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Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "e;imaginary numbers"e;--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "e;numbers"e; in all of mathematics.Some images inside the book are unavailable due to digital copyright restrictions.

Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist. Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption made from a rod and springs? Levi demonstrates how to use physical intuition to solve these and other fascinating math problems. More than half the problems can be tackled by anyone with precalculus and basic geometry, while the more challenging problems require some calculus. This one-of-a-kind book explains physics and math concepts where needed, and includes an informative appendix of physical principles. The Mathematical Mechanic will appeal to anyone interested in the little-known connections between mathematics and physics and how both endeavors relate to the world around us.

Trigonometry has always been the black sheep of mathematics. It has a reputation as a dry and difficult subject, a glorified form of geometry complicated by tedious computation. In this book, Eli Maor draws on his remarkable talents as a guide to the world of numbers to dispel that view. Rejecting the usual arid descriptions of sine, cosine, and their trigonometric relatives, he brings the subject to life in a compelling blend of history, biography, and mathematics. He presents both a survey of the main elements of trigonometry and a unique account of its vital contribution to science and social development. Woven together in a tapestry of entertaining stories, scientific curiosities, and educational insights, the book more than lives up to the title Trigonometric Delights.Maor, whose previous books have demystified the concept of infinity and the unusual number "e;e,"e; begins by examining the "e;proto-trigonometry"e; of the Egyptian pyramid builders. He shows how Greek astronomers developed the first true trigonometry. He traces the slow emergence of modern, analytical trigonometry, recounting its colorful origins in Renaissance Europe's quest for more accurate artillery, more precise clocks, and more pleasing musical instruments. Along the way, we see trigonometry at work in, for example, the struggle of the famous mapmaker Gerardus Mercator to represent the curved earth on a flat sheet of paper; we see how M. C. Escher used geometric progressions in his art; and we learn how the toy Spirograph uses epicycles and hypocycles.Maor also sketches the lives of some of the intriguing figures who have shaped four thousand years of trigonometric history. We meet, for instance, the Renaissance scholar Regiomontanus, who is rumored to have been poisoned for insulting a colleague, and Maria Agnesi, an eighteenth-century Italian genius who gave up mathematics to work with the poor--but not before she investigated a special curve that, due to mistranslation, bears the unfortunate name "e;the witch of Agnesi."e; The book is richly illustrated, including rare prints from the author's own collection. Trigonometric Delights will change forever our view of a once dreaded subject.

Einstein said that the most incomprehensible thing about the universe is that it is comprehensible. But was he right? Can the quantum theory of fields and Einstein's general theory of relativity, the two most accurate and successful theories in all of physics, be united into a single quantum theory of gravity? Can quantum and cosmos ever be combined? In The Nature of Space and Time, two of the world's most famous physicists-Stephen Hawking (A Brief History of Time) and Roger Penrose (The Road to Reality)-debate these questions.The authors outline how their positions have further diverged on a number of key issues, including the spatial geometry of the universe, inflationary versus cyclic theories of the cosmos, and the black-hole information-loss paradox. Though much progress has been made, Hawking and Penrose stress that physicists still have further to go in their quest for a quantum theory of gravity.

The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as earthquakes, global warming, demographic patterns, financial crises, and the failure of materials. In this book, Didier Sornette boldly applies his varied experience in these areas to propose a simple, powerful, and general theory of how, why, and when stock markets crash. Most attempts to explain market failures seek to pinpoint triggering mechanisms that occur hours, days, or weeks before the collapse. Sornette proposes a radically different view: the underlying cause can be sought months and even years before the abrupt, catastrophic event in the build-up of cooperative speculation, which often translates into an accelerating rise of the market price, otherwise known as a "e;bubble."e; Anchoring his sophisticated, step-by-step analysis in leading-edge physical and statistical modeling techniques, he unearths remarkable insights and some predictions--among them, that the "e;end of the growth era"e; will occur around 2050. Sornette probes major historical precedents, from the decades-long "e;tulip mania"e; in the Netherlands that wilted suddenly in 1637 to the South Sea Bubble that ended with the first huge market crash in England in 1720, to the Great Crash of October 1929 and Black Monday in 1987, to cite just a few. He concludes that most explanations other than cooperative self-organization fail to account for the subtle bubbles by which the markets lay the groundwork for catastrophe. Any investor or investment professional who seeks a genuine understanding of looming financial disasters should read this book. Physicists, geologists, biologists, economists, and others will welcome Why Stock Markets Crash as a highly original "e;scientific tale,"e; as Sornette aptly puts it, of the exciting and sometimes fearsome--but no longer quite so unfathomable--world of stock markets.

The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with only a modest mathematical background, this biography brings out the central importance of e to mathematics and illuminates a golden era in the age of science.

John Tyler Bonner, one of our most distinguished and creative biologists, here offers a completely new perspective on the role of size in biology. In his hallmark friendly style, he explores the universal impact of being the right size. By examining stories ranging from Alice in Wonderland to Gulliver's Travels, he shows that humans have always been fascinated by things big and small. Why then does size always reside on the fringes of science and never on the center stage? Why do biologists and others ponder size only when studying something else-running speed, life span, or metabolism?Why Size Matters, a pioneering book of big ideas in a compact size, gives size its due by presenting a profound yet lucid overview of what we know about its role in the living world. Bonner argues that size really does matter-that it is the supreme and universal determinant of what any organism can be and do. For example, because tiny creatures are subject primarily to forces of cohesion and larger beasts to gravity, a fly can easily walk up a wall, something we humans cannot even begin to imagine doing.Bonner introduces us to size through the giants and dwarfs of human, animal, and plant history and then explores questions including the physics of size as it affects biology, the evolution of size over geological time, and the role of size in the function and longevity of living things.As this elegantly written book shows, size affects life in its every aspect. It is a universal frame from which nothing escapes.

This is a rare and much-needed book: a concise but comprehensive account of quantum mechanics for popular science readers written by a respected physicist. Sam Treiman--internationally renowned for his work in particle physics--makes quantum mechanics accessible to nonspecialists. Combining mastery of the material with clear, elegant prose and infectious enthusiasm, he conveys the substance, methods, and profound oddities of the field. Treiman begins with an overview of quantum mechanics. He sketches the early development of the field by Einstein, Bohr, Heisenberg, Schrodinger, and others, and he makes clear how the quantum outlook flies in the face of common sense. As he explains, the quantum world is intrinsically probabilistic. For example, a particle is not in general in some particular place at a given instant, nor does it have a definite momentum. According to the Heisenberg uncertainty principle, there is a limit to how well both location and momentum can be specified simultaneously. In addition, particles can move through barriers and otherwise move in regions of space that are forbidden by classical mechanics. If a particle has a choice of different paths, it pursues all of them at once. Particles display wave-like characteristics and waves show particle-like characteristics. Treiman pays special attention to the more fundamental wave outlook and its expression in quantum field theory. He deals here with the remarkable fact that all the particles of a given species are strictly identical, and with the unnerving fact that particles can be created and destroyed. As Treiman introduces us to these and other wonders, he also touches--without resolution--on some of the deep philosophical problems of quantum mechanics, notably how probabilities become facts. Weaving together impeccable and up-to-date science, engaging writing, and a talent for clear explanation honed over Treiman's distinguished career as a physicist and teacher, The Odd Quantum is a remarkable survey of a field that changed the course of modern scientific and philosophical thought.

The marvelous microbes that made life on Earth possible and support our very existenceFor almost four billion years, microbes had the primordial oceans all to themselves. The stewards of Earth, these organisms transformed the chemistry of our planet to make it habitable for plants, animals, and us. Life's Engines takes readers deep into the microscopic world to explore how these marvelous creatures made life on Earth possible-and how human life today would cease to exist without them.Paul Falkowski looks "e;under the hood"e; of microbes to find the engines of life, the actual working parts that do the biochemical heavy lifting for every living organism on Earth. With insight and humor, he explains how these miniature engines are built-and how they have been appropriated by and assembled like Lego sets within every creature that walks, swims, or flies. Falkowski shows how evolution works to maintain this core machinery of life, and how we and other animals are veritable conglomerations of microbes.A vibrantly entertaining book about the microbes that support our very existence, Life's Engines will inspire wonder about these elegantly complex nanomachines that have driven life since its origin. It also issues a timely warning about the dangers of tinkering with that machinery to make it more "e;efficient"e; at meeting the ever-growing demands of humans in the coming century.

An entertaining and enlightening history of irrational numbers, from ancient Greece to the twenty-first centuryThe ancient Greeks discovered them, but it wasn't until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define-and why so many questions still surround them. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.

Intended for all readers--including magicians, detectives, musicians, orthopedic surgeons, and anthropologists--this book offers a thorough account of that most intriguing and most human of appendages: the hand. In this illustrated work, John Napier explores a wide range of absorbing subjects such as fingerprints, handedness, gestures, fossil remains, and the making and using of tools.

On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history--one that would confound thousands of puzzlers for more than a century. This is the amazing story of how the "e;map problem"e; was solved. The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron. It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm. Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map. This new edition features many color illustrations. It also includes a new foreword by Ian Stewart on the importance of the map problem and how it was solved.

"The most remarkable history of biology that has ever been written."--Michel Foucault Nobel Prize-winning scientist François Jacob's The Logic of Life is a landmark book in the history of biology and science. Focusing on heredity, which Jacob considers the fundamental feature of living things, he shows how, since the sixteenth century, the scientific understanding of inherited traits has moved not in a linear, progressive way, from error to truth, but instead through a series of frameworks. He reveals how these successive interpretive approaches--focusing on visible structures, internal structures (especially cells), evolution, genes, and DNA and other molecules--each have their own power but also limitations. Fundamentally challenging how the history of biology is told, much as Thomas Kuhn's Structure of Scientific Revolutions did for the history of science as a whole, The Logic of Life has greatly influenced the way scientists and historians view the past, present, and future of biology.

"I found Gordon's writing style fascinating; his book reads like a novel, and the technical content is superb."--Enoch J. Durbin, Princeton University

This classic work in the philosophy of physical science is an incisive and readable account of the scientific method. Pierre Duhem was one of the great figures in French science, a devoted teacher, and a distinguished scholar of the history and philosophy of science. This book represents his most mature thought on a wide range of topics.

Using shells to explore major areas of biology, the author of this celebration of the shell examines such issues as the evolution of shells and their function. He demonstrates how shells provide insight into the lives of animals of this era, as well as those of the distant past.

A thrilling tour of the sea's most extreme species, written by one of the world's leading marine scientistsThe ocean teems with life that thrives under difficult situations in unusual environments. The Extreme Life of the Sea takes readers to the absolute limits of the ocean world-the fastest and deepest, the hottest and oldest creatures of the oceans. It dives into the icy Arctic and boiling hydrothermal vents-and exposes the eternal darkness of the deepest undersea trenches-to show how marine life thrives against the odds. This thrilling book brings to life the sea's most extreme species, and tells their stories as characters in the drama of the oceans. Coauthored by Stephen Palumbi, one of today's leading marine scientists, The Extreme Life of the Sea tells the unforgettable tales of some of the most marvelous life forms on Earth, and the challenges they overcome to survive. Modern science and a fluid narrative style give every reader a deep look at the lives of these species.The Extreme Life of the Sea shows you the world's oldest living species. It describes how flying fish strain to escape their predators, how predatory deep-sea fish use red searchlights only they can see to find and attack food, and how, at the end of her life, a mother octopus dedicates herself to raising her batch of young. This wide-ranging and highly accessible book also shows how ocean adaptations can inspire innovative commercial products-such as fan blades modeled on the flippers of humpback whales-and how future extremes created by human changes to the oceans might push some of these amazing species over the edge.An enhanced edition is also available and includes eleven videos.

What is life? Is it just the biologically familiar - birds, trees, snails, people - or is it an infinitely complex set of patterns that a computer could simulate? This book outlines many of the challenges and controversies involved in the dynamic and curious science of artificial life.

Astronomers believe that a supernova is a massive explosion signaling the death of a star, causing a cosmic recycling of the chemical elements and leaving behind a pulsar, black hole, or nothing at all. This book tells how early astronomers identified supernovae, and how later scientists came to their current understanding.

Presenting the story of Poincare's work, this book traces the history of attempts to solve the problems of celestial mechanics posed in Isaac Newton's "Principia" in 1686. It introduces the people whose ideas led to the field called nonlinear dynamics.

Chemists have created superconducting ceramics for brain scanners, designed liquid crystal flat screens for televisions and watch displays, and made fabrics that change color while you wear them. This book lets the lay reader into the world of modern chemistry.

Do genes explain life? Can advances in evolutionary and molecular biology account for what we look like, how we behave, and why we die? This intervention into biological thinking argues that such genetic reductionism has limits. It shows how an understanding of the self-organizing patterns of networks is necessary for making sense of nature.

The concept of symmetry has widespread manifestations and many diverse applications-from architecture to mathematics to science. Yet, as twentieth-century physics has revealed, symmetry has a special, central role in nature, one that is occasionally and enigmatically violated. Fearful Symmetry brings the incredible discoveries of the juxtaposition of symmetry and asymmetry in contemporary physics within everyone's grasp. A. Zee, a distinguished physicist and skillful expositor, tells the exciting story of how contemporary theoretical physicists are following Einstein in their search for the beauty and simplicity of Nature. Animated by a sense of reverence and whimsy, Fearful Symmetry describes the majestic sweep and accomplishments of twentieth-century physics-one of the greatest chapters in the intellectual history of humankind.

Can virtuous behavior be explained by nature, and not by human rational choice? "e;It's the animal in us,"e; we often hear when we've been bad. But why not when we're good? Primates and Philosophers tackles this question by exploring the biological foundations of one of humanity's most valued traits: morality. In this provocative book, renowned primatologist Frans de Waal argues that modern-day evolutionary biology takes far too dim a view of the natural world, emphasizing our "e;selfish"e; genes and reinforcing our habit of labeling ethical behavior as humane and the less civilized as animalistic. Seeking the origin of human morality not in evolution but in human culture, science insists that we are moral by choice, not by nature. Citing remarkable evidence based on his extensive research of primate behavior, de Waal attacks "e;Veneer Theory,"e; which posits morality as a thin overlay on an otherwise nasty nature. He explains how we evolved from a long line of animals that care for the weak and build cooperation with reciprocal transactions. Drawing on Darwin, recent scientific advances, and his extensive research of primate behavior, de Waal demonstrates a strong continuity between human and animal behavior. He probes issues such as anthropomorphism and human responsibilities toward animals. His compelling account of how human morality evolved out of mammalian society will fascinate anyone who has ever wondered about the origins and reach of human goodness. Based on the Tanner Lectures de Waal delivered at Princeton University's Center for Human Values in 2004, Primates and Philosophers includes responses by the philosophers Peter Singer, Christine M. Korsgaard, and Philip Kitcher and the science writer Robert Wright. They press de Waal to clarify the differences between humans and other animals, yielding a lively debate that will fascinate all those who wonder about the origins and reach of human goodness.

What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area? By combining the mathematical history of extrema with contemporary examples, Paul J. Nahin answers these intriguing questions and more in this engaging and witty volume. He shows how life often works at the extremes--with values becoming as small (or as large) as possible--and how mathematicians over the centuries have struggled to calculate these problems of minima and maxima. From medieval writings to the development of modern calculus to the current field of optimization, Nahin tells the story of Dido's problem, Fermat and Descartes, Torricelli, Bishop Berkeley, Goldschmidt, and more. Along the way, he explores how to build the shortest bridge possible between two towns, how to shop for garbage bags, how to vary speed during a race, and how to make the perfect basketball shot. Written in a conversational tone and requiring only an early undergraduate level of mathematical knowledge, When Least Is Best is full of fascinating examples and ready-to-try-at-home experiments. This is the first book on optimization written for a wide audience, and math enthusiasts of all backgrounds will delight in its lively topics.

In It's About Time, N. David Mermin asserts that relativity ought to be an important part of everyone's education--after all, it is largely about time, a subject with which all are familiar. The book reveals that some of our most intuitive notions about time are shockingly wrong, and that the real nature of time discovered by Einstein can be rigorously explained without advanced mathematics. This readable exposition of the nature of time as addressed in Einstein's theory of relativity is accessible to anyone who remembers a little high school algebra and elementary plane geometry. The book evolved as Mermin taught the subject to diverse groups of undergraduates at Cornell University, none of them science majors, over three and a half decades. Mermin's approach is imaginative, yet accurate and complete. Clear, lively, and informal, the book will appeal to intellectually curious readers of all kinds, including even professional physicists, who will be intrigued by its highly original approach.

Explores the complex problem of mental life. This book argues that the roots of creativity for lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness.