Thursday, September 13, 2012

A great explainer makes all the difference

kw: book reviews, nonfiction, mathematics, equations

I've been told I ought to show a book's cover in these reviews. I almost never do. In this case, I just had to. The cover illustration by Jennifer Carrow is incomparable! It is said that Euler's Identity (we will come to it) is the most beautiful equation ever devised, and it ranks high among equations printed on T-shirts for math lovers. This circle of 17 of the most influential equations belongs on one of those T-shirts.

I admit this mathematical soup bowl doesn't have the concise heft of the Maxwell System (4 very short equations, and probably the "equation" most commonly found on a geek's T-shirt). But it does show that a major chunk of modern technology depends on a working understanding of a few deep principles.

The equations chosen by Ian Stewart for In Pursuit of the Unknown: 17 Equations that Changed the World span the gamut of mathematical understanding. And mathematical understanding makes nearly everything work upon which we depend: auto engines, electrical power transmission, cell phones, the water and sewage system, airplanes, and the radios and TVs we dote upon.

Some of those equations go way, way back. The venerable Pythagorean Theorem, that the sum of the squares of the legs of a right triangle equals the square of the long side (c²=a²+b²), is possibly 500 years older than Pythagoras, pushing it back some 3,000 years. Much more recently the Logistic Equation (new x=kx(1-x)) has codified the "butterfly effect", illustrating that in unstable systems, and nature is full of them, a butterfly that decides to flap twice instead of once during a particular moment can make the difference between a clear day two weeks later, or a thunderstorm.

The title embodies a lovely pun. A mathematical equation is a mental machine, used to combine things we know into discovery of what we don't know. It is usually written with some symbols on the right, the "knowns", and you have to solve it to get an "answer", symbolized by whatever is on the left, the "unknown". In the case of Pythagoras's equation, we know a and b, and use the equation to find a value for c.

For the Euler Identity, embodied in the seven characters shown here, there is nothing to "solve". Rather, it expresses a deep relationship between quantities that once were thought to have nothing to do with one another. The first symbol, e, the exponential number, is the basis of natural logarithms (a large subject all in itself). The second, named pi, is the circumference of a circle whose diameter is 1. This is not one of the 17 equations, but shows up in the chapter on the square root of -1, which is often written i, and originally called "the imaginary number". When I took engineering math courses, though, we used j, both to use a larger symbol that we would be less likely to misplace, and to get our minds used to the fact that this number is very, very real. All of modern electronics, and indeed, all wave phenomena, would not work without it.

For those that know a smidgen of trigonometry, the first cluster of symbols above, the exponential function of pi times i, represents a particular combination of the sine and cosine function, which adds up to -1. Until i was understood, the others had no known relationship.

I have known some of the equations in Dr. Steward discusses, but others were new to me. In all cases, his explanations may not have made me any kind of expert, but they did convey an understanding of what the subject was about. For example, I had only a glancing contact with Maxwell's Equations in the past. There is really no way for me to get into these; I'll just say that their chapter got me a little closer to understanding the relationship between electrical and magnetic fields that wrap together into light, radio waves and x-rays. That little pair of symbols after the fourth equal sign, mu-zero times epsilon-zero, works out to the reciprocal square of the speed of light. When you solve the two lower equations together, you get the speed of light, an insight that led Maxwell and his colleagues to realize that light is an electromagnetic wave.

Most of us will never know much about such things as the way air and gasoline vapor mix inside an auto engine. We just want to drive the car. Fortunately, some very smart people do know a great deal about vapor mixing and burning, and also about the way the steering wheel affects the direction the tires point, and how to do so stably, plus how to size the brakes so you can safely stop. We don't have to know digital signal compression theory to operate a cell phone. But a few folks have to know it to design its circuitry. The equations that lie beneath all our gadgets can be beautiful to a mathematician, and wholly opaque to most of us. But if you read this book, you'll appreciate the power of a skillful explainer such as the author, to crack open the lid on our minds. It can take years to learn to work some of these mental machines. With Ian Stewart's help, it takes just a few minutes to appreciate their effects.

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