number. or
16x .36
Similarly. hence these numbers are relatively prime and the equation (2. each throwing a coconut to the monkeys.yl) of the equation To this end. Hence
where y is some integer. fourth. respective1y.1). the convergents of the continued fraction
.32) has integral solutions.DIOPHANTINE EQUATIONS
49
of the nuts into five equal piles. Multiplying both sides by 3126 we obtain the indeterminate equation Factoring into primes we find 1024 = 2l0 and 15625 = 56. let x be the original nurnher of coconuts.1) coconuts and left $(x . I n order to solve this problem. In their turn the other three sailors did the same thing.
50
CONTINUED FRACTIONS
are calculated:
The convergent c9 yields the particular solution x l q g = 10849. we search for the vaiue of t which gives the smallest positive value of x and which at the same time makes y positive. From (2. = yo = 8404~1 .
. 5975244 will be a particular solution of equation = (2. Hence xo = 8 4 0 4 ~ ~91174996.32). published by the National Council of Teachers of Mathematics. For an interesting discussion of this and related proble& see the article entitled "Mathematical Games" by Martin Gardner in ScientiJic American. A Guide to the Literature.33). April. by William L. we finally obtain
which means that the original number of coconuts was 3121 and each sailor received 204 in the final distribution. y1 = p9 = 711 of equatlon (2.34) we find that t must be an integer satisfying the two inequalities
Hence the required vaiue is t = . Recreationat Mathematics. The genera1 solution is
-
Since both x and y must be positive. Schaaf.34). 1938.5833. One should also keep in mind the excellent collection of references. Introducing this value of t into equations (2.
and where D is a positive integer not a perfect square. an'd we shall see that these fractions do not terminate but go on forever.CHAPTER THREE
Expansion of Irrational Numbers
3. and.
. is irrational. We proved that a rational number can be expanded into a finite simple continued fraction. & are integers. The numbers
are al1 irrational.1 Introduction
So far our discussion has been limited to the expansion of rational numbers. Any number of the form
where P. A number of this form is called a quudratic irrational or quadratic surd since it is the root of the quadratic equation
Our discussion will be limited fo the expansion of quadratic irrationals. An irrational number is one which cannot be represented as the ratio of two integers. every finite simple continued fraction represents a rational number. D. conversely. This chapter will deal with the simple continued fraction expansion of irrational numbers.
and express 2 in the form
where the number
is irrational.2 Prellminary Examples
The procedure for expanding an irrational number is fundamentally the same as that used for rational numbers.14159 . for.e. Calculate al. namely algebraic irrational numbers and transcendental numbers. .
. using decimal approximations to these numbers. The irrational number 4%the solution of the algebraic equation x2 . .
3. an equation of the form
are integers. and to study the deeper properties of each should read the first monograph in the NML (New Mathematical Library) series: Numbers: Rational and Irrational. the largest integer less than x2.2 = 0. Let x be the given irrational number. To continue. by Ivail Niven. al. but the methods of obtaining the expansions of x and e given in Appendix 1 are beyond the scope of this monograph. such as x = 3. An algebraic number is a number x whch satisfies an algebraic equation. if an integer is subtracted from sil irrational number. algebraic is called a transcendental number. 1 Those who wish to learn &out the two classes of irrational numbers. and e = 2. is one example. the greatest integer less than x. It is quite difñcult to expand transcendental numhers into continued f ractions . calculate a2. we can calculate a few of the first terms of their continued fractioii expansions. It can be proved that x is transcendental.. not al1 zero. and
t See 1.71828 . the result and the reciprocal of the result are irrational.t The number e is also transcendental. Niveri
[8]. The irrational number x = 3.14159 . but this not easy to d0.52
CONTINUED FRACTIONS
There are irrational numbers which are not quadratic surds. i. is and is therefore called an "algebraic number". A number which is not where a*.
az.. then XJ from the third into this result.. which is impossible since each successive z is i irrational. for x. x2.1) into the first equation. a. producing in succession the equations
where al. to be equal to x.inued fractlion
.. produces the required infinite simple cont. This calculation may be repeated indefinitely.x3. . . . 2 4 . and so on. This process cannot terminate. the number
is irrational.IRRATIONAL NUMBERS
express
x2
in the form
where. again. the only way this could happen would be for sorne integer a. are al1 integers and where the numbers are al1 irrational. . Substituting z2 from the second equation in (3.
e
+
The bar over the 2 on the right indicates that the number 2 ie repeated over and over. so The z
EXAMPLEExpand 1.
d into an infinite simple continued fraction. . Thus al1 the subsequent partial quotients will be equal to 2 and the infinite expansion of 4 will be 2
+
+
x4. the calculations of .
Solving this equation for xz. an example or two should be worked to make sure the expansion procedure is understood. we get
Consequently.414 . will al1 produce the same result.414 . is
aa = 2.54
CONTINUED FRACTIONS
where the three dots indicate that the process is continued indefinitely. namely -\/S 1. .
The largest integer
< xs
=2 <
+ 1 = 2. = . z So~unorr.\/S 1. is al = 1. x6. largest integer < d = 1. Before discussing some of the more "theoretical" aspects of injtnite simple continued fractiom. so
where
At this stage we know that
Since xa = 2 < 1 is the same as x.
.
S] actually represents the irrational number 4 5 ? Certainly there is more to this than is evident a t first glance. and it will be one of the more difficult questions to be discussed in this chapter. we write
hence or 1 = 1. roughly speaking. that we go through certain manipulations. With this understanding. We can. which tells us nothing about x. we can write
from which we see that
Thus
.IRRATIONAL NUMBERS
55
Imrnediately some questions are raised. give a formal answer to this question. is it possible to prove that the infinite continued fraction [l. For example. 2. however. A fomnal answer means. ] = [l. but no claim is made that every move is necessarily justified. using the same idea. 2. However. .
6 3 -fi-3 2 X-1 3+4533-453
> 1. Then
where
x*=-.
SOLUTION. This theorem will be proved in Chapter 4. These examples are illustrations of a theorem first proved by Lagrange in 1770 to the effect that the cmttinued fraction expansion of any quadratic irrational ia perz'odic after a certain stage.
EXAMPLE Find the infinite continued fraction expansion for 2. so
where
Thus
x4
=
23. proceed exactly as in Example 1. Since fiis between We 7 and 8.
.CONTINUED FRACTIONS
Some additional examples of a similar sort are:
I n each of these examples the numbers under the bar form the periodic part of the expansion. the number 4% having quite a long period.
and so the last caloulation will repeat over and over again.
The largest integer
< xl
is a9 = 2. the largest integer < x is al = 1. so
where
x3re-= 1 X4-2 2
=
453-7
aa
=
dE+7
2
The largest integer
< xa is
7.1 22 3 .
we find
7+dG. the required expansion is
so that finally we obtain
=
[l. .
Now let us reverse the process. l
. let us start with the infinite expansion and try to get back to the original value of x.we obtain
which ia the o r i ~ i n a value of x. I t is convenient to replace
where
Then y satisfies the equation Solving for y (by the quadratic formula) a.2. 2
Hence
Simplifying the right-hand side.nd noting that y that
0 =
> O. 71.IRRATIONAL NUMBERS
Hence.
. we define p-1 = O. The convergent cn = pn/qn is calculated by the same formulas
= 1. and q.C.. . = 7" = 3. but greater than 3 8 = 3. this is quite a remarkable result considering
. where. ) stated that the ratio of the circumference of any circle to its s diameter i less than 31. less accurate than the above Egyptian value.
Calculate the first five convergents. Archimedes (c. The computational scheme is the a m e .l
are calculated in exactly the same way as before. The out as follows:
'A =
'A =
3. was used by the Babylonians.1604.7. Translated into our decimal notation. 2
-.14159
starts
[3. . The approximation -U = 3. as before.292. 15. 1. SOLUTION.
E~AMPLE infinite continued fraction for 1. for al1 n 2 1. = O. . 1.14084 . 9-1 = 1. 225 B . .l .C O N T I N U E D FRACTIONS
3.3 Convergents
The convergents to the infinite continued fract ion x=at+a2
1
+G +
1
= [al. table of convergents is as follows: The
In this connection it is interesting t o note that the earliest approximation to a is to be found in the Rhind Papyrus preserved in the British Museum and dated about 1700 B. the value of u stated there is 3. 1. These convergents give successively better approximations t o 'A. . C . a 9 as.14285 .
Discussion Problern. 4 . 41
=
d 6
(b) [5. The following is one of the classical straight-edge and compass problems. 5.4s in the second half of Example 2. it possible t o construct the length n we could then construct l/?r 113' the following means: Let AB = r . Construct. . 2. . Then x = BD = prove this use the similar triangles ABU and CBD. 1)raw BU perpendicular to AC. 1. 2 . SO a square with the same area would have a side Were equal to 6. square equal in area to a circle of radius 1. see To Figure 2 . A circle of radius 1 has an area A = r r 2 = A.IRRATIONAL NUMBERS
the very limited means at his disposal. 101
=
6
3.2.
-
(a) [2. a. 2 .
Figure 2
.V 1 5
17
=
[1.
Problem Set 9
1.
G] (d)
24 . 1. Section 3.
4 . using only a straightedge and cornpass. More information on the use of continued fractions to give rational approximations t o irrational niimbers will be taken u p in C h a p t e r 5. 4 . The approximation
is correct to six decimal places. verify that the following continued fractions represent the irrational numbers written on the right. Verify the following expansions and calculate the first five convergents:
7
(a)
d = [ 2 .6
] = [2. BC = 1 and draw a semicircle with center a t O and passing through A and C . 1.
fi]
2.
and also verify that the con-
both numerators and denominators being formed from the sequence of Fibonacci numbers Each of these numbers is the sum of the preceding two.4F = 2.141592 . Then prove that . 1.
+ +
Figure 3
4. However. FG parallel to E 0 and FH ~)arallelto DG. there are many interesting approximate constructions. 2. Show that vergents are
: d+ 1) = [l. and i t only remains to construct a line equal in length to 3 AH.3. During the second. can be 5. and so on. The Fibonacci numbers Fi = 1. F3 = 2. Skctcli such a trec after a five-year growing
. Let O D = i. A discussion of these interesting numbers will be given in Section 3. Let . then branches again. discussed a t the end of Section 3. 1 .CONTINUED FRACTIONS
It can be proved that a length equal to ñ cannot be constructed with straightedge and compass. then "rests" for ayear. F = 3. 4 into this formula.10. F 2 = 1. 3. Since
e
•
the approximation to a can easily be constructed as follows: Let O be the center of ti circle with radius OE = 1. Imagine that each branch of a certain tree has the following pattern of growth. I t produces no new branches during its first year of growth.
6. (i
. it puts forth one braiich.
Verify this by substituting n = 1.1. For example. 3. d iii the general formula reproduced by substituting n = 1. Jakob de Gelder in 1849 gave the following construction using the convergent +#= 3.4H = 42/(72 82).iB be a diameter perand . see Figure 3. 2. Draw pendicular to OR. 1.
if we regard the trunk and its extensions a s branches..4 Additional Theorems on Convergents
The nurnerators p.3. in the second year two branches.
satisfy the f u n d a m e n t a l recurrence relation
(3. he \vil1 win the game.
m
7. and in general the number of branches will reproduce the Fibonacci numbers 1. So unless A makes a mistake. 19 (1953). a n d denominators qn of the convergents c.
8. 8. construct a point G on a line segment AB such that (AG) = dGB). Using only a straightedge and compass. S. of the infinite simple continued fraction
[al.4). Use tlie results of Problem 8 to show hoiv to construct a rcgiilar pentagon using only a straightedgc and compass. then he must remove a n equal number of counters from each. = p.IRRATIONAL NUMBERS
61
period and show that.
+(di +
+
9. M. t h e proof given there being independent of
. If he wishes to take counters from both heaps.
3. vol. az. .l)". then in the first year of the tree's growth it has one branch (the trunk). * .4.
=
-1
(. Scripta Mathematica. Alternately two players A and B remove counters from two heaps according t o the follo~ving rules: At his turn a player may take any number of counters from the first or froin the second heap. For more details about this game and related subjects see H. Wythoff). Phyllotaxis. 2. I t can be proved that the nth pair of numbers forming a safe combination is given by
where í = 1) and where { z ] stands for the greatest integer less than or equal to x. 2. pp./q. A. and TVythofs Garne. 3. where T = i(1 4).2)
pnqn-i . Coxeter: The Golden Section. and A can always convert this back into a safe combination (safe for . The player who takes the Iast counter from the table wins. In order for player A to win he should.
m
.
proved in Thcorern 1. IYythoJ's game (invented in 1907 by W. a.Pn-lQn
n 2 0. leave one of the folIowing safe combinations (safe for -4):
Then no matter what B does in the next move he \vil1 leave a n unsafe combinatio~i(unsafe for B). after his move. 135-143. 5. Verify this statement for n = 1.
equation (3. upon dividing both sides by qnqn-i.2.
. we see that
respectively.un(-
1).2. This proves Theorem 3. These theorems give us important information as to how the convergents cn change as n increases.~ n . we find that
Since cn
=
pn/qn. These inequalities show that
(3-4)
CI
< c2
and that
c3
< cq. and recall that the qn's are positive.1.3) can be stated as
Similarly we can prove THEOREM 3.62
CONTINUED FRACTIONS
whether the continued fraction was finite or infinite.-l Qnqn-2
t
n
2 3.
Cn
. substitute
obtaining
where the last equality follows from equation (3.
PROOFClearly .2 .1.2) with n replaced by n .
In the numerator on the right. From this equation. If we set 7% = 2 and then n = 3 in Theorem 3.
in succession.1. Eies between the two preceding cmvergents. f o r m a decrea* sequence.2. using n = 3. setting n = 3 in Theorem 3.4) proves that
Similarly. q ~ a3 are al1 positive numbers. Moreouer. n 2 3. continued fraction fornz a n increasing sequence. Hence c~ < cg.2 shows that
.
Problem Set 10
1. as we have seen.
The conversion of an irrational number x into an infinite cont inued fraction gave. we see that
C3
< C 4 < C2.IRRATIONAL NUMBERS
63
On the other hand. and every odd convergent i s lesa than a n y even convergent.+l of u n inJinzte simple 3.
. followed by n = 4 in Theorem 3. Give a numerical verification of Theorem 3. and the even convergents cz. since qa.. and combining this result with those in (3.3 using the convergents
to d . each convergent c. we obtain the inequalities
Combining these inequalities we obtain the fundamental result
We state it as a theorem: THEOREM T h e odd convergents c2. i
36 Some Notions of a L M . then n = 4 in Theorem 3.
Proceeding step by step in this fashion..3.
The implication is that we can somehow carry out an infinite number of operations and thereby arrive a t a certain number which is asserted to be x. if we add the numbers 1. To make this clear. is irrational. We shall see. Realizing this. that the only way to attach a rnathe~naticalvzeaning to such an infinite process i s to introduce the notion of a linzit. . Which of the follolving infinite surns have meaning?
Clearly. so we say the sum A becornes infinite as the number of terms added increases indefinitely. one is tempted to write (as we did)
which irnplies that the infinite continued fraction on the right actually represents the irrational number x. and such a result is not of much use to us. I t is advisable to reflect on the fneaning of such a statement.1)st calculation we had
where x. On the other hand. +! 1 1 T ?8.CON'I'INU'ED F R A C T I O N S
so that. let us first go back to ordinary addition. we get i succession the partial sutns n
. the given irrational number. we can make the "sum" as large as we please. a t the end of the (n . and we saw that the process could be continued indefinitely. however. if we add 1 to itself over and over.
that is.a.IRRATIONAL NUMBERS
which can be represented graphically as shown in Figure 4. as n .
1
In order to prove that they continua!ly approsch this upper limit 2. (+)"-l .
. or approaches 2 as a limit. they are al1 bounded above by the constant 2. where so that the partial sums continua2ly increase. we w ~ t e
so that
Subtracting the second line from the first. But each partial sum sn is less than 2 . approaches zero. that is. we obtain
which implies that
As n increases indefinitely. and so S. geb closer and closer to 2.
increasing sequence of numbers al1 bounded above by the convergent c 2 = U. c3. then they hable a linzit l ~ where EL 2 L. the limit Ev must be a number less than al1 the even convergents. since al1 odd convergents are less than al1 t..
We say that
converges to the value 2 as n -+ e .c2.4. and i. On the other hand. 53.
a
t For a discussion of limits of sequences. form a decreasing sequence of numbers al1 bounded below by the convergent cl = L. 2s Zess than U. S?. hence they will converge to a limit Eu 5 U.66
CONTINUED FRACTIONS
S. Zippin [15].
Our task is to attach a meaning to the infinite continued fraction
form an Theorem 3. . . that is. S Z . sa.3 states that the odd convergents cl. It also illustrates a fundamental theorem of analysis whích we state but do not attempt to pr0ve.
We then assign this limit 2 as the valw of the infinite sum in question and we write
This illustrates. see L.heeven convergents.
. S .t
THEOREM I f a sequence o j numbers si. cb. cs. which alao treats
this fundamental theorem of analysis (Theorem 3. . the mathematical notion of a limit needed to attach meaning to an infinite continued fraction. continually decrease but are all .f jor each n. or in symbols. s3. continually increases. Moreover.haue a limit Eu. 3.
36 Infinite Continued Fractions . that is.c4. then the numbers SI. where U is some $sed number. 8 2 . in an admittedly rough fashion.4). greater than L. lim
n+
S. where 1~ 5 U. =
2.
We return to the discussion of infinite simple continued fractions. the even convergents c2. If the nunzbers SI.
that the q.
. If lu # EL we would be in trouble. (.5. the fraction 1/ ~ ~ approaches~xero as k approaches infinity. Biit then from ~ q ~ .
>
Figure 5
To this end.1 by 2k . where ZL is a number greater than every odd convergent.c2k-1 approaches zero as h.IRRATIONAL NUMBERS
67
so that even convergents approach the limit EL L.~ equlttion (3.e) the denominator q2kq2k-1 of the fraction in (3. however.6) increases ivithout bound as k increases. Looking a t the convergents graphically (see Figure S ) .n 2 1) is a positive integer.'s increase without bound as n iricreases.he a m e limiting value 2 = lu = ZL. we see that all we have proved so far is that the even convergents have a lirnit ZI. We have proved:
3.
How far have we progressed? 1s this limit E the same niimbcr x which gave rise to the continued fraction in the first place? Actually it is. Every infinite simple continued fraction converges THEOREM lo a limit E which is greater than any odd convergent and Eess than any even convergent. that is. retuin to Theorem 3. Weget
The numbers q.6) we conclude that the difference C2k . but this must be proved. are calculated by means of the recurrence relation therefore it follows. and the only way this can happen is for both C28 and c2k-~ to have t. 13enc. and the odd convergents havc a fmit Eu. that ZU = EL.1.1 and replace n by 26 and 3 n . since each a. We can prove. (n 2 2) and each q . approaches infinity.
51.
where s. that is. is the "rest" of the fraction. (3.9 it follows that
xn+ 1 anf 1
so.8) shows that
n Again.+i. Similarly. To this end. according to the second line i (3. we see that
The next step in the proof is to show that x lies between cn and c. and return to the expansion (3. we compare the three expressions:
.
=an+-
I
9
Zn+i
where again
The second line in (3.68
CONTINUED FRACTIONS
To do so.7). combining these results.
1 1 and since . let z he the givcn irrational niimber.< -.7) shows that
since
Xn+l
is positive.
On the other hand. we see = al 1 / x 2 . and since that c i < x.9)
cl
=
+
Thus
.9) gives al < x l . hence
al and x l = x. narnely
1 -r an
1 -r
Xn
and
1
a. (3.9) we know that
and we can conclude frorn (3. where b y (3. either
A direct calculation shows that
for.IRRATIONAL NUMBERS
We first observe that these expressions have the term
in common so that it is necessary only to compare the terms in which they differ. x a2 < 2 2 or l / x z < l / a 2 .
1 +-an+i
But by (3.10) that x will always lie between two consecutive convergents cn and cn+1. that is.
between C J and c4> between cr and Ca. Therefore it is permissible to write
and we have proved
THEQREM If un irrational number x i s expanded i n t o un 3.. . a. .6. a2. approach x from the left. This is true.7 Approximation Theorems
Our experience with continued fractions and in particular our study of Theorern 3. Before stating such a result as a theorem we make some preliminary remarks.6 have supplied ample evidence that each convergent in the continued fraction expansion of an irrational numher x is nearer to the value of x than is the preceding cohvergent. equations (3. and so on. then the tirnit to which the conuergmts ci. Since al1 odd convergents are less than al1 even convergents. in expanded forrn. c4. that Thus we see that the convergents cl. ca. .
.
3. and c2.of the fraction [al.9
m m
Cn. hence x and E must be one and the %me. But we know that as k increases indefinitely. a2.. U n . infinik simple continued fraction [ai.70
CONTINUED FRACTIONS
Similarly. and the reader will find it impossible to expand. as explained. ' . any given irrational numbers in two different ways. c2.
This theorem should be followed by an additional theorem stating that the expansion of any irrational number into an infinite simple continued fraction is unique.10) show that x lies between c2 and cs. a ] according to the rules described. Let the expansion of the irrational number x be
. approach x fronl the right. . ive are forced to the conclusion that or.
. the odd convergents ctk-1 and the even convergents C2k approach a lirnit 1. -
] converge i s the number x
which gaue rise to the j r a c t i m in the first place.
. equation (3. an+2. pn. a?. however. the given irrational number. . if we calculate convergents in the usual manner. a. we write (3. as before. by analogy with our study of finite continued fractions.-1. . Then. and consequently we have no right to treat it as though it were a legitimate partial quotient. when n = 0. are al1 positive numbers. this should be equal to x. yn-l depend only upon the integers al. p. In particular. qn. (3. the last "convergent" (in Theorem 1. Thus it seems reasonable to write
where.I+Ias a legitimate partial quotient. Suppose.+1 contains an infinite number of integral partial quotients un+*.
When n
=
I. . also note that xl = x.12) gives
and by definition. While x. .IRRATIONAL NUMBERS
where
We assume that ~ 2 ~ x 3 . it should be stressed. take i = n 1 and an+1 = xn+l) woiild be
+
@
and.12) gives
. it need not itself be an integer.3.11) in the f o m of a "finite" continued fraction aild treat X. .
according to (3. 11 = 7.
. 1-71 7
-
if a
< 0.t hence
We know that for n '_ 2. then la( = lb1 tcl.12) holds for al1 n can be proved in exactly the same way as we proved Theorem 1.12))
and from this we obtain or.+~> 1. rneans
a
ifa > O . hence
> q n . r e d "absoiute value of
tal
c
a". the expansion of the given irrational number x be Let
where Then.tinuedfraction than is the preceding convergent.72
C O N T I N U E D FRACTIONS
That (3.a
For example. and 1-al
=
lal. we obtain
Now if a = b .
simple con.
t The syrnbol la[. x. we have for n 2 2
Dividing through by xn+iqn. the successive steps being nearly identical.3.~> O .
la1 = . We are now ready to state the main theorem of this section:
THEOREM Each convergeni is nearer to the value of un infinite 3.
7
.c . rearranging.
PROOF.7. and that q.
nqn+t < 1/~2.7 that x is closer t.IRRATIONAL NUMBERS
and so
Thus (3. we know already from Theorem 3.13) shows that
or.+i than it is to cn.
. and c. This shows that cn is closer to x than is c.+1. that
+
Taking the absolute value of both sides. or estanate.o c. we know from Theorem 3. so that c. of just how closely c. this tells us that
Figiire G
Moreover. and it follows that the absolute value of the difference between x and cn will always be greater than one-half the absolute value of the difference between c.+i > q n J Hence we can state
qnqn+i
> 4R
and
SO
I/q. is to the left of c. In fact. with n repIaced by n 1.
It would be interesting to have some measure. and the theorem is proved..1. Clearly AB < AC < AD.-1. This becomes clear if the situation is studied graphícally. what is the same thing. approximates x.+I. or
Since q. Figure 6 shows the case when n i s odd.
such that
This is the beginning of the theory of rational approximation t o irrational numbers. ( p .
actiially approaches a limit can be suggested by numerical evidence:
The number e ia taken as the base of the system of natural logarz'thms.74
C O N T I N U E D FRACTIONS
If x is irrational. satisfying Theorem 3. Thus we have the following theorem :
T W E ~ R E M I f x i s irrationul.
Commenta. a decimal fraction which approximates e correctly to six decimal places.just as 10 is used as the base for cmrnon logarithms.718282
*
-
give better and better approximations to this number.
EXAMPLEShow that the first few convergents to the number 1..
. there exists an infinite number of convergents
pn/q.9. q > O. q ) = 1. The continued fraction expansion of e is the proof is quite difficult. These convergents should be calculated by finding the first few convergents to 2.718282. a subject we shall discuss briefly in Chapter 5. there exists un injinite number of 3.
t!
= 2.8. rational fractions p / q . The irrational niimber e arises quite naturally in the atudy of calculus and is defined as
That the sequence of numbers (1
+
(1
+)
(1+ !)') .
--e
The inequality
o Theorem 3. Convert convergen&. S ] . 1. .
find a fraction with a smaller EXAMPLE Given the fraction +&#ea6-. rnight be considerably improved.IRRATIONAL NUMBERS
75
SOLUTION. 2. that in succession these give better and better approximations to e . The tsblc givcs the numerical resulta:
w-
. and this sugvalue of e gests that Theorem 3. I n the next f example we s h l l approximate a rational number. 1. notice that p7/q7 = hence it should be true that
w. We shall see in Chapter 5 that this is indeed the case. As a check on Theorem 3. or being content with the approximation
we find that
e = [2.00065746.
The corresponding convergents are
and a conversion to decimals shows. 2. 4 . We observe that the and this is certainly less than 1/39" is approximately one-half that of 1/39%. numerator and a smaller denominator whose value approximates that of the given fraction correctly to t hree decimal places. into a continued fraction and calculate the SOL~ION.9.9. 1. Assurning the above expansion for e.
.
A numerical calculation shows that
0.8 is true for rational or irrational x. 1 .. indeed. regarded as an approximation theorem.
.+.P5 --197 q5
will suffice.q.CONTINUED FRACTIONS
Referring now to Theorem 3. Expand fiinto an infinite simple eontinued fraction and find a fraction which will approximate 6 9 mith accuracy to four decimal places. 7 .
3.+i > 1 / e .
qn
Problem Set 11
1./q. = pn/qn and c+i = pn+l/q. Given the fraction find a fraction with a smaller numerator and a smaller denominator whose value approsimates that of the given fraction correctly to three decimal places.Note that if we had worked with the fraction l / q : instead of l/qnq. Use Theorem 3. 1 . < e .q.8.
M+.y
87 38
p4
q4
451 . that is. 1 . 1 . witl-i an error less thau half a unit in the fourth decimal place. we wish to approximttte Z#'$J.+i our answer would have been the next convergent +S.= .8 to investigate how closely the first four convcrgcnts approximate A. A little esperimentation soon shows that
. 15.+l which will make
That is. with an error of less than 5 units in the fourth place. where e is any given number.
a]. 292. folloning this by an use a table of squares and first check that additional check to see if q. such that l/q. for
Hence the required fraction is 3%. The continued fraction expansion of n is [3.0005.q. In order to find values of q.
2. we could > 1 / e . 1 .by p. since 1/38' is not less than 0. we search for two convergents e.+.
1907. called lattice points.he line.IRRATIONAL NUMBERS
3. and a = y/x would be a rational number. to t. I we move the thread f away from the line in the other direction. some of whose works are available today in reprint form. .]. See Figure 7. . toward the left. At these points.
t F.he thread in our hand. a?. respectively. . if it did there would be a point (x? y) with integral coordinates satisfying the equation y = ax.
17-25. We pul1 the thread taut so that the end in oiir hand is at the origin. we move our hand away from the origin. . the thread will catch on certain pegs above t.he odd convergents. . The pegs contacted by the thread on the lower side are situated at tlhe lattice points with coordinates and correspond. 9 Cn. and that we hold the other end of t.8 Geometrical Interpretation of Continued Fractions
A striking geometrical interpretation of the manner in which the
convergents ci. . pp. . Felix Klein was not only a prominent mathematician but a most popular mathematical expositor. imagine that pegs or pins are inserted. a. assume a positive. This is impossible since a! is irrational. Klcin : A ~trycrodhlleKapitel def. Keeping the thrcad taitt. .. Zahledheorie. Let a be an irrational niimber whose expansion is [al. and whose convergents are
1:or simplicity. Kext plot the line
p
=
ax. and on graph paper mark with dots al1 points (x.
This line does not pass through any of the lattice points. c2. it wilI catch on certain other pegs. of a continued fraction for an irrational number converge to the value of the given number was given by FeIix Kleint in 1897. Now imagine that a piece of thin blactk thread is tied to an infinitely remote point on the line y = ax.
. for.. y) whose coordinates x and ' 1 ~ are positive integers. Teubner.
The pegs contacted above the line are sitilated a t the lattice points
.78
CONTINUED FRACTIONS
O
T
2
3
4
5
6
Figure 7
which are al1 less than a.
(5. Those points corresponding to the even convergents are (1. 1 Y3!4 (Chapter 8).IRRATIONAL NUMBERS
corresponding t o the even convergents.
EXAMPLE. p4) = (3. .
8
R. New York: The MaoiniIIan Compaiiy. for example. or 5/3 > g/3 = a. 13). I n fact. y) marked in Figure 7. 5). 21.
t See
. 1. and are al1 below the line. (8. 3).lost of the elementary properties of continued fractions have geometrical interpretations. and are above the line. Draw a Klein diagram for the continued fraction espansion of
1
1 + d 5 a= = [1. Hancock. 2
-
-1. that the point (qr. l). the theory of simple continued fractions can bc developed geometrically. Let us show.
(di -
2. (2. Development of the Minkowski Geomelry of Numhers. or a = 4/3. we see that y = a 3. Since it is on the line y = ax. see Figure 7. Consider the point (3. convergents are The
The points or pegs corresponding to the odd convergents are (1. The point (3. Construct a Klcin diagram for the continued fraction expansion of 1)/2.?
Problem Set 12
1. 8). (3. 1. 1. hence the corivergent 5/3 > a.
al1 of which are greater than a. . Construet a Klein diagrarn for the eontinued fraction expansion of 2/3. 5) corresponds to the even convergent p4/q4 =. Each of the two positions of the string forms a polygonal path which approaches the line y = ax more and more closely the farther out we go. . H. 5) is above the line so 5 > 4. which is greater than a.
SOLUTION. 5/3.
1. 1. getting
so that
For example. when a = 1.3.
(a) x2 . provided.]. better and better approximations to the actual solution i ( 1 See also Problem 4 of Section 3. we have only to divide both sides of (3. We shall now examine the quadratic polynomial equation
the positive root of any quadratic equation of the form (3. . that it has such a root. . of course.14) has the continued fraction expansion
If a
> 0.5x
-1=0
.1
=
O
(1)) x2 .
and the successive convergei~ts this continued fraction will give to 45).
+
Problem Set 13
1. A more detailed discussion of this particular number foIlows in the next section. Use the quadratic fo~mmulato find the positive roots o the following f equations and compare the exact solutions with the approximate solutions obtained by computing the first few convergents to the continiied fraction expansioiis of these positive roots. the equation
x2=x+1
has a positlive root
x
= [1.
To see this.3x .CONTINUED FRACTIONS
3.14) by x.9 Solution of the Equation x2 = ax
+1
Continued fractions can be used to approximate the positive root of any polynomial equation.
1.
p n / ~ n . by giving the positive integers a and b particular values.
and has the va. Suppose that
and that b is a multiple of a.10 Fibonacci Wumbers
The simplest of al1 infinite simple continued fractions is
T
=
[ l .2.2/q. that is b Show that then x satisfies the equation
22-bx-c-O
= ac
(where c is an integer).liie
3. p n + 2 / q n + 2. and by selecting particular convergents pn.IRRATIONAL NUMBERS
2. that if
3.
where
T
satisfies the equation
which has the positive root
The convergents to
T
are
both numerators and denominators being formed from t h e sequence hof integers
... l ?1 . Verify.
Thus a line segment is said to be divided according to the golden mean if one part is T times the other.. after the first two. In geometry. especially in the printing and advertising crafts. which has many forms. although he was not the first to use them. The Greeks claimed that the creations of nature and art owed their beauty to certain underlying mathematical patterns. it arises from what some cal1 the "most pleasing" division of a line segment AR by a point C .16) are known as the Fibonacci nu~nbers. The figures and drawings were made by Leonardo da Vinci. Luca Pacioli pubIished a book. The numbers (3.
+
+
If we now let
fc
= b / a . the ratio 3 to 5 is approximately eqiial to the ratio 1 to T.
. In this book Pacioli described thirteen interesting properties of T. in the proportions of the human body. and so on. l'or example. whose sides are in the approximate ratio 1 to T.82
CONTINUED FRACTIONS
Each of these numbers. Witness the popularit$ of the 3 X 5 index card. One of these was the law of the golden nzean. Divina Proportione. aesthetically. The golden mean appears a t many unexpected turns: in the pentagonal symmetry of certain flowers and marine animals. is equal to the sum of the preceding two. devoted to a study of the number T . 3 = 2 1. or golden section. thus 2 = 1 1. 1170-1250). This is said to be attained by selecting a point C such that the ratio of the parts a to b (see I'igure 8) is the same as the ratio of b to the whole segment a + b.
+
I n 1509. Le. we have
so that z = b/a = f (1 1/5) = T. or b = Tu. named after the great thirteenth century mathernatician Leonardo Fibonacci (c. Man has employed the golden mean in the creative arts and in various aspects of conternporary design. the majority of people considers that rectangle to be most pleasing. and so on.
q ( p = q6) as the dirnensions of our square. q6 as the dimensions of our rectangle. while that of the rectangle with what seem to be the same component parts is 5 13 = 65. The number T occurs in connection with many mathematical games. as shown in Figure 9a.D do not lie on a straight line but are the vertices of a parallelogram ABCD (for a n exaggerated picture of the situation see Figure 9b) whose area is exactly equaI to the "extra" unit of area.
o
Figure 9a
This puzzle is based on the fbllowing facts: The convergents (3.
. the obtuse a n d e s ADC and ABC differ from straight angles by less than f +O. and p6. which. B. The most familiar. and the above rela6 ~ tion tells us that the areas of these figures differ by only one unit.8 = 64. for n
=
6. C.15) have the property that the denominator of each is the numerator of the previous one. is the one involving a square 8 units by 8. Ir1 case of the rectangle of Figure 9a. I n particular.IRRATIONAL NUMBERS
83
In geometry. The area ~f the square is 8 . can seemingly be broken up and fitted together again to form a rectangle 5 by 13. and the convergents to T also occur in connection with certain geometrical deceptions. the points A. perhaps. so that somehow the area has been increased by 1 unit.
Now consider the relation which in this case. the golden mean is the key to the construction of the regular pentagon. becomes
We have chosen p6. Actually.
I Ftn is large (say F 2 n = 144. and
Fk
=
Fk-2
+
Fk-1
for k
> 2. pp. . if the Fibonacci numbers are defined by the relations
F x = 1. then it can be shown that when the parts are reassembled to form a rectangle. a hole in the shape of a parallelogram ABCD of unit area will appear and the altitude of this parallelogram is 1 / 1 / ~ : .
t This section is rather technical and may be omitted without loes of
continuity. No. 8. . then the hole is so f narrow that it is difficult indeed to detect it.
+
Fzn
Figure 9c
3. fMathematica2 Tab2es and Other Aids to Computation. 45. describes a method for calculating logarithms which is worth recording because of its adaptability to high-speed computing machines. 60-641. April 1954. FZn-2 = 55). in a journal devoted to numerical comput~tions. F 2 = 1. Vol.CONTINUED FRACTIONS
Figure 9b
More generally.~.
. F . (with even siibscript) is divided into parts as shown in Figure 9c.11 A Method for Calcuiating Logaritbmst
1
Daniel Shanks.
and if a square with a side equal to a Fibonacci number F2.
Calculate
.
and the sequence of positive integers
where the numbers n l . be.17) where l / x l
bo
=
b1
ni+. nz.
we then calculate
bz
=
-
bo bf'
and determine a n integer nz for which
b$Qbl<b2
ntf 1
. bz.l. then
The procedure is now continued.
If n2 is such an integer.+l.
This shows that (3. we first find an integer n l such that
o. the relations
are determined by means of
Thus.IRRATIONAL NUMBERS
85
To calculate the logarithm loa.' < bo < b.
1
)
< 1. bl to the base bo of a number bi (where 1 < bl < bo) we compute two sequences:
bz) b3. ba.
18) we have
On the other hand. Solving equation (3. bl notice that from equations (3. from (3. To see that we are actually calcuhting lo&.
and hence we can write
Similarly we can show that
and so on.17) and (3.86
CONTINUED FRACTIONS
and find an integer n3 such that
b i a < b o < b 3na+l .
. by the deíinition of a logarithm.
whence
and so on.19).17) for bi and using these results we have
and so.
we find that
so that ni = 3 and bg = 10/28 = 1. The paper by Shanks gives the following resulta:
This shows that log 2 =
3+3+9+2+2+***
1
1
1
1
1
=
[O.024.the value of log 2 to 11 places is 0. 2.
. 3.25)Q = 1. b~ = 2.
SOLUTXON.
Next we calculate the convergents:
The convergent cs gives the approximation 0. . 3.30103093 . we see that
Thus n2 = 3 and ba = 2/(1. With bo = 10.IRRATIONAL NUMBERS
EXAMPLE. -1. 2. Subsequent calculations become more difficult but can easily be done with the aid of a desk calculator. 9. in general.30102999566.
. each convergent approxirnafes log 2 to one more correct decimal place than does the previous convergent.25. I t can be shown that. Using a table of powers. Calculate loglo 2.
and that irrational numbers have non-terminating. or quadratic surds. It is not hard to show that
. i.1 Introduction
Our study so far has shown that rational numbers have finite continued fraction expansions. In aH the examples considered.
+
where. For example . the bar over the partial quotients indicates those numbers which are repeated indefinitely.CHAPTER FOUR
Periodic Continued Fractions
4. as before. expansions. with irrational numbers of the forrn ~ k v 5 where P. Q. or they were periodic from some point onward. In Chapter 3 we dealt kainly with the expansion of qvadratic irrationals.e. Iike the expansion of f (1 d%)below. the expansions of such numbers were either purely periodic. or infinite.. D are integers and where D is positive and not a perfect square.
1 W ) . I n 1657 Fermat stated that equation (4.1) was given by Lagrange about 1766. every periodic continued fraction represents a quadratic irrational.1) is known as Pe2E7sequation. The most interesting
proposed by Fermat as a challenge to English mathematicians of the time. The first complete discussion of (4. 2. at the age of thirteen. 1 1 i u wtw c*lccteda fellow of the Roya1 Society in 16&3. p. fe11 had mastered eight languages before he wsil twcnty.1) has infinitely many solutions. 3411.4 supplies a deeper study of reduced quadratic irrationals.5. equation (4.
.1652).1 Many authors refer to the equation a s Fermat's equation.PERIODIC CONTINUED FRACTIONS
89
any purely periodic continuedfraction. This will be accomplished in several stages. References to indeterminate equations of the Pell type occur throughout the history of mathematics. t Lord Brouncker in the same year gave a systematic method for solving the equation. The aim of this chapter is the presentation of the proofs of these theorems. a t Rreda ( 1040. Carnbridge. and Section 4. and. was first proved by Lagrange in 1770. This is followed by the proof of Lagrange's theorem which states that the continued fraction expansion of any quadratic irrational is periodic from some point on. and where N is a given integer not a perfect square.2 and is followed by the proof for the general case. For a complete history of the subject see Dickson [4. and he was Crornwell's representative in Switzerland (10454. The chapter will end with a brief discussion of the indeterminate equation
where x and y are unknown integers. an example is presented a t the beginning of Section 4.
t Actually
1John Pell(1611-1685) was a great teacher and scholar. called a reduced quadratic irrational. The more difficult theorem. These sections contain the tools necessary for proving. conversely. that any quadratic irrational has a continued fraction expansion which i s periodic after a certain stage. vol. represents a quadratic irrational. but this is unjustified since Pell did not make any independent contribution to the subject. He waa profeasor of mathemat ica a t Amsterdam (1643lt)46). but he did not supply the proof. or any fraction which i s perz'odic frum so?12e point onward. Commonly. First it will be shown that a purely periodic continued fraction represents a quadratic irrational of a special kind.3 furnishes a more detailed discussion of quadratic irrationals. Section 4. that any reduced quadratic irrational has a purely periodic continued fraction expansion. Admitted to Trinity College. in Section 4.
See Heath f6.f The solution of this problem contains eight unknowns (each representing the nurcber of cattle of various kinds) which satisfy certain equations and conditions.
4. p. Eumbers represented by purely periodic continued fractions are quadratic irrationals of a particular kind. I n fact some historians doubt that the problem had any connection with Archimedes. Others. The smallest solution of the cattle problem corresponding to these values of x and y consists of numbers with hundreds of thousands of digits. New York: Sirnon and Rchuater. 3421.
( a ) A nu~nerical exa?nple.
. Newrnan. 1211.90
CONTINUED FRACTIONS
example arises i connection with the so-called "cattle problem" of n Archimedes. The problem can be reduced to the equation whose smallest solution involves numbers x and y with 45 and 41 digits respectively.2 Purely Periodic Continued Fractions
Certain continued fractions. There is no evidence that the ancients came anywhere near to the solution of the problem. 187-188. like
are periodic only after a certain stage. like
are periodic from the beginning on and are called purely periodic continued fractions. vol. pp. Consider some purely periodic continued fraction. 1956. while others are convinced that it was propounded by Archimedes to Eratosthenes. 2. Dickson f4. p. such as
8
Vde can write
t For a statement of the cattle problem see The World of Mdhemalics by James R. and we shall now investigate how these numbers can be distinguished from other quadratic irrationals.
There we showed that if
where
then
where p .~ and pn/qn are the convergents corresponding. respectively. We form the table
al
=
3.. + a2 a3
1
~ ~ ~ 1 . + a. t o the partial quotients anWland a.6) to the special case (4. as = 2. (4. a . I n the case of a purely periodic continued fraction
a = [al.5) shows that a can be calculated from the equation
We now apply (4. and that in calculating CY we can regard as though it were a legitimate partial quotient. using a* = 1.
.+ .3) as though it were a finite continued fraction.~ / q . . I n effect.5) shows that we can treat (4.2). ] = al
1 1 1 1 + . a = 13.P E R I O D I C C O N T I N U E D FRACTIONS
91
It is now necessary to recal a result studied in Section 3.1
we see that
and hence equation (4. 1.7. .a2)
=
. 21.+ .
we obtain
This leads to the quadratic equation
which is the same equation we would have obtained had we worked with equation (4.
We now consider the number /3 obtained from a by reversing the period. 0 > 1.1l).92
C O N T I N U E D FRACTXONS
Hence. where . and so a! and . Moreover. and so . has the positive root a and the negative root a = .
a =
5+dZ
3
and
a = '
5 . This shows that the quadratic equation (4.9) can be written in the form
Comparing (4. or (4.7) has two roots.7).1 < . that is. These roots cannot be equal are positive. the number
Applying (4.1/@ have opposince both a and site signs.1/@.6) to 0.f i 3
.1 < a < 0.2).l/@< O. The quadratic formula shows that (4. ' '
It is easy to check these results numerically.7) and (4.10) we see that the quadratic equation
has solutions x = a and x = -1/@. we get
this leads to the quadratic equation
Equation (4.
1. respectively.. so that .l/qi represent. . a2.].-¡.all. a 11 is the continued fraction for a with the period reversed. ~ ~ . a' lies between .1/@ = a is the '
second. 4. and. a = 3. Moreover.9) is
and hence
' which shows that .-.694 > 1.
PROOF.1)st conver~ents the continued fraction [a.if 8 = [a.361. a. the purely
[al.l / @is equnl to a. a.1...[a. namely that if
. and a = -0. are posz'tive integers. the nth and (n .-1. or conjugate root. al] = Q n 7
J
and
Where p:/qL and pk . . to three decimal places. a2. . then .
y
an]
is greakr ¿han 1 and as ¿he positiue root of a quadratic equation with integral coeficients.
(b) The general case.
a..1 < a < 0. periodic continued fraction
=
.. require t ~ v oresiilts stated in Problem 7 of Set 3. equally inzportant. a.
then
Pn --
pn-I
.a27 . of the quadratic equataon satisfied b y a.1 and 0. a2.
P.PERIODIC CONTINUED FRACTIONS
93
The positive root @ of (4. We shall now prove
THEOREM I f al. We page 26. of .
. ' ' The purely periodic continued fraction a is indeed a quadratic irrational. a. kioreover.= [al.
rln
Pn
. a2. ..
9.15) we can replace (4. a2./qk and p .1 are. we obtain
and again.-l/q. equivalent to the quadratic equation
Revetsing the period in a. . respectively. according to (4.-1
= O. it follows that
Since a is purely periodic we can write it in the form
and. Equation (4.6). Using the results stated in (4. . in the form
where p. .]. respectively.-l are defined. and p.-1.-1)
(. .p.16) is and (n .
.19)
qn
(-
./qk .6). as the nth y a. P a.].94
CONTINUED FRACTIONS
Since convergents are in their lowest terrns.. according to (4.f) .1)st convergents of [al.(p.l)st : convergents to [a./q..18) by
so that fl satisfies the equation which is equivalent to the equation (4. the nth and (n . a. we see that
where p.
though the result was implicit in the earlier work of Lagrange. bz. < ' the root a = . such as a = [al.NOW. a.1//3 lies between 1 and O. while in caee (iii) the second root is necessarily greater than O. and if the second root &' of this quadratic equation lies between .
(ii) Those with an acyclic part consisting of a single quotient al. 0 = [al. ba. .-1.1/@. thus we have p > 1. and the root $2 = .]. . then the continued fraction expansion of a is purely periodic.1 and O. (iii) Those with an acyclic part containing at least two quotients. whose second root a lies between . such as a == [al. a. This remarkable fact was first proved by Galois in 1828. siich as . and hence satisfies a quadratic equation with integral coefficients.1 is also true (and will be proved in Section 4. bz. . In cases (ii) and (iii) it can ' also be proved that a is a quadratic irrational satisfying a quadratic equation with integral coefficients.17) and (4. .-1. bl. al are al1 positive integers.. . . . Simple recurring continued fractions may be grouped as follows:
(i) Fractions which have no acyclic (or non-repeating) part. ' What is to be emphasized is that these few conditions on a and a conzpletely characterixe the nui~zberswhich have purely periodic continued fraction expansions. as. O < l/@ 1. where a. al].PERIODIC CONTINUED FRACTIONS
95
Comparing equations (4.].. We will not prove theae last two results. a. This means that if a > 1 is a quadratic irrational number. a2..19). In other words.
.
-
m
S
We proved. bi. .5). that a is a quadratic irrational which satisfies a quadratic equation with integral coefficients.1 and O. as. for fractions of type (i). . This completes the proof.P stands for the purely periodic continued fraction [a..
a
-
The converse of Theorem 4. and so -1 < -1/@ < O. bn]. we conclude that the quadratic equation has two roots: The root XI = a.. b. but in case (ii) the second root a ' of this quadratic equation is either less than -1 or greater than O.
1 and 0. are irrational. then
=
Bt. (b) that y = 11. not a perfect square. (b) find the equation of which a is a root. and that a therefore lies between -1 and 0. TOprove this.3 Quadratic irrationals
I n this section we shall be concerned mainly with nurnbers of the form where A and R are arbitrary rational numbers. does not lie between . aside from trivial variations such as
+
+
In other words. for an arbitrary but fixed positive integer D. so that . If a = [2.A2 = O or A l = AS.
4. 2. write the above
. First we observe that.Ri)
45. 31 satisfies an equation whose other root. Hence the assumption that B # Rp leads to a contradiction and we must conclude that i B . a'. Verify numerically (a) that a == [l. and therefore.
if and only if Al = A 2 and BI equality in the form
Ai
if B2 f R1.A2
=
(B2 . 1 (a) verify nurnerically that a > 1 and f? > 1.CONTINUED FRACTIONS
Problem Set 14
l. 21. there is only one way of writing the number A B ds. of this equation satisfies the relation a' = -1/@. and where D is a fixed positive integer not a perfect square. 31 satisfies an equation whose other root.
. a'. is positive. A l .\/E. = B2. and hence also A B d D . '
-
-
2.
b
would be rational. (c) show that the other root. y'. 6 and f? = [6. 2. contrary to assumption.
a >O . if a # O. When we speak of quadratic irrationals.4ac. the result is again of this form. we clairn that when numbers of this form are combined by any of the elementary operations of arithmetic (addition. but cal1 attention to the fact that in this connection. i. subtraction. We leave the proofs of these properties to the reader (see Problem 1 of Set 15). division). Clearly if a = O. multiplication.PERIODIC CONTINUED FRACTIONS
97
Next. i s the root of a quadratic equation ax2 bx c = O.
+ + +
+
In order to prove the statement in italics we recall that any quadratic equation ax2 bx c = 0.
+
We prove next that every number x = A B 45. has roots
+ +
where D = b2 . and consequently
Hence. "numbers of the form A B " include those for which R = O. we shall assurne B # O. c are integers and where b2 .e.4ac > O. however. not a perfect square. since otherwise the number under consideration would be rational. where the coe& cients a > O. we can verify by direct substitution that
. b. where A and R # O are rational and D i s a positive integer. ordinary rational numbers. x = -c/b would be rational and hence could not represent the irrational number A R 1/Ti. we can replace a s 2
+ bx + c
=
O by
c .
Conversely.
the common denominator of the rational numbers 2A and A 2 .4ac is not a perfect square. a > 0.4ac of this last equation is positive . satisjies one and only one B quadratic equution as2 bx c = O where a. since D was assumed to be positive.B2D) = A2 k ZAR 45 + B2D .B 0 need not have integral coefñcients. for. c have nofactors in were a root of comrnon. Finally. For. # O.2A2 T 2AB Z/D + A 2 . but if we multiply through by a. Observe also that b2 . provided B tr O. The above discussion leads us to a precise definition of a quadratic irrationai. b = -2aA.ZA(A I B fi)+ (A2 . The numbers A B <D we have been dealing with are therefore al1 quadratic surds according to this definition. we obtain the quadratic equation where the three coefñcients .98
CONTINUED FRACTIONS
=
(and x (A
A-B
a) this last equation: satisfies
A B 0 and A . The equation x2 .
+ B f i 1 2 . and c = a(A2 . or quudratic surd.B2D) = O satisfied by
+
+
A quadrcatic surd A R 0. the discriminant b2 . if x = A R and also of g2(x)
=
+ + + + 4s
a2x2
+ b2x + c2
=
0.R2D) are integers.aib2 # O.
b
a
then it would also be a root of the equation Now if a2bl . b. then this would imply that
.2Ax + (A2 .R2D = 0. it is a number which satisfies a quadratic equation whose coeficients are integers and whose discriminant is positive but not a perfect square.B2D.
leaving the rest as problems. difference. then
2. difference. The conjugate of the conjugate of a quadratic irrational number a is a.
a' also satisfies this equation. or from consequence l. b2 = kbl. or quotient of their conjugates. or quotient of two quadratic surds al and a2 is equal. contrary to the assumption that x is irrational. The conjugate of the sum. and hence that
that a2 = kal. (Why?)
+ bx + c = O. if a2bi .. In symbols.
SO
Every quadratic irrational has a conjugate at=A-Rdii formed by merely changing the sign of the coefñcient B of < . On the other hand. This follows directly from the definition of a conjugate. one being merely a constant multiple of the other. c2 = kcl and the two quadratic equations gl(x) = O and g2(z) = O are actually equivalent.
-
Al
+ Bi fi
and
a 2
= A2
+B 2 6 .
. because a quadratic equation has only two roots. this means that
(al az)'
= al
I
&.
We prove the first assertion. D This definition has a number of useful consequences:
1 If a satisfies the quadratic equation ax2 .
3.alc2 = 0.a1b2 = O then the equation
+
implies that a2c1 . Henoe in this case x = A R 1/D could not satisfy both equations. Thus if
a. product. product.PERIODIC C O N T I N U E D F R A C T I O N S
99
is rational. respectively. to the sum.
or that O < P < fi.
We have shown that if a is a reduced quadratic irrational of the f o m (4. given ' by (4.P < Q < V% and
+ P < 2 .
. fi .1 Finally we that
a
>
1
< O. that
u =
a. >1
and and a '
-1 <a'=
P . D will be as defined by (4. Under these assumptions the quadratic irrational a given by (4. i.1 and 0.\/»>
shows that observe that
-Q.20).23)
O
< P < fi. Q. I we assume that D > O is not a perfect square. Suppose. The implies that P + 0 > Q . Q. then the integers P. that the value of a given by (4. Throughout the rest of this chapter. D satisfy the conditions
(4.21).1 irnply that
a !
+ a' > O.20) i s sazd to be reduced if a i s greater than 1and if its conjugate a.
P + f i
Q
a !
> 1.f i -
Q
< 0. P. then the roots a and u are quadratic surds of the form A f B ' where A = P/Q and B = 1/Q are rational.
we conclude that P
> O.
and since Q
> O. from
and it follows inequality a'> . It is important in what follows to find out more about the form and properties of reduced quadratic irrationals.20) is a reduced quadratic irrational..e.
or
The conditions
>
.
Also.\/D. then.PERIODIC CONTINUED FRACTIONS
101
and
where
f are integers. lies between .22). and the inequality
PP-.
or
fl-P<Q.
for.4ac > O not a perfect square. However. there is only a finite number of positive integers P and Q such that P<1/Dand Q < 2 0 . This follows directly from the inequalities (4. for any given D > 1. a > O. To establish this result. namely
+
where X is the largest integer lessthan With this determination of A. once D is fixed. l/al satisfies the
+ +
+
+
. there exists always at least one reduced quadratic surd assoez'ated wz'th it. This idea. and is intimately related t o the theory of reduced quadratic forms. Q = 2a.20). and where al is again a reduced quadratic surd. The quadratic equation satisfied by a and a' is
+
6.23). however.22). i t muy be expressed in the form
where al i s t h largest integer less than a. the importance of the idea depends upon the fact that for a n y given D there i s only a Jinite number of reduced quadratic surds of the form (4.102
CONTINUED FRACTIONS
The reason for introducing the notion of reduced quadratic surds has not been explained. where al is the greatest integer less than a.0 obeys . c are integers.
It is necessary to have the following result: I f a i s a reduced quadratic surd. not a perfect square. Write cw in the form a = al l/ai.1 < a' < O. Clearly a = a. where a. is a well-established concept in the theory of numbers. For our purpose. b. X 0 = a is clearly greater than 1. P = -b. see (4. Could it happen that there are no reduced quadratic surds of the form ( P 1/D )/Q associated with a given D? If so we might be talking about an empty set of reduced surds. and its conjugate a' = X . let the reduced quadratic surd a be the root
of the equation ax2 bx c = O. and D = b2 .
1 < a < O. QI.23) are automatically inherited by P1. and it only remains to prove that . It will now be shown that al is a reduced quadratic surd. for. as required.
QI = 2(aa:
+ bu1 + c). so al > 1.1/ar is also a reduced quadratic irrational. .(2aa1 + b). therefore O < l/al < 1. we obtain
where
Pi =
and
. we prove that if a is a reduced quadratic irrational. To this end. and hence the inequalities (4.PERIODIC CONTINUED FRACTIONS
103
quadratic equation
Solving for the positive root al. It is also clear that Pi.
. we recall that al is the greatest integer less than a .1 < 4 < O. by hxpotheses. then its asaociatc /3 = . and DI = D are integers.1 < a: < O. we obtain
+
Theref ore
' since al 2 1 and. or . Solving the equation a = al (I/al) for al and taking the conjugate of the result (see page 99).and DI = D. Finally. QI. Thus al is a reduced quadratric surd.
These expressions give us the expficit form of al. and
has the same irrational part t/D as a has. If follows that O < -a: < 1.
+
the fornl
4.23) imply that 1 < a and . then we show that this expansion is necessarily purely periodic . Determine al1 the reduced quadratic irrationals of (P f i 3 ) / & . i f
+
2. This we established in Section 4.1 < a' < O imply that /3 that 0' = . prove that conditions (4. then al is a reduced quadratic irrational.4. .
and
Problem Set 16
1.
PROOF. and where
is again a reduced quadratic irrational associated with D.1 quadratic irrational.
+ ( l l a l ) . In other words.
. so that a > 1 i s the root of a quadratic equation with integral coejiaents whose conjugate root U!' lies between .1
We are now ready to prove
THEOREM (CONVERSE THEOREM ) .
> 1. then the continued fracticm for a i s purely periodic. If a i. where
that.104
CONTINUED FRACTIONS
the inequalities a > 1.1 and 0.
.s a reduced 4. The first step is to express the reduced quadratic irrational U! in the form
-
where a l is the largest integer less than a.6 Converse of Theorem 4. Show
a = al
a = t ( 5 4 1 is expressed in the form 3) al is the largest integer le^ than a . Shoiv that the conditions (4.23) are necessary and sufficient conditions for a [defined by equation (4.1 and 0. < af < o.20)]to be a reduced quadratic irrational.2 OF 4.l / a lies between .
3 .l . first investigate the actual expansion of a into a We
continued fraction.
PERIODIC CONTINUED FRACTIONS
105
Step (4.24) is the first step in converting U! into a continued fraction. and where
Since u is irrational this procesa never comes to an end. Repeating the procesa on al. and
is a reduced quadratic irrational. and
Continuing the process. al. At this stage we have
where a. al. are al1 reduced quadratic irrationals associated with D.we obtain
where a2 is the largest integer less than al. a2. and hence we seemingly are gerierating an infinite number of reduced surds
. a2 are reduced. we generate step-by-step a string of equations
where a0 = a.
For this purpose. from which it follows that To prove (ii) we shall show that
e
=
az+3. It then follows that the recipro~ l cals of aa+l and a ~ are equal and hence that a k + l = QZ+I. we obtain and
and hence
... O<k<l. is repeated.. we have and taking conjugates. we use the conjugates of the equal complete quotients a k and al. al1 associated with D. al.
. and. . we must arrive eventually at a n reduced surd which has occurred before. also yields a[k+2 = a1+2. in other words. a k = a~ implies a k + l = al+'^.. and that al is the first one whose value has occurred before.. l1 .4 that there can only be a finite number of reduced a l a associated with a given D . = a. in other words.. so that al = a k . a = ao.1 = a . al-I are different.
--
. . Thk argument.106
ao. therefore.
(ii) The very first complete quotient. a 2 . Suppose. a. ai. we merely recall that
are the greatest integers less than a k = a*. ak-2 = al-%.
ak+3
for O < k < 1 implies a k . al.
To prove (i). . al1 subsequent completa quotients are repeated. obtaining oi. then.. QO = al-k. It is then possibIe to prove that: (i) Once a complete quotient is repeated. that i the sequence a11 the complete quotients ao. the sequence a = ao. when repeated.
a k
=
al
Now if k # O. a k + 2 = QZ+~. since ak+i and we may conclude tbat a k + = al+l..
.
e
C O N T I N U E D FRACTIONS
But we proved in Section 4. is purely periodic.
ak-1
I
and
. also.t?k = 01.
Pk-1
1
and
o < -al-l
t
=P1-1
1
< 1. it follows that a k = al and hence.27) are the largest integers less than Pkj 01. NOW 7C . if a k ifJ not the very first complete quotient.27)
Since so that
1
B k = a k + Pk-1
ak-1..
This shows that the a k . and since . in expanding the reduced quadratic irrational a into a continued fraction we generate the string of equations
.7 = al .28) is and the right side is al-l..
and
61-1
al-1
are reduced. a 2
6 =al+1
1
1
which is the same as saying
(4..he first a. if i.
until we arrive a t t.
ak-3
QE-3. we have and
-1
-l<ak-i<O
<
a[-l < O
o < -ak-1
t
= -<1. al in (4.
etc.ak
.1 # O.~ = al-l.al-1.e.PERIODIC CONTINUED FRACTIONS
107
t
1
.7 -
ffk
. we may repeat this argument 7c times to prove that
ak-2
=
al-2. that
Since the left side of (4. and obtain Thus. we have shown that a k = al implies a k . respectively.
.
x" ia 1x1. . and then G to F(a.). will also repeat :
Therefore. we have adoptcd thc
t The traditional notation for "largest integer less than
bracee hcre. a. al. a.]
of a purely periodic continued fraction. This completes the proof of Theorem 4. a2. .+l. into l/an+r and G maps I/a.. that is. the reciprocal of the next a . a 2 . such that F maps a. and from this point on the a's repeat. observe that
where ak+ is the largest integer less than a k .. To define the funct'ion F. to every ak. but since this conflicts with oiir notat.-1 are al1 different. and where a. F(x) and G(x). Since for every a k > 1. = a. . Before extending the proof to al1 quadratic irrationals (reduced and not reduced).. we shall obtain a. we present a graphical illustration of the periodic character of the complete quotients al. By first applying F to some a.+l into its reciprocal. it is clear that the sequence al. in the expressions
We shall define two functions.
S
S
.. a. there exists exactly one biggest integer ak less than a k . t'he continued fraction for a has the form
-
a = [al.108
CONTINUED FRACTIONS
where a. a. a ~ .+l. Let the symbol { x ] denote the largest integer less than x.
.ion for continued fract.2. t then we may write
and we define the function F accordingly :
4
We now have a function which assigns. a2. .ions.
The projection of C 0nt. consists of one branch of the equilateral hyperbola y = l/x.PERIODIC CONTINUED FRACTIONS
109
h'ow. the appropriate definition of G is simply so that In other words. plot the functions F(x) = x . We then find the point on the graph of G(x) which has the same ordinate as the point B. since the reciprocal of the reciprocal of a number is the number itself. The graph of F(x) consists of the parallel line segments and the graph of G(x). for positive x. G[F(ak)l
= W+I. Let a be the given quadratic irrational. namely l/crl.{xj and G(x) = l / x on the same graph paper..
Figure 10
In order to apply this scheme graphically. we cal1 this point C. hecause
.e. see Figure 10.o the x-axis represents the value of al. to the point F(a) = B]. We locate it on the horizontal axis (poii~tA ) and find F(a) = l/ai by rneasuring the vertical distance from A so the graph of F ( x ) [i.
a double arrow from a to a 2 . eventually a reduced complete quotient a + is reached. . has a periodic continued fraction expansion.2. are periodic. then there will be a repetition and the a are i periodic. the abscissa of C' represenb the value of a . If. in the course of our path. Observe that the partial quotients al. Any quadratic irrational number EM
a has a continued
fraction expansion which is periodic frorn some point onward. going from A' to B' to C ' . al. then the path will eventually repeat itself . Problem Set 17
1. . we are led t o a l point on the hyperbola which was already covered by the earlier part of the path. if the a.5) we know that
.110
CONTINUED FRACTIONS
Starting with al.
PROOF.we now repeat this procesa. etc. Shoiv that a
2.4].
4. a single arrow leading from a to al. and . respectively. by equation (4. Show that
= 1 d is reduced and verify that its expansion is 2 the purely periodic continued fraction [$l.3. central idea of the proof is to show that when any The
quadratic irrational number a is developed into a continued fraction. [2. a2. Let the expansion of a be
#
Then. although not purely periodic. Conversely. Use tl~e graphical method explained at the end of the last section in
order to show that d5.6 Lagrange's Theorem
T H E ~ R 4. can be determined by recording which segment of F(x) is hit by the part of the path issuing from a.
3.
+
di is not reduced and that ita eontinued fraction expansioii
is not purely periodic.l from then on the fraction will be periodic by Theorem 4. s The arrows in the figure indicate the paths that lead from each a to the next.
but they will be alternately slightly less than 1 and slightly greater than 1.
. < 1. tend to the limit a.1 and O.29) will of necessity líe between .PERIODIC CONTINUED FRACTIONS
111
where a and a.30) slightly less than 1. We know also that the convergents c. solving for
%+l. the values of the fraction (4. are convergents to a. = pn/q. once we have found a value of n which makes the fraction (4. by Theorem 4. and hence eventually. and consequently
a'
. But from our study of convergents in Chapter 3 we know that as n increases indefinitely. we get f
a.2 the continued fraction for a will be periodic from there on.
Factoring the numerator and the denominator.. as n increases. We notice also that in (4.30) will not only get closer and closer to 1.29) the numbers and are both positive integers and (see page 67) that O < qn-1 < q. both en-1 and c. are alternately less than a and greater than a. Thus.+l are quadratic irrationals and conjugates of both sides o this equation.-Jq. so that qp. Thus Lagrange's "theorem has been proved.cR-1 a' cn
-
tends to
a '
01'
-a
-a = 1
as n approaches infinity. This proves that a is redueed .
Taking
or. the wlue o f given by (4.+l
> 1. this gives
where c = n l / l and c.
+. which is reduced. First ilotice that 4 Ñ is greater than 1. and hence its conjugate . < a l . Show that a
= i(8
+ dE) not reduced. Adding al to both sides of (4. the continued fraction for 4 Ñ has ail interesting form.CONTINUED FRACTIONS
Problem Set 18
1. and its expansion
cannot be purely periodic. if is
we eventually come to an a. .N -\/Ñ al is redoced. so N < is no¿ reduced.31) we get
+
+
+
aild since this expansion is purely periodic it must have the forrn
Consequently.
4.4 cannot lie % between . since al is the largest integer less than d í the number 4% al is greater than 1. does lie hetween -1 and 0. h.
. and verify that the expansion is periodic from then on. so and its conjugate.7 The Continued Fraction for
-\r~
If N > O is an int'eger which is not a perfect square.1 and O. For example. On the other hand. the expansion for 1 is 6
where the period starts after the first term and ends mrith the term 2al. but that.
Wc assume that N is not a
. that continued fraction expansions are uniqiie. al.32). ar.dr al is the conjugate of a = -\TN al. a2. al.PERIODIC CONTINUED FRACTIONS
113
Notice that.
-
.35). x2 . for addit'ional examples. subtracting al frorn both sides of this equation yields
and the reciprocal of this expression is
We know.33) .
4. however. recall from Section 4. hence. Hence.. To investigate the symmetrical part.2 that if a' = . un. page 116.
See Tahle 2. and where x and y are unknown integerri whou* vriliic*~wcB are seeking.8 Peii's Equation. we obtain
+
+
On the other haiid. except for the term 2a1.34) and (4. 2~11. an. but with the period reversed. then the expansion of -l/a' is the same as that of a. we conclude that
It follows that the continued fraction for f orm
di necessarily has the
d~ = [al. the periodic part is symnzetrvical. ñ e can obtaiii the expansion for (fial)-L quite easily from (4. The symmetrical part may or may not have a central term.Ny2= -t 1
At the beginning of this chapter we mentioned that the cattle problem of Archimedes reduced t o the solution of the equation In this section we shall discuss the solutions in integers x and y of the equation wheru! N > O in s given integer. reversing the period in (4. comparing (4.
3). Hence the last equation requires that and
alqn
+
+
+ qn-I 'Pn*
. 1)2 . c.l t cn = pn/qn which come immediately before the term 2a1 in (4. otherwise the equation is of little interest. p. multiplying both sides by the denominator. d are integers and ..114
CONTLNUED FRACTIONS
perject square.l / ~ n . since the difference of two perfect squaresis never equal to 1 except in the special cases (+.-. Replacing a. (Why?)
The continued fraction expansion for -\TN supplies al1 the equipment we need to solve Pell's equation x2 . b.\/Ñ is irrational. we get
which is equivalent to
Now this is an equation of the form a bN < =c dd Ñ . qn-l. We know that
-
-al+--+
-
-
+ + an+í
an
-
?
where
We again use the fact that
where p.37).Ny2 = 1. and this implies that a = c and b = d (see Section 4.+l in (4. where a. or x2 .39) by the right side of (4.38) yields
then. provided solutions exist..02. and q. are calculated from the two convergents cn-l = p n .N1~2= 1.
4 we know that and. Notice that
so that the term a.40).Ny2 = 1. If n is odd and we still desire a solution the equation x2 .
But from Theorem 1.41) becomes
and hence a particular solution of Pell's equation x2 .-1 and 9.. then 2 2 fin .1)2n= 1.-1 and qn-I from (4.PERIODIC CONTINUED FRACTIONS
115
Solving these equations for p. is actually the term azn. where it occurs for the second that time. when it occurs again. we move ahead t o the second period in the expansion of TN. with the values of p. . is.1 . Y1 = Q2nl gives us again a particiitar uolution of the equation x2
.
. this equation has the form
If n is euen.Ny2 = . out to the term a.
Y1
=
qn
gives a particular solutioii of the equation x2 . equation (4.Ny2 = 1. then
and
X1 =
Pn.-1 we find that
in terms of pn and qn.Nqzn = (.Ny2 = 1 is
If n is odd.and so 5 1 = P2n.
Here we shall confine our examples to eqriations that have solutions.1 and not + l . give x2 . yi = q6 = 13.3 y 2 = .N y 2 = .21 y2 = 1 .
x2
. 81 = [al. A calculation
hence
XI =
55.
2. The next period gives the convergen&
and so.1 can be solved. Here N = 21. Hence. expansion of The
4% is
so that n = 5.
EXAMPLI.1 has no integral solutions. it can be proved (see Appendix I a t the end of this book) thilt the equation x 2 . For example. and the continued fraction expansion given in Table 2 is
fi= [4.1. The first five convergents are
But xl = p6 = 70. if we take
XI
Z
= 9801. 1.96059600 = 1. an even number.13* = . so that and
=
as. = as.29y2 = 1 . h'ot al1 equations of the form x 2 . :
. yl
=
12 is a particular solution of the given equation. an odd number.
6.
SOLUTION. a*.
cvhich shows that a.: Fiiid a particular solution of the equation 1.
we get
X:
. as. 1 . 1. SOLTJTION. we must move on to the next period.2 9 ~ = 96059601 . 2.29y2 the value 702 . as.PERIODIC CONTINUED FRACTIONS
117
The above analysis shows that we can always find particular solutions of the equation and sometimes particular solutions of the equation x 2 . 2a11. so that n shows that c6 = 35.
y1 = 1820. 1 .N y2 = .29 . EXAMPLK Find a particular solution of the equation x2 .
Ny:' = 1 or x: : = -1. y. in turn.9 How to Obtain Other Solutions of Pell's Equation
We have seen that Pell's equation x 2 . which indicates that \ve have to move to the next period to obtain a solution of the equation x2 .1.
-
THEOREM 4. we can systematically generate al1 the other positive solutions... y1 = 5 is a solution of the equation x2 . Show that xl = 8. In this Table.29y2 = + l . but that not al1 equations of the form x 2 . for the Table shows that
-
and gives a solution $1 = 70.1.118
CONTINUED FRACTIONS
The solutions arrived at in Example 1 can be checked against Table 2. that is.1.1.8]..13y2 = . We shall state the main theorems involved and illustrate them by examples. Problem Set 19
l. then al1 the other positive solutions (x. ?ji > O such that x .Ng2 = . These statements will not be proved.2. Once the least positive solution has been obtained. However.13g2 = 1. Likewise we can check Example 2. y1 = 12 of the equation x2 21y2 = 1. = 3 is a solution of the equation z2
.
as indicated in Table 2 . = 13 of the equation x2 .4.
4.1 have solutions. and further to the right we find listed a solution xl = 55.8 will always produce the least positive (minimal) solution.
The values of x. N a positive integer not a perfect square.
. and proceed to the next period to find a solution of the equation x2 . 3. 2. If ( 2 1 . if either of these equations has solutions.N y 2 = 1. yl) i s the least posz'tive solution of x2 . it will always produce the two smallest integers X I > 0.42) by expanding the term (xl y 1 d Ñ ) n by the binomial theorem and equating the
+
. Show that xl = 18. are obtained from (4.7 y 2 = 1. y.
2.N y 2 = I . n = 1. then the method outlined in Section 4. opposite N = 21 we find the expansion o f dE=~=[4.) can be obtuined jrom the equation
by setting. and y . can always be solved.1.29y2 = -1 .
N y 2 = 1. and use (4.42). y1 = 12 is a solution (minimal) of the equation x2 . are solutions of the equation x2 .PERIODIC CONTINUED FRACTIONS
119
rational parts and purely irrational parts of the resulting equation. Using these values.8 we found that X I = 55 and 1. y.43)
y. if (xl.N y 2 = 1. A second solution ( 2 2 . N < = ( X I .y2) can be found by taking n = 2 in (4.42). x. this gives
.43):
-
r'. Since the conjugate of a ~ r o d i i c t t. and y. are calculated b y equation (4.Ny.42).y*) is a solution of x2 . 1/2) can be obtained by setting n = 2 in (4.he product of the conjugates.y1 fl)'. For example. Now we factor xn .
EXAMPLE In Example 1 of Section 4. yl) is the least positive solution of x2 . from (4.N y .
where there are n factors in tbe expression on the right-hand side..21y2 = 1. a
since by assumption ( x l . We have.N?/: = (xn
+ yn1/Ñ)(xn -
l/a
fi)
Thus x.42). . if x. and y2 direct calculation shows that
+
=
2xlyl.42) and (4.N y 2 = 1. This gives
so that $ 2 = x. then x: . then the solution (x2. It is easy to show that. = 1. is this @ves
(4.
x = & l . C .10y2 = 1. equations of the form
Ax2
+ Bxy + C y 2 + D X+ E y + F
=
0. see Figure 1 1 . Table 2 shomrs that xi = 18. Then.
In concluding this section. Now the problem is reduced to finding integral solutions of the equation Solve t h h equation and list the first four solutions of x2 * such that y . Use Theorem 4.n2) = (m2 n2)2.
y
-x
=
m 2 .1 . then the values
+ y2 = z2. This involves an extensive study. 2. y2 = 6.N y 2 = 1 is preliminary to the study of the most general equation of second degree in two unknowns. and where x and y are the unknown integers.2n2 = -+l. find two solutions of the equation x2 . and l g a .
where A .n 2 .13y2 = 1. B . = 17.
+
+
+ y2 = z2
. By means of certain substitutions for the variables x and y.5 to find the next solution. y1 = 4 is the minimal solution of the equation x2 . ure remark that the study of the equation x2 . E and F are integers.
+
+
We now propose the problem of finding right triangles with legs of lengths x and y.
-
3.PERIODIC CONTINUED FRACTIONS
121
This gives $ 2 = 19. 62 = 1 so t h a t these values are solutions of x 2 . $he solutions of this equation (if they exist) can be made to depend upon the corresponding solutions of an equation of the form x2 . Consider the Pythagorean equation x2 integers. Problem Set 20
1. Table 2 indicates that x. Use Theorem4.4 to find the next two solutions. Also. D . yl 5 is the minimal solution of the equation x2 .
Let m . so that x and y are consecutive integers.Ny2 = dd.2mn
=
(nz . so that m = u n = u v.10 .18y2 = 1. +
if ?n and n are
will always give integral solutions of x2 y2 = z 2 because of the identity (2mn) (m2 .n = u.n ) 2 .13y2 = . and so we must be content with this introduction. n = u.
y. the angle e between x a n d z approaches 600. Find sets of integers (x. as these integers increase.
Figure 11
.122
CONTINUED FRACTIONS
4. 2) for the sides of the right triangle of Figure 11 such that.
however.2 Statement of the Problem
Throughout this chapter let a! be a given irrational number. and would lead to the general solution. Proofs of the theorems stated here and of many related theorems can be found in the books by Niven [8]. in integers. and let p/q be a rational fraction. however small.Ny2 = M could be undertaken. if E s is any given number. with positive q. in other words. q such that
.1 Introduction
In this chapter we shall preview some results t h a t can be studied once the first four chapters of this book have been mastered. on theorems related to the approximation of an irrational number by a rational fraction.
6. where p and q have no factors in common. We have already indicated that a complete study of Pell's equation x 2 .CHAPTER FIVE
Epilogue
6. of the equation
Ax2
+ Bxy + Cy2 + Dx + Ey + F = 0. It is clear that we can always find a rational fraction p/q. as close a we please to a. and Hardy and Wright [5]. we can always find relatively prime integers p.
We shall concentrate now.
with q 2 1 . of the form p/q.3 Hurwitz's Theorem
*
Inequality (5. .
5.3) has this interesting feature: I f a is anu irrational number. to the conAny of the convergents p l / q l .1. how small can we make e ? We have already accomplished something along these lines.9..124
CONTINUED FRACTIONS
But this is not the interesting point. is there a number k > 2 such that the inequality
has infinitely many solutions p/q ? If so. then how large can k be?
. p2/4?./q. the inequality (5.2). Theorem 3. It is possible to sharpen the inequality (5. such that
then it can be proved that p/q is necessarily one of ¿he convergents of the simple continued fraction expansion of a. p.+l/q. . tinued fraction expansion of a can serve as the fraction p/q in (5. given a and q. q > O. stated here without proof.2) as shown by the following theorem.
THEOREM Of any tlwo consecutive convergents p.how large must q be? Or. p. there exists an infinite number of rational fractions in lowest terms. We proved in Chapter 3. such that
. that if a is irrational.3) immediately suggests the following question concerning still better approximations. Given an irrational niimber a ./q. at least one (call it p/q) satisfies the inequalit y
Moreover. and if p/q is a rational fraction i n lowest terms.+l to the continued fraction expansion of a.l).. and 5. What we should like to know is : Given a and e in (5. .
2. -1.-1
=
pn and
I t can be shown that.
]
=
[O. a. . if the continued fraction expansion of a is [al. a.. the "simplest" numbers are the worst in the following sense: The "simplest" irrational num ber is
f. .
where each ai has the smallest possible value. ./q. approxima¿ions p/q which satisf y the inequality
The number d is ¿he best possible number. then the rational approximations p. /5 These remarks suggest the truth of the following theorem.EPILOGUE
125
I t can be shown that. On the other hand if there are small numbers in the sequence al. . no matter how far out we go. a. and if p. Iarger number were substituted f o t
4. a2. . 2
m
-
. 5
. then
m
hence we can get very good approximations to U! if the numbers al./qn cannot be too good for small a. From the point of view of stpproximation. the expression
gets closer and closer to 1 d qi..T]. The convergents to are the fractions
SO
that q.. the theorem would become i false if an. get large very fast. a2.1 = [O) 1. . 1... is the nth convergent. for n very large. . THEOREM Any irrational nulnber U! has un in$nity of rational 5. ... .
=
2/5 . first proved by Hurwitz in 1891. . .as) .
not an infinite number. the irrational nunzber O < fl < 1 lies between two consecutive fractions p / q .
+
. Substituting for /3 in the above inequality. we let 8 = a ..6).n where n is the greatest integer less than a. the one important for this study is: If for any n.2 depends upon the fact that. the sequence F . Hunvitz did not use continued fractions. r / s can be used for x / l j in the inequalit y
+
+
In order to make such an inequality valid also for an irrational number a > 1.\/5 is the best possible number in this sense. is the set of rational numbers a/b with O 5 a 5 b 5 n. The first four sequences are:
4 5
These sequences have many useful properties. b) = 1. in the continued fraction expansion of a.2. (a. instead he based his proof on properties of certain fractions known as the Farey seqzdences. r / s of the sequence F. This is the central idea in Hurwitz'e proof. One proof (by means of continued fractions) of Theorem 5. then there exists only afinite number of such rational approximations p/q to a.126
CONTINUED FRACTLONS
By "false" we mean here that if were replaced by any number k > fi. For ' complete details see LeVeque [7]. For any positive integer n. Niven [83 gives an elementary proof that . ( p r ) / ( q S ) . arranged in increasing order of magnitude. I n his original proof of Theorem 5. then at least one of the three ratios p / q . at least one of every three consecutive convergents beyond the first satisfies the inequality (5. we obtaiii
where x = ny x .
that a a. if a real number a! has the contjnued fraction expansion
a = !
[al.
(i) refle. then y x. Now.2. d satisfying the
and siich that
. Such a statement always seems to stimiilate further research.
An equivalence relation divides the set of a11 numbers into equivalente classes in such a way that each number belongs tJoone and only one equivalence class. i.e. I t is equivalence just defined has al1 the properties lente relation. I a certain class of irrationals f were ruled out.q. could this constant perhaps be replaced by a larger number? Indeed this can be done. (5. 6 2 1. then x N Z.
it follows from
p.6). and ud .
N N
-
x). x condition
N
y) if there are integers a. ffn+~]. b = 3.p. Hence if a! and /3 are any two real numbere with coiitiriiied fraction expansions
and from
-
.7) and (5. c.
m
. i.\/2: and x = (2 d because x = (a b)/(c fi d) with d = 1. = ( .+~ [cf. b. (iii) transitive.EPILOGUE
127
Mathematicians-are never content mith a "best possible result". if y = .bc = 2 . and are therefore just as hard to approximate.1) which forced us to accept .-~q.\/5 as the "best possible" 6 constant iTi the inequality (5. if x y and y Z.e.1)" (see Theorem 1 .e. The class of irratjonals to be excliided consists of al1 numbers eqztivalent to the critica1 number l = 4 ( . a n .. We shal show that al1 numbers equivalent t o l have the same periodic part a t the end of their continued fraction expansions as 6 has.3 = -1.2:
can be expressed in terms of y by the fraction
s Forexample. if x y..-1 . x
a = 2 .8)].
DEFINITION a number x is said to be equivalent to a number : Here
y (in symbols.. i. namely that it be
+
+
+ 3)/(*
+ l). 4 . such as the constant & in Theorem 5. c = easy tso see that the reqiiired of an equivaN
-
y 1. every x is equivalent to itself (x (ii) symmetric.cive.
There are infinitely many irrational numbers equivalent to f = *(& . C O . then a an+l 0. -
11..
and since 1 1. C I . . -1. b.
+(a
. SO a N P. and hence al1 these equivalent irrat'ionals play essentially the same role in Hurwitz's theorem as the number E = + ( 4 5 . x y.1/51 has an infinity of rational approximations p/q which satisfu the inequality
There is a chain of theorems similar to this one.
that is. each of these is expanded into a simple continued fraction.ain place on. by Theorem 5.
13 . Then..3. Any irrational number /3 not equivalent ¿o [ = +(1 . and only 2f
0
. each of these expansions will contain the same sequence of quotients. from a cert.+i b. I n particular any two rational numbers x and y are equivalent.1 ) does. . It seems reasonable t o guess that if we rule out the number f and al1 irrationals equivalent to it.bn. let us suppose that. THEOREM 5. b. c2. i f and only i f the sequence of quotients in a afier the mth i s the sume as ¿he sequence ~ T LP after the nth. The question as to when one irrational number is equivalent t o another is answered by the following theorem. stated here without proof .
co. .4. .a2. then the constant 4 5 i1i Hurwitz's theorem could be replaced by a larger number.9) can be replaced b y any number less than or equal lo 4 2 2 1 / 5 .ci.in (5.
[al. . .128
CONTINUED FRACTIONS
and if an+l = Pm+l. In fact the following theorem caii be proved. then the number . THEOREM Two irrational nurnbers a and 0 are equivalent if 5. = [bl.. For example.
Kow let us return t o Hurwitz's theorem. .bz. . a. .. for their expansions can always be written in the form
2 =
Y
N
.1).3.1) or 4 2 . ca.
=
[bi. if 0 is no¿ equivalent to either .
.
No. and a very simple proof using Farey sequences was recently given by Ivan Niven. 9. vol. notice that the lower bound is not just the negative of the upper bound. this is Hurwitz's Theorem. but the study of continued fractions is. and probaibly w i l remain. THEOREM For any real nulnber r 2 0. of course. 121-123. and the expression is unsymmetrical.
6. One coiild. hl. the inequality
has infinitely many ~olut~ions.?
5. 2. Using continued fractions. Robinson (1947) gave a proof of Segre's theorem. go into the subject of continued fractions more deeply by reading such books as Perron [ll]. For example. In recent years severa1 new methods for solving problems in this field have been invented. this monograph can serve as the point of departure for further study of a variety of topics. the basic stepping stone for those wishing to explore this subject. Segre in 1946.side of Hurwitz's ineiuality can be strength eiied without essentially weakening the other. The Michigan Math. the following theorem was proved by B.5. 11i02.
. and also proved that given e > 0. R. The subject has a long history. The field of Diophantine approximations by no means exhausts the avenues of exploration open to the interested student.4 Conclusion
Hurwitz's theorem is an example of a whole class of related theorems and problems studied under the general title of Diophantine approximations. pp.EPILOGUE
129
Recently interest has been showii in "lop-sided" or unsymmetrical approximations to irrational numbers. t On Aayrnmelrtr Diophantiae Approximations. result is interesting since it shows This that one . Journal. a n irrational number a! can be approzimated by infinitelg m a n y rational fractions p/q in szich a way that
When r = 1. For r # 1. yet there are still many challenging prohlems Ieft t o be solved.
If x = f ( l ~ ) show that y = ( . and (iii) transitive.130
CONTINUED FRACTIONS
Alternatively.
Problem Set 21
l. r/cr of the Farey sequence Fp on page 126 and verify that a t least one of the numbers p/q. a subject initiated by Stieltjes and others.
. a t least one satisfies Hurwitz's inequality (5. 24. see Hardy and Wright [53. .
+
2.
+
+
4. there is t h e extension to analytic continued fractions (see Wall [14]). Calculate the first six convergents to a = f (1
4%) show that and of every three of these consecutive convergents beyond the first.6). Expand both x and y into simple continued fractions and use these to give a numerical check on Theorem 5.6). Chapters 3.10s 7)/(72. founded by Minkowski. (ii) symmetric. (p r)/(q S). Locate a = +(di6 2) betseen two successive elements p/q.
3. Fg. Calculate the next row. a n d there is the beautiful and closely related subject of the geometry of numbers. Prove that the equivalence relation defined on page 127 is (i) reflexive. r/s satisfies the inequality (5.5) is equiva.of the Farey sequences given on page 126.
+
+
5.3. lent to s. For an introduction t o t h e geometry of numbers.
if it did. if x2 . y = 2y are both even integers. Then
-
+
and since yl and y1 1 are consecutive integers one of them must be even.3y2 = -1 is to have integral solutions. then we cannot have
-
-
But an integer of the form 41 1 41 = -2. we notice first that x and y cannot be both even or both odd. hence 4y1(y1 1) is divisible by 8. y odd. in the second case.1 is not solvable in integers x. cannot have the value . or x odd.1. Suppose that x is even and y is odd.6n 1 is an integer. y 2y1 1. and therefore 1 = i t to the reader to show that if x odd and y evan. y. Hence. So yl(y1 1) is divisible by 2. then we must have x even. and. from (1)) we conclude that y2 has the form 8n . then
+
is also even (twice an integer) and again could not equal -1. We leave xa Bya -1.3y2 = . would not be an integer.1 Has No Integral Solutions
To show that the equation $2 . Similarly. y even. For. where n is an integer.1. Then
+
+
+
where 2 = xi . if x = 2xl 1. if x = 2xi.3y2 = .APPENDIX 1
Proof That x2 . so that x = 2x1. in the first case. then is even and so could not be equal to .1. y = 2y1+ 1 are both odd integers.
+
-+
.
at. . and others. 2al]. This last equation is closely connected with a famous theorem stated by Fermat in 1640 and proved by Euler in 1754:
+
THEOREM: Every prime p of the form 4c 1 can be expressed as the sum 7 of two squares. in Section 4. then the equation x2 .1 is soluble. if N = p is a prime number of the form 4 c 1. Legendre's method depends upon the fact that the periodic part of the continued fraction for has a symmetrical part a2.
. if x2 . a*. ar. al. whenever N is such that N . that if the symmetrical part has no central term (n odd).a2 followed by 2a1.+I by reversing its period.1 always has solutions. .py2 = .1 has no solutions. is equal to &.8.132
CONTINUED FRACTIONS
Hence there do not exist integral solutions z. Q such that p = P2 Q2. as.. We proved.y of the equation In fact.. .
1 6
This we write in the equivalent form
where.. the equation x2 .
+
+
Once this theorem became known it was natural for mathematicians to search for ways to calculate the numbers P and Q in terma of the given prime p..+l. there exists one and only one pair oj integers P.py2 = .3 is an integral multiple of 4. as. and this representation is unique. The converse is also true.1)
Moreover. Gauss (1825). then the equation 7 x2 . the period in the expansion of a... a3. a3.
Now CY. hence the continued fraction for dc has the form = [al. Without entering the details of the proof. obtained from CY. beginning at the middle of the symmetrical part. . however. Serret (1848). a. On the other hand. . namely. a. we shall present the essential idea of Legendre's construction.py2 = -1 is soluble then there is no central term in the symmetrical part of the period. Constructions were given by Legendre (1808). That is.Ny2 = .~ is a purely periodic continued fraction and hence has the form (see Theorem 4.+~ is symmetrical and hence the number 0.
2.1.3
+ 1.
Problem Set 22
1 . ~ --1 P = that d or
Q
Q
As an illustration.APPENDIX 1
133
But according to Theorem 4. take p = 13 so that Hence all we have to do is calculate
a . i f one soldier is added or taken away from one of the squarca. c Show also that. Q = 3. the conjugate by
of
am+l is
related to /3
so that
This means that
~ + 1 / i . .
. giving
p = 13 = 22
+ 3a. Expanding fiwe obtain
Thus
so that P = 2. 3
p =
P2
+ Qa. thc two cl(btrrclirncntscan sometimos be coinbined into a square.
3. Show that it is impossible to combine the two squnres into a single square of soIdiers. Ex~iressp = 29 as the sum of two squnres. There are two equal detachments of soldiers arranged in two squares.
=
4 . each containing b rows of b soldiers. Espress p = 433 as the sum of two squares.
Bombelli.A P P E N D I X 11
Some Miscellaneous Expansions
The following is a small collection of miscellaneous continued fractions.
1.? The list is not restricted to simple continued fractions. Cataldi. He expmssed the continued fraction expansi*n of in the form
fi
and also in the form
t See D. E. mainly of historical interest. In modern notation he knew essentially that
2.
. 1572.Smith [13]. 1613.
He found the following expansions involving the number
the base of the natural logarithms. about 1658. .
. EuIer.A P P E N D I X 11
3. 1737.
m
4.14159 . both discoveries were important steps in the history of a = 3. Lord Brouncker.
This expansion is closely connected historically with the infinite product
given by Wallis in 1655.
. both numerators and denominators The convergents are f. 5. being formed from the sequence of Fibonacci numbers 1.
10.
. 13. 1. 8. 1833. 2. +. 3. The convergents t o ?r are
m
the fraction
approximates n with an error of at most 3 units in the 7th decimal place.
. 9.
a .A P P E N D I X 11
137
Unlike the espansion of e.1415926536 does not seem to have any regularity. the simple continued fraction expansion of R = 3. Stern. g. S.
8.1.2 In this relation let n have in turn the value . n. 3 . . From pn
= c ~ p . 1..-3
Similarly
The required result is then obtained from these equations by successive substitutions. n .2. we used the fact that p.~ ~ ~ .i is proved in a like rnanner. The resuft for qn/q.142
CONTINUED FRACTIONB
7. 2. This gives the following equatibns:
+
pn-r
=
(n
. = n p . I n constructing our table of convergents.. .1)pn-a + Pn-a
.+Pn-2
+
1
pn-2
P. .i .~ pn-2
+
we see that
and from the fact that pnel = an-~pn12 pn-3 we see that
Pn. n .=a.
With E as center and radius EB describe the arc BF. = 0. Construct the point E such that AE = E D and draw EB = 4 4 5 2 .+I = F.-I
for
n = 2. F.S O L U T I O N S TO P R O B L E M S
Figure 12
Y .3. the next year.
8. Thus
F.
and FI = 1 (because only the trunk was present during the first year) yield the recursion formula for these F'ibonacci numbers. I n symbols.
Figure 13a
. First solutwn: Construct the square ABCD of aide z = A B . that are less than one p a r old.
During
branches..
+ F. see Figure 13a. there are
+ Y. On = Fn-i. Since the number of at-least-one-year-old branches constitutes fhe total number of branches of the previous year.
With A as a center and radius AF draw the where T = arc FG. and that B H is parallel to F I . then I D = x . Hence BG = C D .
x
9. and G D parallel to BE.= .
Second solution: Construct the right triangle B A C such that A B = x . Then A C C D = T X .= -D GB DE
TX
-7. and if we let A D = x. With C as center and radius B C = d5 x construct point D. But C D = BG = A l . . we see that A D / A I = C D / H I .
. Now B C = 1 = A I . hence A D / A I = A I / I D . and x ( x .
so that
x
=
=
+(1 +
4) 5. A C = 2 x . Using similar triangles. Sirnilarly. and so BF = H I .1. Then
+
+
+
AG A .4B = s.1 = O.CONTINUED FRACTIONS
Then
1). or
' X
. see Figure 13b.x . Draw BE. where T = +(1 d5).
and
AG=T(GB). For the regular pentagon A B C D E whose sides have length 1. Construct point E such that D E = . or ( A D ) ( I D ) = ( A I ) 2 . and H I = BF = I D . Clearly AG = X / T and so
: d+ (i
Consequently. first prove that A D is parallel to B C and that BE is parallel to C D . prove that H I is parallel to B E .1 ) = 1.
The only extended account of the subject in English is that given in Vol. Hardy and
E. Davenport. Edwin Beckenbach and Richard Bellman. for he gets to the heart of the matter quickly and with very little fussing. Wa~hington Carnegie Institiite of Washington. 1 of Chrystal's 1 Algebra. vol. The book by Daveilport is very good reading. 11. G . an old-fashioned yet still valuable text. 4th ed. 1919 . An Introduction to the Theory o Numbere. :
New York : Chelsea. The Higker Arithmetic.
2. New York: Random House. London : Hutchinson's University Library. G . 1 1 1. H. vols. 1959. or deal with subject matter that has been referred to in the text. Edinburgh: Adam and Black. 1. 1952. The standard treatise on contiked fractions is Perron's Kettenbrüche. Algebra. 1950. New York : Chelsea. f
.
1. 11. 1961.. E.References
The books listed below either contain chapters on continued fractions. but this book is for the specialist. New Mathematical Library 3.
5. An Introduction to Inqualities. L. M. H. 1889 . Oxford : Clarendon Press. Inc.
3. Chrystal. reprinted. Wright. 1960. reprinted. No attempt has been made to compile a complete bibliography. Historly-ojthe Theory of Numbers..
4. Dickson. | 677.169 | 1 |
This paper proposes an introduction to the study of the Pythagorean Theorem, highlighting the importance of practical demonstrations and the need to establish relations of this content problem-situations experienced by the group of students from 7th grade, on the State High School Teacher Carolina Argemi Vazquez, with the purpose of leading them to a clear understanding and correct application of that knowledge, which are prerequisites for the 8th grade (current 9th grade). The theoretical researched and thorough, critical analysis of textbooks and the use of NCPs (National Curriculum Parameters) and Geogebra software are indispensable tools for the effective teaching-learning process. The methodology is filled with concrete activities, easily absorbed. The interventions listed are valuable for the teacher to promote students' reflection, which will change paradigms and behaviors. Consequently, the teaching-learning process is satisfying because it brings a new meaning practice both for students and for teachers, becoming more attractive and real the teaching of mathematics | 677.169 | 1 |
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Grading: Your course grade will be determined by your cumulative average at the end of
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A-
B+
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B-
C+
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D
97
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We may adjust the scale to be more lenient, but we guarantee that the grade
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Beginning Algebra
9780495118077
ISBN:
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Edition: 8 Pub Date: 2007 Publisher: Thomson Learning
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Customize your Course with ESource: Instructors can adopt this title as is, or use the ESource website to select the chapters they need, in the sequence they want.
Describe the Exceptional Computational and Visualization Capabilities of MATLAB: Students will gain a clear understanding of how to use MATLAB.
Illustrate the Problem-solving Process through a Variety of Engineering Examples and Applications: Numerous examples emphasize the creation of readable and simple solutions to develop and reinforce problem-solving skills.
Keep your Course Current with Discussion of the Latest Technologies: The discussions, screen captures, examples, and problem solutions have been updated to reflect MATLAB Version 8.2, R2013b. | 677.169 | 1 |
meStudying: Algebra 1
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Practice your high school Algebra skills on the go! meStudying: Algebra 1 allows you to easily choose a topic and interact with questions on your mobile device. Reinforce the concepts you've learned and be ready for the next test. Whether you want quick practice or a review of more detailed explanations, meStudying: Algebra 1 is here for you when you want it, where you want it. meStudying: Algebra 1 is brought to you by Florida Virtual School, the leader in K-12 virtual education | 677.169 | 1 |
Math Center
The Math Center is a non-credit, Community Education class which provides assistance
in mathematics as a completely free service. Current Allan Hancock College students
as well as other individuals who are 18 years or older may fill out a simple registration
form and attend as frequently as they want. Registration forms may be found in the
Math Center or at Community Education in Building S.
The goal of the Math Center (sometimes called the Math Lab) is to assist students
in the successful completion of any Allan Hancock College mathematics class by providing
additional instructional resources. The Math Center offers many resources, including
one-on-one, drop-in tutoring by our staff of instructors and student tutors. Please
see the full list of resources below:
Handouts on math topics, including content from various math courses as well as information
on overcoming math anxiety and preparing for and taking math tests
Two private study rooms
Make-up testing
Joining the math center group
Current students may access more detailed information by entering their myHancock
portal and joining the Math Center Group. Details may include information such as
the current schedule of instructors and student tutors who work in the Math Center, helpful
handouts on math topics, upcoming workshops, etc.
To join the Math Center Group:
Enter myHancock
Look at the center of the Home page in the box titled "My Groups." Click on "View
All Groups" at the bottom of the box.
STAFF
Achieve success at the math center | 677.169 | 1 |
97804711540 Combinatorics (Wiley Series in Discrete Mathematics and Optimization)
Combinatorics is the study of how discrete sets are arranged, counted, and constructed. This book details several general theories: the Ramsey theory, the Plya theory of counting, and the probabilistic method. It provides a basic framework in which to introduce fundamental results, discuss interconnections and problem solving techniques, and collect open problems | 677.169 | 1 |
Use Wolfram|Alpha to Solve Calculus Problems and…...
Use Wolfram|Alpha to Solve Calculus Problems and… Everything Else.
Wolfram|Alpha is like Google on crack. However, it is not technically a search engine; it is a "computational knowledge" engine. They use a huge collection of trustworthy, built-in data to get the user the information or knowledge they are looking for. When you search for an item, Wolfram|Alpha gives you all of the relevant knowledge they have on that specific search query. For example, here is the results for the search "when did the Beatles break up?" Not only do you get the date the Beatles broke up, you also get how long away that date is from today and other noteworthy events that occurred on the same day. Here is another example, for the search "carbon footprint driving 536 miles at 32mpg" that tells you the amount of fuel consumed and the amount of c02 and carbon emitted.
Because Wolfram|Alpha is just retrieving answers from its huge database of information and formulas, you have to be specific and ask non-opinionated questions. For example, the website does not know which Lil Wayne song is the best. However, it does know things that are not opinions, like the nutritional facts of 10,000 big macs and how many planes are currently flying directly over you.
I find Wolfram|Alpha to be better than Google when I am quickly looking for specific answers. I just typed in "Countries that border France" on both Wolfram|Alpha and Google. Wolfram|Alpha quickly showed me a list of the 8 countries and a map with of France with its bordering countries highlighted. Google on the other hand sent me over to Yahoo Answers…
Other than a fun search engine, Wolfram|Alpha can also be used as a highly effective tool for college. Like the title mentions, the knowledge engine can in fact solve any calculus problem. It can easily solve any math problem thrown its way, from a basic algebra problem to whatever this is.
Wolfram|Alpha can also be used for many other college courses such as biology, astronomy, history, etc.
As Wolfram|Alpha can be kind of confusing and hard to get the hang of at first, I suggest going through this short tour and looking at some examples to help give you a better sense of how to use it. Even if you find it a little bit confusing at first, keep trying because Wolfram|Alpha really is a great way to "hack college." | 677.169 | 1 |
Mathematics
MAT090 is designed to provide students with the tools they need to achieve a higher level of success in their entry level mathematics courses. Students who have fully participated in but have been unsuccessful in 0-level math courses should take this course. The course is designed to help students understand and learn the skills that are required to be successful in mathematics. Students will learn to be active rather than passive participants in the learning process. Students will work individually and collaboratively throughout the course.
Co-requisite: A co-requisite of at least one MAT course.
Beginning Algebra is intended for students who need a foundation in, or to review the general topics related to Algebra. Topics covered include operations with fractions, signed numbers, solving equations, factoring, linear equations and polynomials. A grade of C or better is required for entrance into MAT 095/099 or 131. MAT 092 (Math Literacy for College Students) is preferred for students going to MAT 109 or 118, but a C in MAT 091 will still serve as a pre-requisite.
List of pre-requisites: Compass Pre-Algebra score of at least 36 OR Compass Algebra Score of at least 23 OR CSM 094 with a grade of C or better.
This course will provide students with the essential quantitative skills and knowledge needed in the workplace, and needed for entrance into BUS 101, MAT 109, MAT 116, MAT 118, or 100-level general education science courses. It will emphasize number sense, percents, computational ability, and basic applications of mathematics including graphs and rate of change.
Pre-requisites: Student never took Regents Algebra 2/Trig exam (if student took this exam, then the student should be placed higher). Regents Geometry score of 1-49 in the last two years, OR Regents Integrated Algebra score of 50-74 in the last two years, OR CSM 094 with grade of C or higher, OR Compass Pre-Algebra score of 36 or more, OR Compass Algebra score of 23-48.
This is the first credit of the three-credit Intermediate Algebra sequence of courses. Topics covered include using function notation, finding domain and range, and identifying basic features of linear, quadratic, and exponential functions. A TI-83 or TI-84 calculator is required. If placed into this course, then a grade of C or higher in this course is required for entrance to the Intermediate Algebra Part II course.
Pre-requisites: Compass Algebra exam score of 49 or higher, or Integrated Algebra Regents exam score of 75-84 within the last 2 years, or MAT 091 with a C or higher. Students who took MAT 092 instead of MAT 091 must have an A- in MAT 092 or receive approval from the department head.
This is the second credit of the three-credit Intermediate Algebra sequence of courses. Topics covered include graphically and numerically solving problems with the calculator, expanding and factoring and the quadratic formula, finding equations of linear functions, and interpreting the real-world meaning of the features of a function. A TI-83 or TI-84 calculator is required. If placed into this course, then a grade of C or higher in this course is required for entrance to Intermediate Algebra Part III.
Pre-requisite: MAT095, Intermediate Algebra Part I, with C or higher.
This is the final credit of the three-credit Intermediate Algebra sequence of courses. Topics covered include fractions without a calculator, exponent rules, systems of equations, and basic applications. A TI-83 or TI-84 calculator is required. If placed into this course, then a grade of C or higher in this course is required for entrance to College Algebra (MAT110) or Math for Elementary School Teachers (MAT107) or Algebra and Trigonometry for Pre-Calculus (MAT184).
Prerequisites: MAT096, Intermediate Algebra Part II, with a C or higher.
MAT099 is intended for students who must bring their mathematics proficiency to the level necessary for entrance into MAT110, 184, or 107. This course cannot be used to satisfy the mathematics requirement of the Associate in Art degree program. MAT109 will fulfill the mathematics requirement for many students in Associate of Arts degree programs. Topics include: Functions, Linear Functions, Quadratic Functions, Exponential Functions, Solving Equations symbolically and graphically and numerically, Systems of Linear Equations, Factoring and Graphing. The TI-83, or TI-83 Plus, or TI-84 or TI-84 Plus is required.
Pre-requisites: Compass Algebra score of at least 49 OR Integrated Algebra Regents within the last 2 years of at least 75 OR MAT 091 with at least a C. Students who took MAT 092 instead of MAT 091 must have an A- in MAT 092 or receive approval from the Department head.
This course meets the Math requirement for students who are enrolled in the Liberal Arts and Sciences: Education, Early Childhood Education (Birth - Grade 2) and Childhood Education (Grade 1-6) dual certification with SUNY New Paltz, A.S. degree program and who plan to transfer to SUNY New Paltz. The emphasis is on problem-solving as it relates to the number system. Probability and statistics are also introducedThe course will allow students the opportunity to explore mathematics through interesting real life applications, as they strengthen their critical thinking and practical problem solving skills. Students will be required to use contemporary technology, perform web research and will work collaboratively throughout the course. Topics will include geometry, probability, statistics, and finance. Other topics may include history of mathematics and modern mathematical systems.
Pre-This course satisfies the SUNY General Education mathematics requirement and is the prerequisite for Business Calculus (MAT125). Topics include applications of linear, reciprocal, exponential, logarithmic, power, and quadratic functions; composition and inverses of functions; systems of equations; regression; and piecewise equations. Students will solve equations both algebraically and graphically. Use of the one of the following graphing calculators will be required: TI-83, 83 Plus, 84 or 84 Plus. Not for students who intend to take MAT185, 221, 222 or 223 gives students the opportunity to explore mathematics through interesting, real life applications. Each semester students will select an area of study such as forensic science, amusement park ride design, encryption, the cellular phone industry, etc. Mathematics will be presented in class, as it is needed, within the context of the problem being explored. The emphasis of this course is on helping students get a better understanding of the links between mathematics and real life applications as they strengthen their critical thinking and practical problem solving skills. Students will be required to do web research and will work collaboratively throughout the course.
Pre-requisites: Compass Algebra Score of at least 49 OR Math A Regents/Integrated Algebra Regents within the last 2 years of at least 65 OR MAT 091 with at least a C .
This course is a requirement for students in Early Childhood Education (Birth-Grade 2) and Childhood Education (Grade 1-6) programs. It emphasizes background information for the teaching of elementary school geometry. Topics include spacial visualization, measurement, coordinate geometry, similarity and congruence, and transformational geometry. Students learn mathematical theory and application, and experience the role of elementary school students through a variety of classroom activities and demonstrations.
Pre-requisite: MAT107 with a grade of C or better
3 Lecture 0 Lab 3 Credit Hours
Satisfies the mathematics requirement of the Associate in Arts degree program. Basic statistical procedures are developed. Topics include descriptive statistics; probability; probability distributions; hypothesis testing; confidence intervals; correlation and regression. Technology (either a graphing calculator from the TI-83/84 family or a statistical analysis software) will be used regularly throughout course
PreA survey of the basic concepts and operations of calculus with business and management applications. Designed for students in the Business Administration Transfer program and should not be taken by mathematics and science majors. Students will use Microsoft Excel extensively throughout the course. No previous knowledge of Excel is required.
Prerequisite: Compass College Algebra Score of at least 46 OR Algebra II and Trigonometry Regents exam score within the last 2 years of at least 85 OR MAT 110 with at least a C or MAT 184 with at least a C.
This is the first course in a two-semester sequence of intermediate algebra and trigonometry with technical applications. Topics include operations in the real number system, functions and graphs, first-degree equations, lines and linear functions, systems of linear equations, right triangle trigonometry, geometry (perimeters, areas, volumes of common figures), rules of exponents, polynomial operations, factoring, operations on rational expressions, quadratic equations, and binary and hexadecimal notation. A calculator and a laptop computer will be used throughout.
This is the second course in a two-semester sequence of intermediate algebra and trigonometry with technical applications. Topics include the operations of exponents and radicals, exponential and logarithmic functions and equations, trig functions of any angle, radians, sinusoidal functions and graphing, vectors, complex numbers and their applications, oblique triangles, inequalities, ratio and proportion, variation, introduction to statistics (optional) and an intuitive approach to calculus. The graphing calculator and laptop computer will be integrated throughout the course.
Prerequisite: MAT128.
This course satisfies the math requirement for the Applied Academic Certificate in ACR. It is designed for those students who need to improve their math proficiency for entrance into MAT 132. Topics include: review of operations on whole numbers, fractions, and decimals; operations using signed numbers; exponents and roots; scientific notation; unit analysis; percentage; algebraic expressions; factoring; linear equations; literal equations; geometry of the triangle, circle and regular polygons; measurement conversions; and introduction to basic trigonometry. Use of a scientific calculator is required.
Prerequisites: Regents Algebra 2/Trig score 1-49 in the last two years, OR Regents Geometry score of 50 or more in the last 2 years, OR Regents Integrated Algebra of 75 or more in the last two years, OR MAT 091 with a C or higher, OR Compass Algebra score of 49 or higher.
This course satisfies the mathematics requirement for students in ARC, CNS, FIR and FTP. Students enrolled in the above curricula may receive credit for MAT 132 or MAT 110, but not both. Topics include a review of right triangle trigonometry, law of sines and cosines, vectors, factoring, literal, fractional and quadratic equations and applications. Use of a scientific calculator is requiredSatisfies the mathematics requirement of the Associate in Arts degree program, and is intended to prepare students for MAT185 (Precalculus). Topics include equations and inequalities, graphing techniques, analysis of a variety of functions, and triangle trigonometry including the Laws of Sines and Cosines is intended primarily for students planning to take calculus. Topics include a review of the fundamental operations; polynomial, rational, trigonometric, exponential, logarithmic, and inverse functions; modeling and data analysis. A graphing calculator from the TI-83/84 family of calculators is required for this course.
Pre-requisites Compass College Algebra Score of at least 46 OR Algebra II and Trigonometry Regents exam score within the last 2 years of at least 65 OR MAT 184 with at least a C OR MAT 132 with at least a C OR MAT 110 with at least an A-.
Intended primarily for students in the CPS, EDM, or LAM curriculum. Students will be introduced to mathematical reasoning and proof techniques through topics in discrete mathematics. The topics selected for this course will be from areas of logic, set theory, combinatorics, number theory and functions. Direct and indirect proof methods will be covered along with the technique of mathematical induction.
Pre-requisite: MAT 221 with a C or better.
This course is the first of a three-semester sequence developing calculus for the student majoring in engineering, mathematics, or the sciences. Topics include the derivative, limits, continuity, differentiability, the definite integral, the Fundamental Theorem of Calculus, techniques of differentiation (including for transcendental functions), applications of differentiation, mathematical modeling and computer applications. A graphing calculator from the TI-83/84 family of calculators is required for this course.
Pre-requisites: MAT 185 with a grade of at least C, OR one year of high school Precalculus with a grade of at least 70 AND Compass Trigonometry Score of at least 46, OR permission of the department.
This course is the second of a three-semester sequence developing calculus for the student majoring in engineering, mathematics or the sciences. Topics include the Fundamental Theorems of calculus, definite and indefinite integrals, techniques of integration, improper integrals, applications of integration, sequences, series and Taylor series, differential equations, mathematical modeling and computer applications. A graphing calculator from the TI-83/84 family of calculators is required for this course.
Prerequisite: MAT 221 with a grade of C or better, or permission of the department.
A continuation of MAT 222. Topics include vectors in the plane, solid analytic geometry, functions of several variables, partial differentiation, multiple integration, line integrals and vector fields, Green's Theorem, Stokes' Theorem, applications. A graphing calculator from the TI-83/84 family of calculators is required for this course.
Prerequisite: MAT 222 with a grade of C or better or advanced placement with the permission of the department.
An introductory course in differential equations for students in mathematics, engineering and sciences. Topics include the theory, solution and estimation of differential equations of the first and second order, Laplace transforms, systems of differential equations, power series and an introduction to Fourier series and partial differential equations.
Prerequisite: MAT 223 with a grade of C or better.
A special learning experience designed by one or more students with the cooperation and approval of a faculty member. Proposed study plans require departmental approval. Projects may be based on reading, research, community service, work experience, or other activities that advance the student's knowledge and competence in the field of mathematics or related areas. The student's time commitment to the project will be approximately 35-50 hours. | 677.169 | 1 |
9780130236 as a foundation for algebra, this comprehensive program motivates students as they build the important skills and confidence they need to take on algebra. Correlated to the NCTM Standards, Pacemaker Pre-Algebra features an attractive, full-color design that offers predictable and manageable two-page lessons that promote student success. Written at a controlled reading level of grades 3 4, students of all abilities are provided with essential preparation for a variety of testing situations, including the most widely used standardized tests. This program teaches the essentials of problem solving using the Polya 4-step approach which provides step-by-step guidance for building successful problem-solving | 677.169 | 1 |
College AlgebraThese effectiveness to not only pass the course, but truly understand the material. | 677.169 | 1 |
9780495561668
ISBN:
0495561665
Edition: 4 Pub Date: 2009 Publisher: Cengage Learning
Summary: Thomas Sonnabend is the author of Mathematics for Teachers: An Interactive Approach for Grades K-8 (Available 2010 Titles Enhanced Web Assign), published 2009 under ISBN 9780495561668 and 0495561665. Three hundred seventy nine Mathematics for Teachers: An Interactive Approach for Grades K-8 (Available 2010 Titles Enhanced Web Assign) textbooks are available for sale on ValoreBooks.com, sixty four used from the cheape...st price of $65.97, or buy new starting at $218 Does not have access card. Ships same day o [more]
Book has signs of cover wear. Inside pages may have highlighting, writing and/or underlining. Used books may have stickers on them. Does not have access card5561712 Has Activity Cards. New and in great condition with no missing or damaged pages. Need it urgently? Upgrade to Expedited. In stock and we ship d [more]
ALTERNATE EDITION: Annotated Instructor's Edition. Has Activity Cards. New and in great condition with no missing or damaged pages. Need it urgently? Upgrade to Expedited. In stock and we ship daily on weekdays & Saturdays.[less5561668
ISBN:0495561665
Edition:4th
Pub Date:2009 Publisher:Cengage Learning
is unbeatable for cheap Mathematics for Teachers: An Interactive Approach for Grades K-8 (Available 2010 Titles Enhanced Web Assign) rentals, or used and new condition books available to purchase and have shipped quickly. | 677.169 | 1 |
A study guide for basic math courses, titled "How to Survive Your College Math Class (and Take Home Something of Value)," includes sections on study skills, reading and understanding mathematics (featuring "An Equation is a Sentence," "Deriving Conditions of Equality," and "Assertion and Proof"), and writing mathematics and solving problems. This guide, as well as a quick review of basic set... | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
volume offers a concise, highly focused review of what high school and beginning college undergraduates need to know to successfully solve the trigonometry problems they will encounter on exams· Rigorously tested examples and coherent, to-the-point explanations are presented in an accessible form and will provide valuable assistance in conquering this challenging subject· | 677.169 | 1 |
Unit 5: Exponential, Logarithmic, and Logistic Functions Goal: The student will demonstrate the ability to investigate exponential, logarithmic, and logistic functions and solve real-world problems, both with and without the use of technology.
a. Sketch and analyze exponential functions and their transformations.
Mathematical Background/Clarifying Examples:
Students will use knowledge from courses and units in this course to graph transformations of exponential functions. It is important to emphasize the definition of an exponential function and its different parts for students to identify a function as a growth or decay. The y-intercept and horizontal asymptote need to be stated with graphing problems.
Resources: Additional Practice: This resource has students identify whether an exponential function is a growth or decay. It also has students practice graphing exponential functions as well as evaluating a half life problem.
c. Express the inverse of an exponential function as a logarithmic function.
Mathematical Background/Clarifying Examples:
Use concepts covered in objectives a and b to identify that a logarithmic function is an inverse of an exponential functions. Students will need rewrite an exponential function as a logarithm and vice-versa. This concept will be used in future objectives.
Resources:
Discovery: This activity assists students in discovering the relationships between exponential and logarithmic functions.
Mathematical Background/Clarifying Examples:
Students will use knowledge of logarithms and exponential functions to evaluate with and without a calculator. Students will need to use knowledge of the zero exponent property, negative exponent property, and square roots.
e. Use and apply the laws of logarithms and the change of base formula.
Mathematical Background/Clarifying Examples: Guide students in recognizing the inverse relationship between logarithms and exponential functions. Use this knowledge to simplify logarithmic expressions and use in later sections to solve logarithmic and exponential equations.
f. Sketch and analyze logarithmic functions and their transformations.
Mathematical Background/Clarifying Examples: Students will need to use their knowledge of exponential functions and graphs of inverses in order to graph logarithmic functions. Knowledge from previous units should be used to analyze transformations.
Mathematical Background/Clarifying Examples: Guide students to use their knowledge of exponential functions to graph and analyze logistic functions. Discuss end behavior to label horizontal asymptotes of logistic functions. Word problems involving restricted growth will help students understanding of this concept.
Resources: Application: This problem has student write a logistic equation that models a set of data. Students are to use the graphing calculator to perform a logistic regression. There are follow up questions.
Mathematical Background/Clarifying Examples: Use knowledge from this unit and apply its concepts to solve application problems.
Once the population of Wilde Lake High School exceeds 1800 students the school will reconsider building a new high school in the area. Currently Wilde Lake High School has 1400 student and the population is growing 4% per year. How many years will it take for the school board to consider building a new school?
Additional student practice: The resource contains practice problems using exponential, logarithmic, and logistic functions. Some problems incorporate the use of technology to identify the what type of function the data models. | 677.169 | 1 |
Combinatorics Topics, Techniques, Algorithms
9780521457613
ISBN:
0521457610
Pub Date: 1995 Publisher: Cambridge University Press
Summary: A textbook in combinatorics for second-year undergraduate to beginning graduate students. The author stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter. The book is divided into two parts, the second at a higher level and with a wider range than the first. More advanced topics are given as projects, and there are a number of exercise...s, some with solutions given.
Cameron, Peter J. is the author of Combinatorics Topics, Techniques, Algorithms, published 1995 under ISBN 9780521457613 and 0521457610. Three hundred fourteen Combinatorics Topics, Techniques, Algorithms textbooks are available for sale on ValoreBooks.com, fifty seven used from the cheapest price of $66.07, or buy new starting at $66.07 | 677.169 | 1 |
Details about Discrete Mathematics with Applications:
Susanna Epp's Discrete Mathematics with Applications, Second Edition provides a clear introduction to discrete mathematics. Epp has always been recognized for her lucid, accessible prose that explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. The text is suitable for many course structures, including one-semester or full-year classes. Its emphasis on reasoning provides strong preparation for computer science or more advanced mathematics courses. | 677.169 | 1 |
Shipping prices may be approximate. Please verify cost before checkout.
About the book:
This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance.
This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach--emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s--was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authorsSunshine Book Store via United States
Softcover, ISBN 0821816462 Publisher: American Mathematical Society, 1999 0821816462821816462 Publisher: American Mathematical Society, 1999 0821816462 Publisher: American Mathematical Society, 1999 New. SoftCover International edition. Different ISBN and Cover image but contents are same as US edition. Customer Satisfaction guaranteed!!
Hardcover, ISBN 0821816462 Publisher: American Mathematical Society, 1999 Paper!
Softcover, ISBN 0821816462 Publisher: American Mathematical Society, 1999 66
Hardcover, ISBN 0821816462 Publisher: Chelsea Publishing Co./American Mathematical Society, 1999 Used - Very Good, Usually dispatched within 1-2 business days, This is the FRENCH EDITION 1973 third reprint edition, VOLUME 1&2, very good condition, excellent study text for any french math studentMATHEMATICS ALGEBRA & TRIGONOMETRY) | 677.169 | 1 |
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myCBSEguide.com has launched FREE online tests for class-5 Mathematics. There are around 1000 MCQs for practice. It covers the whole syllabus as issued by CBSE/NCERT for class-5 Mathematics. The chapters included are : Number System, rounding numbers, estimation, Roman Numbers, Place Value Addition and Subtraction Multiplication and Division LCM and FCF Fractional Numbers (including percentage) […]
myCBSEguide.com is proud to announce CBSE MCQ online test for class 10 Mathematics and Science today on the occasion of 67th Independence Day. This online test program is absolutely free of cost. Even no registration is required to attempt the MCQs there. There are 3-5 MCQ Test papers from each topic. users will find around […] | 677.169 | 1 |
☐ Solve a system of one linear and one quadratic equation in two variables, where only factoring is required.
Note: The quadratic equation should represent a parabola and the solution(s) should be integers.
☐ Understand the following terms:
Member (or element) of a set, subset, Universal set, Null (or empty) set, intersection of sets (no more than three sets), union of sets (no more than three sets), the difference between two sets, the complement of a set
☐ Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse)
Note: Students do not need to identify groups and fields, but students should be engaged in the ideas. | 677.169 | 1 |
accelerated course prepares students for transfer-level Statistics. It covers core concepts from elementary algebra, intermediate algebra, and descriptive statistics. Topics include ratios, rates, and proportional reasoning; arithmetic reasoning using fractions, decimals and percents; evaluating expressions, solving equations, analyzing algebraic forms to understand statistical measures; use of linear, quadratic, absolute value, exponential, and logarithmic functions to model bivariate data; graphical and numerical descriptive statistics for quantitative and categorical data. This course is designed for students who do not want to major in fields such as math, science, computer science, and business. Note: This course is NOT intended for students who plan to study science, technology, engineering, math, as well as business and other non-STEM majors. Please see your counselor. | 677.169 | 1 |
Starting from first principles, this book covers all of the foundational material needed to develop a clear understanding of the Mathematica language, with a practical emphasis on solving problems. Concrete examples throughout the text demonstrate how Mathematica can be used to solve problems in science, engineering, economics/finance, computational linguistics, geoscience, bioinformatics, and a range of other fields. The book will appeal to students, researchers and programmers wishing to further their understanding of Mathematica. Designed to suit users of any ability, it assumes no formal knowledge of programming so it is ideal for self-study. Over 290 exercises are provided to challenge the reader's understanding of the material covered and these provide ample opportunity to practice using the language. Mathematica notebooks containing examples, programs and solutions to exercises are available from less | 677.169 | 1 |
-Mathematics, A Good Beginning - Strategies for Teaching Children by Andria
Troutman and Betty Lichtenberg, softcover college text for elementary math
teachers, ideas for hands on activities, how to teach concepts, more; over
500 pages, good cond but some notes written in some few margins and cover
of book, $5.00 ppd
-Business Mathematics by Thompson and Lowe, Glencoe pub, 1988, High school
student hardback text in excell cond (only used by our one son), odd number
answers in back of text, $5.00 ppd
-Stein's Refresher Mathematics, high school remedial math text that is good
for anyone needing review, covers everything from basic math to
measurement, geometry review, basic algebra, graphs, statistics, and
probability, and a final chapter on everyday, practical life math; very
good cond, Allyn and Bacon pub, $5.00 ppd
Mathematics Its Power And Utility....This is a Jr College general math
with some Consumers math combined..... book that would be used by a
freshman or any highschooler....Covers all including computors beginning
programming, calculators,budgeting, fractions, volume,Algebra
etc.....Answers to odd numbered problems in back of book...$5
Accounting Principles 4th edition Chapters 1-13.....soft cover large
Text (over 600 pages) AND Work book to go with it a few pages out of
over
600 done. ..... LIKE NEW!...1996 edition..$5
Interactive Mathamatics...Geometry...Work Text...answers to odd number
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Understanding College Mathematics.....A Calculator - Based
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Arithmetic and Logic in Computer Systems provides a useful guide to a fundamental subject of computer science and engineering. Algorithms for performing operations like addition, subtraction, multiplication, and division in digital computer systems are presented, with the goal of explaining the concepts behind the algorithms, rather than addressing any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and reasoning behind theories and examples. No other current books deal with this subject, and the author is a leading authority in the field of computer arithmetic. The text introduces the Conventional Radix Number System and the Signed-Digit Number System, as well as Residue Number System and Logarithmic Number System. This book serves as an essential, up-to-date guide for students of electrical engineering and computer and mathematical sciences, as well as practicing engineers and computer scientists involved in the design, application, and development of computer arithmetic units.
Review quote
"...the perfect concise reference for computer arithmetic, and I highly recommended it to anyone involved in the study or implementation of such systems." (Computing Reviews.com, June 7, 2005) "This comprehensive treatment of computer arithmetic is ideally suited for upper-level undergraduate or graduate students." (Computing Reviews.com, May 12, 2004) "Lu has prepared one of the best books this reviewer has read...An Excellent book for graduate and senior undergraduate engineering and computer science students." (Choice, July 2004)
Back cover copy
A practical introduction to fundamentals of computer arithmetic Computer arithmetic is one of the foundations of computer science and engineering. Designed as both a practical reference for engineers and computer scientists and an introductory text for students of electrical engineering and the computer and mathematical sciences, Arithmetic and Logic in Computer Systems describes the various algorithms and implementations in computer arithmetic and explains the fundamental principles that guide them. Focusing on promoting an understanding of the concepts, Professor Mi Lu addresses: Number representations, including the Conventional Radix and Signed-Digit Number Systems as well as Floating Point, Residue, and Logarithmic Number Systems Ripple Carry Adders and high-speed adders Sequential multiplication, parallel multiplication, sequential division, and fast array dividers Floating point operations, Residue Number operations, and operations through logarithms To assist the reader, alternative methods are examined and thorough explanations of the material are supplied, along with discussions of the reasoning behind the theory. Ample examples and problems help the reader master the concepts.
About Mi Lu
Mi Lu received her MS and PhD in electrical engineering from Rice University, Houston. She joined the Department of Electrical Engineering at Texas A&M University in 1987 and is currently a professor. Her research interests include computer arithmetic, parallel computing, parallel computer architectures, VLSI algorithms, and computer networks. She has published over one hundred technical papers, and has served as associate editor of the Journal of Computing and Information and the Information Sciences Journal. She was conference chairperson of the Fifth, Sixth, and Seventh International Conferences on Computer Science and Informatics. She served on the panel of the National Science Foundation, the panel of the IEEE Workshop on Imprecise and Approximate Computation, and many conference program committees. She is the chairperson of sixty research advisory committees for masters and doctoral students. Dr. Lu is a registered professional engineer and a senior member of the IEEE. She has been recognized in Who's Who in America. | 677.169 | 1 |
Student Guide for Stewart's Single Variable Calculus: Early Transcendentals, 6 helpful guide contains a short list of key concepts; a short list of skills to master; a brief introduction to the ideas of the section; an elaboration of the concepts and skills, including extra worked-out examples; and links in the margin to earlier and later material in the text and Study Guide. | 677.169 | 1 |
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Algebra I
Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios.
Introduction to algebra
Did you realize that the word "algebra" comes from Arabic (just like "algorithm" and "al jazeera" and "Aladdin")? And what is so great about algebra anyway?
This tutorial doesn't explore algebra so much as it introduces the history and ideas that underpin it.
Wait, why are we using letters in math? How can an 'x' represent a number? What number is it? I must figure this out!!! Yes, you must.
This tutorial is great if you're just beginning to delve into the world of algebraic variables and expressions.
Using the combined powers of Chuck Norris and polar bears (which are much less powerful than Mr. Norris) to better understand what expressions represent and how we can manipulate them.
Great tutorial if you want to understand that expressions are just a way to express things!
All the symbols you write in math are just a language or short-hand to represent real-world ideas. In this tutorial, we'll get experience writing algebraic expressions to elegantly represent real-life ideas.
The core underlying concepts in algebra are variables, expressions, equations and inequalities. You will see them throughout your math life (and even life after school).
This tutorial won't give you all the tools that you'll later learn to analyze and interpret these ideas, but it'll get you started thinking about them.
In second grade you may have raised your hand in class and asked what you get when you divide by zero. The answer was probably "it's not defined." In this tutorial we'll explore what that (and "indeterminate") means and why the math world has left this gap in arithmetic. (They could define something divided by 0 as 7 or 9 or 119.57 but have decided not to.) | 677.169 | 1 |
Schiller Park Chemistry meet the student at their level of knowledge and build from there. Success in physics requires a balance of rigorous logic with imagination. I can help you develop both by using interactive questioning and real world examples, from an apple falling off a tree to the stars dancing above usLarry M.
...Business Mathematics is an applied branch of Mathematics that centers on financial challenges. Perhaps the most important question in business mathematics is to predict the time value of money given certain assumptions. For example, would it be better to receive $100 today or $110 in eighteen months? | 677.169 | 1 |
12/28Sullivan/Struve/Mazzarella AlgebraSerieswas written to motivate students to "do the math" outside of the classroom through a design and organization that models what you do inside the classroom. The left-to-right annotations in the examples provide a teacherrs"s voice through every step of the problem-solving process. The Sullivan exercise sets, which begin with Quick Checks to reinforce each example, present problem types of every possible derivation with a gradual increase in difficulty level. The new "Do the Math" Workbook acts as a companion to the text and to MyMathLab reg; by providing short warm-up exercises, guided practice examples, and additional "Do the Math" practice exercises for every section of the text. Operations on Real Numbers and Algebraic Expressions; Equations and Inequalities in One Variable; Introduction to Graphing and Equations of Lines; Systems of Linear Equations and Inequalities; Exponents and Polynomials; Factoring Polynomials; Rational Expressions and Equations; Roots and Radicals; Quadratic Equations; Graphs of Quadratic Equations in Two Variables and an Introduction to Functions For all readers interested in beginning algebra.
Author Biography
Mike Sullivan, III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of three children and an avid golfer who tries to spend as much of his limited free time as possible on the golf course. | 677.169 | 1 |
Understanding Our Quantitative World is a textbook intended for use in college-level general education mathematics courses to develop "quantitative literacy." The authors intend to address the question: "What mathematical skills and concepts are useful for informed citizens?" They define quantitative literacy indirectly via the contents of their book, and more directly, though vaguely, through their stated goals for students. In the authors' words these are to:
Realize that mathematics is a useful tool for interpreting information.
See mathematics as a way of viewing the world that goes far beyond memorizing formulas.
Become comfortable using and interpreting mathematics so they will voluntarily use it as a tool outside academics.
I think it's useful to ask more specifically what quantitative literacy means if we are going to assess the ability of this textbook to provide it. After all, quantitative literacy per se only has a meaning by analogy with verbal literacy. There is widespread belief, backed by a large literature, that quantitative literacy is of critical importance. Unfortunately, few writers get specific about what knowledge or skills it entails.
One of the clearest definitions I could find comes from an early report from an MAA committee [1] that offers the following:
A quantitatively literate college graduate should be able to:
Interpret mathematical models such as formulas, graphs, tables, and schematics, and draw inferences from them.
Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.
Recognize that mathematical and statistical methods have limits.
I would argue that practical quantitative literacy for the "informed citizen" should put highest priority on interpretation and evaluation of mathematical and statistical information provided by others, and so concentrate on areas (1) and (4), followed by (3) and (2) in that order. There is obviously a lot of room for difference of opinion here. Consensus anytime soon on the specifics of quantitative literacy seems pretty unlikely.
Let's provisionally adopt the MAA committee's interpretation as our definition of qualitative literacy and take a look at the book in that light. How does it measure up? In summary, it does OK on (1) and (2), but considerably less well on (3) and (4). First, let's discuss what the authors do well.
The authors choose to emphasize the concept of function throughout, and I'll discuss some of the ramifications of this later. One thing they do well is to show how functions can naturally be represented in many forms, including symbolic, graphical, tabular and verbal. A tabular representation of a function makes the only-one-element-of-the-range for each element of the domain a more natural idea; after all, a table of student names versus heights can hardly have more than one height per student. There are several good examples, exercises and class activities. Later on, the authors discuss applications and interpretations of graphs and include examples with a motion detector. These motion detector graphs (especially if they are demonstrated with a simple motion detector in the classroom) can directly connect a student's kinesthetic sense with the more abstract concept of a graph.
Throughout the book there are many excellent examples and exercises that are far better than I have seen in comparable texts. They involve a huge variety of applications likely to be interesting to students, and they are presented in meaningful contexts, not as isolated problems. It's also good to see exercises showing bad graphs from newspapers and magazines. I have found examples like this work very well to awaken awareness and a quantitative critical sense in students. The book could use even more examples like these. (USA Today is a wonderful source.)
Another strength is the presentation of contour graphs in the chapter on multivariable functions. Contour graphs give another opportunity to connect with a student's kinesthetic sense. It would probably help to emphasize this by using the idea of gradually filling the space containing a graph with water and identifying contours with water levels.
Unfortunately, there are several things in the book that don't work very well. Some of them relate specifically to the handling of topics I would agree are important for quantitative literacy. Others arise from the authors' apparently different view of quantitative literacy, one that I think overemphasizes some topics and slights others.
I was struck by the first words of the first chapter: "A function is a mathematical object..." Many students who would take a course using a book like this are people I think of as "math-damaged". However that damage occurred, it makes its subjects very uncomfortable with mathematical language. Those first words from Chapter One would certainly crank up their anxiety. I don't want to overemphasize one phrase, but it definitely got my attention. If the authors felt that they really needed to emphasize that a function is a mathematical object, why not first discuss how mathematical use of the word differs from its use in ordinary discourse? Throughout most of the book the authors use an informal and conversational style that should appeal to students. That first sentence, though, is an odd false step.
Another concern is that prerequisites are unclear. In places, it appears that some facility with algebra is expected; in others, algebraic steps are very carefully articulated. Expectations are uneven. An exercise in the first chapter asks the student to write linear equations for cell phone and phone card usage versus cost. But linear equations are not discussed until Chapter 7. More generally, the order of topics treated is rather odd. After the beginning chapters on graphical and tabular representation of data and one chapter on descriptive statistics, the next chapter describes multivariable functions and contour diagrams! Succeeding chapters then take up — in order — linear, exponential, logarithmic, periodic and power functions, with a chapter on regression and correlation tucked in after linear functions.
My greatest problem with this textbook, however, is its selection and emphasis of topics. While it is important that students understand the differences between linear, polynomial and exponential growth, I simply don't understand the reason for an overwhelming emphasis on functions in a quantitative literacy course. My main concern is what gets left out or slighted, and that includes much of topics (3) and (4) above. Probability and statistics are taken up in the last two chapters of the text, and the treatment is much too brief. A basic understanding of probability and an ability to interpret statistical results are especially important for "informed citizens." Consider the number of newspaper articles over the past year discussing, for example, the probability of developing a disease, of a false positive result in a diagnostic test, or of a catastrophic failure of the space shuttle. Think of our daily bombardment with questionable statistical information and what it takes to evaluate it critically.
Calculator use is another concern. Anyone who has taught a course at this level knows that the use of calculators is problematic. There are many benefits, but the cost is generally a lot of classroom time devoted to the mechanics of calculator use. This text makes fairly extensive use of a graphing calculator. The guidance — especially regarding selection of scale and viewing window — is simply inadequate. The authors seem to suggest that the students need to understand the behavior of the functions they're graphing before they graph them. At the level of this course, that's a bit much. An instructor using this text would have to spend much more time demonstrating practical use of the calculator with a variety of functions.
The chapters on exponential and power functions have several nice examples of such functions with data from realistic examples. They also provide calculator procedures for exponential regression to fit the data. Although it is not a big issue, I am uneasy with this. Regression, when it is presented at this level, is treated as a black box with data going in and a fitting function coming out. Students need to know that there are pitfalls, and perhaps how they might be recognized.
The text has a real gap in the area of quantitative literacy described in (4) above. There is a lot of doing in the book, but not enough does this make sense. There is a worrisome problem in Chapter 3 that deals with the price of DVD players. A graph shows the prices of DVD player as a function of time from 1998 to 2002 showing piecewise linear decreases in price. The problem asks, among other things, what price might be expected in 2005? What is the student to make of this? I could find no discussion in the book about the hazards of extrapolation. The student could reasonably, but unhelpfully, answer that the price of DVD players is likely to be even lower in 2005. A simple linear extrapolation from the graph would suggest that the DVD player would have a negative price in 2005. What did the authors have in mind? Perhaps they mean to use this exercise as a vehicle to discuss extrapolation of data, but surely this is important enough to deserve some serious attention in the text. The exercise by itself is a minor issue, but it does inadvertently identify a big gap.
I would have a lot of misgivings about using this textbook for a course in developing quantitative literacy. The heavy emphasis of the text on functions and the omission or limited coverage of the areas described in areas (3) and (4) above — aspects that I think contribute a level of mathematical "street smarts" — are serious concerns.
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. | 677.169 | 1 |
Precalculus by Robert Blitzer, Instructor's Solutions Manual. 2010, 4th Ed.2 volumes of worked out solutions for the textbook. Any large books may acquire cover wear from warehouse handling and storage.Free domestic shipping
Free Delivery Worldwide : Precalculus : Undefined : McGraw-Hill Education - Europe : 9780077349912 : 0077349911 : 01 Feb 2010 : Suitable for either one or two semester college algebra with trigonometry or precalculus courses, this title introduces a unit circle approach to trigonometry and includes a chapter on limits to provide students with a foundation for calculus concepts. It employs a large number of pedagogical devices that guide a student through the course.This teacher's guide accompanies BJU Press' sold-separately Science Student Text, Grade 5, 4th Edition. Reduced-size student pages have the correct answers overlaid in 1/4 of the two-page spread. A page and a half are devoted to lesson notes, including objectives, background information, project ideas, assessments, how to introduce the lesson, and the lesson itself. Semi-scripted lessons include questions to ask in blue; correct answers in pink; directions for what to explain, display, distribute, or other teacher actions are printed in black. . The included Teacher's Toolkit CD provides instructional aids, rubrics, visuals, National Science Education Content Standards, and more. 305 pages, spiralbound with soft frontcover and hard backcover. This resource is also known as Bob Jones...
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Free Delivery Worldwide : Combo: Precalculus with the Student Solutions Manual : Hardback : McGraw-Hill Education : 9780077942090 : 0077942094 : 23 Feb 2010 : The Barnett, Ziegler, Byleen, and Sobecki College Algebra series is designed to be user friendly and to maximize student comprehension by emphasizing computational skills, ideas, and problem solving as opposed to mathematical theory. Suitable for either one or two semester college algebra with trigonometry or precalculus courses, Precalculus introduces a unit circle approach to trigonometry and includes a chapter on limits to provide students with a solid foundation for calculus concepts. The large...
The Official Scrabble Players Dictionary, 5th Edition is the definitive source for answers. This hardcover Scrabble dictionary is a must-have for serious players. The Official Scrabble Players Dictionary, 5th Edition is endorsed by the National Scrabble Association. The Official Scrabble Players Dictionary, 5th Edition has the final word.That's not a word-is it? There's only one way to officially find out: Merriam Webster's Official Scrabble Dictionary. Endorsed by the National Scrabble Association, it gives the correct spelling and definition for words you didn't even know were words. A must-have for serious players. 6-3/4L x 9-1/2H x 1-3/4W.Benefits of the Official Scrabble Players Dictionary, 5th Edition | 677.169 | 1 |
Professor Schey wrote div, grad, curl and all that to help science and engineering students gain a thorough understanding of those ubiquitous vector operators: divergence, gradient, curl, and Laplacian. Since the publication of the First Edition over thirty years ago, several generations of students have learned vector calculus from this little book. Div, grad, curl and all that has been a successful supplement in a variety of physics and engineering courses, from electromagnetic theory to fluid dynamics. The Fourth Edition preserves Schey's clear, informal style and moderately paced exposition as well as avoids unnecessary mathematical rigor.
New for the Fourth Edition: A dozen new example exercises; Updated notation to bring the text in line with modern usage, switching the roles of the two spherical angles, such that the polar angle is now φ and the azimuth is now θ. The Instructor's Solutions Manual, with step-by-step solutions to all the problems, now available online for faculty to download. | 677.169 | 1 |
Int International Math Olympiad team.
Volume 1 is appropriate for students just beginning in math contests. MATHCOUNTS and novice high school students particularly have found it invaluable. Although the Art of Problem Solving is widely used by students preparing for mathematics competitions, the book is not just a collection of tricks. The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book.
Speaking of problems, the Art of Problem Solving, Volume 1, contains over 500 examples and exercises culled from such contests as MATHCOUNTS, the Mandelbrot Competition, the AMC tests, and ARML. Full solutions (not just answers!) are available for all the problems in the solution manual....more
Community Reviews
math team, or I had the knowledge of math competitions, maybe I would've been set right a long time ago.
Both books combined offer a complete overview of pre-calculus mathematics (although their newer curriculum series goes more in depth with more problems) with hundreds of challenging problems from math competitions. There is no better way to get better at solving problems, and with the exception of writing problems, no better way to learn math, than to struggle with challenging problems that rely on using concepts that you know. These books are pure gold for that purpose.
Thanks Richard Rusczyk and AoPS team, for giving me a new passion and for helping me realize what mathematics is all about....more | 677.169 | 1 |
Basic Analysis: Introduction to Real Analysis This free online textbook is a one semester course in basic analysis. These were my lecture notes for teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in fall 2009. The course is a first course in mathematical analysis aimed at students who do not necessarily wish to continue a graduate study in mathematics. A Sample Darboux sums prerequisite for the course is a basic proof course. The course does not cover topics such as metric spaces, which a more advanced course woul Author(s): No creator set
Introduction to Social Network Methods This on-line textbook introduces many of the basics of formal approaches to the analysis of social networks. The text relies heavily on the work of Freeman, Borgatti, and Everett (the authors of the UCINET software package). The materials here, and their organization, were also very strongly influenced by the text of Wasserman and Faust, and by a graduate seminar conducted by Professor Phillip Bonacich at UCLA. Many other users have also made very helpful comments and suggestions based on the Author(s): No creator set
Helping Your Child Learn Mathematics and Statistics This site features dozens of fun activities parents can use to help children (K-5th grade) have fun learning geometry, algebra, measurement, statistics, probability and other important mathematical concepts. Activities relate math to everyday life and can be done at home, at the grocery store, or while traveling. It includes sections for parents on what math is like in schools today and a parents' booklist for helping children learn math. Author(s): No creator set
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Introduction This unit uses a documentary about artists in Buenos Aires, to help you to develop your listening skills. By assessing various parts of the documentary you will be able to extend your comprehension of spoken Spanish to pronunciation If you already have a working knowledge of the Spanish language this unit will help to improve you ability to describe places, events and routines in the past. By using Medieval Spain as the setting you will learn crucial grammatical points regarding the preterite and imperfect tenses. Author(s): The Open University
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This unit is designed as an introduction to the academic study of the concept of rules, but will also serve as an introduction to a variety of different writing styles that are used in the academic world. It will challenge you to think about why some statements are rules and some are not, and what it is that distinguishes rules from habits and customs. It also looks at more formal rules and how such rules are applied and enforced. Rules shape our lives because they set out what we may and may Rules affect us all, and the way they are made and interpreted could effect how we live. This unit explores how we could interpret and apply rules, and provides you with a basic understanding of rules and rule making within the English legal system. Author(s): The Open University
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The key message of this unit is that different psychologists focus on different aspects of human behaviour in different ways. Take the topic of learning, some psychologists will study what happens in our brain when we learn, while others will consider how we learn within a social context. This unit will first highlight how psychology is now a very visible part of everyday life and then explore its diverse roots in medicine, philosophy, biology, psychoanalysis and e.1 Introduction Do you want to relocate to the UK? This unit will help you with the language difficulties that can arise while providing assistance with the practicalities of moving your company and its relocating its employees. You will also learn how other companies have approached this task.1 Introduction Do you want to relocate to the UK? This unit will help you with the language difficulties that can arise while providing assistance with the practicalities of the decision-making processes involved and the consultation that is necessary to ensure employees are kept informed. Author(s): The Open University
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This unit looks at identity, focusing upon the individual's perception of self in relation to others; the relationships between multi-ethnicity, cultural diversity and identity; and the effects of inequality and social class upon identity. It also looks at inequality and social class as they relate to perceived identity.
This material is from our archive and is an adapted extract from Introducing the social sciences (DD100) which is no longer taught by The Open University. If you you the opportunity to put in some early practice Introduction you the opportunity to put in some early practice. Author(s): The Open University
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Licensed under a Creative Commons Attribution - NonCommercial-ShareAlike 2.0 Licence - see - Original copyright The Open University | 677.169 | 1 |
Understanding Modern Mathematics / Edition 1
Paperback
Temporarily out of stock online.
Overview and symmetry can be found on the text's web site, providing students the opportunity to see the 3-dimensional geometric figures in full color. The text provides students with an understanding of how these important mathematical topics are relevant in their everyday lives while emphasizing the history of mathematics. Understanding Modern Mathematics is the perfect complement to any Liberal Arts Mathematics course. | 677.169 | 1 |
3 text helps bridge computationally oriented mathematics with more theoretically oriented mathematics, preparing readers for more advanced courses that require understanding proofs. It covers logic, set theory, axiomatics, number systems, and reading, evaluating, and creating proofs. This third edition includes some of the more modern topics from theoretical computer science, such as the P/NP problem, Boolean algebra, and Church's thesis. Along with new problems and examples, it also illustrates logic in action and discusses topics from the real number system and topology. | 677.169 | 1 |
Worksheets and Projects in Maths for Economists
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Part of an online course on mathematics and statistics, this module presents text and activities on the topics of price indices, inflation, averages, ratios, percentages and proportion. The intended duration of the module is 20 hours. The content is available for download into a VLE.
Fowad Murtaza, University of Essex, Domenico Tabasso, University of Essex
This course webpage supports an introductory module on quantitative economics as taught by Fowad Murtaza and Domenico Tabasso at the University of Essex in 2009/10. It introduces students to the methods of quantitative economics, i.e. to how data are used in economics. Beginning from an elementary level (assuming no background in statistics), the course shows how economic data can be described and analysed. The elements of probability and random variables are introduced in the context of economic applications. The probability theory enables an introduction to elementary statistical inference: parameter estimation, confidence intervals and hypothesis tests. With these foundations, students are then introduced to the linear regression model that forms a starting point for econometrics. It includes a course outline / handbook, lecture presentations, lecture notes, coursework assignments, problem sets with solutions and statistical data.
From an online course on mathematics and statistics, this module introduces ratios, proportions and percentages in turn, using text and activities. The intended duration of the module is five hours. The content is available for download into a VLE under a Creative Commons Attribution Non-commercial Share Alike licence.
This series of fifteen case studies, downloadable as .doc files, illustrates the use of mathematical concepts in economic and business contexts. They are intended to help students see the importance of maths in economic reasoning. Each case study is four sides long and makes use of graphs and algebra. Most include questions at the end. Permission is given for educators to redistribute and alter these materials as much as they like. These materials were produced by an Economics Network project with funding from the JISC. | 677.169 | 1 |
books.google.com - OrthogonalPolynomialsandSpecialFunctions(OPSF)is a veryoldbranchof mathematics having a very rich history. Many famous mathematicians have contributed to the subject: Euler's work on the gamma function, Gauss's and Riemann's work onthe hypergeometricfunctions andthe hypergeometric di?erentialequation,... Polynomials and Special Functions | 677.169 | 1 |
An alternative to the TI-83, the Datexx® Silver Graphing Scientific Calculator is ideal for high school and college students. Perfect for algebra 1 and 2, trigonometry, calculus, biology, chemistry, physics and statistics it features step-by-step instruction manual for learning scientific calculations. It also includes a 2-line LCD display and 889 easy-to-use functions | 677.169 | 1 |
Wear to covers and pages Liberal Arts Math course, a common question students ask is, "Why do I have to know this?" A Survey of Mathematics with Applications continues to be a best-seller because it shows students how we use mathematics in our daily lives and whythis is important. The Ninth Edition further emphasizes this with the addition of new "Why This Is Important" sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and a wide range of exercises help students to develop their problem-solving and critical thinking skills.
Angel, Abbott, and Runde present the material in a way that is clear and accessible to non-math majors. The text includes a wide variety of math topics, with contents that are flexible for use in any one- or two-semester Liberal Arts Math course.
Note: This is a standalone book, if you want the book/access card please order the ISBN listed belowAllen Angel received his BS and MS in mathematics from SUNY at New Paltz. He completed additional graduate work at Rutgers University. He taught at Sullivan County Community College and Monroe Community College, where he served as chairperson of the Mathematics Department. He served as Assistant Director of the National Science Foundation at Rutgers University for the summers of 1967 - 1970. He was President of The New York State Mathematics Association of Two Year Colleges (NYSMATYC). He also served as Northeast Vice President of the American Mathematics Association of Two Year Colleges (AMATYC). Allen lives in Palm Harbor, Florida but spends his summers in Penfield, New York. He enjoys playing tennis and watching sports. He also enjoys traveling with his wife Kathy.
Christine Abbott received her undergraduate degree in mathematics from SUNY Brockport and her graduate degree in mathematics education from Syracuse University. Since then she has taught mathematics at Monroe Community College and has recently chaired the department. In her spare time she enjoys watching sporting events, particularly baseball, college basketball, college football, and the NFL. She also enjoys spending time with her family, traveling, and reading
Dennis Runde has a BS degree and an MS degree in Mathematics from the University of Wisconsin--Platteville and Milwaukee respectively. He has a PhD in Mathematics Education from the University of South Florida. He has been teaching for more than fifteen years at State College of Florida–Manatee-Sarasota and for almost ten years at Saint Stephen's Episcopal School. Besides coaching little league baseball, his other interests include history, politics, fishing, canoeing, and cooking. He and his wife Kristin stay busy keeping up with their three sons--Alex, Nick, and Max purchased the book with ISBN:978-0321828040 which DID include the access code needed for my math course (no additional fee). I see that these reviews also include the book with ISBN:978-0321639325 which apparently does not include the access code.
Therefore if you need the access code AND the book, make sure you purchase ISBN:978-0321828040.
I had to get this math book for my MAT142 class. According to my math teacher this edition is not much different from the previous edition. The page numbers are different and there are only a small number of differences in the problems between the eighth edition and ninth edition. I would recommend you buy the eighth edition since it will probably be much more cheaper.
This is an excellent resource for those you have a hard time with math. I am using this book for a class I am taking in college. This book along with the instructor have really brought a lot to life for me. I am not good at math, but it has helped bring some very needed understanding which makes me more comfortable with the subject. The book is very self explanatory and it helps to work through the examples in the instruction part of the book to gain a better hold of the subject matter. Also, the Student's Solutions Manual needs to be purchased because it helps to fill in the holes.
It has been many years since I had to take a math class from my previous degree. With my career change, I had to take a college level class since they felt my previous algebra class was more of an intermediate class. This book was required for my college. I have to say - I REALLY like this book. It has made math very easy to pick back up again. We are not doing all the chapters in the book, but it has been really good at explaining the concepts, demonstrating how to get to the answers with many examples and then having practice exercises to do.
I don't know if it is because I have had college math in the past, but I think this textbook did an excellent job in clarifying how and why you perform a problem.
There were some problems that there were incomplete instructions. For example, Chapter 3: Logic, example 6. This was an example of an argument with 3 premises using a Truth Table. I had understood everything up until then. I even got most of the truth table done, BUT there were no instructions on what to do with the table results from the added third premise. Which column do I take to complete the truth table argument with the consequent, since there are three premises in the antecedent? It's like it is missing a step in explaining what to do with the third antecedent to complete the equation.
I also found a gap in my understanding in the Statistics chapter dealing with normal curves and deviations (Chapter 13.7). For some reason I could not "get" the higher level concepts and had a little trouble understanding parts of it. This was not necessarily the textbook, but my own mental block. ;-)
Overall I really liked the book. I think there were those couple of issues I noticed. My nephew is in 9th grade and I think I will pass it on to him to help him with areas he is having some challenges because it explains most concepts so well. | 677.169 | 1 |
Maranao Crafts and Mathematics
Maranao Crafts and Mathematics
Unit Summary
This Unit investigates transformations of the graph of linear, quadratic and exponential functions. It serves as an enrichment activity for students where they learn to appreciate local crafts in Lanao del Norte such as the Malong and at the same time learn about transformation concepts.
In the course of the unit, students used graphing calculators and made generalizations on the effects of parameter changes in the equation of the function to its graph. Working collaboratively in groups, students conducted information searches about famous Muslim mathematicians and their contributions to mathematics. Each group was also asked to choose a Maranao textile and analyzed the transformation concepts used in its design. As students did this, they learned to relate geometric concepts of transformation to transformations of graphs in algebra as well as relate the study of mathematics to crafts. The students also developed a variation of the design of the Maranao craft they had chosen. Students presented their work through a multimedia presentation. They also wrote journals on their learning experiences in relating mathematics to the real world. As a culminating activity, students take on the role as textile designers, where they created designs applying the transformation concepts they learned and promoted their own designs in a brochure. The students' reports in multimedia presentations and their promotional materials in brochures, were evaluated using rubrics.
Curriculum-Framing Questions
Essential Question
Is everything related to everything else?
Unit Questions
How are designs created?
How does mathematics relate to craft?
Content Questions
What is a function?
How do we graph functions?
What is the domain and range of the graphs?
What are the contributions of Muslim mathematicians?
What are the types of geometric transformation?
What are the geometric transformations found in a Maranao textiles?
Instructional Procedures
Orient them on what the unit will cover and what are expected from them as proof of their learnings.
Show a Malong to class and ask the following questions:
What designs can you see in this malong?
Can you find any mathematical concept in this design?
How was this design created?
Are these designs somehow connected to the culture of the people making these designs?
As you ask them these questions, make sure that students' answers to these questions will be processed. Inform them also that the activities they will go through in this unit will help them answer these questions.
Day 1-3. Discussion on Functions and Their Graphs
Students will be given activity sheets and graphing papers in doing an activity on understanding graphs of functions. The activity also covers a discussion of the domain and range of functions using graphical representations.
To validate students answers, a discussion on graphing functions follows, where students will be guided in answering the following questions:
What is a function?
How do we graph functions?
What is the domain and range of a function?
How do we identify domain and range from the graph of the function?
Day 4-5. Research on a Famous Muslim Mathematician
The students worked in groups of five. Each group researched on famous Muslim mathematicians and their contributions to mathematics.
Expected Outputs: (a) Accomplished Activity sheets and (b) biography of the famous Muslim Mathematician and (c) summary of his/her contributions to Mathematics.
Day 6-10. Exploring Graphs of Functions Using a Graphing Calculator. (Connecting Mathematics Skills to Muslim Textiles)
Students will do the activity on horizontal and vertical translation(doc). In this activity students will make a generalization on the translation of y = x that gives the graph y = x ± k, if k is a positive real number. They will then make generalizations on the translation of y = x2 that gives the graph y = x2 ± k.
In addition, students also write their description on the translation of figures they will find in the various Batik patterns from Mindanao. Each group will choose a Maranao textile and investigate the mathematical concepts used in the design.
In this activity students reflect back on the answers to the questions, How are designs created? How does mathematics relate to craft? and also the content questions What are the geometric transformations found in a Maranao textile?
Expected Outputs: (a) Accomplished Activity Sheets and (b) Initial Results of their investigation of transformation concepts found in a design
Day 11-12. More Activities on Exploring Graphs of Functions Using Graphics Calculator.(Connecting Mathematics Skills to Muslim Textiles)
Students will do activity on exploring exponential function(doc), logarithmic function and its inverse(doc) using graphics calculator. In this two activities students will make conjectures on the effect of 'P', 'a', and 'k' on the graph y = Pakx, and the effect of 'a' on the graph y = logax. The concept of reflection was used on solving the inverse of a function.
During these days, each group will create their own designs. To do this, the students will further research on the concepts of transformation and consult an art teacher in making their suggested designs.
With their initial investigation on the mathematics concepts found in their chosen Maranao textile, the student will writejournals(doc)on How does mathematics relate to craft?
Expect Output: Journal on How does mathematics relate to craft?
Day 13- 15. Creating Multimedia Presentation and Brochure.
Students will present their storyboard before they can use the computer laboratory. Their multimedia presentations(ppt) must have the following content.
Life of their chosen Muslim Mathematician and his/her contributions to Mathematics.
Analysis of the transformation concepts found in the chosen Maranao Craft.
Their brochure on the other hand must showcase the design they created applying the transformation concepts they learned. The figure they need to transform may note necessarily be functions. Creating designs by transforming graphs can be an activity for advanced students.
Expected Outputs: Multimedia presentation and brochure.
Prerequisite Skills
The students must have prior knowledge on:
classifying angles and polygons.
finding relationship among angles, sides and diagonals of parallelogram.
locating points on a coordinate plane.
graphing functions using graphing paper.
Technology-related Skills:
Students must know how to
use a graphing calculator;
use an encyclopedia on CD ROM or surf for information using the Internet;
scan, edit, and save images;
use presentation or publication software in coming up with a multimedia presentation or website; and
use a word processing application.
Differentiated Instruction
Resource Student
Resource students can focus more on identifying transformation concepts in Maranao crafts. As soon as they master to do this, they can then be allowed to develop their own designs using transformation concepts.
Gifted Student
The students can investigate further on the use of arithmetic sequence, fibonacci and geometric sequence on arts. Students can explore functions and make designs by transforming graphs of functions.
Student Assessment
To evaluate technology-based outputs of students, the teacher will use the following rubrics: | 677.169 | 1 |
Details about Introductory Algebra:
Features AMATYC/NCTM standards of content and pedagogy - integrated into applications, marginal exercises, and pretests. This book includes graphics, models and illustrations which clarify and reinforce concepts with the frequent integration of bar charts, line graphs, applications, illustrations, calculator screens and geometric figures.
Back to top
Rent Introductory Algebra 2nd edition today, or search our site for K. Elayn textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson. | 677.169 | 1 |
Business Math Procedures Brief Editionis a comprehensive introduction to the concepts and applications of mathematics to personal and commercial business problems. The text uses basic arithmetic and problem solving techniques and illustrates their use in retailing, interest and loans, banking, payroll, taxes, investments, insurance, and a variety of other business situations. The text is well known for the motivating integration of interesting real world examples and photos from the Wall Street Journal, Kiplinger's, and many other business journals.Slateris the most popular and widely used book for this course and is carefully written and developed to support students with little math experience with practice quizzes, thousands of exercises, color coded procedures and diagrams, supporting tutorial videos on DVD, and the highest standards of reliability and cleanliness.
Practical Business Math Procedures Brief Edition, 10th Edition
Chapter 1: Whole Numbers: How to Dissect and Solve Word Problems
Chapter 2: Fractions
Chapter 3: Decimals
Chapter 4: Banking
Chapter 5: Solving for the Unknown: A How-To Approach for Solving Equations | 677.169 | 1 |
Main menu
Algebraic Thinking and Misconceptions
Algebra is considered the "gateway" to higher education. Failure rates in Algebra courses are staggering in school districts across the country. This lack of success disproportionately affects high needs students and impacts high school graduation rates, success in higher mathematics, and college enrollment. A new, innovative approach that prepares new teachers to stem the tide of failure is needed that takes into account why students struggle in algebra. Over 800 studies examine students' struggles in algebra and potential strategies to help students overcome conceptual obstacles. Yet, the size of this resource makes it essentially inaccessible to preservice teachers.
The mission of the Center is to help open the algebra gateway for students. We aim to help restructure preservice math teacher education and professional development of veteran teachers by utilizing research on students' algebraic thinking. After initial funding by the Fund for the Improvement of Post-Secondary Education (FIPSE), we are now a non-profit organization. The innovative Center includes
We also have two new books published to help teachers use our iOS apps. You can get the iBook, Teaching Algebra through iPad Apps, through the iTunes store (try searching by the title or one of the authors: Doug Neill or Steve Rhine) or through Gumroad. The Kindle version and student PDF pages are also available at Gumroad ( | 677.169 | 1 |
Introduces readers to the theory and application of modern numerical approximation techniques. Providing an accessible treatment that only requires a calculus prerequisite, the authors explain how, why, and when approximation...
Taalman and Kohn's Calculus offers a streamlined, structured exposition of calculus combining the clarity of classic textbooks with a modern perspective on concepts, skills, applications, and theory. Its uncluttered design | 677.169 | 1 |
Student Ambassadors
Algebra 2 is a course where students expand on their knowledge of algebraic thinking and symbolic reasoning to better understand the structure of Algebra. Functions and equations are used for analysis and understanding relationships among algebraic concepts. The Applications in Algebra 2 Series gives the students the opportunity to make connections between algebra and geometry and use one to help solve the other.
Geometry, Algebra 2, Precalculus Students will apply mathematics formulas to calculate various geometric measures in the relationships between the height of a lunar communication tower and the tower's communications range. | 677.169 | 1 |
Facts on File Dictionary of Mathematics
9780816056514
ISBN:
081605651X
Edition: 4 Pub Date: 2005 Publisher: Facts On File, Incorporated
Summary: Encompassing every mathematical term and concept of interest, this book conveys information to students and general readers in a proven, accessible format. This fourth edition contains approximately 320 new entries, dozens of new photographs, new pronunciation symbols, a list of websites, and a bibliography. Current entries and back matter have been revised as needed. Relating significant information in a non-special...ist manner, this dictionary is an invaluable resource.
Daintith, John is the author of Facts on File Dictionary of Mathematics, published 2005 under ISBN 9780816056514 and 081605651X. Three Facts on File Dictionary of Mathematics textbooks are available for sale on ValoreBooks.com, and three used from the cheapest price of $2.39.[read more] | 677.169 | 1 |
Overview
REA's Math Workbook for Algebra and Functions is perfect for high school exams, including end-of-course exams and graduation/exit exams.
This math workbook will help high school math students at all learning levels understand algebra and functions. Students will develop the skills, confidence, and knowledge they need to succeed on high school math exams with emphasis on passing high school graduation exams.
REA's Algebra & Functions Math Workbook includes:
Over 20 review lessons with many step-by-step examples
Each lesson builds on the students' past progress
Drills reinforce essential skills
Skill scorecard measures progress and success
"Math Flash" feature provides tips and strategies
Quizzes measure subject mastery
Answer key with detailed explanations
The Algebra & Functions Math Workbook will help students master the basics of algebra—and help them face their next math test—with confidence!
Related Subjects
Read an Excerpt
About This Book
This book will help high school math students at all learning levels understand basic algebra. Students will develop the skills, confidence, and knowledge they need to succeed on high school math exams with emphasis on passing high school graduation exams.
More than 20 easy-to-follow lessons break down the material into the basics. In-depth, step-by-step examples and solutions reinforce student learning, while the "Math Flash" feature provides useful tips and strategies, including advice on common mistakes to avoid.
Students can take drills and quizzes to test themselves on the subject matter, then review any areas in which they need improvement or additional reinforcement. The book concludes with a final exam, designed to comprehensively test what students have learned.
The Ready, Set, Go! Algebra & Functions Workbook will help students master the basics of mathematics—and help them face their next math test—with confidence! | 677.169 | 1 |
academic news
The Midwestern Higher Education Compact has named the campus the statewide leader in both areas.
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A Freshman Math Course-which one is right for you?
Getting started in the correct math course for you is important, so we have put together some resources to:
help you understand the math courses at Morris,
explain our Math Diagnostic (which helps you determine which course is right for you), and
assist in your preparation for the Math Diagnostic Exam.
Freshman Mathematics Courses at Morris
Here are the courses a Freshman typically enrolls in:
The arrows indicate prerequisites, so if you needed Calculus I for your major you would need to have mastered:
Precalculus I: Functions
Precalculus II: Trigonometry
before taking Calculus I.
Survey of Math
Survey of Math has no prerequisites other than high school math—this course meets the M/SR General Education requirement, but relies only minimally on the algebra you learned in high school. It is typically taken by Elementary Education majors who are not pursuing the Math Sub-plan, or any student who wishes to meet a general education requirement but are not planning to take any further math courses (this course does not prepare you to take Calculus I, for example).
Survey of Calculus
Survey of Calculus is an introduction to calculus that does not involve any trigonometry. It is an option for students who major in Biology, or any student who wants to meet the M/SR General Education requirement by learning some more about what calculus is all about.
Calculus I
Calculus I is an introduction to calculus that will use trigonometry, and is required for students who plan to major in Chemistry, Economics, Environmental Science, Financial Management, Geology, Math, Physics, or Statistics. Computer Science majors typically take Calculus I, and Biology students may also take Calculus I for the Biology major.
The courses Basic Algebra, PreCalculus I: Functions, and PreCaluclus II: Trigonometry are also frequently taken by Freshmen who need to build their mathematical skills before proceeding to Survey of Calculus or Calculus I.
Basic Algebra
Basic Algebra reviews the algebra you will use in PreCalculus I and PreCalculus II. It generally moves at a slower pace than the precalculus courses.
PreCalculus I
Functions reviews the concept of functions and properties of functions you will use in Survey of Calculus, Calculus I and beyond.
PreCalculus II
Trigonometry reviews the trigonometry you will use in Calculus I and beyond.
You can find more detailed descriptions of all our courses in the Morris Catalog.
Choosing the Initial Mathematics Course That is Best for You
Morris offers exemption from Calculus I and/or Calculus II to those who do well in AP calculus, according to the following scale of performance.
Automatic exemption from Calculus I and II for AP calculus B/C score of 4 or 5.
Automatic exemption from Calculus I for AP calculus B/C score of 3 or AP calculus A/B score of 5.
An AP calculus A/B score of 3 does not gain an exemption for Calculus I.
Automatic exemption from Survey of Calculus for AP calculus A/B score of 3 or 4.
For all other students intending to do a mathematics course in their first year, advice is given based on a combination of their performance in a diagnostic 'placement' test, their ACT math subscore and information on previous math courses they have done.
The Diagnostic 'Placement' Test in Mathematics
The Test
This computer-based, diagnostic test consists of 40 short multiple-choice questions. It tests the level of your command in three areas of precalculus mathematics - basic algebra (14 questions), functions (13 questions), and trigonometry (13 questions). Each section of the test is timed at 25 minutes. The purpose is to assess the level of your command of the material in each section. A preliminary recommendation is made, based on various combination of scores, taken in conjunction with your ACT mathematics subscore, of which courses you are deemed to have gained equivalence. A table of these equivalencies is below. This preliminary recommendation is conveyed to your advisor, who considers it in conjunction with other relevant information before giving a further recommendation.
The levels at which the preliminary recommendations are made are based on correlation, made over several years, with student success rate in the subsequent course entered. They are chosen to anticipate a 90% success rate at the grade of 'C' or above in the course a student enters.
Practice
In order that your true level of mastery may be determined it is recommended that you review material before doing the test. This advice is especially true for those who may not have taken any mathematics for several months or even a year.
After a student has an account as an incoming student, on the Morris computer system, access is available to a practice test from the Morris Test Center.
When the test is available
A usual time to do the placement test is at the beginning of the day you come to UMM to register for classes. However you may also arrange with Jane Kill in the Testing Center, to do the test remotely online. It is required that your testing be proctored by an approved teacher or counselor, or at a testing center at another tertiary institution (for which you may have to pay a small fee).
Repeat testing
If you are dissatisfied with the your initial performance, you have the opportunity to complete a repeat test. This may be done at UMM before classes begin in the fall or remotely, as above.
A difficult choice
Students who are recommended to start in Basic Algebra have a difficult choice as the credits for this course do not count towards graduation, though they do count for financial aid and athletic activity requirements. Some students given this recommendation attempt to start at the next level, Precalculus I. The success rate for these students at the grade of 'C' or above is historically at the 50% level, with very, very few students gaining a grade higher than the 'C' level in precalculus. The Mathematics Discipline's strong recommendation is that you should build a firm foundation in your mathematics skills before proceeding!
Information on the Mathematics Diagnostics
The introductory mathematics courses and their prerequisites are:
Course
Prerequisite
Calculus I
Precalculus I
Precalculus II
Survey of Calculus
Precalculus I
Precalculus II
Basic Algebra
Precalculus I
Basic Algebra
Based on the score in each section of the math diagnostic test (Trigonometry, Functions, Algebra) and on the ACT math subscore, each row of the table below indicates which prerequisite a student is deemed to have satisfied.
The table should be read line by line. For example, suppose a student has an ACT math subscore of 29 and in the placement test has scores of Basic Algebra 8, Functions 5, Trigonometry 5. By line: (a) the student has NOT tested out of Precalculus II (Trigonometry). However, reading down to the Precalculus I section, the student scores in Functions and ACT satisfy line (c) and so the student is deemed to have tested out of Precalculus I (Functions) - and hence, automatically, to have tested out of Basic Algebra.
Necessary Scores
Prerequisite Satisfied
Trigonometry
Functions
Algebra
ACTMath
Precalculus II
7 - 13
(a)
Precalculus I
9 - 13
(b)
Precalculus I
5 - 8
29 - 36
(c)
Precalculus I
7 - 13
5 - 8
20 - 28
(d)
Basic Algebra
8 - 14
(e)
Basic Algebra
9 - 13
(f)
Basic Algebra
29 - 36
(g)
Basic Algebra
5 - 8
20 - 28
(h)
Basic Algebra
7 - 13
20 - 28
(i)
Lines (h) and (i) are deemed borderline situations, and you may wish to discuss which math course to take with one of the math faculty.
A simple way to see your provisional placement from the above table is a follows:
If your scores satisfy line (a) , count 4 points. If not, count 0.
If further your scores satisfy any one of lines (b), (c), (d) add 2 more points (only once).
If your scores satisfy any one of lines (e), (f), (g), (h), (i) add 1 more point (only once).
Then
if your total is 6 or 7, you would be placed in Calculus I,
if your total is 4 or 5, you would be placed into Precalculus I (exempt Precalculus II),
if your total is 2 or 3, you would be placed in Precalculus II or Survey of Calculus,
if your score is 1, you would be placed in Precalculus I and Precalculus II,
if your score is 0 you would be placed in Basic Algebra | 677.169 | 1 |
Description
KEY BENEFIT: This trusted reference offers an intellectually honest, thought-provoking, sound introduction to linear algebra. Enables readers to grasp the subject with a challenging, yet visually accessible approach that does not sacrifice mathematical integrity. Adds over 400 new exercises to the problem sets, ranging in difficulty from elementary to more challenging. Adds new historical problems taken from ancient Chinese, Indian, Arabic, and early European sources. Strengthens geometric and conceptual emphasis. A comprehensive, thorough reference for anyone who needs to brush up on their knowledge of linear algebra | 677.169 | 1 |
Programs
Math Learning Center
The Math Learning Center (MLC) is a room where a math teacher and/or student tutor can help you out if you have questions.
Some students go to the MLC to "talk out" their ideas about problems and to clarify questions from class. Other students go regularly to have a quiet place to do their math homework and like the security of having a teacher there in case of questions.
Going to the MLC is one way to show that you have initiative and determination for learning math, and that you are taking responsibility for your own learning. It is a popular room where students get to know teachers and peer tutors better.
The Math Learning Center is on the second floor of Weaver Hall in room 207 | 677.169 | 1 |
Resources
Math Learning Lab
The Learning Lab is located in Room 222 of the Doggett Math and Behavioral Social Sciences Building. The purpose of the Learning Lab is to assist students in their mathematical studies at Walters State . The Mathematics Learning Lab offers free tutoring for all your math classes and other math related questions. Beth Dixon is the Learning Lab Technician for Mathematics. Beth and student workers provide tutoring in the Math Lab. No appointments are necessary; students may drop in during any time that the Lab is open. Help can be obtained through e-mail and by phone for students unable to make it to the main campus during lab hours. Please contact Mrs. Dixon if you have any questions or need any assistance.
Free Tutoring
Are you struggling with your Mathematics course? Would you like to improve your math grades? Do you need help with a single question, a few problems, a whole chapter, or the entire course? Tutoring can help!
Tutoring is a service offered to Walters State students. Students can drop by the Math Lab in Room 222 in the Dogget Math and Behavioral and Social Sciences Building on the Morristown campus to receive tutoring. Free tutoring is available each semester.
Podcasts
The Mathematics Learning Lab has provided podcasts on various mathematics topics to assists students in their math courses. The podcasts are organized by course and topic and links are provided below. The podcasts are not intended to replace teacher instruction but instead to provide additional practice to supplement classroom time. Classroom attendance is vital.
Students needing additional help may come to the Mathematics Learning Lab located on the Morristown campus in MBSS room 222. Tutors and Mrs. Beth Dixon are available to help students. | 677.169 | 1 |
Join the Conversation
Mathworks Curriculum Samples
Mathworks Math Explorations
Math Explorations is a series of three textbooks that cover the math TEKS for 6th grade, 7th grade, 8th grade, and Algebra I. The textbooks integrate research from the laboratory of our summer math programs that have been held for more than 25 years.
Using this curriculum, young students are engaged in using algebraic ideas, and these ideas are built upon throughout their middle school years. Math Explorations weaves algebra and algebraic ideas with hands-on, inquiry-based explorations for students working independently and in groups. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Find a La HondaThis equation relates a function with its derivative(s). It can represent physical properties as a function of time As an example, a differential equation can be used to describe the velocity of a ball falling through the air considering gravity and air resistance. My background in Civil Engi...
...In addition to teaching content, my primary intentions in every lesson are: to enable the student to feel like the experience is personal, relevant, and empowering; to guide him or her to discovery by training the mind to be open, flexible, and mostly to develop critical reasoning skills necessar... | 677.169 | 1 |
Developed for the "transition" course for mathematics majors moving beyond the primarily procedural methods of their calculus courses toward a more abstract and conceptual environment found in more advanced courses, A Transition to Mathematical Proofs emphasizes mathematical rigor and helps students learn how to develop and write mathematical proofs. The author takes great care to develop a text that is accessible and readable for students at all levels. It addresses standard topics such as set theory, number system, logic, relations, functions, and induction in at a pace appropriate for a wide range of readers. Throughout early chapters students gradually become aware of the need for rigor, proof, and precision, and mathematical ideas are motivated through examples. | 677.169 | 1 |
Better student preparation needed for university maths: UK study
August 1, 2012
Moving from sixth form, or college, into higher education (HE) can be a challenge for many students, especially those who start mathematically demanding courses. Life prior to university focuses on achieving maximum examination success to be sure of a place. Faced with this pressure, school and college maths courses pay little attention to preparing students to use maths in other areas of study according to a project funded by the Economic and Social Research Council (ESRC).
A student's ability to apply mathematical reasoning is critical to their success, especially in HE courses like science, technology, engineering and medicine. The study, undertaken by Professor Julian Williams, Dr Pauline Davis, Dr Laura Black, Dr Birgit Pepin of the University of Manchester and Associate Professor Geoffrey Wake from the University of Nottingham, shows that it is important to understand how students can prepare for the 'shock to the system' they face and how they can be given support at school, college and university to help in the transition.
The researchers found that students were not fully aware of the importance of the mathematical content in the courses they had joined at university, and particularly how to apply maths in practice.
Associate Professor Geoffrey Wake states, "Different teaching styles of university lecturers and the need for autonomously-managed learning, where students need to learn some mathematical content of their courses on their own without input from lecturers, also came as a bit of a shock for many students. On the other hand, some of the lecturers had limited knowledge of the exam-driven priorities of A-level maths courses and were not aware of the techniques students had been taught prior to attending their university courses."
The researchers also found significant problems in motivating students to engage with the mathematics within their chosen university coursewhere mathematics was not their main area of study. Generally, schools and colleges were found not to be preparing students for university learning practices, and the level of learning-skills support was variable once students arrived at university.
"Many students felt that they would benefit from student-centred learning and greater opportunity for dialogue with their lecturers," says Associate Professor Wake. "Unfortunately, the efficiencies required of university teaching resulting in lecturing of large numbers of students makes developing such a learning culture unlikely."
The findings led the researchers to consider the implications for the policies and practices of schools, colleges and universities recommending a better two-way flow of information between schools and colleges and universities to address the issues of preparation and expectation.
They concluded that the sixth-form curriculum should provide 'learning to learn' skills and mathematical modelling for students following A-level maths coursesA recent study reports that high school students who study fewer science topics, but study them in greater depth, have an advantage in college science classes over their peers who study more topics and spend less time on ...
Engineering students with average grades from upper secondary school can manage difficult courses just as well as students with high grades. At least, if a group of them meet an older student once a week during the first ...
College students participating in a new study on online courses said they felt less connected and had a smaller sense of classroom community than those who took the same classes in person – but that didnt keep online ...
Introductory science courses – in biology, chemistry, math and physics – can be challenging for first-year college, CEGEP and university students. Science 101 courses can make or break a student's decision to venture ...
What is the difference between e-learning, online learning and distance learning? University of Missouri researchers have found that even educators can't agree on what different forms of learning environments entail and | 677.169 | 1 |
I'm getting really tired in my math class. It's integration by parts calculator, but we're covering higher grade syllabus . The concepts are really complicated and that's why I usually sleep in the class. I like the subject and don't want to fail , but I have a big problem understanding it. Can someone help me?
I suggest that you try out Algebrator. I have been using this program for a few months now and I can frankly say that it is what helped me save my grades this semester. Algebrator offers amazing ways to deal with difficult problems. You will definitely love it, I can guarantee. | 677.169 | 1 |
DVD Review: The Basic Math Word Problem Tutor and The Algebra 2 Tutor
Math has never been a particularly strong suit of mine. Everything started going downhill after long division, and I was well lost by the time multiplying and dividing fractions rolled around, let alone algebra. It was with some fear and trembling that I found The Algebra 2 Tutor – 6 Hour Video Course from MathTutorDVD in the mail. The Basic Math Word Problem Tutor – 8 Hour Video Course I felt confident in my ability to conquer, but the algebra — it scared me. It wasn't even Algebra 1, but an advanced level!
Thankfully instructor/owner Jason Gibson's highly visual, common sense approach to math made algebra seem within even my limited grasp. In each video course Gibson breaks down mathematical concepts by dividing and conquering. Each math topic, whether it be algebra, word problems or his other DVDs on calculus, physics, etc. divides the topic into sub-topics, each of which is worked through in an ascending level of complexity through abundant problems drawn out on a white board. Avoiding lengthy lectures, Gibson gets down to basics and explores the necessary connections while working out problems at the board. In fact, most of what you'll see is the back of his head; he turns to instruct and give detail, but most of his explanations are interwoven with real problems.
His thorough breakdown of concepts is best demonstrated by sharing the outline of the courses.
There is nothing flashy about the discs, they consist simply of Gibson at the whiteboard as he instructs. Gibson is not only the star of the show, but he's in fact a one-man show – providing direction, production etc. As a result, the production quality is nothing to write home about, but let's be honest; this is about math, not about entertainment.
I was so strongly reminded of my own experiences in math class that I was surprised to find that Gibson is not himself a math teacher, but a physicist working with NASA. Hey, that works for me, we all know physicists need to know a lot of math – I'm just surprised that Gibson is able to make it so real. Even when relating the most basic concepts of word problems he's never condescending, and always patient and thorough. With his extensive personal understanding of mathematics he's able to take students from basic math problems through to advanced courses of study.
Algebra 2 can be supplemented with a CD containing a complete set of worksheets, making for a complete course of study. A good understanding of basic mathematical facts, pre-algebra and Algebra 1 (or equivalent) is necessary before embarking on this DVD course. Basic Math Word Problems does not have any supplementary worksheets, and is better suited to a student who has mastered basic mathematical operations and needs review, remediation, test preparation or just a different angle or explanation for approaching word problems.
Gibson's low-key yet thorough approach to mathematical concepts are rapidly placing him as a favourite in the realm of live-action math instruction and at an average price of $27/course there is absolutely no comparison to an in-person tutor. The Math Tutor DVDs are an economical option for adults who need to brush up on their math, homeschooled students who have surpassed their parents' math knowledge, and students from late elementary through college who are struggling with understanding their math courses.
For a complete listing of course DVDs and supplementary worksheet CDs visit the MathTutorDVD website. | 677.169 | 1 |
A step-by-step visual guide to Hands-On Equations®. Each of the 26 lessons of the program is clearly demonstrated. The DVD is designed to be viewed one lesson at a time. Ideal for classroom introduction or as a student self-introduc..
• Hands-on, concrete approach to algebraic linear equations• Enhances student interest in mathematics• Does not require any algebraic prerequisitesDeveloped by Dr. Henry Borenson, this is an innovative approach...
More than 250 verbal problems with solutions! Included in this resource are number, age, coin, and distance problems. Problems are provided for Levels I, II, and III. 203 pages. Grades 3-8.Click here for...
For teachers of Hands-On Equations® who are also using the SMART Board. A slide is included for each teaching example of the red, blue, and green manuals. To maintain the effectiveness of the program, the teacher and students must also use the ..
Prices listed are U.S. Domestic prices only and apply to orders shipped within the United States.
Orders from outside the United States may be charged additional distributor, customs, and shipping charges. | 677.169 | 1 |
books.google.com - An undergraduate-level text which challenges the student throughout with the development of topics in linear algebra. A study guide, instrutor's edition and instructor's technology resource manuals are also available.... Algebra and Its Applications
Linear Algebra and Its Applications
An undergraduate-level text which challenges the student throughout with the development of topics in linear algebra. A study guide, instrutor's edition and instructor's technology resource manuals are also available.
From inside the book
Review: Student Study Guide for Linear Algebra and Its Applications
User Review - Gabriel Mintzer - Goodreads
This study guide did indeed guide me through my study of Linear Algebra. Not only did it present detailed solutions to the textbook exercises; it also provided chapter summaries and useful notes.Read full review
Review: Student Study Guide for Linear Algebra and Its Applications
User Review - Goodreads
This study guide did indeed guide me through my study of Linear Algebra. Not only did it present detailed solutions to the textbook exercises; it also provided chapter summaries and useful notes.Read full review | 677.169 | 1 |
This learning object from Wisc-Online covers simplifying algebraic expressions using multiplication. The unit's activities include defining the terminology associated with algebraic operations, using the fundamental laws of algebra to simplify those expressions, removing the symbols of grouping and changing the signs of the appropriate terms to simplify expressions. Practice questions are also...
This learning object from Wisc-Online covers the properties of equality as related to algebraic equations. The unit's activities include defining the terminology and properties of equality associated with algebraic operations and solving simple equations using these properties. Practice questions are also included.
This learning object from Wisc-Online covers solving systems of linear equations using the addition or subtraction method. The unit looks at the common solution to simultaneous linear equations (also referred to as "system of linear equations"). Practice questions are also included.
This learning object from Wisc-Online covers solving systems of linear equations using the substitution method. The unit looks at the common solution to two or more linear equations in two variables. Practice questions are also included.
This learning object from Wisc-Online covers simplifying algebraic expressions using division. The unit's activities include defining the terminology associated with algebraic operations, using the fundamental laws of algebra in simplifying algebraic expressions, removing the symbols of grouping and changing the signs of the appropriate terms to simplify algebraic expressions. Practice questions... | 677.169 | 1 |
AMS Grad Blog
By and for Math Grad StudentsSat, 28 Nov 2015 20:25:28 +0000en-UShourly1 I help my Students overcome their Fears, create a Supportive Classroom, and get Students to ask Good Questions
28 Nov 2015 20:22:47 +0000 reading →]]>During my five years of teaching experience as a teaching assistant including teaching, grading and math tutoring at Washington State University (WSU) and American University of Sharjah (AUS), I have noticed that many students are not motivated because they are scared of subjects that deal with mathematics, and because there is a common belief that math is complicated and boring subject. When they go to any math class in general and freshmen and sophomore levels in particular, they already have a prejudice against the subject matter. One of my major obstacles as a teacher is to create a friendly environment. With patience and encouragement, I then proceed to build their self-confidence in learning mathematics. A successful math teacher must make the students feel that mathematics is learnable, applicable and enjoyable. The following is a list of two examples that I do in my Calculus II class to help my students overcome their fears from topics such as tests of convergence and divergence for series, and absolutely and conditionally convergent series:
Tests of Convergence and Divergence for Series: students do not usually like this important section of Calculus II curriculum because when they go to the exam, they get confused with two or more tests due to the similarity between some tests. As a result, they do not do well in exams and eventually in the course itself. Therefore, I decided to help them all by creating a table that contains all tests together, and I also added a "notes" section so they understand my notes and comments about each test. This table is available on my course webpage. Here is a sample of my table:
Absolutely and Conditionally Convergent Series: students usually consider deciding whether series is absolutely convergent or conditionally convergent is one of the most difficult things in Calculus II. However, I created for them a table, and I called it "Binary Method for Alternating Series Test". The name of this method stems from the fact that binary numbers are 0 0, 0 1, 1 0, and 11, and they represent divergence-divergence, divergence-convergence, convergence-divergence, and convergence-convergence, respectively. This table is available on my course webpage. The following is the Binary Method for Alternating Series Test table:
In conclusion, I also believe that students must engage with this learning environment during class discussions by asking challenging questions that make us both (teacher and student) think about these questions. For example, when I create a handout for my Calculus II students, I usually like to include one challenging question and ask my students to think about it. Then, we can start the class discussions about that challenging question. A perfect example of one of my challenging questions in Calculus II is the following:
To see more examples like this and other handy methods, please see my course webpage. Best of luck and feel free to reach out if you have questions!
My plant, Dave, and I have earned 161 fake internet points, plus 6 more special fake internet points. Voldemort currently has more fake points than we do.
As a fourth-year grad student in math at the University of Minnesota, I spend a lot of time thinking about math problems, but I get worn out when I think about the same problem for too long. Sometimes it can be helpful to take a break and work on something more fun, such as…other math problems. Easier problems. Problems you know how to do. Doing this can feel like you are procrastinating and accomplishing something at the same time. In my first post here I want to tell you about a great resource for endless math problems of every sort of difficulty: Math Stack Exchange.
This website has the added benefit that in return for your hard work spent answering math problems (i.e. procrastination), you receive "reputation points." The more points you get the smarter you feel, even if you could have spent that time on your own research.
The Stack Exchange website is a child of Stack Overflow, a site created in 2008 as a forum for professional programmers to request and share answers to the many questions they come across while coding. The best answers are voted up and move higher in the list of answers so you can easily find them. Poor answers and questions that are off topic are voted down so they won't have a negative impact on the forum. The site is somewhat like Yahoo Answers, except users will say intelligent things. If someone writes a nice solution and you vote it up, they will be awarded more reputation points. As it turns out, fake points on the internet are a huge motivator. Hundreds of people are waiting at all hours of the day, hoping they know the answer to your question. Hardly ever am I faced with a coding problem that hasn't been answered already on Stack Overflow.
Math Stack Exchange (Math.SE) is a forum for any sort of math question. There is a similar site called Math Overflow for questions related to open research problems, but on Math.SE any question is fair game, as long as you have put in some effort to figure it out yourself. If you are looking for an endless feed of calculus questions, this is the website for you. The first question I answered on Math.SE was related to circle maps, which is something I'm currently studying. Often in math it can feel like few people care about the highly specific abstract problems you are solving. However, there might be people out there who do, and there's a good possibility they are on Stack Exchange. Some of them might even need your help; they will be sure give you fake points in return.
]]> nature, how we teach math, and the birthday problem
25 Nov 2015 19:31:01 +0000 reading →]]>I've spent a few weeks wondering what I can write about for my first post here. I'm a first-year PhD student with an endless supply of questions but without much wisdom or insight to share yet about my short graduate life. As a recent college graduate, however, I have spent years thinking about how my friends and peers perceive my mathematical interests and my career choices. And while I'm still learning about graduate life, I have a wealth of opinions about how we approach communicating math and statistics to students, clients, and colleagues interested primarily in other areas.
In college, I asked many other students to clarify why they don't like math. I got two answers over and over again- it's boring, and it's too hard. The chain rule and the shell method of integration do not strike them as relevant to their future careers or to their broader understanding of the world around them. Moreover, they have been told from a young age that math is too hard, so why bother trying?
As mathematicians, we appreciate beautiful math for its own sake and do not question that a theorem's truth accords it value. If we incorrectly assume that our friends and our students automatically share this appreciation for mathematical truth and elegance, however, we miss the opportunity to reintroduce them to math in a way that challenges their views on it.
What got you interested in math in the first place? For me, an important part was questions that have surprising answers, questions that make me think differently and re-examine the world around me. A simple but incredible example of such a question is the famous birthday problem. Given 20 people in a room, what is the probability that at least two have the same birthday? Intuitively, most people would think the probability is fairly low. Ignoring leap years for simplicity, there are 365 days in a year, so 20 people seems like a very small number comparatively; it should be pretty unlikely for 2 to have the same birthday.
As you probably know, an easy way to actually calculate the probability is to compute the probability that no two people have the same birthday, and subtract it from one. Take the first person in the room- let's say it's me. My birthday is August 18th, so we can rule out August 18th for everybody else. Then consider the second person in the room; we must have different birthdays, so he can have a birthday on any of the remaining 364 days, which happens with probability 364/365. Say his birthday is September 26th. Now take the third person; she can have a birthday on any of the remaining 363 days, which occurs with probability 363/365, and so it continues. We multiply each of these probabilities to get our answer for the probability that no two students have the same birthday: ∏ (365-i)/365, from i=1 to i=19. Call this probability A; then, the probability we want is then 1-A. I (okay, Wolfram Alpha) computed this as about 0.411438.
Who would have guessed? With only 20 people there's a 2/5 chance that a pair of them have the same birthday out of 365 possible birthdays! This seemingly paradoxical answer is mathematically satisfying, but also reveals some interesting ideas about how we think (Stamp, Mark. Information Security: Principles and Practice. Jon Wiley & Sons, Inc. Hoboken, New Jersey, 2011). Why are our first guesses so far from the truth?
My first instinct, a common one, is to think about the probability that somebody else has the same birthday as I do. That's 1/365 * 19, about a 5% chance. The less obvious but far more important question is: do any of the people in the room share a birthday with each other, that may not be August 18th? When we consider that, we open not just 19 comparisons but 20 choose 2, or 190 comparisons. In that light, it seems far more likely that we could get a match; now 41% sounds quite reasonable.
The birthday problem is a fairly straightforward math exercise. However, the seemingly paradoxical answer to it highlights our nature to think selfishly, to insert ourselves into every comparison even though we should mostly be considering pairs of other people that do not contain ourselves. A counting problem concerning days of the year may be difficult, but it is far more approachable than taking anti-derivatives of complicated functions. And considering the selfish nature of human beings, and how it manifests itself in our attempts at problem solving, can hardly be called boring or worthless. If we can present math and statistics in a framework like this, we can engage far more students. Math is surprising and amazing sometimes, and while elegant proofs have merit, so do tricky and surprising problems. Both perspectives on math were central for developing my excitement about the field, and have motivated me to pursue a graduate degree in statistics.So here I am, and for everybody with wisdom to share about qualifying exams, choosing an advisor, and getting that NSF GRFP, I'll be reading.
The Institute for Advanced Study in Princeton, NJ is one of many possible hosts for your math staycation. (Photo by Alexi Hoeft)
Thanksgiving is this week and the holidays are right around the corner, which means most of us will be getting several weeks off from formal grad school requirements. But the time off is good for much more than just plentiful eating, quality family time, and Netflix binge-watching (a verb which, if you missed it, was recently added to the dictionary). A fun holiday activity to add to the list: a mathstaycation! (Shockingly enough, this marvelous term [according to Google] is not yet in use.)
A math staycation consists of remotely (in space and/or time) attending a math conference by watching the video lectures from the convenience of wherever you might find yourself during the holidays.
After deciding that this sounds like the most exciting use of your holiday time that you ever could have imagined, the first step toward planning your math staycation is choosing the math conference that you want to follow. The quantity, quality, and variety of recorded conferences has skyrocketed in the last decade, and you will likely have many more good candidate workshops than time allows.
There are two main styles of math staycation:
video combo plate (custom-build a playlist of math video lectures/recorded seminars that are of interest to you, even if they originate from different conferences) and
traditional conference entrée (watch an entire conference more or less in its entirety and in the original order).
Search for your math staycation materials using Google or begin with the extensive lists I've gathered for you below:
Next, you will want to plan your math staycation (to ensure it actually happens). Set aside about a week of time for your staycation. Schedule your staycation for a quieter portion of your holiday season, e.g. after the New Year if that is when you might have a bit more uninterrupted time to yourself. Mark it in your calendar to help make it psychologically "official" that you are "attending" the conference. If you also want more guidance on the day-to-day schedule of your math staycation, feel free to use as a guide the posted program on the corresponding conference site or that of a similar conference.
If you thrive with accountability, you can use these last remaining weeks of the semester to find one or two graduate students in your area of interest who might want to embark on a math staycation with you; your group could maintain an email chain throughout the math staycation to help everyone stay connected and engaged. You could even choose students outside your field and mutually commit to presenting what you learned from the math staycation to one another over lunch soon after the holidays are over. Alternatively, you can sign yourself up to give a talk next semester through one of the graduate student seminar series at your department in order to give yourself a focused goal during your math staycation.
Are there other great web pages of archived math conference videos that you would recommend? Have you chosen your math staycation? Leave a comment below!
I hope you enjoy your math staycation, and come back after break to let us know how it all went!
And if you're done applying for jobs, consider some of these tips to keep busy for these next few stressful weeks of waiting.
Tips for the Post-Application Doldrums
If you have any social media accounts, you may want to double check their privacy settings. I recently realized that many of my old Facebook posts were publicly available. I hadn't posted anything particularly incriminating, but I also didn't expect potential employers to have access to those comments. I followed these steps from Gizmodo to lock down old posts.
At tea a few weeks ago, a fellow job seeker mentioned a a cold calling technique I hadn't previously considered. For those jobs for which you're particularly interested, write a short note saying as much. Do your research in order to direct the message to the correct person. I would imagine that the head of the search committee is probably the last person who wants to hear from applicants outside of the approved channels. Instead, look for any personal or research connections to the faculty; try to warm up that cold call. The person who suggested this tactic said that it netted them a handful of speaking invitations.
To lessen the job search anxiety, try to have some well-defined research tasks scheduled. I find collaboration the easiest way to stay on track. I currently have a weekly Skype meeting with my advisor's previous student, as well as an email conversation with another. I would be much less productive without this correspondence. Moreover, the Skype discussion doubles as a mentoring session as the other person has a tenure-track position and shares timely advice about the hiring process. If you don't have any collaborators, find some! The blog post Building Your Research Army contains links to many programs which support small groups of researchers. And don't forget to make the most of the Joint Meetings by emailing researchers closely "related" to you. Suggest a meeting to discuss potential projects (which you should have just written about for your research statement).
My next suggestion is to prepare your job talk. Once the interviews start rolling in, you will find yourself visiting a new campus, giving a talk in front of a very important audience (in terms of your future employment). The AMS Sectional Meetings provide a proving ground for you to test your presentation skills. I wasn't aware that there are two levels of talks: special sessions and contributed talks. You should submit for a 20 minute talk in a special session, but you will be considered for a 10 minute contributed talk if the session is full. Ideally, you would attend the Sectional Meeting in the region to which you're applying for jobs, though this may not be possible. Giving a talk is a great way to get invited to give more talks, which is a great way to get your job application on the top of the stack.
There's plenty of job talk advice floating around, so I will only link to this article and another, in addition to this essay of Paul Halmos. Ok and this excellent slide deck of advice. Remember to practice early and often. You may gather all of the soon-to-be graduates in your department for a "Job Talk Seminar". Invite everyone to view the talks and distribute a grading rubric so attendees may provide anonymous feedback to the speaker.
Finally, you need to prepare for upcoming screening interviews, both over the phone/Skype and at the Joint Meetings. There are a number of questions you can expect to be asked (see here and here), so open up a new notebook and start writing down answers. Of course you don't want to sound robotically rehearsed, but it's not a bad idea to think of yourself as a campaigning for an election versus applying for a job. I actually already had a phone interview and was asked to explain my research to a non-math person in twenty seconds or less. Twenty seconds! I had prepared a 2-3 minute explanation and was able to condense it down, but it would have been disastrous had I not prepared anything at all! Afterwards, I immediately recorded a debrief in my notebook to help improve my responses. If, like me, you haven't had a phone interview recently, let me remind you how grueling they can be. Especially panel-type interviews where the interviewees are situated around a conference table dishing out questions. But stay positive and remember that if you made it this far, you're hirable on paper!
Not every math PhD program has preliminary exams (aka written qualifiers) and/or master's exams. But for the programs that do, these exams can seem daunting to first and second year students. Both prelims and master's exams are long in duration (varies by program, but around 3 hours from my knowledge) and span the topics of multiple courses. They require endurance, mental agility, and a thorough understanding of the test topics.
Prior to grad school, I did not have experience with exams of this nature. Of course I had experience taking 3 hour final exams for a single course, but I had no experience with preparing for an equally long exam that tests my mastery in multiple graduate-level courses. I sought preparation advice from more experienced graduate students and professors and also learned through trial and error. Here are some tips that I've found effective in preparing for these types of exams.
1.) Make a study plan several weeks (if not more) prior to your exam date. Include topics you need to review on which days. Check off the days as you complete the study assignments as this will help to motivate you and build a sense of accomplishment. A long-term plan like this will guide you through a steady and thorough review of the material, ensuring that you do not resort to cramming at the last minute.
2.) Prioritize prelim/master's exam courses as you take them. Don't take shortcuts in these courses as your success on the prelims and master's exams depend on a deep understanding of these topics. Stay organized in these courses and make an effort to take excellent notes so you can study from them when preparing for your exam.
3.) Schedule full-length, timed practice exams. This is a particularly useful tip for those who are not naturally great test takers. Schedule mock exams for yourself in the exam setting. For example, if your exam is 3 hours long in a quiet room, schedule a 3 hour block where you will go to a quiet room and do a full length practice exam (without your notes!). This will get you comfortable with the exam setting. The practice under timed pressure will also train you to think on your feet, which you'll need on the exam.
4.) Find a study group to meet with regularly at least several weeks prior to the exam. Studying math with a group of classmates is always fun and it has many benefits for prelim/master's exam preparation. Discussing concepts with a study group can help you to absorb concepts more deeply. Talking out loud about your understanding can also highlight weak areas in your understanding; it is better to determine your weak areas of knowledge sooner rather than later. You can also use this as an opportunity to learn from others – perhaps a classmate is strong where you are weak and vice versa. Solving new practice problems together also gives you practice with thinking on your feet but in a stress-free setting.
5.) Complete all learning of exam topics at least a couple of weeks before the exam. This means that, ideally, two weeks prior to the exam, you should not be learning a required concept or topic for the first time. The two weeks prior to the exam should be reserved for practice and review. This two week period of reinforcement and practice will help to solidify the concepts in your brain, releasing your potential for mental agility on test day.
I hope you find some of this tips helpful! Please feel free to add to these tips by commenting below. It would be interesting to see what approaches others have found effective in preparing for these types of exams.
Congratulations to Ian Agol for being awarded the 2016 Breakthrough Prize in Mathematics, the so-called, "Oscars of Science" and mathematics [1]! Tech entrepreneurs Mark Zuckerberg and Yuri Milner created the Breakthrough Prize in Mathematics in 2014 to, "Reward[s] significant discoveries across the many branches of the subject." The prize carries a 3 million dollar award, and was announced during a live televised red carpet ceremony complete with celebrities like Kate Hudson, Pharrell Williams, and Christina Aguilera.
Agol is a topologist with much of his work focusing on the topology of three manifolds. In addition to many of other contributions Agol, together with Daniel Groves and Jason Manning, proved the Virtual Haken Conjecture [2]. I am no expert in low dimensional topology, yet alone the Virtual Haken Conjecture, and I will just point people to Quanta's very nice article giving a non-technical overview. Danny Calegari also has an excellent series of blog posts getting into the more technical aspects of Agol's work.
In addition to Agol, Larry Guth and André Arroja Neves were awarded the New Horizon in Mathematics Prize. This prize, also funded by Zuckerberg and Milner, recognizes, "Junior researchers in the field of mathematics who have already produced important work." In perhaps one of the more interesting aspects of the evening Peter Scholze turned down the New Horizons prize, according to Michael Harris and The Guardian [2]. Congratulations to all those recognized tonight!
]]> Calculus: Two Handy Techniques
05 Nov 2015 05:43:53 +0000 reading →]]>In my five years of teaching calculus, I've noticed that students often struggle with partial fractions and integration by parts. Therefore, here are two alternative methods I use that are faster and easier than the traditional methods that most authors of calculus textbooks use and work on most types of problems:
The Cover Method: Many of my students struggle with partial fractions because of the complexity of the equations they have to solve. I decided to help them by introducing a new method called the cover method, which can solve about half of the partial fraction decomposition problems:
The Table Method: Integration by parts is considered one of the difficult topics for students of Calculus II because some students either do not memorize the standard form of integration by parts or they do not know how to derive it. To make it easy for my students, I decided to introduce a method called the table method. This method does not require memorization, and it can solve integration by parts problems faster than the traditional method. However, it only works for some integration by parts problems such as ones involving polynomials, exponential functions, and trigonometric functions. The example below is from my textbook A Friendly Introduction to Differential Equations
To see more examples like this and other handy techniques, please see my textbook. Best of luck and feel free to reach out if you have questions!
]]> Few Math Riddles
31 Oct 2015 00:32:03 +0000 reading →]]>Hello, and welcome! I wanted to make my first post for this blog about something light, so I thought I would share three of my favorite "over-the-dinner-table" math riddles/problems. These are all in the folklore, but you might not have heard of them. (I certainly didn't make them up myself – they were told to me at various times during casual conversation.) I have posted the problems sans solutions to increase discussion!
I'll begin with a classic:
THE HATS PROBLEM
Ten mathematicians are captured by a madman and are imprisoned in a cell. The madman tells them that tomorrow, they must play a (possibly fatal) game. The game proceeds as follows:
The ten mathematicians stand in a line, one behind the other, all facing the same direction. The madman places a single black or white hat on each person's head. The line is so arranged that each person can see everyone in front of him or her (as well as their hat colors) but cannot turn around and cannot see their own hat. Starting from the back of the line (i.e., the person who can see nine people) and proceeding to the front, the madman gives each mathematician the opportunity to say either "black" or "white". If the spoken color matches the color of hat on their head, then that mathematician goes free; otherwise, he or she is immediately executed.
To heighten the suspense, the madman declares that everyone will be able to hear the "black" or "white" choices (as well as if the person in question lives or dies). The question is: how can the mathematicians devise a strategy to guarantee the safety of some or most of their group?
For example, the mathematicians could pair up into five groups of two: the 10th and 9th places in line, the 8th and 7th places, and so on, with the rearmost person in each pair saying the color of the hat in front of him. The frontmost person in each pair then knows their hat color and can say it in order to go free. With this strategy, five people are guaranteed to live. Can you do better?
Now, another problem:
A DICE PROBLEM
A hundred computer scientists are playing a game. (They are computer scientists because I heard this riddle at a computer science conference dinner.) Each person simultaneously rolls a regular six-sided die. The participants are sitting in a circle in such a way so that each of them cannot see their own die roll, but can see the rolls of all ninety-nine of their co-workers. After all the dice rolls, each of the hundred people writes down a number from one through six. The participants may choose their number based on the dice rolls of everyone else, but are not allowed to communicate in any way or to see what the others are writing.
The hundred numbers are then simultaneously examined. The group wins if every person correctly guessed the number of his or her own die; if even one person writes down a number that does not mach their die roll, the group loses. Can the group devise a strategy which gives them a decent shot at winning?
At first glance it might seem that the group can't do better than guessing randomly (after all, seeing everyone else's dice rolls doesn't help with guessing your own). To convince you that the all-random strategy can be beaten, let's take a simpler version of the game with only two people, in which guessing randomly evidently gives a (1/6)^2 = 1/36 chance of winning. Consider, however, the following perverse strategy: each participant writes down the die roll of the other person. Since this strategy wins exactly when the two dice rolls are the same, we now have a 1/6 chance of winning the game!
This two-person strategy isn't the one that generalizes the most easily to the 100-person game. Can you find – with proof! – an optimal strategy in the general case?
Finally, my favorite:
THE FIVE-CARD TRICK
You and your friend are attempting to work out how to perform a magic trick. The setup of the trick is as follows: you (the assistant) will be given five cards, randomly selected from a standard deck of 52 cards. Initially, your friend (the magician) is not allowed to see any of the five cards. You are allowed to select any four of the cards in your hand and lay them down on the table, in any order you choose. (That is, you remove four of the five cards in your hand and show them, one after one, to your friend.) Your friend is then supposed to ("magically") guess the remaining fifth card. How can you devise a communication strategy by which your friend can always guess correctly?
For example, one way to approach the problem might be to agree on an ordering of the 52 cards beforehand. Then showing your friend a sequence of four cards is the same as communicating a permutation of (1, 2, 3, 4) (simply by mapping the lowest of the four cards to 1, the second lowest to 2, and so on). Unfortunately this doesn't seem to be enough information, because there are 24 such permutations and 48 possibilities for the fifth card! (Your friend knows the fifth card is not any of the four cards already shown, giving the number 52 – 4 = 48.) What to do?
Hint: you can get the solution started by observing that there are five cards but four suits, and thus two cards of the same suit. Since you (the assistant) can choose which card is the fifth card, you can choose one of these two as the fifth card and use the other card to communicate the suit.
]]> to start planning your summer!
26 Oct 2015 04:58:44 +0000 reading →]]>As the days grow shorter and pumpkin-flavored everythings begin to inundate our lives, we are just starting to accept that summer is really over. But believe it or not, it's already time to start planning next summer! In this post, I have gathered some of my favorite advice on how to build your ideal mathematical summer. First of all, here are the two big reasons why sooner is better with summer planning:
Monotonically decreasing funding availability: As a general rule (with the exception of occasional opportunities which suddenly crop up), funding availability for summer math activities decreases monotonically from here on out. A good number of summer funding deadlines are in December and January (and some even earlier), so browsing for conferences and workshops now can help protect you from missing the perfect one.
Visualizing the large-scale structure of your summer: Your summer schedule probably has a lot of moving parts which might include teaching, research, visiting family and friends, conference and workshop travel, and (maybe) even some relaxing. Now is the time to start thinking about your summer goals and priorities and figuring out how the major blocks of time will fit together. You may still have several overlapping possibilities, and some of these time conflicts may not get settled until last-minute administrative or funding decisions are made, but visualizing your summer options now on a month-by-month desk calendar or on Google Calendar can be very helpful. And then when that email arrives from your university asking who is interested in which summer teaching assignments, you'll be ready to hit reply immediately with a well-informed decision so you can be first in line for your top choice!
Even if you are not ready to present any original results, attending math conferences over the summer can be a great opportunity to meet leaders in your field and to get inspired by hearing about the latest developments. Alternatively, summer schools and workshops are often geared towards graduate students and can be an efficient and fun way to learn new technical skills. Below are some tricks for finding the workshops, conferences, and summer schools that are best for you.
AMS Mathematics Calendar: Perhaps the most comprehensive compilation of math conference titles, dates and links anywhere on the web is found on the website of the American Mathematical Society on the Mathematics Calendar page. It is a fantastic place to start browsing through upcoming possibilities with the least amount of effort. (And it's fascinating to see how far in advance some conferences are planned!) There is also an analogous list provided by the European Mathematical Society.
Google: With so many conference organizers depending on word of mouth to announce their conferences, some conference websites stay hidden in the dustiest and most obscure corners of cyberspace, seen only by visitors with direct links. But armed with your expert googling abilities and some patience and care, you can find some real gems. Be ready to do up to a few dozen searches with slight variations of keywords; include perhaps only one specific math subject or term at a time, along with any subset of {graduate, math, summer, 2016, workshop, conference, funding}. You can also try adding in specific locations, names, or universities if you want to further narrow things down. Also try some searches where you leave out "2016"; this will cause you to get more outdated workshop pages in your results, but sometimes the workshops you find are annual and you can find the most recent workshop page either by deleting part of the URL or trying a more targeted Google search with keywords specific to that event. You can also send an email to the listed organizer of a past conference that you would have loved to attend, and politely ask them if they know of any comparable upcoming events this summer.
Word of mouth: A targeted and effective approach for combing through the first list of options you may have built from steps one and two above (and for adding events you may have missed) is by simply asking professors and grad students at your department. Stop by your department's tea time this week and ask others (both in your field and not) about conferences and workshops they have attended or organized in the past or what they are looking forward to for this summer. Even if some people can't think of anything at that moment, they may think of you the next time they get a conference email, and they might forward it to you. Another idea is to roam the halls of your department looking for relevant conference posters on professors' doors and to ask the corresponding professors about the posters that seem particularly interesting to you (even if they happened in the past). Also be sure to visit the bulletin boards where math event announcements are hung up in your department.
Sign up for relevant email lists: If you sign up for the right email lists for your math interests, you will find that conference and workshop invitations will begin to land effortlessly in your email inbox without any advanced googling. You will, however, actually need to open and read your emails for this technique to work. Here is an excellent list of math-related listservs to get you started. The idea is to sign up for as few of the most relevant lists as possible, so choose carefully! You might also want to ask the organizers of your favorite weekly seminars at your department to add you to their internal list, as conference announcements are sometimes circulated that way. You may also enjoy signing up for newsletters and announcements from a few major math institutes, e.g. this mailing list for the Fields Institute, in order to get early information of upcoming programs.
What are you favorite summer math events, conferences, or workshops that you have attended in the past or are looking forward to attending in the future? Have you found any great online lists of conferences that you would like to share? Do you have any tips or questions on how to build the ideal mathematical summer? Leave a comment below! | 677.169 | 1 |
In most mathematics textbooks, the most exciting part of mathematics--the process of invention and discovery--is completely hidden from the reader. The aim of Groups and Symmetry is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: Its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you. <P>Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirementNicely produced and concentrates on the informal analysis of geometrical patterns with the emphasis on informality ... could serve as a useful collection of activities to precede a formal course and would provide a range of intuitive experiences to which the more formal treatment could refer." ---- The Mathematical Gazette
"On the basis of this book it is possible to tailor a good course for high school students to really discover mathematics ... for anyone who is working with high school students in an advanced level the book is really recommended." ---- Zentralblatt MATH
"Written in a lively conversational style ... entertaining, and sometimes provoking, and will doubtlessly prove useful to its intended audience." ---- Mathematical Reviews
Most Helpful Customer Reviews
This book was the foundational textbook for a 100-level class in symmetry at my university. I recommend it highly to anyone who wants to get a better feel for what mathematicians actually do and think about and work with. Folks who never got into the higher math classes often have a different idea of what mathematics is all about than mathematicians. At the level of introductory algebra and geometry and even some calculus, math education often seems to be mainly about memorizing formulas and recognizing in which situations to apply them. That's an important thing to learn, but it is not useful for imparting an idea and a feel of the field of mathematics as a whole. Farmer's book brings home the understanding that mathematics is, at its heart, about patterns and that mathematics is not so much about memorization and application as it is about discovery.
The level of mathematical understanding required to get something useful out of this book is low. I believe the professor required beginning algebra as the prerequisite. If you can count to six, recognize the difference between a square and a pentagon, and understand that variables like n, m, or x can be used as substitutes for numbers then you probably have enough mathematical sophistication to work your way through this book and gain insights into the beauty of higher math.
Groups are the first structures encountered in abstract algebra and form the foundation for most of the others. Fortunately, they are also the easiest to physically represent, so in some sense they are the most concrete. In this book, groups are introduced as the motions and structures of geometric figures, so the presentation is largely by diagram rather than formula. Very little previous knowledge of mathematics is required and after reading the book, you will have a solid understanding of what a group is. The first topic is the moving of a complete figure to a different location of the plane defined by a grid of points. By keeping the figure rigid and fixed in orientation, a set of legal moves is defined. After that, some of the rules are relaxed and that allows for additional moves to be added. Exercises and problems are put forward here and throughout the book, and with the accent on figures, often give the appearance of a game. The next steps are then to allow for all possible rotations, translations and reflections of the objects, using these to explain the structure of a group. This is an effective way to introduce group theory, and is how I will do it if I teach abstract algebra again. Permutation and plane tiling symmetry groups are then introduced and examined, and their relationship to the previous groups discussed, which introduces the concept of isomorphism. Basic group theory is something that everyone can understand, as humans have a natural affinity for patterns and recognizing them despite "trivial" alterations. This book is an excellent primer on group theory and I strongly recommend it to anyone either learning or teaching abstract algebra. Published in Journal of Recreational Mathematics, reprinted with permission. | 677.169 | 1 |
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