It describes how the laws and discoveries of information theory now support controversial revisions to Darwinian evolution, begin to unravel the mysteries of language, memory and dreams, and stimulate provocative ideas in psychology, philosophy, art, music, computers and even the structure of society. Perhaps its most fascinating and unexpected surprise is the suggestion the order and complexity may be as natural as disorder and disorganization. Contrary to the entropy principle, which implies that order is the exception and confusion the rule, information theory asserts that order and sense can indeed prevail against disorder and nonsense. From the simplest forms of organic life to the words used to express our most complex ideas, from our genes to our dreams, from microcomputers to telecommunications, virtually everything around us follows simple rules of information. Life and the material world, like language, remain "grammatical." Grammatical man inhabits a grammatical universe.

    Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

    The stories show that, as James Gleick puts it in the foreword, "inspiration strikes willy-nilly." One researcher thinks of quantum chaotic systems at a bus stop; another suddenly realizes a path to proving a theorem of number theory while in a friend's backyard; a statistician has a "bathroom sink epiphany" and discovers the key to solving the Gaussian correlation inequality. Readers of The Prime Number Conspiracy, says Quanta editor-in-chief Thomas Lin, are headed on "breathtaking intellectual journeys to the bleeding edge of discovery strapped to the narrative rocket of humanity's never-ending pursuit of knowledge."Quanta is the only popular publication that offers in-depth coverage of the latest breakthroughs in understanding our mathematical universe. It communicates mathematics by taking it seriously, wrestling with difficult concepts and clearly explaining them in a way that speaks to our innate curiosity about our world and ourselves. Readers of this volume will learn that prime numbers have decided preferences about the final digits of the primes that immediately follow them (the "conspiracy" of the title); consider whether math is the universal language of nature (allowing for "a unified theory of randomness"); discover surprising solutions (including a pentagon tiling proof that solves a century-old math problem); ponder the limits of computation; measure infinity; and explore the eternal question "Is mathematics good for you?"ContributorsAriel Bleicher, Robbert Dijkgraaf, Kevin Hartnett, Erica Klarreich, Thomas Lin, John Pavlus, Siobhan Roberts, Natalie WolchoverCopublished with Quanta Magazine

    Rare Book

    presidential elections have been won by the second most popular candidate. The reason was a "spoiler"--a minor candidate who takes enough votes away from the most popular candidate to tip the election to someone else. The spoiler effect is more than a glitch. It is a consequence of one of the most surprising intellectual discoveries of the twentieth century: the "impossibility theorem" of Nobel laureate economist Kenneth Arrow. The impossibility theorem asserts that voting is fundamentally unfair--a finding that has not been lost on today's political consultants. Armed with polls, focus groups, and smear campaigns, political strategists are exploiting the mathematical faults of the simple majority vote. In recent election cycles, this has led to such unlikely tactics as Republicans funding ballot drives for Green spoilers and Democrats paying for right-wing candidates' radio ads. Gaming the Vote shows that there is a solution to the spoiler problem that will satisfy both right and left. A systemcalled range voting, already widely used on the Internet, is the fairest voting method of all, according to computer studies. Despite these findings, range voting remains controversial, and Gaming the Vote assesses the obstacles confronting any attempt to change the American electoral system. The latest of several books by William Poundstone on the theme of how important scientific ideas have affected the real world, Gaming the Vote is a wry exposé of how the political system really works, and a call to action.

    Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing the mathematical model) and how to solve the equation (along with initial and boundary conditions). Written for advanced undergraduate and graduate students, as well as professionals working in the applied sciences, this clearly written book offers realistic, practical coverage of diffusion-type problems, hyperbolic-type problems, elliptic-type problems, and numerical and approximate methods. Each chapter contains a selection of relevant problems (answers are provided) and suggestions for further reading.

    His Incompleteness Theorem turned not only mathematics but also the whole world of science and philosophy on its head. Equally legendary were Gö's eccentricities, his close friendship with Albert Einstein, and his paranoid fear of germs that eventually led to his death from self-starvation. Now, in the first popular biography of this strange and brilliant thinker, John Casti and Werner DePauli bring the legend to life. After describing his childhood in the Moravian capital of Brno, the authors trace the arc of Gö's remarkable career, from the famed Vienna Circle, where philosophers and scientists debated notions of truth, to the Institute for Advanced Study in Princeton, New Jersey, where he lived and worked until his death in 1978. In the process, they shed light on Gö's contributions to mathematics, philosophy, computer science, artificial intelligence -- even cosmology -- in an entertaining and accessible way.

    This book will offer students a broad outline of essential mathematics and will help to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential and analytical geometry, real analysis, point-set topology, probability, complex analysis, set theory, algorithms, and more. An annotated bibliography offers a guide to further reading and to more rigorous foundations.

    Erno Rubik inspires us with what he’s learned in a lifetime of creating, curiosity, and discovery.Erno Rubik was a child when he first became obsessed with puzzles of all kinds. “Puzzles,” he writes, “bring out important qualities in each of us: concentration, curiosity, a sense of play, the eagerness to discover a solution.” To Rubik puzzles aren’t just games—they’re creativity machines. He encourages us to embrace our inner curiosity and find the puzzles that surround us in our everyday lives. “If you are determined, you will solve them,” he writes. Rubik’s own puzzle, the Cube, went on to be solved by millions worldwide for over forty years, become one of the bestselling toys of all time, and to be featured as a global symbol of intelligence and ingenuity.In Cubed, Rubik covers more than just his journey to inventing his eponymous cube. He makes a case for always being an amateur—something he has always considered himself to be. He discusses the inevitability of problems during any act of invention. He reveals what it was like to experience the astonishing worldwide success of an object he made purely for his own play. And he offers what he thinks it means to be a true creator (hint: anyone can do it). Steeped in the wisdom and also the humility of a born inventor, Cubed offers a unique look at the imperfect science of creation.

    Providing a concise introduction to abstract algebra, this work unfolds some of the fundamental systems with the aim of reaching applicable, significant results.

    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.

    It aims to help you to take the plunge to realize your potential, so that you do not have to remain stuck in a social and lifestyle rut as if there is no alternative.

    joyfully shows you how to make nature's numbers dance."--Bill Nye (the science guy)The Magic of Math is the math book you wish you had in school. Using a delightful assortment of examples-from ice-cream scoops and poker hands to measuring mountains and making magic squares-this book revels in key mathematical fields including arithmetic, algebra, geometry, and calculus, plus Fibonacci numbers, infinity, and, of course, mathematical magic tricks. Known throughout the world as the "mathemagician," Arthur Benjamin mixes mathematics and magic to make the subject fun, attractive, and easy to understand for math fan and math-phobic alike."A positively joyful exploration of mathematics."-Publishers Weekly, starred review"Each [trick] is more dazzling than the last."-Physics World

    It is for everyone who cares about education in an uncertain world and explains how teachers, parents and grandparents can cultivate confidence, curiosity, collaboration, communication, creativity, commitment and craftsmanship in children, at the same time as helping them to do well in public examinations. Educating Ruby shows, unequivocally, that schools can get the right results in the right way, so that the Rubys of tomorrow will emerge from their time at school able to talk with honest pleasure and reflective optimism about their schooling. Featuring the views of schoolchildren, parents, educators and employers and drawing on Guy Claxton and Bill Lucas’ years of experience in education, including their work with Building Learning Power and the Expansive Education Network, this powerful new book is sure to provoke thinking and debate. Just as Willy Russell’s Educating Rita helped us rethink university, the authors of Educating Ruby invite fresh scrutiny of our schools.

    Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be reasoned out--from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft--indeed, brilliant--instructions on stripping away irrelevancies and going straight to the heart of the problem.