16-minute read

Audio available

In this book, Ian Stewart focuses on 17 famous equations in mathematics and physics history, highlighting their impact on society. Stewart gives a brief history of the wonders of scientific discovery, and peppers it with vivid examples and anecdotes.

- Anyone interested in how mathematical ideas affect science and society
- Anyone who’s always been scared of complicated-looking formulas
- Anyone fascinated by the beauty and hidden power of formulas

Stewart is professor emeritus of mathematics at the University of Warwick, England, and fellow of the Royal Society. His main mathematical interests are Lie algebras and the theory of dynamical systems and its connection to chaos theory and biology. He is the author of several popular science books including *Does* *God* *Play* *Dice?* *The* *New* *Mathematics* *of* *Chaos* and *Why* *Beauty* *Is* *Truth:* *A* *History* *of* *Symmetry*.

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- Contains 10 key ideas

On the surface, the Pythagorean theorem looks simple. Made up of the equation a²+b²=c², it explains the relationship between the three sides of a right-angled triangle where a and b represent the sides that form the right angle and c is the longest side.

Despite its simplicity, the theorem was a vital first step in developing the geometric techniques needed for making accurate maps: *triangulation*. Since all polygons can be cut into triangles, and all triangles can be cut into two right-angled triangles, the equation allows us to calculate the sides and angles of any polygon. In mapmaking, the area being surveyed is covered with a network of triangles, which allows us to measure distances and angles between stretches of land.

Because of technological advancements in measuring tools, explicit triangulation is no longer used. However, it’s still there in the methods used to deduce locations from the satellite data.

Generalizations of Pythagoras’s theorem also enable us to measure the shape of the universe. The Earth has the shape of a sphere, sure, but what about the universe? Finding the answer to this question seems like an impossible task: after all, we can’t step outside to see what shape it is.

However, thanks to generalizations of Pythagoras’s theorem, there is something else we can do.

Pythagoras’s theorem applies to triangles in a *flat* plane. But generalizations of it can apply to triangles in spaces that are *curved*. For example, applying triangles onto a sphere makes it possible to measure our planet. But there’s more: these generalizations can also tell us *how* curved the spaces are. So by measuring triangles within space, we can try to compute its curvature.

- The Pythagorean theorem shaped our understanding of geography and Earth’s place in the solar system.
- Euler’s formula revolutionized the world of math – and helps us understand DNA.
- The normal distribution is one of the most important tools for analyzing social, medical and scientific data.
- The Navier-Stokes equation provides solutions for innumerable scientific and technological problems.
- Newton’s law of gravitation enables predictions of the solar system and space mission orbits.
- Einstein’s theory of relativity changed our view of space and time – and our everyday lives.
- Schrödinger’s equation changed our notion of matter and is ubiquitous in modern electronics.
- Chaos theory pervades most areas of science, and understanding chaos can help in everyday life.
- The Black-Scholes equation led to the massive growth of the financial sector – and the risks of financial crisis.
- Final summary