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Mathematics
Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences
Recently Reviewed Titles
The Statistical Mind in Modern Society: The Netherlands 1850-1940, 2 volumes by Jacques van Maarseveen (Editor), Paul Klep (Editor), Ida Stamhuis (Aksant Academic Publishers) The contributions in this first volume, produced by experts from various disciplines, cover a great diversity of topics. In addition to the institutionalisation and internationalisation of official governmental statistics, attention is paid to statistics sup-porting policies for modernising society, in areas like agriculture, social legislation, education and justice. The application of statistics in trade and industry (such as banking and insurance, and the railways) is also discussed, as well as the growth of state power to combat social and economic problems such as child labour, the fight against alcoholism and economic crises.
The contributions in the second volume, produced by experts from various disciplines, cover a great diversity of topics. The application of statistics in the sciences (demography, geography, genetics, economic historiography, agricultural and medical sciences) is discussed in an international context. Special attention is given to the general emergence of thinking in terms of probabilities and the influence of mechanisation in statistics, as well as to the way Dutch scholars and scientists tried to solve statistical measurement problems (in meteorology, demographic forecasting, business-cycle research and unemployment).
In this review the following topics are discussed. First we examine the notion of the 'statistical mind'. Then we look the place of this publication within the historiography of statistics and brief summaries present the various contributions. More
Bayesian Methods: A Social and Behavioral Sciences Approach, Second Edition by Jeff Gill (Statistics in the Social and Behavioral Sciences: Chapman and Hall/CRC) The first edition helped pave the way for Bayesian approaches to become more prominent in social science methodology. While the focus remains on practical modeling and basic theory as well as on intuitive explanations and derivations without skipping steps, this second edition incorporates the latest methodology and recent changes in software offerings. More
Handbook of Granular Computing by Witold Pedrycz, Andrzej Skowron, and Vladik Kreinovich (Wiley) Although the notion is a relatively recent one, the notions and principles of Granular Computing (GrC) have appeared in a different guise in many related fields including granularity in Artificial Intelligence, interval computing, cluster analysis, quotient space theory and many others. Recent years have witnessed a renewed and expanding interest in the topic as it begins to play a key role in bioinformatics, e-commerce, machine learning, security, data mining and wireless mobile computing when it comes to the issues of effectiveness, robustness and uncertainty. More
Artificial Intelligence with Uncertainty by Deyi Li, Yi Du (Chapman & Hall/CRC) The information deluge currently assaulting us in the 21st century is having profound impact on our lifestyles and how we work. We must constantly separate trustworthy and required information from the massive amount of data we encounter each day. Through mathematical theories, models, and experiment. computations, Artificial Intelligence with Uncertainty explores the uncertainties of knowledge and intelligence that occur during the cognitive processes of human beings. The authors focus on the importance of natural language—the carrier of knowledge and intelligence—for artificial intelligence (Al) study. More
Monte Carlo Methods For Applied Scientists by Ivan T. Dimov
(World Scientific Publishing Company) Stochastic optimization refers
to the minimization (or maximization) of a function in the presence
of randomness in the optimization process. The randomness may be
present as either noise in measurements or Monte Carlo randomness in
the search procedure, or both. The study of random geometric
structures. Stochastic geometry leads to modelling and analysis
tools such as Monte Carlo methods.
Common methods of stochastic optimization include direct search
methods (such as the Nelder-Mead method), stochastic approximation,
stochastic programming, and miscellaneous methods such as simulated
annealing and genetic algorithms.
The Monte Carlo method is inherently parallel and the extensive and rapid development in parallel computers, computational clusters and grids has resulted in renewed and increasing interest in this method. At the same time there has been an expansion in the application areas and the method is now widely used in many important areas of science including nuclear and semiconductor physics, statistical mechanics and heat and mass transfer. More
Numerical Modeling of Water Waves, Second Edition, includes CD-ROM by Charles L. Mader (CRC Press) is a well-written, comprehensive treatise of the evolving science of computer modeling of waves. In a very skillful and methodical manner, the Dr. Charles Mader provides new insights on the subject and updates the reader with what is being done with state-of-the-art, high-performance computers which allow for the adaptation of new codes that can result in even more accurate simulations of waves generated from a variety of source mechanisms - whether generated by earthquakes, landslides, explosions, or the impact of asteroids. The book is an outstanding work of scholarship and a valuable reference for any researcher involved or interested in the numerical modeling of waves.
Statistical Mechanics by Donald A. McQuarrie (University Science Books) is the extended version of my earlier text, Statistical Thermodynamics. The present volume is intended primarily for a two-semester course or for a second one-semester course in statistical mechanics. Whereas Statistical Thermodynamics deals principally with equilibrium systems whose particles are either independent or effectively independent, Statistical Mechanics treats equilibrium systems whose particles are strongly interacting as well as nonequilibrium systems. The first twelve chapters of this book also form the first chapters in Statistical Thermodynamics, while the next ten chaptèrs, 13-22, appear only in Statistical Mechanics. Chapter 13 deals with the radial distribution function approach to liquids, and Chapter 14 is a fairly detailed discussion of statistical mechanical perturbation theories of liquids. These theories were developed in the late 1960s and early 1970s and have brought the numerical calculation of the thermodynamic properties of simple dense fluids to a practical level. A number of the problems at the end of the Chapter 14 require the student to calculate such properties and compare the results to experimental data. Chapter 15, on ionic solutions, is the last chapter on equilibrium systems. Section 15-2 is an introduction to advances in ionic solution theory that were developed in the 1970s and that now allow one to calculate the thermodynamic properties of simple ionic solutions up to concentrations of 2 molar. More
The Structural Stabilization of Polymers: Fractal Models by G.
V. Kozlov, G. E. Zaikov (New Concepts in
Polymer Science: VSP International (Brill) This monograph deals with
the structural aspects of transport processes of gases, physical
ageing and thermo-oxidative degradation of polymers in detail.
Fractal analysis, cluster models of the polymer structure's
amorphous state as well as irreversible aggregation models are used
as main structural models. It is shown that the polymer structure is
often a more important parameter than its chemical construction.
Another significant aspect is the structural role in polymer melts
oxidation.
The basis for understanding of structural stabilization gives
anomalous diffusion of oxidant molecules on the fractal structure
for both solid state polymers and polymeric melts. The important
part of this problem is structure connectivity characterized by its
spectral dimension. Therefore branched (cross-linked) polymers have
smaller diffusivity in comparison with linear polymers. Fractal
mathematics is used throughout to sharpen measures and tighten
explanations. The volume could have used an English-language editor.
More
Mind-Bending Math and Science Activities for Gifted Students (For Grades K-12) by Callard-Szulgit Rosemary (Rowman & Littlefield Education) Here is a reference and guide for teachers and parents that covers many aspects of gifted thinking in relation to math and science. It features competitions and curricula that can be easily adapted to students' lifestyles outside of the classroom and the materials are accessible to adults with limited scientific backgrounds. Advice, vignettes, and cartoons are included. Intended for grades K-12. More
Mathematical Development in Young Children: Exploring Notations by Barbara M. Brizuela (Teachers College Press) (Hardcover) Using data from interviews and in depth conversations with children from five to nine years of age, Brizuela's study examines how children understand and learn mathematical notations in their development as mathematics learners. Each chapter focuses on a different notational system--written numbers, commas and periods, fractions, data tables, number lines, graphs, and student-invented systems drawing from established conventions. The chapters are organized chronologically in terms of the ages of the children described in each chapter, so the resulting order is by both increasing age of the children and increasing complexity of the mathematical content. More
Teaching Mathematics in Primary Schools by Robyn Zevenbergen, Shelley Dole, Robert J. Wright (Allen & Unwin) A systematic, research-based introduction to the principles and practice of teaching mathematics at the primary school level, this inquiry moves beyond traditional lockstep approaches to teaching mathematics to emphasize how students can learn to think mathematically in terms of globalization and new technologies. More
Handbook of Discrete and Computational Geometry, Second Edition edited by Jacob E. Goodman, Joseph O'Rourke (CRC Press) Comprehensive handbook for professionals in the field of Mathematics, Computer Science, and Computational Mathematics. A complete reference volume. The second edition of the Handbook of Discrete and Computational Geometry is a thoroughly revised version of the bestselling first edition. With the addition of 500 pages and 14 new chapters covering topics such as geometric graphs, collision detection, clustering, applications of computational geometry, and statistical applications, this is a significant update. This edition includes expanded coverage on the topics of mesh generation in two and three dimensions, aspect graphs, center points, and probabilistic roadmap algorithms. It also features new results on solutions of the Kepler conjecture, and honeycomb conjecture, new bounds on K sets, and new results on face numbers of polytopes. More
Probability Demystified by Allan G. Bluman (McGraw-Hill Professional) Don't Roll The Dice Learning Probability! Now anyone who ever flipped a coin, played cards, or placed a bet can grasp the principles that govern probability -- without formal training, unlimited time, or an Einstein IQ. In Probability Demystified, experienced math instructor Allan Bluman provides an illuminating and entertaining way to master chance, odds, and predictability. More
Probability for Electrical and Computer Engineers by Charles W. Therrien, Murali Tummala (CRC Press) Written specifically for electrical and computer engineers, this book provides an introduction to probability and random variables. It includes methods of probability that deal with computing the likelihood of uncertain events for scientists and engineers who predict the outcome of experiments, extrapolate results from a small case to a larger one, and design systems that will perform optimally when the exact characteristics of the inputs are unknown. Electrical and computer engineers seeking solutions to practical problems will find it a valuable resource in the design of communication systems, control systems, military or medical sensing or monitoring systems, and computer networks. More
Introductory Linear Algebra: An Application-Oriented First Course (8th Edition) by Bernard Kolman, David R. Hill (Prentice Hall) This book presents an introduction to linear algebra and to some of its significant applications. It is designed for a course at the freshman or sophomore level. There is more than enough material for a semester or quarter course. By omitting certain sections, it is possible in a one-semester or quarter course to cover the essentials of linear algebra (including eigenvalues and eigenvectors), to show how the computer is used, and to explore some applications of linear algebra. It is no exaggeration to say that with the many applications of linear algebra in other areas of mathematics, physics, biology, chemistry, engineering, statistics, economics, finance, psychology, and sociology, linear algebra is the undergraduate course that will have the most impact on students' lives. The level and pace of the course can be readily changed by varying the amount of time spent on the theoretical material and on the applications. Calculus is not a prerequisite; examples and exercises using very basic calculus are included and these are labeled "Calculus Required." More
Algebraic Theory of Automata & Languages by Masami Ito (World Scientific Publishing Company) Although there are some books dealing with algebraic theory of automata, their contents consist mainly of Krohn–Rhodes theory and related topics. The topics in the present book are rather different. For example, automorphism groups of automata and the partially ordered sets of automata are systematically discussed. Moreover, some operations on languages and special classes of regular languages associated with deterministic and nondeterministic directable automata are dealt with. The book is self-contained and hence does not require any knowledge of automata and formal languages. More
Beginning Algebra (4th Edition) by K. Elayn Martin-Gay (Prentice Hall) Beginning Algebra, Fourth Edition was written to provide a solid foundation in algebra for students who might have had no previous experience in algebra. Specific care has been taken to ensure that students have the most up-to-date and relevant text preparation for their next mathematics course, as well as to help students to succeed in nonmathematical courses that require a grasp of algebraic fundamentals. I have tried to achieve this by writing a user-friendly text that is keyed to objectives and contains many worked-out examples. The basic concepts of graphing are introduced early, and problem solving techniques, real-life and real-data applications, data interpretation, appropriate use of technology, mental mathematics, number sense, critical thinking, decision-making, and geometric concepts are emphasized and integrated throughout the book. More
Advanced Computer Applications: An Information Technology Approach includes CD-ROM by Daphne Press (EMC/Paradigm Publishing) In this book/CD-ROM package, Press (Ozarks Technical Community College) presents an extended case that allows students to make decisions as IT professionals working in the IT department of a landscaping company. Designed for a semester-long course, the text requires the use of advanced Microsoft Office functions and tools such as forms, templates, macros, and VBA. It also requires integration of productivity and system tools such as data import and export and file management across platforms. Students should be familiar with Office XP or a later version. The CD-ROM includes data files, a project planning template, and testing templates. More
An Introduction to Random Sets by Hung T. Nguyen (Chapman & Hall/CRC) Random sets as models for set-valued observations, are a new type of data proving useful in areas such as survey sampling, biostatistics, and intelligent systems. This is the first text to explore the topic in depth, using extended probability theory to provide a framework and tools for statistical analysis of random sets. With an abundance of examples, it highlights the basic role random sets play in a variety of statistical settings, links their study to fuzzy logic, fully develops the theory, and concludes with a variety of applications. Written by an author of the best-selling A First Course in Fuzzy Logic, this book is rigorous yet readable and fills a significant need for a textbook treatment of the subject. More
The Mathematics of Infinity: A Guide to Great Ideas by Theodore G. Faticoni (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts: Wiley-Interscience) addresses infinite cardinals and is appropriate for readers at any level. Inviting the reader to imagine constructing an infinite chain infinities, which are called cardinals, the author successfully prepares and motivates readers for topics covered within the book. The most unique feature of the book is that set theoretic depth is achieved without losing the target audience. Complementing existing popular books on infinity by actually doing the mathematics involved in addition to talking about the mathematics, the reader is gently led into the world of mathematical proofs. More
Math Refresher for Scientists and Engineers, 3rd edition by John R. Fanchi (Wiley-IEEE Press) Expanded coverage of essential math, including integral equations, calculus of variations, tensor analysis, and special integrals. Math Refresher for Scientists and Engineers, Third Edition is specifically designed as a self-study guide to help busy professionals and students in science and engineering quickly refresh and improve the math skills needed to perform their jobs and advance their careers. The book focuses on practical applications and exercises that readers are likely to face in their professional environments. All the basic math skills needed to manage contemporary technology problems are addressed and presented in a clear, lucid style that readers familiar with previous editions have come to appreciate and value. More
Using Multivariate Statistics (5th Edition) by Barbara G. Tabachnick, Linda S. Fidell (Allyn & Bacon) provides advanced students with a timely and comprehensive introduction to today’s most commonly encountered statistical and multivariate techniques, while assuming only a limited knowledge of higher level mathematics. This long-awaited revision reflects extensive updates throughout, especially in the areas of Data Screening (Chapter 4) Multiple Regression (Chapter 5) and Logistic Regression (Chapter 12). A brand new chapter (Chapter 15) on Multilevel Linear Modeling explains techniques for dealing with hierarchical data sets. Also included are syntax and output for accomplishing many analyses through the most recent releases of SAS and SPSS. More
Multivariate Statistical Methods: A Primer, Third Edition by Bryan F. J. Manly (Chapman & Hall/CRC) This is a thoroughly revised, updated edition of a best-selling introductory textbook and primer. The third edition retains the author's trademark concise and clear style and its focus on examples in the biological and environmental sciences. Topics new to this edition include confirmatory factor analysis, handling missing values, and the emerging techniques of data mining and neural networks. While not linking the book to any specific software package, the book now includes an appendix comparing and contrasting various statistical software packages, such as Stata, Statistica, SAS, and Genstat. More
Computers: Understanding Technology 2nd Edition, with CD-ROM by Floyd Fuller, Brian Larson (EMC/Paradigm Publishing) describes the role of computers in our lives and in society, and covers various aspects of computer hardware (including input, processing, output, and storage), system and application software, telecommunications and networks, databases and information management, applications design and programming, security and ethics, and careers. A companion CD-ROM contains videos illustrating key points, projects and tutorials, self-tests, and a chronology of computer development. Fuller teaches at the Appalachian State University; Larson, at California State University- Stanislaus. More
The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Marcus Giaquinto (Oxford University Press) (Hardcover) The author has performed an impressive balancing act. He manages to treat details precisely without being pedantic. He does not shortchange history, but he also does not permit the pursuit of historical authenticity to interfere with clarity of exposition.
In the early decades of the twentieth century, mathematicians showed an unprecedented concern for the foundations of their subject, not just in expressions of disquiet but also in attempts to find a secure basis. This search for certainty and the crisis that sparked it off is the central subject of this book. First Giaquinto shows mathematical setting of this story to see how the foundational accomplishments grew out of the nineteenth-century quest for clarity and rigor in mathematics.
The clarification of basic properties and relations of analysis set the tone for the search for certainity. The objects of analysis—real numbers and more generally points, classes of points, and functions on classes of points—were taken for granted. But in the later decades of the nineteenth century, according to Giaquinto, mathematicians came to feel that an explicit account of real numbers was needed. also an account of the transfinite numbers discovered—or, some would say, invented by Cantor is explored.
Giaquinto first explains the two best-known accounts of real numbers. Next, he presents a sketch of the way in which the ideas for the transfinite ordinal and cardinal number systems grew out of the study of classes of points, and the rudiments of those number systems are presented. After which he looks at accounts of the natural numbers.
Towards the end of the nineteenth century, the drive for clarity and rigor seemed to be reaching a successful conclusion. Among its fruits were precise accounts of the real and natural numbers, the first general theory of transfinite classes and numbers, and a first account of quantifier logic—no meager harvest. But celebrations had barely begun when certain paradoxes were found in the general theory of classes, which was the basis for all supposedly rigorous accounts of the number systems. This defeat in the hour of triumph made foundational research a major area of concern for mathematicians.
Deeper excavation was needed, and the younger mathematicians who took up the task intended to reach bedrock. So the drive to find sure foundations for mathematics issued largely from problems internal to mathematics, together with the conviction that, if certainty is to be found anywhere, it is to be found in mathematics. In this way, the mathematical concern was tied to a philosophical one: how can we be certain that the theorems of mathematics are trustworthy? The bulk this book examines the attempts to meet this challenge.
The central concern of this book is the epistemic status of non-finitary mathematics. Epistemology is not the only concern in foundational studies, though it has been dominant. The nature and intrinsic organization of mathematics has also been a major concern. Later developments in mathematics show that set theory is not the only basis for this kind of inquiry. Of course, those who think that true mathematics must be constructed will reject not only classical set theory but also the nineteenth-century mathematics out of which it grew. In this regard the development of constructive analysis can be regarded as partial fulfillment of an alternative foundational program. Giaquinto is not able to evaluate the success and significance of this program, and perhaps we are too close to see all of what needs to be seen. For those who accept classical mathematics, category theory has been offered as an alternative to set theory for its catholic reach. Mathematics is definitely not just logic, not just higher-order logic, not just set theory. The old picture of a single fundamental theory to which all else must be reduced has faded. If pure mathematics is the study of abstract structures, set theory is just one framework among others for thinking about that subject matter, and it may not be the best. Universes of sets are themselves structures, and these may be instances of something more general, as is suggested by topos theory. In addition, topos theory sheds new light on the intrinsic organization of mathematics, revealing a surprising unity between apparently disparate fields, topology and algebraic geometry on the one hand, and logic and set theory on the other.
The initial impulse for foundational study was the need to clarify our understanding of the continuum and the basis of infinitesimal calculus. The standard set-theoretic account is an explication that has served well — witness the use made of it in classic textbooks on analysis. But now there are other explications of the basic intuitions. Robinson's non-standard analysis rehabilitated infinitesimals. Non-classical accounts include intuitionistic analysis and Bishop's constructive analysis. The development of synthetic differential geometry gives yet another perspective on the continuum and a novel theory of infinitesimals. Thus we now have a plurality of mathematical ways of refining and abstracting from what are originally spatial intuitions. This too is a way in which foundational study has spread out and away from the monolithic view.
If new developments within mathematics advance our understanding of the nature and intrinsic organization of mathematics, epistemological advances are likely to come from developments outside. In the period covered by Giaquinto in this book, the central epistemological concern has been to justify a body of mathematics. Another concern is to explain how it is possible for an individual to have mathematical knowledge and understanding. This inquiry needs fine-grained information about how we actually acquire our mathematical beliefs and abilities; then we can investigate how best to evaluate those modes of cognitive growth in epistemic terms. The empirical input must come primarily from cognitive sciences. Investigations of simple numerical abilities have already proved fruitful, aided by a recent confluence of evidence from different sources: experiments on healthy adults, children, and even infants, clinical tests on brain-damaged patients, brain imaging studies, and animal studies. There is still a long way to go. The history of mathematics is another source of information.
Arts of Calculation: Numerical Thought in Early Modern Europe by David Glimp (Editor), Michelle R. Warren (Palgrave Macmillan) The essays in this volume focus primarily on 16th and 17th century Europe and are broadly interdisciplinary. They answer questions such as: what kinds of cultural work do numbers do?; What roles does calculation play in colonial, imperial, and/or national projects or ideologies?; What are the relationships between aesthetic practices and bureaucratic modes of calculation (such as accounting, census taking, and demography)?; What kinds of agencies and subjectivities do numbers and numbering enable and foreclose?; How do different kinds of economic strategies (eg exchange, gift, capital, debt, interest) affect representational strategies and vice versa?; What cultural dynamics inform spatial measurements, such as cartography?; and what do analyses of counting practices tell us about the production of knowledge more generally? More
A.D. Alexandrov Selected Works: Intrinsic Geometry of Convex
Surfaces edited by S. S. Kutateladze (Classics of Soviet
Mathematics: Gordon & Breach Publishing Group) The 2-volume set
includes Alexandrov: Selected Works, Part 1, Hb ISBN 2881249841,
1996, 332 pp, and Alexandrov, Selected Works, Part 2, Hb 041529802
4, 2003, 504 pp. The 2-volume set provides definitive sources for
the development of intrinsic geometry. A classic in the field that
remains unsurpassed in its clarity and scope, this text would be of
great value to graduate students seeking a better understanding of
this area of geometry.
A.D. Alexandrov: Selected Works by A. D. Aleksandrov (Classics
of Soviet Mathematics: Gordon & Breach Publishing Group) This
monograph offers any student or professional working in the field a
unique access to a selection of Aleksandrov's best quality work,
published here for the first time in English translation.
This volume contains some of the most important papers by this world
class mathematician, who exerted such a powerful influence on the
development of modern mathematics. Aleksandrov was particularly
noted for his work in geometry, and the 16 papers included in this
volume represent some of his most seminal work. Topics treated
include convex polyhedrons and closed surfaces, an elementary proof
and extension of Minkowski's theorm, a chapter in Riemannian
geometry and general method for majorizing the solutions of
Dirichlet problems. At present there is no such collection of this
extremely well respected geometer's work.
A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of
Convex Surfaces by S. S. Kutateladze (Classics of Soviet
Mathematics: Gordon & Breach Publishing Group) A.D. Alexandrov's
contribution to the field of intrinsic geometry was original and
very influential. This text is a classic that remains unsurpassed in
its clarity and scope. It presents his core material, originally
published in Russian in 1948, beginning wth an outline of the main
concepts and then exploring other topics, such as general
propositions on an intrinsic metric; angles and curvature; existence
of a convex polyhedron with prescribed metric; curves on convex
surfaces; and the role of specific curvature. This text provides
Adefinitive source for the development of intrinsic geometry and is
indispensable for graduate students who want a better understanding
of this subject.
Pi: A Biography of the World's Most Mysterious Number by Alfred
S Posamentier, Ingmar Lehmann (Prometheus Books) Among its many
attributes, mathematicians call pi (p) a
"transcendental number" because its curious value cannot be
calculated by any combination of addition, subtraction,
multiplication, division, or square root extraction. More curious
still, regardless of the number of decimal places to which you
extend the value of pi, the decimal never repeats itself. In 2002 a
Japanese professor using a supercomputer calculated the value to
1.24 trillion decimal places! Nonetheless, in this huge string of
decimals there is no periodic repetition.
This enlightening, intriguing, and stimulating approach to mathematics will entertain and fascinate readers while honing their mathematical literacy.
pi—this seemingly mundane number—holds a world of mystery, which has fascinated mathematicians from ancient times to the present. What is pi? What is the real value of ir? How do mathematicians determine the value of pi? In what ways is pi used? How was it calculated in ancient times? Its elusive nature has led investigators over the years to ever-closer approximations.
In this delightful introduction to one of math's most interesting phenomena, Drs. Posamentier and Lehmann review pi's history from prebiblical times to the twenty-first century and the many amusing and often mind-boggling attempts to estimate its precise value. They show how this ubiquitous number comes up when you least expect it, such as in the calculation of probabilities and in biblical scholarship. In addition, they present some quirky examples of obsessing about pi over the centuries—including an attempt to legislate its exact value, and even a pi song—as well as useful applications of pi in everyday Life.
Surely the title makes it clear that this is a book about it, but you may be wondering how a book could be written about just one number. We will hope to convince you throughout this book that n is no ordinary number. Rather, it is special and comes up in the most unexpected places. You will also find how useful this number is throughout mathematics. We hope to present it to you in a very "reader-friendly" way—mindful of the beauty that is inherent in the study of this most important number.
You may remember that in the school curriculum the value that took on was either 3.14, 3 , or . For a student's purposes, this was more than adequate. It might have even been easier to simply use pi = 3. But what is pi? What is the real value of pi? How do we determine the value of pi? How was it calculated in ancient times? How can the value be found today using the most modern technology? How might pi be used? These are just some of the questions that we will explore as you embark on the chapters of this book.
We will begin our introduction of pi by telling you what it is and roughly where it came from. Just as with any biography (and this book is no exception), we will tell you who named it and why, and how it grew up to be what it is today. The first chapter tells you what pi essentially is and how it achieved its current prominence.
In chapter 2 we will take you through a brief history of the evolution of pi. This history goes back about four thousand years. To understand how old the concept of pi is, compare it to our number system, the place value decimal system, that has only been used in the Western world for the past 802 years!' We will recall the discovery of the pi ratio as a constant and the many efforts to determine its value. Along the way we will consider such diverse questions as the value of pi as it is mentioned in the Bible and its value in connection with the field of probability. Once the computer enters the chase for finding the "exact" value of pi, the story changes its complexion. Now it is no longer a question of finding the mathematical solution, but rather how fast and how accurate can the computer be in giving us an ever-greater accuracy for the value of pi.
Now that we have reviewed the history of the development of the value of it, chapter 3 provides a variety of methods for arriving at its value. We have chosen a wide variety of methods, some precise, some experimental, and some just good guessing. They have been selected so that the average reader can not only understand them but also independently apply them to generate the value of pi. There are many very sophisticated methods to generate the value of it that are well beyond the scope of this book. We have the general reader in mind with the book's level of difficulty.
With all this excitement through the ages centered on pi, it is no wonder that it has elicited a cultlike following in pursuit of this evasive number. Chapter 4 centers on activities and findings by mathematicians and math hobbyists who have explored the value of Pi and related fields in ways that the ancient mathematicians would never have dreamed of. Furthermore, with the advent of the computer, they have found new avenues to explore. We will look at some of these here.
As an offshoot of chapter 4, we have a number of curious phenomena that focus on the value and concept of pi. Chapter 5 exhibits some of these curiosities. Here we investigate how Pi relates to other famous numbers and to other seemingly unrelated concepts such as continued fractions. Again, we have limited our presentation to material that would require no more mathematical knowledge than that of high school mathematics. Not only will you be amused by some of the it equivalents, but you may even be inspired to develop your own versions of them.
Chapter 6 is dedicated to applications of pi. We begin this chapter with a discussion of another figure that is very closely related to the circle but isn't round. This Reuleaux triangle is truly a fascinating example of how Tc just gets around to geometry beyond the circle. From here we move on to some circle applications. You will see how pi is quite ubiquitous—it always comes up! There are some useful problem-solving techniques incorporated into this chapter that will allow you to look at an ordinary situation from a very different point of view—which may prove quite fruitful.
In our final chapter, we present some astonishing relationships involving Pi and circles. The situation that we will present regarding a rope placed around the earth will surely challenge everyone's intuition. Though a relatively short chapter, it will surely surprise you.
Math Word Problems Demystified by Allan G. Bluman (Demystified: McGraw-Hill Professional) Now anyone, even those whose palms begin to sweat at the first sight of math problems that begin "A train left the station going 65 mph..." can overcome anxiety and learn to solve word problems. In Math Word Problems Demystified, experienced math instructor Allan G. Bluman provides an effective, tension-free, approach to conquering the word problems on the SATs and many other standardized tests, in algebra, and in other mathematics and science classes.
With Math Word Problems Demystified, you master the subject one simple step at a time -- at your own speed. This unique self-teaching guide offers practice problems, a quiz at the end of each chapter to pinpoint weaknesses, and a 40 question final exam to reinforce the methods and material presented in the book.
If you want to master math word problems, here's the self-teaching course that will get it done. Get ready to --
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Transform word problems into solvable equations
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Master a 4-step strategy that empowers you to understand and solve the most common word problems
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Conquer word problems involving distances, mixtures, levers, finance, work, speed, percents, coins, ages, and more
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Obtain the right answers using formulas and proportions
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Score better on science and math exams as well as standardized tests
A fast, effective, and fun way to master word problems, Math Word
Problems Demystified is the perfect shortcut to gain the confidence
and develop the necessary skills to solve these tough test
questions.
Summary:
Word problems are the most difficult part of any math course –- and
the most important to both the SATs and other standardized tests.
This book teaches proven methods for analyzing and solving any type
of math word problem.
Pythagorean Triangles by Wadaw Sierpiriski (Dover) unabridged republication of the edition published by the Graduate School of Science, Yeshiva University, New York, 1962. Translated by Dr. Ambikeshwar Sharma.
The Pythagorean Theorem is one of the fundamental theorems of elementary geometry, and Pythagorean triangles—right triangles whose sides are natural numbers—have been studied by mathematicians since antiquity. In this classic text, a brilliant Polish mathematician explores the intriguing mathematical relationships in such triangles.
Starting with "primitive" Pythagorean triangles, the text examines triangles with sides less than 100, triangles with two sides that are successive numbers, divisibility of one of the sides by 3 or by 5, the values of the sides of triangles, triangles with the same arm or the same hypotenuse, triangles with the same perimeter, and triangles with the same area. Additional topics include the radii of circles inscribed in Pythagorean triangles, triangles in which one or more sides are squares, triangles with natural sides and natural areas, triangles in which the hypotenuse and the sum of the arms are squares, representation of triangles with the help of the points of a plane, right triangles whose sides are reciprocals of natural numbers, and cuboids with edges and diagonals expressed by natural numbers.
The Skeleton Key of Mathematics: A Simple Account of Complex Algebraic Theories by Dudley Ernest Littlewood (Dover) unabridged republication of the edition published by Harper & Brothers, New York, 1960 (original publication 1949). As the title promises, this helpful volume offers easy access to the abstract principles common to science and mathematics. It eschews technical terms and omits troublesome details in favor of straight-forward explanations that will allow scientists to read papers in branches of science other than their own, mathematicians to appreciate papers on topics on which they have no specialized knowledge, and other readers to cultivate an improved understanding of subjects employing mathematical principles.
The broad scope of topics encompasses Euclid's algorithm; congruences; polynomials; complex numbers and algebraic fields; algebraic integers, ideals, and p-adic numbers; groups; the Galois theory of equations; algebraic geometry; matrices and determinants; invariants and tensors; algebras; group algebras; and more.
"It is refreshing to find a book which deals briefly but competently with a variety of concatenated algebraic topics, that is not written for the specialist," enthused the Journal of the Institute of Actuaries Students' Society about this volume, adding "Littlewood's book can be unreservedly recommended."
Appraising Lakatos: Mathematics, Methodology, and the Man edited
by George Kampis, Ladislav Kvasz, Michael
Stoltzner (Vienna Circle Institute Library: Kluwer Academic
Publishers) a critical re-evaluation of the ideas of Imre
Lakatos, a leader in the shaping of what is called the new
philosophy of science. The 17 contributions (the result of a joint
venture between the
CRC Concise Encyclopedia of Mathematics, Second Edition by Eric W. Weisstein (Chapman & Hall CRC) Allows readers to implement the formulas presented, perform calculations, construct geographical displays of results, and generate remarkable mathematical illustrations. More than 1000 new pages of terms defined, illustrated, and referenced. More
The Millennium Problems: The
Seven Greatest Unsolved Mathematical Puzzles of Our Time by
Keith J. Devlin (Basic) The definitive lay reader's account of the
Everests of mathematics--the seven unsolved problems that definethe
state of the art in contemporary math.
In 2000, the Clay Foundation of
Finite Mathematics and Its Applications, Eighth Edition by Larry
Joel Goldstein, David I. Schneider, Martha J. Siegel, T. E. Graedel
(Prentice Hall) This work is the eighth edition of our text for the
traditional finite mathematics course taught to first- and
second-year college students, especially those majoring in business
and the social and biological sciences. Finite mathematics courses
exhibit tremendous diversity with respect to both content and
approach. Therefore, in revising this book, we incorporated a wide
range of topics from which an instructor may design a curriculum, as
well as a high degree of flexibility in the order in which the
topics may be presented. For the mathematics of finance, we even
allow for flexibility in the approach of the presentation.
Analyzing Multivariate Data by Jim Lattin, Doug Carroll, Paul E. Green (Brooks/Cole) Offering the latest teaching and practice of applied multivariate statistics, this text is perfect for students who need an applied introduction to the subject. Lattin, Green, and Carroll have created a text that speaks to the needs of applied students who have advanced beyond the beginning level, but are not yet advanced statistics majors. Their text accomplishes this through a three-part structure. First, the authors begin each major topic by developing students' statistical intuition through geometric presentation. Then, they are providing illustrative examples for support. Finally, for those courses where it will be valuable, they describe relevant mathematical underpinnings with matrix algebra. More
The A to Z of Mathematics: A Basic Guide by Thomas H. Sidebotham (Wiley Interscience) a guide that makes math simple without making it simplistic. Invaluable resource for parents and students, home schoolers, teachers, and anyone else who wants to improve his or her math skills and discover the amazing relevance of mathematics to the world around us. More
The Applicability of Mathematics As a Philosophical Problem by
Mark Steiner (Harvard University Press) analyzes the different ways
mathematics is applicable in the physical sciences, and presents a
startling thesis - the success of mathematical physics appears to
assign the human mind a special place in the cosmos.
This book has two separate objectives. The first is to examine in
what ways mathematics can be said to be applicable in the natural
sciences or to the empirical world. Mathematics is applicable in
many senses, and this ambiguity has bred confusion and error--even
among "analytic" philosophers: because there are many senses of
"application" and "applicability," there are many questions about
the application of mathematics that ought to be, but have not been,
distinguished by philosophers. As a result, we do not always know
what problem they are dealing with.
Cryptanalysis of Number Theoretic Ciphers by Samuel S. Wagstaff, Jr., edited by Mikhail J. Atallah (Chapman & Hall/CRC) First book to take readers all the way from basic number theory through the inner workings of ciphers and protocols to their strengths and weaknesses. Presents cryptosystem as practical, workable algorithms, not just as oversimplified mathematical objects. More
Finite Mathematics by Bill Armstrong, Don Davis (Prentice Hall)
modern in its writing style as well as in its applications, contains
numerous exercises—both skill oriented and applications—, real data
problems, and a problem solving method. The book features exercises
based on data form the World Wide Web, technology options for those
who wish to use a graphing calculator, review boxes, strategic
checkpoints, interactive activities, section summaries and projects,
and chapter openers and reviews. For anyone who wants to see and
understand how mathematics are used in everyday life.
Fuzzy Topology by
The key features are
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Large number of examples
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Counterexamples, characterizations, implications
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References to original sources
Elementary Differential Equations by W. E.
Kohler, Lee W. Johnson (Addison Wesley) This book is designed
for the sophomore differential equations course taken by students
majoring in science and engineering. We assume the reader has had a
course in elementary calculus.
The authors have integrated the underlying theory, the solution
procedures, and the numerical and computational aspects of
differential equations as seamlessly as possible. For exa
Discrete Mathematics (5th Edition) by Kenneth A. Ross, Charles
R. B. Wright (Prentice Hall) Presenting conceptual chains in an
orderly and gradual fashion, this informal but thorough introduction
to discrete mathematics offers a careful treatment of the basics
essential for computer science such as relations, induction,
counting techniques, logic, and graphs. It also covers the more
advanced topics of Boolean algebra and permutation groups, and comes
with a wealth of examples to reinforce material and to allow readers
to view topics from several perspectives. The book includes new
coverage of probability that examines such areas as random variables
and distributions and new sections on the Euclidean algorithm and
loop invariants, providing a powerful tool for designing algorithms
and verifying their correctness.
Invitation to Linear Operators: From Matrix
to Bounded Linear Operators on a Hilbert Space
by Takayuki Furuta (Gordan & Beach: Taylor & Francis)
Essential guide explains in easy to follow steps, the newest
essential and fundamental results on linear operators as based on
matrix theory. Serves as a reference book for advanced readers in
mathematics. Written for non-specialists with a good grasp of matrix
theory, this introductory guide to linear operators situates its
theory and recent results in the context of matrix theory. Furuta
begins by describing the basic properties of a Hilbert space and
then arranges the fundamental properties of bounded linear operators
on a Hilbert space, and ends with a discussion of current research.
Furuta teaches applied mathematics at the Science University of
Tokyo.
More
What the Numbers Say: A Field Guide to Mastering Our Numerical World by Derrick Niederman & David Boyum (Broadway Books) Our society is churning out more numbers than ever before, whether in the form of spreadsheets, brokerage statements, survey results, health risks, the numbers on the sports pages, probabilities at the roulette table, and the list goes on. Unfortunately, people’s ability to understand and analyze numbers isn’t keeping pace with today’s whizzing data streams. And the benefits of living in the Information Age are available only to those who can process the information in front of them. More
Basic College Mathematics: A Text/Workbook (Book with CD-ROM and Infotrac) by Charles Patrick McKeague (Brooks/Cole) Exceptionally clear and accessible, Pat McKeague’s text, Basic College Mathematics, offers all the review, drill, and practice you need to develop rock-solid mathematical proficiency and confidence. McKeague’s attention to detail, exceptional writing style, and organization of mathematical concepts make learning accessible and enjoyable. More
Algebra and Trigonometry, Second Edition by Robert Blitzer
(Prentice Hall) was written Algebra and Trigonometry, Second Edition
do help diverse students, with different backgrounds ant future
goals do succeed. The book has three fundamental goals:
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To help students acquire a solid foundation in algebra ant trigonometry, preparing them for other courses such as calculus, business calculus, ant finite mathematics.
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To show students how algebra ant trigonometry can motel and solve authentic real-world problems.
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To enable students do develop problem-solving skills, while fostering critical thinking, within an interesting setting. More
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