. . Nothing less than a major contribution to the scientific culture of this world." — The New York Times Book ReviewThis major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and including material on both elementary and advanced levels, encompasses 20 prime subject areas in mathematics in terms of their simple origins and their subsequent sophisticated developement. As Professor Morris Kline of New York University noted, "This unique work presents the amazing panorama of mathematics proper. It is the best answer in print to what mathematics contains both on the elementary and advanced levels."Beginning with an overview and analysis of mathematics, the first of three major divisions of the book progresses to an exploration of analytic geometry, algebra, and ordinary differential equations. The second part introduces partial differential equations, along with theories of curves and surfaces, the calculus of variations, and functions of a complex variable. It furthur examines prime numbers, the theory of probability, approximations, and the role of computers in mathematics. The theory of functions of a real variable opens the final section, followed by discussions of linear algebra and nonEuclidian geometry, topology, functional analysis, and groups and other algebraic systems.Thorough, coherent explanations of each topic are further augumented by numerous illustrative figures, and every chapter concludes with a suggested reading list. Formerly issued as a three-volume set, this mathematical masterpiece is now available in a convenient and modestly priced one-volume edition, perfect for study or reference."This is a masterful English translation of a stupendous and formidable mathematical masterpiece . . ." — Social Science

    This book upends the conventional approach to math, inviting you to think creatively about shape and dimension, the infinite and infinitesimal, symmetries, proofs, and how these concepts all fit together. What awaits readers is a freewheeling tour of the inimitable joys and unsolved mysteries of this curiously powerful subject.Like the classic math allegory Flatland, first published over a century ago, or Douglas Hofstadter's Godel, Escher, Bach forty years ago, there has never been a math book quite like Math Without Numbers. So many popularizations of math have dwelt on numbers like pi or zero or infinity. This book goes well beyond to questions such as: How many shapes are there? Is anything bigger than infinity? And is math even true? Milo Beckman shows why math is mostly just pattern recognition and how it keeps on surprising us with unexpected, useful connections to the real world.The ambitions of this book take a special kind of author. An inventive, original thinker pursuing his calling with jubilant passion. A prodigy. Milo Beckman completed the graduate-level course sequence in mathematics at age sixteen, when he was a sophomore at Harvard; while writing this book, he was studying the philosophical foundations of physics at Columbia under Brian Greene, among others.

    This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.

    Heath's translation of the thirteen books of Euclid's Elements. In keeping with Green Lion's design commitment, diagrams have been placed on every spread for convenient reference while working through the proofs; running heads on every page indicate both Euclid's book number and proposition numbers for that page; and adequate space for notes is allowed between propositions and around diagrams. The all-new index has built into it a glossary of Euclid's Greek terms.Heath's translation has stood the test of time, and, as one done by a renowned scholar of ancient mathematics, it can be relied upon not to have inadvertantly introduced modern concepts or nomenclature. We have excised the voluminous historical and scholarly commentary that swells the Dover edition to three volumes and impedes classroom use of the original text. The single volume is not only more convenient, but less expensive as well.

    The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understandingof paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questionsabout the mathematical community (such as Is it true that mathematicians burn out at the age of 25?) It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.About the Series: Combining authority with wit, accessibility, and style, Very Short Introductions offer an introduction to some of life's most interesting topics. Written by experts for the newcomer, they demonstrate the finest contemporary thinking about the central problems and issues in hundredsof key topics, from philosophy to Freud, quantum theory to Islam.

    With his trademark clarity and humor, Holt probes the mysteries of quantum mechanics, the quest for the foundations of mathematics, and the nature of logic and truth. Along the way, he offers intimate biographical sketches of celebrated and neglected thinkers, from the physicist Emmy Noether to the computing pioneer Alan Turing and the discoverer of fractals, Benoit Mandelbrot. Holt offers a painless and playful introduction to many of our most beautiful but least understood ideas, from Einsteinian relativity to string theory, and also invites us to consider why the greatest logician of the twentieth century believed the U.S. Constitution contained a terrible contradiction--and whether the universe truly has a future.

    Admittedly real life wasn’t like that. But only, they argued, because we didn’t have enough data to be certain.Then the cracks began to appear. It proved impossible to predict exactly how three planets orbiting each other would move. Meteorologists discovered that the weather was truly chaotic – so dependent on small variations that it could never be predicted for more than a few days out. And the final nail in the coffin was quantum theory, showing that everything in the universe has probability at its heart.That gives human beings a problem. We understand the world through patterns. Randomness and probability will always be alien to us. But it’s time to plunge into this fascinating, shadowy world, because randomness crops up everywhere. Probability and statistics are the only way to get a grip on nature’s workings. They may even seal the fate of free will and predict how the universe will end.Forget Newton’s clockwork universe. Welcome to Dice World.

    His feverish and singular pursuit of this goal has come to define his life. Now an old man, he is looked on with suspicion and shame by his family-until his ambitious young nephew intervenes.Seeking to understand his uncle's mysterious mind, the narrator of this novel unravels his story, a dramatic tale set against a tableau of brilliant historical figures-among them G. H. Hardy, the self-taught Indian genius Srinivasa Ramanujan, and a young Kurt Gödel. Meanwhile, as Petros recounts his own life's work, a bond is formed between uncle and nephew, pulling each one deeper into mathematical obsession, and risking both of their sanity.

    However, according to Hofstadter, the formal system that underlies all mental activity transcends the system that supports it. If life can grow out of the formal chemical substrate of the cell, if consciousness can emerge out of a formal system of firing neurons, then so too will computers attain human intelligence. Gödel, Escher, Bach is a wonderful exploration of fascinating ideas at the heart of cognitive science: meaning, reduction, recursion, and much more.

    Underlining the importance of critical thinking in all matters numerical, Best illustrates his points with examples of good and bad statistics about such contemporary concerns as school shootings, fatal hospital errors, bullying, teen suicides, deaths at the World Trade Center, college ratings, the risks of divorce, racial profiling, and fatalities caused by falling coconuts."More Damned Lies and Statistics" encourages all of us to think in a more sophisticated and skeptical manner about how statistics are used to promote causes, create fear, and advance particular points of view.

    This is the classic introduction for the educated lay reader to the richly diverse world of mathematics: its history, philosophy, principles, and personalities.

    In The Universe and the Teacup, K. C. Cole demystifies mathematics and shows us-with humor and wonderfully accessible stories-why math need not be frightening. Using the O. J. Simpson trial, the bell curve, and Emmy Noether, the nineteenth-century woman scientist whose work was essential for Einstein's theory of relativity, Cole helps us see that more than just being a tool, math is a key to understanding the beauty of everything from rainbows to relativity.

    This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. * The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible.

    They speak of it in awed terms and consider it to be an even more difficult problem than Fermat's last theorem, which was finally proven by Andrew Wiles in 1995.In The Riemann Hypothesis, acclaimed author Karl Sabbagh interviews some of the world's finest mathematicians who have spent their lives working on the problem--and whose approaches to meeting the challenges thrown up by the hypothesis are as diverse as their personalities.Wryly humorous, lively, accessible and comprehensive, The Riemann Hypothesis is a compelling exploration of the people who do math and the ideas that motivate them to the brink of obsession--and a profound meditation on the ultimate meaning of mathematics.

    In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory.Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples. Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated..