Saturday, November 12, 2016

Major mathematical MEGO moments

kw: book reviews, nonfiction, mathematics, mathematical games

Is it possible to be too enthusiastic about something? We might say, Yes, look at fanatics. I would say that Mathematics Professor Arthur Benjamin falls somewhere between an "ordinary" enthusiast and a fanatic, in his book The Magic of Math: Solving for x and Figuring out Why.

Perhaps the title of this post is a bit hyped—I do love alliteration—but I can only describe the style of the book as Barrage. A barrage of facts, tricks, methods and tips along a spectrum that I call "ordinary" mathematics, the string of mathematical disciplines that most of us encounter if we were paying attention for some parts of all twelve grades of an El-Hi education. Plus a few side trips into less-well-known things such as Fibonacci Numbers and why (or how) π is irrational.

Now, I am a math enthusiast, and working as a scientific programmer for decades, I frequently corrected the algebra or calculus derivations of the scientists I worked with. So I was already familiar with many of the tips and tricks scattered throughout the book. Others were new, and some were beyond me. I remembered some advice once given to Stephen Hawking, which he reports in A Brief History of Time: Keep equations to a minimum, because each equation in a book cuts its potential audience in half. At the rate of several equations per page, the audience for Magic of Math could be tiny indeed!

A bit of math trickery to emphasize the point: ½ times ½ times ½ just ten times yields less than one in 1,000; 1/1024 to be exact. Can ten equations in a book really reduce the readership to a tenth of a percent? Possibly. But the word "half" above is a bit hyped. Let us instead suppose that, while a single equation might drive away half an audience, beyond that, you lose only 1% per equation. So what is 0.5x0.99x0.99 and so forth for a total of, for example, 20 equations? There is a neat little yx button on many calculators that lets me push 0.99 yx 19 x 0.5 and get 0.413… But this book has 300 pages. Let's be optimistic and suppose one per page (there are more): What is the effect of 300 equations? I do it again for 299 instead of 19, to get 0.0247 and some more digits. That's not the end of the world for a writer. It is about a fortieth of the original audience.

I am tempted to say that the twelve chapters of the book correspond to grades in school, but the subject of Chapter 2 is Algebra, which few of us encountered before Grade 6 (where I had it 55 years ago, but more recently it's a Middle School subject, and Common Core has now pushed it to Grade 9). The third chapter is about the number 9, including "casting out nines", which was once taught as a way to check the addition of a column of numbers. Our calculators and other machinery don't make mistakes in arithmetic, so it hasn't been taught for decades. What else could he possibly say about 9? Well, the chapter has 20 pages. He says a lot! For example, casting out nines is a type of checksum that is easy for humans to calculate. Similar schemes, some of which only a coder could love, are used for various reasons, including the checksum that is the last digit of the ISBN or ISBN-13 found in every book and many other publications. It is the remainder in a calculation using the prior 9 or 12 digits, to see if someone has phonied up an ISBN, or if in some human process digits were transposed in writing it down.

I know I sound like I am down on the book and the author, but his writing is really quite good and engaging. It just didn't all "reach" me, particularly Chapter 6 about Proofs. I don't have the right kind of mind for formally proving theorems, and getting my axioms (or propositions) in the right order. That is where My Eyes Glazed Over, big time. Yet the following chapter, on Geometry, has many interesting examples of the way we think geometrically, and the way we sometimes err consistently in certain ways. Again the proofs slid by me, but near the end of the chapter he shows how Geometry and Algebra are equivalent and can be used to check each other. I assure you, this will help students who might find one or the other more congenial, to bootstrap their understanding. Few teachers even know this.

I was amazed that Dr. Benjamin had the audacity to write a chapter on Calculus (#11). But if a mathophobe is capable of learning calculus, this chapter can get him or her started learning it. Its clear explanations show how calculus is a logical extension of simple algebra. In one lovely example, we find the shortest route from point A to point B, with point C to be chosen along some line to the side (say, you want to go by way of yon brook). One more step shows that it is the same as using the "brook" as a mirror and reflecting points A and B to A' and B'; drawing a line from A to B' shows that the shortest path is a mirror reflection, with equal angles on both sides.

This is not a book to sit down and read like a novel. Neither is it a textbook. It is somewhere between, and rewards a reader who reads it at a desk, with paper and pencil handy, to try out things as they are presented. If you are familiar with many of the items, you'll soon encounter something new, something that is probably both interesting and fun. Hey, maybe you'll even like proofs much better than I do! The world needs a few such folk. So the book still gets a big Like from me. Maybe one day I'll look back at Chapter 6, go more slowly, and learn a smidgen more about proofs, that has to date eluded me.

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