
A Most Elegant Equation
Euler’s Formula and the Beauty of Mathematics
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Narrado por:
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Sean Pratt
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De:
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David Stipp
Acerca de esta escucha
Bertrand Russell wrote that mathematics can exalt "as surely as poetry". This is especially true of one equation: ei(pi) + 1 = 0, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity, or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections. It ties together everything from basic arithmetic to compound interest, the circumference of a circle, trigonometry, calculus, and even infinity. In David Stipp's hands, Euler's identity becomes a contemplative stroll through the glories of mathematics. The result is an ode to this magical field.
©2017 David Stipp (P)2017 HighBridge, a division of Recorded BooksLos oyentes también disfrutaron...
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A Good Read Spoiled
- De David A. Donnelly en 12-23-16
De: David Wootton
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Intuition Pumps and Other Tools for Thinking
- De: Daniel C. Dennett
- Narrado por: Jeff Crawford
- Duración: 13 h y 22 m
- Versión completa
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Over a storied career, Daniel C. Dennett has engaged questions about science and the workings of the mind. His answers have combined rigorous argument with strong empirical grounding. And a lot of fun. Intuition Pumps and Other Tools for Thinking offers seventy-seven of Dennett’s most successful “imagination-extenders and focus-holders” meant to guide you through some of life’s most treacherous subject matter: evolution, meaning, mind, and free will.
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Loved it, but some philosophy background needed.
- De LongerILiveLessIKnow en 11-14-13
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Undeniable
- How Biology Confirms Our Intuition That Life Is Designed
- De: Douglas Axe
- Narrado por: Neil Hellegers
- Duración: 7 h y 14 m
- Versión completa
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Throughout his distinguished and unconventional career, engineer-turned-molecular-biologist Douglas Axe has been asking the questions that much of the scientific community would rather silence. Now, he presents his conclusions in this brave and pioneering book. Axe argues that the key to understanding our origin is the "design intuition" - the innate belief held by all humans that tasks we would need knowledge to accomplish can be accomplished only by someone who has that knowledge.
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Seductively Challenge what are consider facts
- De Rafael Vila en 10-08-16
De: Douglas Axe
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Uncertainty
- Einstein, Heisenberg, Bohr, and the Struggle for the Soul of Science
- De: David Lindley
- Narrado por: Robert Blumenfeld
- Duración: 7 h y 2 m
- Versión completa
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Historia
Werner Heisenberg's "uncertainty principle" challenged centuries of scientific understanding, placed him in direct opposition to Albert Einstein, and put Niels Bohr in the middle of one of the most heated debates in scientific history. Heisenberg's theorem stated that there were physical limits to what we could know about sub-atomic particles; this "uncertainty" would have shocking implications.
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fascinating insight into the real drama of physics
- De Ryan en 09-07-10
De: David Lindley
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Spooky Action at a Distance
- The Phenomenon That Reimagines Space and Time-and What It Means for Black Holes, the Big Bang, and Theories of Everything
- De: George Musser
- Narrado por: William Hughes
- Duración: 8 h y 37 m
- Versión completa
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Historia
What is space? It isn't a question that most of us normally stop to ask. Space is the venue of physics; it's where things exist, where they move and take shape. Yet over the past few decades, physicists have discovered a phenomenon that operates outside the confines of space and time. The phenomenon - the ability of one particle to affect another instantly across the vastness of space - appears to be almost magical.
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Rambling but Asks Good Questions
- De Michael en 12-19-15
De: George Musser
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Infinite Powers
- How Calculus Reveals the Secrets of the Universe
- De: Steven Strogatz
- Narrado por: Bob Souer
- Duración: 10 h y 41 m
- Versión completa
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General
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Narración:
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Historia
Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greece and bringing us right up to the discovery of gravitational waves. Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes "backwards" sometimes; how to turn the tide in the fight against AIDS.
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Not written to be read aloud
- De A Reader in Maine en 02-21-20
De: Steven Strogatz
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A Beginner’s Guide to Reality
- Exploring Our Everyday Adventures in Wonderland
- De: Jim Baggott
- Narrado por: Victor Bevine
- Duración: 9 h y 16 m
- Versión completa
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Historia
A unique fusion of philosophy and metaphysics set against the backdrop of contemporary culture. Have you ever wondered if the world is really there when you're not looking? We tend to take the reality of our world very much for granted. This book will lead you down the rabbit hole in search of something we can point to, hang our hats on, and say this is real.
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A real great listen on the nature of reality
- De Patrick Mabry, Jr. en 07-30-14
De: Jim Baggott
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To Explain the World
- The Discovery of Modern Science
- De: Steven Weinberg
- Narrado por: Tom Perkins
- Duración: 10 h y 43 m
- Versión completa
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In this rich, irreverent, and compelling history, Nobel Prize-winning physicist Steven Weinberg takes us across centuries, from ancient Miletus to medieval Baghdad and Oxford, from Plato's Academy and the Museum of Alexandria to the cathedral school of Chartres and the Royal Society of London. He shows that the scientists of ancient and medieval times not only did not understand what we understand about the world--they did not understand what there is to understand or how to understand it.
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How the world created a Newton
- De Gary en 03-02-15
De: Steven Weinberg
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The Upright Thinkers
- The Human Journey From Living in Trees to Understanding the Cosmos
- De: Leonard Mlodinow
- Narrado por: Leonard Mlodinow
- Duración: 12 h y 29 m
- Versión completa
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In this fascinating and illuminating work, Leonard Mlodinow guides us through the critical eras and events in the development of science, all of which, he demonstrates, were propelled forward by humankind's collective struggle to know. From the birth of reasoning and culture to the formation of the studies of physics, chemistry, biology, and modern-day quantum physics, we come to see that much of our progress can be attributed to simple questions - why? how? - bravely asked.
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10/10 Got What I Wanted.
- De Austin en 09-22-15
De: Leonard Mlodinow
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The Clockwork Universe
- Isaac Newton, The Royal Society, and the Birth of the Modern World
- De: Edward Dolnick
- Narrado por: Alan Sklar
- Duración: 10 h y 4 m
- Versión completa
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The Clockwork Universe is the story of a band of men who lived in a world of dirt and disease but pictured a universe that ran like a perfect machine. A meld of history and science, this book is a group portrait of some of the greatest minds who ever lived as they wrestled with natures most sweeping mysteries. The answers they uncovered still hold the key to how we understand the world.
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Calculus Ergo Modernity
- De Nelson Alexander en 07-09-11
De: Edward Dolnick
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What Is Real?
- The Unfinished Quest for the Meaning of Quantum Physics
- De: Adam Becker
- Narrado por: Greg Tremblay
- Duración: 11 h y 45 m
- Versión completa
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Historia
Every physicist agrees quantum mechanics is among humanity's finest scientific achievements. But ask what it means, and the result will be a brawl. For a century, most physicists have followed Niels Bohr's Copenhagen interpretation and dismissed questions about the reality underlying quantum physics as meaningless. A mishmash of solipsism and poor reasoning, Copenhagen endured, as Bohr's students vigorously protected his legacy, and the physics community favored practical experiments over philosophical arguments.
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Good, "light" "read"... potential caveat below...
- De James S. en 03-31-18
De: Adam Becker
Las personas que vieron esto también vieron...
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Significant Figures
- The Lives and Work of Great Mathematicians
- De: Ian Stewart
- Narrado por: Roger Clark
- Duración: 11 h y 39 m
- Versión completa
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General
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Narración:
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Historia
In Significant Figures, acclaimed mathematician Ian Stewart introduces the visionaries of mathematics throughout history. Delving into the lives of twenty-five great mathematicians, Stewart examines the roles they played in creating, inventing, and discovering the mathematics we use today. Through these short biographies, we get acquainted with the history of mathematics.
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Beware
- De Anton Kurtz en 12-08-18
De: Ian Stewart
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The Joy of x
- A Guided Tour of Math, from One to Infinity
- De: Steven Strogatz
- Narrado por: Jonathan Yen
- Duración: 6 h y 9 m
- Versión completa
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General
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Narración:
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Historia
Many people take math in high school and promptly forget much of it. But math plays a part in all of our lives all of the time, whether we know it or not. In The Joy of x, Steven Strogatz expands on his hit New York Times series to explain the big ideas of math gently and clearly, with wit, and insight.
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Great listen
- De cameron en 08-16-19
De: Steven Strogatz
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Infinite Powers
- How Calculus Reveals the Secrets of the Universe
- De: Steven Strogatz
- Narrado por: Bob Souer
- Duración: 10 h y 41 m
- Versión completa
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General
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Narración:
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Historia
Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greece and bringing us right up to the discovery of gravitational waves. Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes "backwards" sometimes; how to turn the tide in the fight against AIDS.
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Not written to be read aloud
- De A Reader in Maine en 02-21-20
De: Steven Strogatz
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A Mathematician's Lament
- How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form
- De: Paul Lockhart
- Narrado por: Matthew Josdal
- Duración: 2 h y 58 m
- Versión completa
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General
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Narración:
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Historia
A brilliant research mathematician reveals math to be a creative art form on par with painting, poetry, and sculpture, and rejects the standard anxiety-producing teaching methods used in most schools today. Witty and accessible, Paul Lockhart's controversial approach will provoke spirited debate among educators and parents alike, altering the way we think about math forever.
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Beautiful call to action
- De Ben Haley en 03-11-25
De: Paul Lockhart
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Journey to the Edge of Reason
- The Life of Kurt Gödel
- De: Stephen Budiansky
- Narrado por: Bob Souer
- Duración: 8 h y 52 m
- Versión completa
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General
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Narración:
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Historia
Nearly a hundred years after its publication, Kurt Gödel's famous proof that every mathematical system must contain propositions that are true - yet never provable - continues to unsettle mathematics, philosophy, and computer science. Yet unlike Einstein, with whom he formed a warm and abiding friendship, Gödel has long escaped all but the most casual scrutiny of his life.
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Interesting story of a great mathematician
- De James Orlin en 04-28-22
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Mathematics
- A Very Short Introduction
- De: Timothy Gowers
- Narrado por: Craig Jessen
- Duración: 5 h y 21 m
- Versión completa
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General
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Narración:
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Historia
The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and listeners of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought.
De: Timothy Gowers
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Significant Figures
- The Lives and Work of Great Mathematicians
- De: Ian Stewart
- Narrado por: Roger Clark
- Duración: 11 h y 39 m
- Versión completa
-
General
-
Narración:
-
Historia
In Significant Figures, acclaimed mathematician Ian Stewart introduces the visionaries of mathematics throughout history. Delving into the lives of twenty-five great mathematicians, Stewart examines the roles they played in creating, inventing, and discovering the mathematics we use today. Through these short biographies, we get acquainted with the history of mathematics.
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Beware
- De Anton Kurtz en 12-08-18
De: Ian Stewart
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The Joy of x
- A Guided Tour of Math, from One to Infinity
- De: Steven Strogatz
- Narrado por: Jonathan Yen
- Duración: 6 h y 9 m
- Versión completa
-
General
-
Narración:
-
Historia
Many people take math in high school and promptly forget much of it. But math plays a part in all of our lives all of the time, whether we know it or not. In The Joy of x, Steven Strogatz expands on his hit New York Times series to explain the big ideas of math gently and clearly, with wit, and insight.
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Great listen
- De cameron en 08-16-19
De: Steven Strogatz
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Infinite Powers
- How Calculus Reveals the Secrets of the Universe
- De: Steven Strogatz
- Narrado por: Bob Souer
- Duración: 10 h y 41 m
- Versión completa
-
General
-
Narración:
-
Historia
Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greece and bringing us right up to the discovery of gravitational waves. Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes "backwards" sometimes; how to turn the tide in the fight against AIDS.
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Not written to be read aloud
- De A Reader in Maine en 02-21-20
De: Steven Strogatz
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A Mathematician's Lament
- How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form
- De: Paul Lockhart
- Narrado por: Matthew Josdal
- Duración: 2 h y 58 m
- Versión completa
-
General
-
Narración:
-
Historia
A brilliant research mathematician reveals math to be a creative art form on par with painting, poetry, and sculpture, and rejects the standard anxiety-producing teaching methods used in most schools today. Witty and accessible, Paul Lockhart's controversial approach will provoke spirited debate among educators and parents alike, altering the way we think about math forever.
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Beautiful call to action
- De Ben Haley en 03-11-25
De: Paul Lockhart
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Journey to the Edge of Reason
- The Life of Kurt Gödel
- De: Stephen Budiansky
- Narrado por: Bob Souer
- Duración: 8 h y 52 m
- Versión completa
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General
-
Narración:
-
Historia
Nearly a hundred years after its publication, Kurt Gödel's famous proof that every mathematical system must contain propositions that are true - yet never provable - continues to unsettle mathematics, philosophy, and computer science. Yet unlike Einstein, with whom he formed a warm and abiding friendship, Gödel has long escaped all but the most casual scrutiny of his life.
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Interesting story of a great mathematician
- De James Orlin en 04-28-22
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Mathematics
- A Very Short Introduction
- De: Timothy Gowers
- Narrado por: Craig Jessen
- Duración: 5 h y 21 m
- Versión completa
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General
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Narración:
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Historia
The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and listeners of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought.
De: Timothy Gowers
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A Brief History of Mathematics
- Complete Series
- De: Marcus du Sautoy
- Narrado por: Marcus du Sautoy
- Duración: 2 h y 13 m
- Versión completa
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General
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Historia
This 10-part history of mathematics reveals the personalities behind the calculations: the passions and rivalries of mathematicians struggling to get their ideas heard. Professor Marcus du Sautoy shows how these masters of abstraction find a role in the real world and proves that mathematics is the driving force behind modern science.
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not a book
- De bob en 06-22-21
De: Marcus du Sautoy
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Paradox
- The Nine Greatest Enigmas in Physics
- De: Jim Al-Khalili
- Narrado por: Matthew Waterson
- Duración: 6 h y 54 m
- Versión completa
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Throughout history, scientists have come up with theories and ideas that just don't seem to make sense. These we call paradoxes. The paradoxes Al-Khalili offers are drawn chiefly from physics and astronomy and represent those that have stumped some of the finest minds. With elegant explanations that bring the listener inside the mind of those who've developed them, Al-Khalili helps us to see that, in fact, paradoxes can be solved if seen from the right angle.
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Almost Useless
- De Michael en 06-19-19
De: Jim Al-Khalili
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No Calculator? No Problem!
- Mastering Mental Math
- De: Art Benjamin, The Great Courses
- Narrado por: Art Benjamin
- Duración: 5 h y 46 m
- Grabación Original
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General
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Historia
No Calculator? No Problem! Mastering Mental Math , award-winning professor of mathematics and celebrated “mathemagician” Arthur T. Benjamin delivers 10 fun-filled lessons on how to do math in your head with confidence, accuracy, and speed - sometimes faster than a calculator. By the end of Professor Benjamin’s lessons, you’ll be able to add, subtract, multiply, and divide numbers faster than ever before. And with your newfound skills, you’ll soon find yourself amazing other people and, perhaps more important, yourself.
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Excellent but need PDF
- De Majeed en 10-15-19
De: Art Benjamin, y otros
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Faraday, Maxwell, and the Electromagnetic Field
- How Two Men Revolutionized Physics
- De: Nancy Forbes, Basil Mahon
- Narrado por: Patrick Lawlor
- Duración: 10 h y 15 m
- Versión completa
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General
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Historia
Two of the boldest and most creative scientists of all time were Michael Faraday (1791-1867) and James Clerk Maxwell (1831-1879). This is the story of how these two men - separated in age by 40 years - discovered the existence of the electromagnetic field and devised a radically new theory which overturned the strictly mechanical view of the world that had prevailed since Newton's time.
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Amazing narration of an incredibly well told story
- De Paul de Jong en 03-01-21
De: Nancy Forbes, y otros
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The Joy of Numbers
- De: Dr. Arthur Benjamin, The Great Courses
- Narrado por: Art Benjamin
- Duración: 5 h y 2 m
- Grabación Original
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General
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Historia
Numbers. Like the alphabet, they’re one of the most elementary of concepts learned and memorized at a young age; but outside of figuring out tips and taxes, you probably haven’t given much thought to them since then. To a mathematician, every number has its own unique properties and personality - and when studied, played with, and manipulated, numbers can actually be tons of fun.
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Very entertaining
- De Mariam en 03-19-22
De: Dr. Arthur Benjamin, y otros
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Why Does E=MC2 and Why Should We Care
- De: Brian Cox, Jeff Forshaw
- Narrado por: Jeff Forshaw
- Duración: 7 h y 3 m
- Versión completa
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Historia
In one of the most exciting and accessible explanations of The Theory of Relativity in recent years, Professors Brian Cox and Jeff Forshaw go on a journey to the frontier of 21st century science to consider the real meaning behind the iconic sequence of symbols that make up Einstein's most famous equation, exploring the principles of physics through everyday life.
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Needs a few Diagrams
- De Roy en 06-13-11
De: Brian Cox, y otros
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Conquering the Electron
- The Geniuses, Visionaries, Egomaniacs, and Scoundrels Who Built Our Electronic Age
- De: Derek Cheung, Eric Brach
- Narrado por: Eric Jason Martin
- Duración: 14 h y 8 m
- Versión completa
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General
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Historia
Want to know how AT&T's Bell Labs developed semiconductor technology - and how its leading scientists almost came to blows in the process? Want to understand how radio and television work - and why RCA drove their inventors to financial ruin and early graves? Conquering the Electron offers these stories and more, presenting each revolutionary technological advance right alongside blow-by-blow personal battles that all too often took place.
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Tech, science, engineering & the people behind it.
- De James S. en 05-29-20
De: Derek Cheung, y otros
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An Introduction to Information Theory
- Symbols, Signals and Noise
- De: John R. Pierce
- Narrado por: Kyle Tait
- Duración: 10 h y 12 m
- Versión completa
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Historia
Behind the familiar surfaces of the telephone, radio, and television lies a sophisticated and intriguing body of knowledge known as information theory. This is the theory that has permitted the rapid development of all sorts of communication, from color television to the clear transmission of photographs from the vicinity of Jupiter. Even more revolutionary progress is expected in the future.
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Not bad, but...
- De Jane Doe en 06-26-20
De: John R. Pierce
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The Complete (Short) Guide to Absolutely Everything
- Adventures in Math and Science
- De: Adam Rutherford, Hannah Fry
- Narrado por: Hannah Fry, Adam Rutherford
- Duración: 7 h y 2 m
- Versión completa
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General
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Historia
Geneticist Adam Rutherford and mathematician Hannah Fry guide listeners through time and space, through our bodies and brains, showing how emotions shape our view of reality, how our minds tell us lies, and why a mostly bald and curious ape decided to begin poking at the fabric of the universe.
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Humour and understandability.
- De Chris B en 09-08-24
De: Adam Rutherford, y otros
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Beyond Weird
- De: Philip Ball
- Narrado por: Jonathan Cowley
- Duración: 9 h y 20 m
- Versión completa
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An exhilarating tour of the contemporary quantum landscape, Beyond Weird is a book about what quantum physics really means - and what it doesn't. Science writer Philip Ball offers an up-to-date, accessible account of the quest to come to grips with the most fundamental theory of physical reality, and to explain how its counterintuitive principles underpin the world we experience.
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A difficult listen
- De Ray en 03-17-19
De: Philip Ball
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A Mind at Play
- How Claude Shannon Invented the Information Age
- De: Rob Goodman, Jimmy Soni
- Narrado por: Jonathan Yen
- Duración: 11 h y 51 m
- Versión completa
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Claude Shannon was a tinkerer, a playful wunderkind, a groundbreaking polymath, and a digital pioneer whose insights made the Information Age possible. He constructed fire-breathing trumpets and customized unicycles, outfoxed Vegas casinos, and built juggling robots, but he also wrote the seminal text of the Digital Revolution. That work allowed scientists to measure and manipulate information as objectively as any physical object. His work gave mathematicians and engineers the tools to bring that world to pass.
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I wanted more information about Information Theory
- De Bonny en 05-08-18
De: Rob Goodman, y otros
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Just Six Numbers
- The Deep Forces That Shape the Universe
- De: Martin J. Rees
- Narrado por: John Curless
- Duración: 6 h y 44 m
- Versión completa
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There are deep connections between stars and atoms, between the cosmos and the microworld. Just six numbers, imprinted in the "Big Bang", determine the essential features of our entire physical world. Moreover, cosmic evolution is astonishingly sensitive to the values of these numbers. If any one of them were "untuned", there could be no stars and no life. This realization offers a radically new perspective on our universe, our place in it, and the nature of physical laws.
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Old Fine-Tuning Book
- De Michael en 12-16-18
De: Martin J. Rees
Lo que los oyentes dicen sobre A Most Elegant Equation
Con calificación alta para:
Reseñas - Selecciona las pestañas a continuación para cambiar el origen de las reseñas.
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Ejecución
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- Kathleen Stevens
- 05-02-22
Bravo!
A beautifully accessible book showing the connections between five seemingly unrelated numbers. Highly recommended for anyone who enjoys math, even if they’ve forgotten most of it.
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- Anyone
- 07-27-22
High level math
The theme of the book is best absorbed in the first few chapters and the closing one. The middle would be better read, I'm sure, rather than listened to in an audiobook. Still, this book opens our minds to the ideas of Euler and the potential of mathematics. The narrator is fantastic.
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- Talistan
- 12-19-18
super. amazing insight into the equation.
glad i bought it. learned something new and it was great siding a drive. audible
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- Amy
- 10-30-22
Unfortunately, the math was still above my head.
I wanted to enjoy this book more than I did. Unfortunately the math was still love of my head and I couldn’t make sense I have some things he tried to explain. But I really enjoyed many of the human aspect he talked about such as the stories of various peoples lives.
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- JH
- 11-16-23
Too dense for Audio only
I love the story and the narration. Very catchy.
However, this audiobook needs an accompanying PDF to follow the mathematical derivations. Euler may have been able to do all that math in his head and visualize the formulae, but I am not Euler and I cannot visualize and follow the derivations at the narrated speed. And rewinding the audio 5 times to write down the formulae by hand so as to retrace the steps of their derivations cannot be the purpose.
The absence of that PDF is the only reason why I am giving this only 3 stars overall. Otherwise, the book is fascinating.
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- Jeb
- 02-17-24
Best and most engaging book on math I've read
Stipp does a great job of explaining the background and relationship between five of the most interesting numbers in math (i, pi, e, 0 and 1). He doesn't overdo the biography of Euler, but sketches a very happy genius all the same. Highly recommended.
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- Andreas Zenker
- 06-05-18
Great math book to listen to
Not all math books translate well to audio but in this case it was a really enjoyable listen.
Writing and narration were quite good and the math was approachable even in this format.
The only thing missing is a pdf of the formulas and images from the last few chapters where I found myself wanting to work along with the author. This is not recommended while driving ;) but I may just re-listen to that section with a pad of paper so I can scratch out the ideas on paper
Great book!
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esto le resultó útil a 20 personas
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- Richard
- 07-24-18
Not well suited to audiobook
I don't think that audiobooks are a suitable medium for the algebraic manipulation of trigonometric expressions.
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Historia
- Aaron
- 02-05-22
Hard to follow math in audio
hard to follow all the math in an audiobook. great story, good history, but too many verbal equations.
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Historia
- Kindle Customer
- 04-09-18
Good treatment of the subject
Overall a good journey through Euler's formula, with nice side trips through intriging relevant stories that were integral to Euler's great, beautiful equation. However, the author, who is a writer who once majored in math, epresses his sometimes fleeting grasp on the subject with the literary awkwardness. He has a journalist's understanding of mathematics, and the journalistic skill of a mathematician.
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esto le resultó útil a 25 personas