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.

    For many years Powers's nearly 600-page Introduction to Tibetan Buddhism has served as the field's most authoritative and comprehensive overview of Tibet's distinctive Buddhist tradition. A Concise Introduction to Tibetan Buddhism explains the core Buddhist doctrines and the practices of meditation and tantra and provides a survey of the four main schools of Tibetan Buddhism.

    Lewis", Kreeft provides a look at the nature of heaven. A refreshingly clear, theologically sound glimpse of the "undiscovered country". Kreeft speaks to the heart and the mind for an unexcelled look at one of the most popular, yet least understood, subjects in religion.

    Yet psychologists and philosophers agree that these commonplace approaches to creating happiness, success, and fulfillment make critical mistakes. Stop Chasing Carrots reveals how these mistakes begin with misconceptions about happiness and then presents a philosophy of life to achieve better results.Here is the rare book that creates a philosophy of life based on scientific evidence. Replacing self-help materialism, Eastern spiritualism/minimalism, and complex but insightful psychological studies with an accessible, balanced, and realistic concept, Stop Chasing Carrots empowers readers to lead lives based on proven ideas.

    But is philosophy really so complicated? And is it really as irrelevant as it sometimes seems? "I Think, Therefore I Am" is the ideal way to take the fear out of philosophy. Written in an accessible and highly entertaining style, it explains how and why philosophy began, and how, from Greek democracy to Communism, the ways in which we live, learn, argue, vote and even spend our money have their origins in philosophical thought.Philosophers certainly like to make life sound awfully complicated. But is philosophy really so complicated? And is it really as irrelevant as it sometimes seems? "I Think, Therefore I Am" is the ideal way to take the fear out of philosophy. Written in an accessible and highly entertaining style, it explains how and why philosophy began, and how, from Greek democracy to Communism, the ways in which we live, learn, argue, vote and even spend our money have their origins in philosophical thought.

    Includes thought-provoking stories and perspectives from all walks of life, including Nation elders, media celebrities, and international contributors. Do you know who - and what - you are? Do you know who you're meant to be? Do you know how to find the answers to questions like these? Knowledge of Self is the result of a process of self-discovery, but few of us know where to begin when we're ready to start looking deeper. Although self-actualization is the highest of all human needs, it is said that only 5% of people ever attain this goal. In the culture of the Nation of Gods and Earths, commonly known as the Five Percent, students are instructed that they must first learn themselves, then their worlds, and then what they must do in order to transform their world for the better. This often intense process has produced thousands of revolutionary thinkers in otherwise desperate environments, where poverty and hopelessness dominate. Until now, few mainstream publications have captured the brilliant, yet practical, perspectives of these luminary men and women. Knowledge of Self: A Collection of Writings on the Science of Everything in Life presents the thoughts of over 50 Five Percenters, both young and old, male and female, from all over the globe, in their own words. Through essays, poems, and even ‘how-to’ articles, this anthology presents readers with an accurate portrait of what the Five Percent study and teach. Featuring essays from Nation elders, as well as Lord Jamar of Brand Nubian, Cappadonna, Popa Wu, and several others.

    Topics include the development of counting and numbers systems, the emergence of zero, cultures that don’t have numbers, algebra, solid geometry, symmetry and beauty, perspective, riddles and problems, calculus, mathematical logic, friction force and displacement, subatomic particles, and the expansion of the universe. Great mathematical thinkers covered include Napier, Liu Hui, Aryabhata, Galileo, Newton, Russell, Einstein, Riemann, Euclid, Carl Friedrich Gauss, Charles Babbage, Montmort, Wittgenstein, and many more. The book is beautifully illustrated throughout in full color.

    The selections range from the earliest economic thought to such towering volumes as Adam Smith's The Wealth of Nations, Thomas Malthus's Essay on the Principle of Population, David Ricardo's Principles of Political Economy, and John Maynard Keynes's The General Theory of Employment, Interest, and Money. Acting as "a docent, not merely an editor," he takes the reader through the core arguments with "brilliantly clear commentary" (New York Times Book Review).

    Yet geometry is so much more than shapes and numbers; indeed, it governs much of our lives—from architecture and microchips to car design, animated movies, the molecules of food, even our own body chemistry. And as Siobhan Roberts elegantly conveys in The King of Infinite Space, there can be no better guide to the majesty of geometry than Donald Coxeter, perhaps the greatest geometer of the twentieth century.Many of the greatest names in intellectual history—Pythagoras, Plato, Archimedes, Euclid— were geometers, and their creativity and achievements illuminate those of Coxeter, revealing geometry to be a living, ever-evolving endeavor, an intellectual adventure that has always been a building block of civilization. Coxeter's special contributions—his famed Coxeter groups and Coxeter diagrams—have been called by other mathematicians "tools as essential as numbers themselves," but his greatest achievement was to almost single-handedly preserve the tradition of classical geometry when it was under attack in a mathematical era that valued all things austere and rational.Coxeter also inspired many outside the field of mathematics. Artist M. C. Escher credited Coxeter with triggering his legendary Circle Limit patterns, while futurist/inventor Buckminster Fuller acknowledged that his famed geodesic dome owed much to Coxeter's vision. The King of Infinite Space is an elegant portal into the fascinating, arcane world of geometry.

    Why?In an era of growing cultural pessimism, many thoughtful individuals believe that different civilisations – especially Islam and the West – cannot live together in peace. The ten countries of ASEAN provide a thriving counter-example of civilizational co-existence. Here 625m people live together in peace. This miracle was delivered by ASEAN.In an era of growing economic pessimism, where many young people believe that their lives will get worse in coming decades, Southeast Asia bubbles with optimism. In an era where many thinkers predict rising geopolitical competition and tension, ASEAN regularly brings together all the world’s great powers.Stories of peace are told less frequently than stories of conflict and war. ASEAN’s imperfections make better headlines than its achievements. But in the hands of thinker and writer Kishore Mahbubani, the good news story is also a provocation and a challenge to the rest of the world."This excellent book explains, in clear and simple terms, how and why ASEAN has become one of the most successful regional organizations in the world."George Yeo"A powerful and passionate account of how, against all odds, ASEAN transformed the region and why Asia and the world need it even more today."Amitav Acharya“Kishore and I have written that the world is coming together in a Fusion of Civilisations. This book documents beautifully how ASEAN has achieved this fusion. The ASEAN story is hugely instructive and this book tells it very well.”Larry SummersKishore Mahbubani is Dean of the Lee Kuan Yew School of Public Policy, National University of Singapore, and author of The New Asian Hemisphere: The Irresistible Shift of Global Power to the East. Jeffery Sng is a writer and former diplomat based in Bangkok, co-author of A History of the Thai-Chinese.

    Readers will progress from building a basic physics engine to creating intelligent moving objects and complex systems, setting the foundation for further experiments in generative design. Subjects covered include forces, trigonometry, fractals, cellular automata, self-organization, and genetic algorithms. The book's examples are written in Processing, an open-source language and development environment built on top of the Java programming language. On the book's website (http://www.natureofcode.com), the examples run in the browser via Processing's JavaScript mode.

    Before this explosive growth into the modern age took place, an all-but-forgotten genius strove for forty years to win acceptance for the atomic theory of matter and an altogether new way of doing physics. Ludwig Boltz-mann battled with philosophers, the scientific establishment, and his own potent demons. His victory led the way to the greatest scientific achievements of the twentieth century. Now acclaimed science writer David Lindley portrays the dramatic story of Boltzmann and his embrace of the atom, while providing a window on the civilized world that gave birth to our scientific era. Boltzmann emerges as an endearingly quixotic character, passionately inspired by Beethoven, who muddled through the practical matters of life in a European gilded age. Boltzmann's story reaches from fin de siecle Vienna, across Germany and Britain, to America. As the Habsburg Empire was crumbling, Germany's intellectual might was growing; Edinburgh in Scotland was one of the most intellectually fertile places on earth; and, in America, brilliant independent minds were beginning to draw on the best ideas of the bureaucratized old world.Boltzmann's nemesis in the field of theoretical physics at home in Austria was Ernst Mach, noted today in the term Mach I, the speed of sound. Mach believed physics should address only that which could be directly observed. How could we know that frisky atoms jiggling about corresponded to heat if we couldn't see them? Why should we bother with theories that only told us what would probably happen, rather than making an absolute prediction? Mach and Boltzmann both believed in the power of science, but their approaches to physics could not have been more opposed. Boltzmann sought to explain the real world, and cast aside any philosophical criteria. Mach, along with many nineteenth-century scientists, wanted to construct an empirical edifice of absolute truths that obeyed strict philosophical rules. Boltzmann did not get on well with authority in any form, and he did his best work at arm's length from it. When at the end of his career he engaged with the philosophical authorities in the Viennese academy, the results were personally disastrous and tragic. Yet Boltzmann's enduring legacy lives on in the new physics and technology of our wired world.Lindley's elegant telling of this tale combines the detailed breadth of the best history, the beauty of theoretical physics, and the psychological insight belonging to the finest of novels.

    He is patriotic—a veteran who loves his country. He is independent—choosing his own path on his motorcycle, living life on his own terms. He is outspoken—he has boldly criticized injustices in American law and society despite the backlash this has evoked from the establishment. Yet the element that he finds most important, most sacred, most American, is freedom.In Freedom, Sonny articulates many of the principles he employs in his own life. Whether he is regarded as a leader, a rebel, a revolutionary, a criminal, or a soldier, Sonny's outlook has been influenced not just by school but by the military, prison, and his experiences riding with the world's most notorious motorcycle club. It was on these various journeys that he learned the lessons that are most important in his life and the qualities he respects when he sees them in others:IndependenceCustomize Yourself; Originals Don't Come Off an Assembly LineToughnessTemper the Steel to Forge a Strong BladeFairness Treat Me Good, I'll Treat You Better; Treat Me Bad, I'll Treat You WorsePresented in the form of fifty credos, this book gives Sonny Barger's perspective on how to live a life that embodies the most fundamental of American virtues: freedom.

    Short-Cut Math is a concise, remarkably clear compendium of about 150 math short-cuts — timesaving tricks that provide faster, easier ways to add, subtract, multiply, and divide.By using the simple foolproof methods in this volume, you can double or triple your calculation speed — even if you always hated math in school. Here's a sampling of the amazingly effective techniques you will learn in minutes: Adding by 10 Groups; No-Carry Addition; Subtraction Without Borrowing; Multiplying by Aliquot Parts; Test for Divisibility by Odd and Even Numbers; Simplifying Dividends and Divisors; Fastest Way to Add or Subtract Any Pair of Fractions; Multiplying and Dividing with Mixed Numbers, and more.The short-cuts in this book require no special math ability. If you can do ordinary arithmetic, you will have no trouble with these methods. There are no complicated formulas or unfamiliar jargon — no long drills or exercises. For each problem, the author provides an explanation of the method and a step-by-step solution. Then the short-cut is applied, with a proof and an explanation of why it works.Students, teachers, businesspeople, accountants, bank tellers, check-out clerks — anyone who uses numbers and wishes to increase his or her speed and arithmetical agility, can benefit from the clear, easy-to-follow techniques given here.

    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.