Review of Steven Strogatz – Infinite Powers: How Calculus Reveals the Secrets of the Universe
This book is really, really good. It should be required supplemental reading for math teachers, who should assign relevant portions to their students. Most math pedagogy consists of memorizing procedures. It’s mostly how, with only a little bit of what or why. There is rarely much of any unifying theme that ties the separate problem-solving procedures together in a way that makes sense. Strogatz provides all that, and in a compelling way, complete with examples ranging from medicine to astronomy.
Strogatz also explains terminology, which is another common weak spot in classrooms. Why are calculus’ two main concepts called derivative and integrals? I didn’t learn that in undergrad. Nor in a high-quality graduate economics program. Instead, I learned it from Strogatz’s popular-level book in my late 30s.
Another fun bit of etymology is that the word “calculus” is derived some the world for rock. It shares a root with calcium, chalk, calcite, and other similar words. This is because in ancient times, people did their counting by sliding stones along an abacus’ strings.
The concept of infinity is key. Calculating the area of a circle is hard because of the curves. Slicing it into quarters, like a pizza, makes it a little easier. The wedges are kind of triangle-like, but there is still plenty of curved surface on the outside. Cutting into 8, 16, and 32 slices makes the curve progressively less important. Tending the number of slices towards infinity sends that tricky curved area towards zero. Long before infinity, it reaches deep decimal territory, where the accuracy of the calculation is good enough to satisfy even the most exacting engineers. Infinite parts are simpler than a complex whole. This view of infinity is the key to understanding calculus.
Differentiating is taking a complex whole, like a circle, and converting into many different parts, which are easier to calculate accurately. Derivatives are parts derived from a larger whole. Integrals take these differentiated parts and integrate them back together. Calculus is essentially the math of moving from a whole to its parts and back, as needed to accomplish the task at hand.
This is simple stuff that is so obvious to veteran instructors that they never bother to teach it to rookie students. This kind of larger context and purpose should be taught on day one of any course, and regularly reinforced as new material is introduced.
In high school, I spent months memorizing procedures for calculating sines and cosines, but never really learned much about their significance, or knew that they had anything to do with calculus. Moreover, why does it matter that the same curved shape is shifted horizontally? More than twenty years later, I finally learned why. The sine wave is interesting because of its continually changing slope. And a sine wave’s derivative is… it’s cosine. And now I have a greater appreciation of everything from the changing length of daylight during the seasons to how sound waves interact with each other. The rate of change in daylight as the calendar moves from solstice to equinox is a sine wave. The rate of change is slowest at the solstice (about 40 seconds), and fastest at the equinox (more than two minutes). Figuring out the rate of this change at any given point can be figuring out the derivative. In the special sine wave case, this is simple—just figure out the cosine.
Again, this is basic stuff that high schoolers deserve to know. GPAs would likely be measurably higher, and understanding measurably greater, by teaching a little bit more of this big-picture context and a little less rote memorization.
Needless to say, I will be reading Strogatz’s other books in short order. Infinite Powers would pair well with David Salsburg‘s The Lady Tasting Tea, which accomplishes a similar task with statistics.