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.

https://www.stevenstrogatz.com/books

Infinite Powers ​ is​ ​ wonderful in the many ways it can be read by different readers. It is both an easy-to-read exegesis on “the importance of Calculus” for casual but curious readers (like parents and their high school children) as well as an informative and interesting historical reference for college students and instructors of mathematics. I believe that even people who (incorrectly) view themselves as “not math” people will be able to generally follow the arguments Strogatz makes and they will definitely enjoy the surprising stories he tells about the many different ways mathematical discoveries are made, and how they have changed our world.

As an applied mathematician, one of the joys of this book is the way Strogatz makes a compelling argument about how connected mathematics is to many aspects of everyday life and society itself, from cell phones to microwave ovens to the Declaration of Independence. As someone who thinks the history of mathematics should be a required course for all undergraduate mathematics majors, I am happy to see Strogatz present so many interesting details about the history of the Calculus, which ends up being a sneaky way of telling the history of mathematics, as well.

The key idea in ​ Infinite Powers ​ is presented early in the Introduction to the book as The Infinity Principle:

“To shed light on any continuous shape, object, motion, process, or phenomenon—no matter how wild and complicated it may appear—reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.”

For most people familiar with Calculus, this representation of its main idea will not be surprising. But for other readers, Strogatz does a beautiful job of summarizing and explaining this central concept of Calculus. In fact, it’s very possible that for students who have only seen Calculus from the perspective of an Advanced Placement curriculum or in a somewhat mechanical or long-forgotten college Calculus course, Strogatz’s description may be enlightening. He uses the evocative term “harnessing infinity” as a shortened version of the Infinity Principle throughout the rest of the text. One of the most interesting ways of experiencing Infinite Powers is as a history of Calculus. However, if you’re expecting a(nother) treatise on the notorious dispute between Sir Isaac Newton and Gottfried Wilhelm Leibniz about who should receive credit for its invention, Strogatz’s book will surprise you. First, he starts his history of Calculus by discussing the history of infinity, so he begins with Zeno and his paradoxes. The discussion of infinity continues with a deep dive into the work of Archimedes (described as ``the man who harnessed infinity”), all the time making it clear that the reason infinity is being harnessed is to solve specific, practical problems. Fans of Newton or Leibniz don’t need to be worried about their hero being dissed, since Strogatz includes plenty of material from each genius that validates each one’s claim to greatness. He then takes the unusual position that neither Leibniz or Newton “invented” Calculus; Strogatz’ view is that calculus (in the form of “harnessing infinity”) has been used for centuries, by Archimedes and others to solve problems and provide answers to important questions.

In the latter sections of the book, Strogatz goes on to describe several ways the infinity principle can be used to analyze natural phenomena and has led to important scientific developments. He links differential equations to the discovery of life-saving HIV treatments, Fourier analysis to the invention of the microwave oven and provides many more examples of the “unreasonable effectiveness” of mathematics. These are equal parts informative and inspiring for anyone who considers mathematics an important part of their life.

Overall, Strogatz’s ​ Infinite Powers ​ is​ ​ a compelling, engaging read, one which readers of all different levels of interest in, aptitude for, or aversion to mathematics will enjoy.

https://maa.org/book-reviews/infinite-powers-how-calculus-reveals-the-secrets-of-the-universe/