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.

    Inspired by and expanding on a series of lectures presented by Leonard Peikoff, David Harriman presents a fascinating answer to the problem of induction-the epistemological question of how we can know the truth of inductive generalizations.Ayn Rand presented her revolutionary theory of concepts in her book Introduction to Objectivist Epistemology. As Dr. Peikoff subsequently explored the concept of induction, he sought out David Harriman, a physicist who had taught philosophy, for his expert knowledge of the scientific discovery process.Here, Harriman presents the result of a collaboration between scientist and philosopher. Beginning with a detailed discussion of the role of mathematics and experimentation in validating generalizations in physics-looking closely at the reasoning of scientists such as Galileo, Kepler, Newton, Lavoisier, and Maxwell-Harriman skillfully argues that the inductive method used in philosophy is in principle indistinguishable from the method used in physics.

    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.

    He was trying to find an equation that explained all physical reality - a theory of everything. He failed, but others have taken up the challenge in a remarkable quest that is shedding light on unsuspected secrets of the cosmos.Experimental physicist and award-winning educator Dr. Don Lincoln of the Fermi National Accelerator Laboratory takes you on this exciting journey in The Theory of Everything: The Quest to Explain All Reality. Suitable for the intellectually curious at all levels and assuming no background beyond basic high-school math, these 24 half-hour lectures cover recent developments at the forefront of particle physics and cosmology, while delving into the history of the centuries-long search for this holy grail of science.You trace the dream of a theory of everything through Newton, Maxwell, Einstein, Bohr, Schrödinger, Feynman, Gell-Mann, Weinberg, and other great physicists, charting their progress toward an all-embracing, unifying theory. Their resulting equations are the masterpieces of physics, which Dr. Lincoln explains in fascinating and accessible detail. Studying them is like touring a museum of great works of art - works that are progressing toward an ultimate, as-yet-unfinished masterpiece.Listening Length: 12 hours and 21 minutes

    You can use these ideas and methods as-is, or better yet, tweak them and create your own enticing educational meals. The message the authors share is that, with imagination and preparation, every teacher can be a Classroom Chef.

    Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofs are classics, but many are new and brilliant proofs of classical results. ...Aigner and Ziegler... write: ..". all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999..". the style is clear and entertaining, the level is close to elementary ... and the proofs are brilliant. ..." LMS Newsletter, January 1999This third edition offers two new chapters, on partition identities, and on card shuffling. Three proofs of Euler's most famous infinite series appear in a separate chapter. There is also a number of other improvements, such as an exciting new way to "enumerate the rationals."

    Using playing cards, a newspaper, the back of an envelope, a Sudoku, some pennies and of course a pair of socks, Rob Eastaway shows how maths can demonstrate its secret beauties in even the most mundane of everyday objects. Among the many fascinating curiosities in these pages, you will discover the strange link between limericks and rabbits, an apparently 'fair' coin game where the odds are massively in your favour, why tourist boards can't agree on where the centre of Britain is, and how simple paper folding can lead to a Jurassic Park monster. With plenty of ideas you'll want to test out for yourself, this engaging and refreshing look at mathematics is for everyone.

    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.

    The Kingfisher Science Encyclopedia is the one reference that includes all of the information students need to know in today's fast-paced world. Clearly written and illustrated articles provide in-depth insight and concise authoritative information. An impressive reference section at the end of the book contains minibiographies of famous scientists, plus an illustrated time line of key inventions and discoveries. Arranged thematically into ten chapters, with reference summaries at the end of each chapter and a full index, the encyclopedia does more than merely provide facts about science and technology—it helps the reader think for him or herself, develop an enquiring mind, pose challenging questions, and explore new topics.

    But from the perspective of mathematics, groupings of ten are arbitrary, and can have serious shortcomings. Twelve would be better for divisibility, and eight is smaller and well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.Paul Lockhart reveals arithmetic not as the rote manipulation of numbers--a practical if mundane branch of knowledge best suited for balancing a checkbook or filling out tax forms--but as a set of ideas that exhibit the fascinating and sometimes surprising behaviors usually reserved for higher branches of mathematics. The essence of arithmetic is the skillful arrangement of numerical information for ease of communication and comparison, an elegant intellectual craft that arises from our desire to count, add to, take away from, divide up, and multiply quantities of important things. Over centuries, humans devised a variety of strategies for representing and using numerical information, from beads and tally marks to adding machines and computers. Lockhart explores the philosophical and aesthetic nature of counting and of different number systems, both Western and non-Western, weighing the pluses and minuses of each.A passionate, entertaining survey of foundational ideas and methods, Arithmetic invites readers to experience the profound and simple beauty of its subject through the eyes of a modern research mathematician.

    Using Everett's unpublished papers (recently discovered in his son's basement) and dozens of interviews with his friends, colleagues, and surviving family members, Byrne paints, for the general reader, a detailed portrait of the genius who invented an astonishing way of describing our complex universe from the inside. Everett's mathematical model (called the "universal wave function") treats all possible events as "equally real," and concludes that countless copies of every person and thing exist in all possible configurations spread over an infinity of universes: many worlds. Afflicted by depression and addictions, Everett strove to bring rational order to the professional realms in which he played historically significant roles. In addition to his famous interpretation of quantum mechanics, Everett wrote a classic paper in game theory; created computer algorithms that revolutionized military operations research; and performed pioneering work in artificial intelligence for top secret government projects. He wrote the original software for targeting cities in a nuclear hot war; and he was one of the first scientists to recognize the danger of nuclear winter. As a Cold Warrior, he designed logical systems that modeled "rational" human and machine behaviors, and yet he was largely oblivious to the emotional damage his irrational personal behavior inflicted upon his family, lovers, and business partners. He died young, but left behind a fascinating record of his life, including correspondence with such philosophically inclined physicists as Niels Bohr, Norbert Wiener, and John Wheeler. These remarkable letters illuminate the long and often bitter struggle to explain the paradox of measurement at the heart of quantum physics. In recent years, Everett's solution to this mysterious problem-the existence of a universe of universes-has gained considerable traction in scientific circles, not as science fiction, but as an explanation of physical reality.

    Earth, born in a nuclear explosion, is a radioactive planet; without radiation, life would not exist. And while radiation can be dangerous, it is also deeply misunderstood and often mistakenly feared. Now Robert Peter Gale, M.D,—the doctor to whom concerned governments turned in the wake of the Chernobyl and Fukushima disasters—in collaboration with medical writer Eric Lax draws on an exceptional depth of knowledge to correct myths and establish facts. Exploring what have become trigger words for anxiety—nuclear energy and nuclear weapons, uranium, plutonium, iodine-131, mammogram, X-ray, CT scan, threats to the food chain—the authors demystify the science and dangers of radiation, and examine its myriad benefits, from safely sterilizing our food to the relatively low-risk fuel alternative of nuclear energy. This is the book for all readers who have asked themselves questions such as: What kinds of radiation, and what degree of exposure, cause cancer? What aftereffects have nuclear accidents and bombs had? Does radiation increase the likelihood of birth defects? And how does radiation work? Hugely illuminating, Radiation is the definitive road map to our post-Chernobyl, post-Fukushima world.

    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.

    To understand algebra is to possess the power to grow your skills and knowledge so you can ace your courses and possibly pursue further study in math. Algebra II For Dummies is the fun and easy way to get a handle on this subject and solve even the trickiest algebra problems. This friendly guide shows you how to get up to speed on exponential functions, laws of logarithms, conic sections, matrices, and other advanced algebra concepts. In no time you'll have the tools you need to:Interpret quadratic functions Find the roots of a polynomial Reason with rational functions Expose exponential and logarithmic functions Cut up conic sections Solve linear and non linear systems of equations Equate inequalities Simplifyy complex numbers Make moves with matrices Sort out sequences and sets This straightforward guide offers plenty of multiplication tricks that only math teachers know. It also profiles special types of numbers, making it easy for you to categorize them and solve any problems without breaking a sweat. When it comes to understanding and working out algebraic equations, Algebra II For Dummies is all you need to succeed!