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Infinite Powers (How Calculus Reveals the Secrets of the Universe)

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English · United States of America · Houghton Mifflin Harcourt · 2 Nisan 2019 · Karton kapak · 9781328879981

“Marvelous . . . an array of witty and astonishing stories . . . to illuminate how calculus has helped bring into being our contemporary world.”—The Washington Post
From preeminent math personality and author of The Joy of x, a brilliant and endlessly appealing explanation of calculus – how it works and why it makes our lives immeasurably better.
Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket.
Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous.
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 (a phenomenon predicted by calculus). 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 make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS.
As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.

Given that 90 is a little less than 100, and 100 equals 10², it seems like 90 should equal 10 raised to some number slightly less than 2. But raised to what number, exactly? Logarithms were invented to answer such questions.

Sayfa 138

In contrast to a mild power function x or x², an exponential function like 2^x or 10^x describes a much more explosive kind of growth, a growth that snowballs and feeds on itself. Instead of adding a constant increment at each step as in linear growth, exponential growth involves multiplying by a constant factor.

Sayfa 134

The pivotal moment in the history of calculus occurred in the middle of the seventeenth century when the mysteries of curves, motion, and change collided on a two-dimensional grid, the xy plane of Fermat and Descartes. Back then, Fermat and Descartes had no idea what a versatile tool they had created. They intended the xy plane as a tool for pure mathematics. Yet from the start, it too was a crossroads of sorts, a place where equations met curves, algebra met geometry, and the mathematics of the East met that of the West. Then, in the next generation, Isaac Newton built on their work as well as on the work of Galileo and Kepler and brought geometry and physics together in a great synthesis. Newton's spark set off the fire that lit the Enlightenment and caused a revolution in Western science and mathematics.

Sayfa 131

Fermat had applied his embryonic version of differential calculus to physics. No one had ever done that before. And in so doing, he showed that light travels in the most efficient way--not the most direct way, but the fastest. Of all the possible paths light can take, it somehow knows, or behaves as if it knows, how to get from here to there as quickly as possible. This was an early clue that calculus was somehow built into the operating system of the universe.

Sayfa 126

Where there is circular motion, there are sine waves.

Sayfa 118

In those days, when algebra was still all about word problems, solutions were given as recipes, step-by-step routes to answers,as elucidated in the famous textbook by Muhammad Ibn Musa al-Khwarizmi (c. 780–850 CE), whose last name lives on in the step-by-step procedures called algorithms.

Sayfa 103

In particular, algebra came from Asia and the Middle East. Its name derives from the Arabic word al-jabr, meaning “restoration” or “the reunion of broken parts.”

Sayfa 102

Calculus, for its part, uses the infinite to study the finite, the unlimited to study the limited, and the straight to study the curved. The Infinity Principle is the key to unlocking the mystery of curves, and it arose here first, in the mystery of pi.

Sayfa 48

In ancient Egypt, the measurement of lines and angles was of paramount importance. Each year surveyors had to redraw the boundaries of farmers’ fields after the summer flooding of the Nile washed the borderlines away. That activity later gave its name to the study of shape in general: geometry, from the Greek gē, “earth,” and metrēs, “measurer.”

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Coping with curved shapes means coping with infinity, one way or another.

Sayfa 62

Before Galileo, Aristotle had proposed that heavy objects fall because they are seeking their natural place at the center of the cosmos. Galileo thought these were empty words. Instead of speculating about why things fell, he wanted to quantify how they fell.

Sayfa 79

(...) the ancient Greeks had defined ellipses as the oval-shaped curves formed by cutting through a cone with a plane at a shallow angle, less steep than the slope of the conical surface itself.

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