e: The Story of a Number

Princeton Science Library

Book 41
Princeton University Press
6
Free sample

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.
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About the author

Eli Maor is the author of Beautiful Geometry (with Eugen Jost), Venus in Transit, Trigonometric Delights, To Infinity and Beyond, and The Pythagorean Theorem: A 4,000-Year History (all Princeton).
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4.2
6 total
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Additional Information

Publisher
Princeton University Press
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Published on
Oct 12, 2011
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Pages
248
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ISBN
9781400832347
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Language
English
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Genres
Mathematics / History & Philosophy
Science / General
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Content Protection
This content is DRM protected.
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Available on Android devices
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Eligible for Family Library

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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 "imaginary numbers"--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 "numbers" in all of mathematics.

Some images inside the book are unavailable due to digital copyright restrictions.

The human impact on Earth's climate is often treated as a hundred-year issue lasting as far into the future as 2100, the year in which most climate projections cease. In The Long Thaw, David Archer, one of the world’s leading climatologists, reveals the hard truth that these changes in climate will be "locked in," essentially forever.

If you think that global warming means slightly hotter weather and a modest rise in sea levels that will persist only so long as fossil fuels hold out (or until we decide to stop burning them), think again. In The Long Thaw, David Archer predicts that if we continue to emit carbon dioxide we may eventually cancel the next ice age and raise the oceans by 50 meters. A human-driven, planet-wide thaw has already begun, and will continue to impact Earth’s climate and sea level for hundreds of thousands of years. The great ice sheets in Antarctica and Greenland may take more than a century to melt, and the overall change in sea level will be one hundred times what is forecast for 2100. By comparing the global warming projection for the next century to natural climate changes of the distant past, and then looking into the future far beyond the usual scientific and political horizon of the year 2100, Archer reveals the hard truths of the long-term climate forecast.

Archer shows how just a few centuries of fossil-fuel use will cause not only a climate storm that will last a few hundred years, but dramatic climate changes that will last thousands. Carbon dioxide emitted today will be a problem for millennia. For the first time, humans have become major players in shaping the long-term climate. In fact, a planetwide thaw driven by humans has already begun. But despite the seriousness of the situation, Archer argues that it is still not too late to avert dangerous climate change--if humans can find a way to cooperate as never before.

Revealing why carbon dioxide may be an even worse gamble in the long run than in the short, this compelling and critically important book brings the best long-term climate science to a general audience for the first time.

With a new preface that discusses recent advances in climate science, and the impact on global warming and climate change, The Long Thaw shows that it is still not too late to avert dangerous climate change—if we can find a way to cooperate as never before.

The Extravagant Universe tells the story of a remarkable adventure of scientific discovery. One of the world's leading astronomers, Robert Kirshner, takes readers inside a lively research team on the quest that led them to an extraordinary cosmological discovery: the expansion of the universe is accelerating under the influence of a dark energy that makes space itself expand. In addition to sharing the story of this exciting discovery, Kirshner also brings the science up-to-date in a new epilogue. He explains how the idea of an accelerating universe--once a daring interpretation of sketchy data--is now the standard assumption in cosmology today.

This measurement of dark energy--a quality of space itself that causes cosmic acceleration--points to a gaping hole in our understanding of fundamental physics. In 1917, Einstein proposed the "cosmological constant" to explain a static universe. When observations proved that the universe was expanding, he cast this early form of dark energy aside. But recent observations described first-hand in this book show that the cosmological constant--or something just like it--dominates the universe's mass and energy budget and determines its fate and shape.


Warned by Einstein's blunder, and contradicted by the initial results of a competing research team, Kirshner and his colleagues were reluctant to accept their own result. But, convinced by evidence built on their hard-earned understanding of exploding stars, they announced their conclusion that the universe is accelerating in February 1998. Other lines of inquiry and parallel supernova research now support a new synthesis of a cosmos dominated by dark energy but also containing several forms of dark matter. We live in an extravagant universe with a surprising number of essential ingredients: the real universe we measure is not the simplest one we could imagine.

The impact on climate from 200 years of industrial development is an everyday fact of life, but did humankind's active involvement in climate change really begin with the industrial revolution, as commonly believed? Plows, Plagues, and Petroleum has sparked lively scientific debate since it was first published--arguing that humans have actually been changing the climate for some 8,000 years--as a result of the earlier discovery of agriculture.

The "Ruddiman Hypothesis" will spark intense debate. We learn that the impact of farming on greenhouse-gas levels, thousands of years before the industrial revolution, kept our planet notably warmer than if natural climate cycles had prevailed--quite possibly forestalling a new ice age.

Plows, Plagues, and Petroleum is the first book to trace the full historical sweep of human interaction with Earth's climate. Ruddiman takes us through three broad stages of human history: when nature was in control; when humans began to take control, discovering agriculture and affecting climate through carbon dioxide and methane emissions; and, finally, the more recent human impact on climate change. Along the way he raises the fascinating possibility that plagues, by depleting human populations, also affected reforestation and thus climate--as suggested by dips in greenhouse gases when major pandemics have occurred. While our massive usage of fossil fuels has certainly contributed to modern climate change, Ruddiman shows that industrial growth is only part of the picture. The book concludes by looking to the future and critiquing the impact of special interest money on the global warming debate. In the afterword, Ruddiman explores the main challenges posed to his hypothesis, and shows how recent investigations and findings ultimately strengthen the book's original claims.

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 "imaginary numbers"--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 "numbers" in all of mathematics.

Some images inside the book are unavailable due to digital copyright restrictions.

How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century

Music is filled with mathematical elements, the works of Bach are often said to possess a math-like logic, and Igor Stravinsky said "musical form is close to mathematics," while Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen went further, writing music explicitly based on mathematical principles. Yet Eli Maor argues that music has influenced math at least as much as math has influenced music. Starting with Pythagoras, proceeding through the work of Schoenberg, and ending with contemporary string theory, Music by the Numbers tells a fascinating story of composers, scientists, inventors, and eccentrics who played a role in the age-old relationship between music, mathematics, and the sciences, especially physics and astronomy.

Music by the Numbers explores key moments in this history, particularly how problems originating in music have inspired mathematicians for centuries. Perhaps the most famous of these problems is the vibrating string, which pitted some of the greatest mathematicians of the eighteenth century against each other in a debate that lasted more than fifty years and that eventually led to the development of post-calculus mathematics. Other highlights in the book include a comparison between meter in music and metric in geometry, complete with examples of rhythmic patterns from Bach to Stravinsky, and an exploration of a suggestive twentieth-century development: the nearly simultaneous emergence of Einstein's theory of relativity and Schoenberg's twelve-tone system.

Weaving these compelling historical episodes with Maor's personal reflections as a mathematician and lover of classical music, Music by the Numbers will delight anyone who loves mathematics and music.

The human impact on Earth's climate is often treated as a hundred-year issue lasting as far into the future as 2100, the year in which most climate projections cease. In The Long Thaw, David Archer, one of the world’s leading climatologists, reveals the hard truth that these changes in climate will be "locked in," essentially forever.

If you think that global warming means slightly hotter weather and a modest rise in sea levels that will persist only so long as fossil fuels hold out (or until we decide to stop burning them), think again. In The Long Thaw, David Archer predicts that if we continue to emit carbon dioxide we may eventually cancel the next ice age and raise the oceans by 50 meters. A human-driven, planet-wide thaw has already begun, and will continue to impact Earth’s climate and sea level for hundreds of thousands of years. The great ice sheets in Antarctica and Greenland may take more than a century to melt, and the overall change in sea level will be one hundred times what is forecast for 2100. By comparing the global warming projection for the next century to natural climate changes of the distant past, and then looking into the future far beyond the usual scientific and political horizon of the year 2100, Archer reveals the hard truths of the long-term climate forecast.

Archer shows how just a few centuries of fossil-fuel use will cause not only a climate storm that will last a few hundred years, but dramatic climate changes that will last thousands. Carbon dioxide emitted today will be a problem for millennia. For the first time, humans have become major players in shaping the long-term climate. In fact, a planetwide thaw driven by humans has already begun. But despite the seriousness of the situation, Archer argues that it is still not too late to avert dangerous climate change--if humans can find a way to cooperate as never before.

Revealing why carbon dioxide may be an even worse gamble in the long run than in the short, this compelling and critically important book brings the best long-term climate science to a general audience for the first time.

With a new preface that discusses recent advances in climate science, and the impact on global warming and climate change, The Long Thaw shows that it is still not too late to avert dangerous climate change—if we can find a way to cooperate as never before.

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