The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination.

    Increasingly, the decisions that affect our lives--where we go to school, whether we can get a job or a loan, how much we pay for health insurance--are being made not by humans, but by machines. In theory, this should lead to greater fairness: Everyone is judged according to the same rules.But as mathematician and data scientist Cathy O'Neil reveals, the mathematical models being used today are unregulated and uncontestable, even when they're wrong. Most troubling, they reinforce discrimination--propping up the lucky, punishing the downtrodden, and undermining our democracy in the process.

    An impassioned critique of K 12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. "Measurement "offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can do the math in a way that brings emotional and aesthetic rewards. "Measurement" is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work."

    The insurance industry’s actuarial tables and the gambler’s roulette wheel both yield to the tools of probability theory. Most situations in life, however, involve a deeper kind of uncertainty, a radical uncertainty for which historical data provide no useful guidance to future outcomes. Radical uncertainty concerns events whose determinants are insufficiently understood for probabilities to be known or forecasting possible. Before President Barack Obama made the fateful decision to send in the Navy Seals, his advisers offered him wildly divergent estimates of the odds that Osama bin Laden would be in the Abbottabad compound. In 2000, no one—not least Steve Jobs—knew what a smartphone was; how could anyone have predicted how many would be sold in 2020? And financial advisers who confidently provide the information required in the standard retirement planning package—what will interest rates, the cost of living, and your state of health be in 2050?—demonstrate only that their advice is worthless.The limits of certainty demonstrate the power of human judgment over artificial intelligence. In most critical decisions there can be no forecasts or probability distributions on which we might sensibly rely. Instead of inventing numbers to fill the gaps in our knowledge, we should adopt business, political, and personal strategies that will be robust to alternative futures and resilient to unpredictable events. Within the security of such a robust and resilient reference narrative, uncertainty can be embraced, because it is the source of creativity, excitement, and profit.

    What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences.This sixtieth anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and reviews and articles on the book that appeared at the time of its original publication in the New York Times, tthe American Economic Review, and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come.

    There was a time when the stock market was a mechanism for growing businesses to raise money, playing a large role in the industrial revolution-boosting America to a global superpower. Today the stock market has morphed into a high-tech system of fluctuating arbitrary numbers which are used by individuals and industries alike to find profit opportunities by placing bets, masqueraded as sophisticated financial maneuvers with fancy labels and acronyms. Nowhere is this more evident than with the tendencies observed today. There is a shocking trend by today's Millennial generation to shamelessly and unapologetically find ways to use the stock market to place very high-risk bets. And unlike formal Wall Street investment institutions, these gamblers, of sorts, don't attempt to disguise the game: they are proud to call Wall Street a casino. Jaime Rogozinski combs through various elements of how reckless investors play Wall Street similar to a casino. He illustrates these often in playful ways, using entertaining and compelling real-world anecdotes. His stories are taken straight from Reddit's r/wallstreetbets community which Jaime founded in 2012, and currently has more than 800,000 followers in addition to 3 million unique visitors a month. WallStreetBets is a forum based gathering where people are notoriously known for taking a brazen and public approach at gambling with the stock market.

    In this work, eminent pollster John Zogby uses literally thousands of polls conducted over the past five decades to reveal a new and developing American character, one that will influence everything from the politicians they vote for to the goods and services they buy to the way they conduct our daily lives.

    Peter Bernstein has written a comprehensive history of man's efforts to understand risk and probability, beginning with early gamblers in ancient Greece, continuing through the 17th-century French mathematicians Pascal and Fermat and up to modern chaos theory. Along the way he demonstrates that understanding risk underlies everything from game theory to bridge-building to winemaking.

    In A Brief History of Mathematical Thought, Luke Heaton explores how the language of mathematics has evolved over time, enabling new technologies and shaping the way people think. From stone-age rituals to algebra, calculus, and the concept of computation, Heaton shows the enormous influence of mathematics on science, philosophy and the broader human story. The book traces the fascinating history of mathematical practice, focusing on the impact of key conceptual innovations. Its structure of thirteen chapters split between four sections is dictated by a combination of historical and thematic considerations. In the first section, Heaton illuminates the fundamental concept of number. He begins with a speculative and rhetorical account of prehistoric rituals, before describing the practice of mathematics in Ancient Egypt, Babylon and Greece. He then examines the relationship between counting and the continuum of measurement, and explains how the rise of algebra has dramatically transformed our world. In the second section, he explores the origins of calculus and the conceptual shift that accompanied the birth of non-Euclidean geometries. In the third section, he examines the concept of the infinite and the fundamentals of formal logic. Finally, in section four, he considers the limits of formal proof, and the critical role of mathematics in our ongoing attempts to comprehend the world around us. The story of mathematics is fascinating in its own right, but Heaton does more than simply outline a history of mathematical ideas. More importantly, he shows clearly how the history and philosophy of maths provides an invaluable perspective on human nature.

    Hand argues that extraordinarily rare events are anything but. In fact, they’re commonplace. Not only that, we should all expect to experience a miracle roughly once every month.     But Hand is no believer in superstitions, prophecies, or the paranormal. His definition of “miracle” is thoroughly rational. No mystical or supernatural explanation is necessary to understand why someone is lucky enough to win the lottery twice, or is destined to be hit by lightning three times and still survive. All we need, Hand argues, is a firm grounding in a powerful set of laws: the laws of inevitability, of truly large numbers, of selection, of the probability lever, and of near enough.     Together, these constitute Hand’s groundbreaking Improbability Principle. And together, they explain why we should not be so surprised to bump into a friend in a foreign country, or to come across the same unfamiliar word four times in one day. Hand wrestles with seemingly less explicable questions as well: what the Bible and Shakespeare have in common, why financial crashes are par for the course, and why lightning does strike the same place (and the same person) twice. Along the way, he teaches us how to use the Improbability Principle in our own lives—including how to cash in at a casino and how to recognize when a medicine is truly effective.     An irresistible adventure into the laws behind “chance” moments and a trusty guide for understanding the world and universe we live in, The Improbability Principle will transform how you think about serendipity and luck, whether it’s in the world of business and finance or you’re merely sitting in your backyard, tossing a ball into the air and wondering where it will land.

    He had gambled and won, but his volatility and reluctance to pay his debts were making him enemies. He lived the high life as only a young man would, buying luxury homes, private jets, sports cars and football teams, and splurging massively to build a horseracing empire.But Tinkler’s dreams had extended beyond even his resources, and his business model worked only in a rising market. When coal prices slumped in 2012, Tinkler had no cash flow to service his massive borrowings and no allies to help him recover. Within months he was trying desperately to stave off his creditors, large and small, and fighting to save his businesses and his fortune.In this impressive biography, leading business writer Paddy Manning tells the story of Tinkler’s meteoric rise to wealth, and captures the drama of his equally rapid downfall.‘Some might see it as a handbook on how to go from broke to billionaire in a matter of years … Others might see it as a morality tale about the canker at the heart of the consumerist, aspirational politics peddled by our leaders for the past few decades.’ —Sydney Morning Herald‘Paddy Manning’s Boganaire: The Rise and Fall of Nathan Tinkler is a rollicking tale, which works on our sense of schadenfreude.’ —Chloe Hooper, the Monthly‘Boganaire is much more than a book for readers of business literature. It provides an insight into a bigger and more important subject than Nathan Tinkler: it shows how the easy prosperity from resource riches might be changing our culture and values for the worse.’ — the Australian‘The richly detailed book lays Tinkler’s life bare, from his early days working in Newcastle’s coal mines through to his triumphant deals with Macarthur Coal and Whitehaven, and his spectacular fall from grace.’ —BRWPaddy Manning is one of Australia’s most respected journalists, with fifteen years’ experience including a decade on the business desks of the Sydney Morning Herald, the Australian Financial Review and the Australian – a period in which he was highly commended in the business category of the Walkley Awards and was a three-time category winner in the prestigious Citigroup Journalism Awards for Excellence. His first book, What the Frack: everything you need to know about coal seam gas, was published in 2012.

    . . that will get your heart pumping and your blood boiling and that will remind you why we’re in these fights—add this one to your list.” —Senator Elizabeth Warren on David Dayen’s Chain of Title Over the last forty years our choices have narrowed, our opportunities have shrunk, and our lives have become governed by a handful of very large and very powerful corporations. Today, practically everything we buy, everywhere we shop, and every service we secure comes from a heavily concentrated market.This is a world where four major banks control most of our money, four airlines shuttle most of us around the country, and four major cell phone providers connect most of our communications. If you are sick you can go to one of three main pharmacies to fill your prescription, and if you end up in a hospital almost every accessory to heal you comes from one of a handful of large medical suppliers.Dayen, the editor of the American Prospect and author of the acclaimed Chain of Title, provides a riveting account of what it means to live in this new age of monopoly and how we might resist this corporate hegemony.Through vignettes and vivid case studies Dayen shows how these monopolies have transformed us, inverted us, and truly changed our lives, at the same time providing readers with the raw material to make monopoly a consequential issue in American life and revive a long-dormant antitrust movement.

    In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory.Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples. Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated..

    In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.

    Whether it is a visualization of the Catalan numbers or an explanation of how the Fibonacci numbers occur in nature, there is something in here to delight everyone. The diagrams and pictures, many of which are in color, make this book particularly appealing and fun. A few of the discussions may be confusing to those who are not adept mathematicians; those who are may be irked that certain facts are mentioned without an accompanying proof. Nonetheless, The Book of Numbers will succeed in infecting any reader with an enthusiasm for numbers.