Shows how any proof can be understood as a sequence of techniques. Covers the full range of techniques used in proofs, such as the contrapositive, induction, and proof by contradiction. Explains how to identify which techniques are used and how they are applied in the specific problem. Illustrates how to read written proofs with many step-by-step examples. Includes new, expanded appendices related to discrete mathematics, linear algebra, modern algebra and real analysis.

    The Times called it elegant, discursive, and littered with quotes and allusions from Aquinas via Gershwin to Woolf and The Philadelphia Inquirer praised it as absolutely scintillating. In this delightful new book, Robert Kaplan, writing together with his wife Ellen Kaplan, once again takes us on a witty, literate, and accessible tour of the world of mathematics. Where The Nothing That Is looked at math through the lens of zero, The Art of the Infinite takes infinity, in its countless guises, as a touchstone for understanding mathematical thinking. Tracing a path from Pythagoras, whose great Theorem led inexorably to a discovery that his followers tried in vain to keep secret (the existence of irrational numbers); through Descartes and Leibniz; to the brilliant, haunted Georg Cantor, who proved that infinity can come in different sizes, the Kaplans show how the attempt to grasp the ungraspable embodies the essence of mathematics. The Kaplans guide us through the Republic of Numbers, where we meet both its upstanding citizens and more shadowy dwellers; and we travel across the plane of geometry into the unlikely realm where parallel lines meet. Along the way, deft character studies of great mathematicians (and equally colorful lesser ones) illustrate the opposed yet intertwined modes of mathematical thinking: the intutionist notion that we discover mathematical truth as it exists, and the formalist belief that math is true because we invent consistent rules for it. Less than All, wrote William Blake, cannot satisfy Man. The Art of the Infinite shows us some of the ways that Man has grappled with All, and reveals mathematics as one of the most exhilarating expressions of the human imagination.

    The Fourth Edition builds on this strength with new examples, additional applications and increased cryptology coverage. Up-to-date information on the latest discoveries is included.Elementary Number Theory and Its Applications provides a diverse group of exercises, including basic exercises designed to help students develop skills, challenging exercises and computer projects. In addition to years of use and professor feedback, the fourth edition of this text has been thoroughly accuracy checked to ensure the quality of the mathematical content and the exercises.

    The text offers a flexible organization, enabling instructors to adapt the book to their particular courses. The book is both complete and careful, and it continues to maintain its emphasis on algorithms and applications. Excellent exercise sets allow students to perfect skills as they practice. This new edition continues to feature numerous computer science applications-making this the ideal text for preparing students for advanced study.

    While such visualizations can better inform us, they can also deceive by displaying incomplete or inaccurate data, suggesting misleading patterns—or simply misinform us by being poorly designed, such as the confusing “eye of the storm” maps shown on TV every hurricane season.Many of us are ill equipped to interpret the visuals that politicians, journalists, advertisers, and even employers present each day, enabling bad actors to easily manipulate visuals to promote their own agendas. Public conversations are increasingly driven by numbers, and to make sense of them we must be able to decode and use visual information. By examining contemporary examples ranging from election-result infographics to global GDP maps and box-office record charts, How Charts Lie teaches us how to do just that.

    With a new foreword by Richard Wilkinson and Kate Pickett, authors of The Spirit Level and a new Afteword by the author examining developments during 2010, this is hard-hitting and uncompromising in its call to action and continues to make essential reading for everyone concerned with social justice.

    This new translation of Volume One, the only volume to be completed and edited by Marx himself, avoids some of the mistakes that have marred earlier versions and seeks to do justice to the literary qualities of the work. The introduction is by Ernest Mandel, author of Late Capitalism, one of the only comprehensive attempts to develop the theoretical legacy of Capital.

    The story only ends there, however, if you’re fortunate enough to live in an adequate civilization.Eliezer Yudkowsky’s Inadequate Equilibria is a sharp and lively guidebook for anyone questioning when and how they can know better, and do better, than the status quo. Freely mixing debates on the foundations of rational decision-making with tips for everyday life, Yudkowsky explores the central question of when we can (and can’t) expect to spot systemic inefficiencies, and exploit them.

    His pages have become an almost indispensable link between the knowing of exotic and remote people with theoretical knowledge about humankind. This volume--originally published in 1922--can be considered the founding document of economic anthropology, and remains the best one to read. It emphasizes the great significance of primitive economics by singling out the notable exchange system of the Trobriand Islands for special consideration. Although the main theme is economic, constant reference is made in this milestone of anthropological research and interpretation to social organization, life and meaning, the power of magic, and to mythology and folklore.

    From the early Greek influences to the Middle Ages and the Renaissance to the end of the 19th century, trace the fascinating foundation of mathematics as it developed through the ages. Aristotle, Galileo, Kepler, Newton: you know the names. Now here's what they really did, and the effect their discoveries had on our culture, all explained in a way the layperson can understand. Begin with the basis of arithmetic (Plato and the introduction of geometry), and discover why the use of Arabic numerals was critical to the development of both commerce and science. The development of calculus made space travel a reality, while the abacus prefigured the computer. The greats examined in depth include Leonardo da Vinci, a brilliant mathematician as well as artist; Pascal, who laid out the theory of probabilities; and Fermat, whose intriguing theory has only recently been solved.

    'Big data', 'data science', and 'machine learning' have become familiar terms in the news, as statistical methods are brought to bear upon the enormous data sets of modern science and commerce. How did we get here? And where are we going? This book takes us on an exhilarating journey through the revolution in data analysis following the introduction of electronic computation in the 1950s. Beginning with classical inferential theories - Bayesian, frequentist, Fisherian - individual chapters take up a series of influential topics: survival analysis, logistic regression, empirical Bayes, the jackknife and bootstrap, random forests, neural networks, Markov chain Monte Carlo, inference after model selection, and dozens more. The distinctly modern approach integrates methodology and algorithms with statistical inference. The book ends with speculation on the future direction of statistics and data science.

    No previous knowledge of statistics is assumed, and mathematical background is assumed to be minimal (lowest-level high-school algebra). This text may be used in a one or two course sequence. Such sequences are commonly required of social science graduate students in sociology, political science, and psychology. Students in geography, anthropology, journalism, and speech also are sometimes required to take at least one statistics course.

    In A Theory of Everything, Wilber uses clear, non-technical language to present complex, cutting-edge theories that integrate the realms of body, mind, soul, and spirit. He then demonstrates how these theories and models can be applied to real world problems and incorporated into readers' everyday lives.Wilber begins his study by presenting models like "spiral dynamics"--a leading model of human evolution--and his groundbreaking "all-level, all-quadrant" model for integrating science and religion, showing how they are being applied to politics, medicine, business, education, and the environment. He also covers broader models, explaining how they can integrate the various worldviews that have been developed around the world throughout the ages. Finally, Wilber proposes that readers take up an integral transformative practice--such as meditation--to help them apply and develop this integral vision in their personal, daily lives. A fascinating and easy-to-follow exploration of the "M Theory," this book is another tour-de-force from one of America's most inventive minds.

    But your pleasure and prowess at games, gambling, and other numerically related pursuits can be heightened with this entertaining volume, in which the authors offer a fascinating view of some of the lesser-known and more imaginative aspects of mathematics.A brief and breezy explanation of the new language of mathematics precedes a smorgasbord of such thought-provoking subjects as the googolplex (the largest definite number anyone has yet bothered to conceive of); assorted geometries — plane and fancy; famous puzzles that made mathematical history; and tantalizing paradoxes. Gamblers receive fair warning on the laws of chance; a look at rubber-sheet geometry twists circles into loops without sacrificing certain important properties; and an exploration of the mathematics of change and growth shows how calculus, among its other uses, helps trace the path of falling bombs.Written with wit and clarity for the intelligent reader who has taken high school and perhaps college math, this volume deftly progresses from simple arithmetic to calculus and non-Euclidean geometry. It “lives up to its title in every way [and] might well have been merely terrifying, whereas it proves to be both charming and exciting." — Saturday Review of Literature.

    Gonick’s The Cartoon Guide to Calculus is a refreshingly humorous, remarkably thorough guide to general calculus that, like his earlier Cartoon Guide to Physics and Cartoon History of the Modern World, will prove a boon to students, educators, and eager learners everywhere.