## Mathematics

The American standard system of measurement is a unique and odd thing to behold with its esoteric, inconsistent standards: twelve inches in a foot, three feet in a yard, sixteen ounces in a pound, one hundred pennies to the dollar. For something as elemental as counting and estimating the world around us, it seems like a confusing tool to use. So how did we end up with it?

Most of the rest of the world is on the metric system, and for a time in the 1970s America appeared ready to make the switch. Yet it never happened, and the reasons for that get to the root of who we think we are, just as the measurements are woven into the ways we think. John Marciano chronicles the origins of measurement systems, the kaleidoscopic array of standards throughout Europe and the thirteen American colonies, the combination of intellect and circumstance that resulted in the metric system's creation in France in the wake of the French Revolution, and America's stubborn adherence to the hybrid United States Customary System ever since. As much as it is a tale of quarters and tenths, it is a human drama, replete with great inventors, visionary presidents, obsessive activists, and science-loving technocrats.

Anyone who reads this inquisitive, engaging story will never read Robert Frost's line "miles to go before I sleep” or eat a foot-long sub again without wondering, Whatever happened to the metric system?

In 1913, a young unschooled Indian clerk wrote a letter to G H Hardy, begging the preeminent English mathematician's opinion on several ideas he had about numbers. Realizing the letter was the work of a genius, Hardy arranged for Srinivasa Ramanujan to come to England. Thus began one of the most improbable and productive collaborations ever chronicled. With a passion for rich and evocative detail, Robert Kanigel takes us from the temples and slums of Madras to the courts and chapels of Cambridge University, where the devout Hindu Ramanujan, "the Prince of Intuition," tested his brilliant theories alongside the sophisticated and eccentric Hardy, "the Apostle of Proof." In time, Ramanujan's creative intensity took its toll: he died at the age of thirty-two and left behind a magical and inspired legacy that is still being plumbed for its secrets today.

Biologists have long dismissed mathematics as being unable to meaningfully contribute to our understanding of living beings. Within the past ten years, however, mathematicians have proven that they hold the key to unlocking the mysteries of our world--and ourselves. In The Mathematics of Life, Ian Stewart provides a fascinating overview of the vital but little-recognized role mathematics has played in pulling back the curtain on the hidden complexities of the natural world--and how its contribution will be even more vital in the years ahead. In his characteristically clear and entertaining fashion, Stewart explains how mathematicians and biologists have come to work together on some of the most difficult scientific problems that the human race has ever tackled, including the nature and origin of life itself.

Most people are familiar with history's great equations: Newton's Law of Gravity, for instance, or Einstein's theory of relativity. But the way these mathematical breakthroughs have contributed to human progress is seldom appreciated. In In Pursuit of the Unknown, celebrated mathematician Ian Stewart untangles the roots of our most important mathematical statements to show that equations have long been a driving force behind nearly every aspect of our lives.

Using seventeen of our most crucial equations--including the Wave Equation that allowed engineers to measure a building's response to earthquakes, saving countless lives, and the Black-Scholes model, used by bankers to track the price of financial derivatives over time--Stewart illustrates that many of the advances we now take for granted were made possible by mathematical discoveries.

An approachable, lively, and informative guide to the mathematical building blocks of modern life, In Pursuit of the Unknown is a penetrating exploration of how we have also used equations to make sense of, and in turn influence, our world.

Math on Trial tells the story of ten trials in which mathematical arguments were used—and disastrously misused—as evidence. Despite years of math classes, most people (and most jurors) fail to detect even simple mathematical sophistry, resulting in such horrors as a medical expert’s faulty calculation of probabilities providing the key evidence for a British mother’s conviction for the murder of her two babies. The conviction was later overturned, but three years in prison took its toll—Sally Clark died of acute alcohol intoxication in March of 2007. Mathematicians Leila Schneps and Coralie Colmez use a wide range of examples, from a mid-19th-century dispute over wills that became a signal case in the forensic use of mathematics, to the conviction and subsequent exoneration of Amanda Knox, to show how the improper application of mathematical concepts can mean the difference between walking free and life in prison.

The cases discussed include:

-The Case of Amanda Knox (How a judge’s denial of a second DNA test may have destroyed a chance to reveal the truth about Meredith Kercher’s murder)

-The Case of Joe Sneed (How a fabricated probability framed a son for his parents’ grisly killing)

-The Case of Sally Clark (How multiplying non-independent probabilities landed an innocent mother in jail for the murder of her children)

-The Case of Janet Collins (How unjustified estimates combined with a miscalculated probability convicted an innocent couple of violent robbery)

A colorful narrative of mathematical abuse featuring such characters as Charles Ponzi, Alfred Dreyfus, Hetty Green, and Oliver Wendell Holmes, Math on Trial shows that legal expertise isn’t everything when it comes to proving a man innocent.

What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren't even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry.

In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we've never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space.

Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man's journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century's leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat's last theorem, that had seemed intractable before.

At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.

The authors, a pair of noted mathematicians, start with a discussion of divisibility and proceed to examine Gaussian primes (their determination and role in Fermat's theorem); polynomials over a field (including the Eisenstein irreducibility criterion); algebraic number fields; bases (finite extensions, conjugates and discriminants, and the cyclotomic field); and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture (concluding with discussions of Pythagorean triples, units in cyclotomic fields, and Kummer's theorem).

In addition to a helpful list of symbols and an index, a set of carefully chosen problems appears at the end of each chapter to reinforce mathematics covered. Students and teachers of undergraduate mathematics courses will find this volume a first-rate introduction to algebraic number theory.

The book is divided into four chapters, with two useful appendices, an excellent bibliography, and an index. A section of exercises enables the student to check his progress. Contents include Volterra Equations, Fredholm Equations, Symmetric Kernels and Orthogonal Systems of Functions, Types of Singular or Nonlinear Integral Equations, and more.

Professor Tricomi has presented the principal results of the theory with sufficient generality and mathematical rigor to facilitate theoretical applications. On the other hand, the treatment is not so abstract as to be inaccessible to physicists and engineers who need integral equations as a basic mathematical tool. In fact, most of the material in this book falls into an analytical framework whose content and methods are already traditional.

Nobel Laureate Eugene Wigner once wondered about “the unreasonable effectiveness of mathematics” in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that—mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is “a product of human thought that is independent of experience,” how can it so accurately describe and even predict the world around us?

Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them.

In the wake of his enormously popular books The Armchair Economist and More Sex Is Safer Sex, Steven Landsburg uses concepts from mathematics, economics, and physics to address the big questions in philosophy: What is real? What can we know? What is the difference between right and wrong? And how should we live?

Widely renowned for his lively explorations of economics, in his fourth book Landsburg branches out into mathematics and physics as well—disciplines that, like economics, the author loves for their beauty, their logical clarity, and their profound and indisputable truth—to take us on a provocative and utterly entertaining journey through the questions that have preoccupied philosophers through the ages. The author begins with the broadest possible categories—Reality and Unreality; Knowledge and Belief; Right and Wrong—and then focuses his exploration on specific concerns: from a mathematical analysis of the arguments for the existence of God; to the real meaning of the Heisenberg Uncertainty Principle and the Godel Incompleteness Theorem; to the moral choices we face in the marketplace and the voting booth.

Stimulating, illuminating, and always surprising, The Big Questions challenges readers to re-evaluate their most fundamental beliefs and reveals the relationship between the loftiest philosophical quests and our everyday lives.

Martin Nowak, one of the world’s experts on evolution and game theory, working here with bestselling science writer Roger Highfield, turns an important aspect of evolutionary theory on its head to explain why cooperation, not competition, has always been the key to the evolution of complexity. He offers a new explanation for the origin of life and a new theory for the origins of language, biology’s second greatest information revolution after the emergence of genes. SuperCooperators also brings to light his game-changing work on disease. Cancer is fundamentally a failure of the body’s cells to cooperate, Nowak has discovered, but organs are cleverly designed to foster cooperation, and he explains how this new understanding can be used in novel cancer treatments.

Nowak and Highfield examine the phenomena of reciprocity, reputation, and reward, explaining how selfless behavior arises naturally from competition; how forgiveness, generosity, and kindness have a mathematical rationale; how companies can be better designed to promote cooperation; and how there is remarkable overlap between the recipe for cooperation that arises from quantitative analysis and the codes of conduct seen in major religions, such as the Golden Rule.

In his first book written for a wide audience, this hugely influential scientist explains his cutting-edge research into the mysteries of cooperation, from the rise of multicellular life to Good Samaritans. With wit and clarity, Nowak and Highfield make the case that cooperation, not competition, is the defining human trait. SuperCooperators will expand our understanding of evolution and provoke debate for years to come.

sounds simple, doesn't it? Yet this familiar expression is a gateway

into the riotous garden of mathematics, and sends us on a journey of

exploration in the company of two inspired guides, acclaimed authors

Robert and Ellen Kaplan. With wit, verve, and clarity, they trace the

life of the Pythagorean theorem, from ancient Babylon to the present,

visiting along the way Leonardo da Vinci, Albert Einstein, President

James Garfield, and the Freemasons-not to mention the elusive Pythagoras

himself, who almost certainly did not make the statement that bears his

name.

How can a theorem have more than one proof? Why does this one have

more than two hundred-or is it four thousand? The Pythagorean theorem

has even more applications than proofs: Ancient Egyptians used it for

surveying property lines, and today astronomers call on it to measure

the distance between stars. Its generalizations are stunning-the theorem

works even with shapes on the sides that aren't squares, and not just

in two dimensions, but any number you like, up to infinity. And perhaps

its most intriguing feature of all, this tidy expression opened the door

to the world of irrational numbers, an untidy discovery that deeply

troubled Pythagoras's disciples.

Like the authors' bestselling The Nothing That Is and Chances Are . . .-hailed as "erudite and witty," "magnificent," and "exhilarating"-Hidden Harmonies makes the excitement of mathematics palpable.

The Art of the Infinite takes infinity, in its countless guises, as a touchstone for understanding mathematical thinking. Robert and Ellen Kaplan guide us through the "Republic of Numbers,” where we meet both its upstanding citizens and its more shadowy dwellers; and transport us across the plane of geometry into the unlikely realm where parallel lines meet. The journey is enriched by deft character studies of great mathematicians (and equally colorful lesser ones). And as we go deeper into infinity, we explore the most profound mystery of mathematics: Are its principles eternal truths that we discover? Or ones that we invent?

Who hasn't feared the math Minotaur in its labyrinth of abstractions? The subject can seem convoluted and forbidding. Yet to do mathematics is to wrestle with "accessible mysteries”-and Out of the Labyrinth shows how exhilarating the challenge can be. Robert and Ellen Kaplan are founders of the Math Circle, a pioneering learning program begun at Harvard in 1994 and now spreading around the world. In their classrooms students ages six to sixty have discovered mathematics as the highest form of intellectual play, while exploring topics that range from Roman numerals to quantum mechanics.

The Kaplans reveal the secrets of their highly successful approach, leading readers out of the labyrinth and into the joyous embrace of mathematics. Stocked with puzzles, colorful anecdotes, and insights from the authors' own teaching experience, Out of the Labyrinth is both an engaging and practical guide for parents and educators, and a treasure chest of mathematical discoveries. For any reader who has felt the excitement of mathematical discovery-or tried to convey it to someone else-this volume will be a delightful and valued companion.

The young Italian, Leonardo of Pisa (better known today as Fibonacci), had learned the Hindu number system when he traveled to North Africa with his father, a customs agent. The book he created was Liber abbaci, the "Book of Calculation," and the revolution that followed its publication was enormous. Arithmetic made it possible for ordinary people to buy and sell goods, convert currencies, and keep accurate records of possessions more readily than ever before. Liber abbaci's publication led directly to large-scale international commerce and the scientific revolution of the Renaissance.

Yet despite the ubiquity of his discoveries, Leonardo of Pisa remains an enigma. His name is best known today in association with an exercise in Liber abbaci whose solution gives rise to a sequence of numbers--the Fibonacci sequence--used by some to predict the rise and fall of financial markets, and evident in myriad biological structures.

One of the great math popularizers of our time, Keith Devlin recreates the life and enduring legacy of an overlooked genius, and in the process makes clear how central numbers and mathematics are to our daily lives.

While recounting memorable episodes such as "Bart the Genius” and "Homer3,” Singh weaves in mathematical stories that explore everything from p to Mersenne primes, Euler's equation to the unsolved riddle of P v. NP; from perfect numbers to narcissistic numbers, infinity to even bigger infinities, and much more. Along the way, Singh meets members of The Simpsons' brilliant writing team-among them David X. Cohen, Al Jean, Jeff Westbrook, and Mike Reiss-whose love of arcane mathematics becomes clear as they reveal the stories behind the episodes.

With wit and clarity, displaying a true fan's zeal, and replete with images from the shows, photographs of the writers, and diagrams and proofs, The Simpsons and Their Mathematical Secrets offers an entirely new insight into the most successful show in television history.