The mathematician Ian Stewart knows the famous story about equations versus book sales. Stephen Hawking's publisher told him that every equation published in _A Brief History of Time_ would halve the number of books sold. One equation got in: E = mc^2, and maybe it really did cut the sales of the book by half. If this rule is true, Stewart is in real trouble with his newest book, _In Pursuit of the Unknown: 17 Equations That Changed the World_ (Basic Books). Readers who know his work, however, know that they are in good hands. Stewart has undoubtedly written mathematical papers that would be over the heads of us other mortals, but his books for the public on the problems, range, and philosophy of mathematics are clear, funny, entertaining, and educational. His seventeen chapters include some simple equations that everyone knows; that E = mc^2 is here, simple to write and to memorize, but pointing to complexities that most of us cannot easily comprehend, even a hundred or so years after it was developed. Some of the equations, like Schrödinger's Equation, are full of Greek letters and only physics experts will recognize them. Throughout the book, however, Stewart shows that these are equations that run our lives in our technical age. The equations may be used professionally just by the egghead experts, but in a wider sense, we all use them, every day.
The first chapter here is on the old familiar a^2 + b^2 = c^2, the Pythagorean Theorem. This is pure math, straight from Euclid, and not (as are many of the equations here) from applied mathematics or mathematical physics. But that does not mean the Pythagorean Theorem is forever locked within the mathematicians' ivory tower; it led to trigonometry. When he does get to E = mc^2, Stewart reflects about how Pythagoras helps understand relativity, because of light paths understood as sides of triangles. All of the equations here have improved our understanding of how nature works, and have supplied reason to wonder at how consistently mathematics underlies everything. The basic equation for calculus is here, which is responsible for most of mathematical physics. Among the simplest of equations here is Euler's formula that shows how faces, edges, and corners of a solid shape are related. Like each simple formula, it has created complications, including the powerful pure mathematics of topology, which has implications for how DNA works and why planets may move in a chaotic way. Another simple one is i^2 = -1, indicating that minus one, which should have no square root since it is negative, does have one, the imaginary number i. Although i may be called imaginary, it is essential for understanding waves and electricity, not to mention quantum mechanics. Chaos theory is here, with an equation that helps show how the flapping of that butterfly's wing may lead to a tornado later on; that's the most famous effect, but the equation models, for instance, how a population of creatures changes over generations if they have to be curtailed by limited resources or predators. Stewart gives clear explanations, but they are relatively deep for the non-mathematician. Many people who read this book will want to take long breaks between its pithy chapters, each of which has been expanded elsewhere into many volumes. Equations are useful for explaining the world, but like any tool they can be misused. Stewart's final chapter is on the Black-Scholes Equation, invented in 1973 and since then used to analyze the changes of price of a financial derivative. Derivatives could thus be traded before they matured. It was a useful formula as long as it was applied only when market situations fit, but it was abused. Stewart makes clear that the formula didn't cause the 2008 - 2009 financial crisis, but abuse of it, along with financial and political ineptitude and lax regulation, made for a crash that didn't have to happen.
In an epilogue, Stewart reflects that most equations aren't important: "I write them down all the time, and believe me, I know." Here are some important ones, though, equations that run our world, always in ways that the inventors of the equations could never have predicted. It may be, Stewart writes at the end, that the cellular automata famously championed by Stephen Wolfram do a better job of explaining the universe than equations do, and maybe algorithms are going to be more important than equations. His engrossing book, showing the vital importance of equations not just for explanation but as causation of historic and social change, makes clear that it will be a long time before any other modeling becomes more important.