Before he ever took a formal algebra class, Arthur Benjamin got a lesson he never forgot. The future mathematician's father said, “Son, doing algebra is just like arithmetic, except you substitute letters for numbers. For example, 2x + 3x = 5x and 3y + 6y = 9y. You got it?” The young Arthur replied that he grasped the concept. After which his dad said, “Okay, then what is 5Q + 5Q?” Arthur replied “10Q.” And his father said, “You're welcome!”
That terrible, wonderful gag appears early in Benjamin's new book, The Magic of Math: Solving for x and Figuring Out Why. Benjamin is now a professor at the small and prestigious Harvey Mudd College in Claremont, Calif. If your alma mater's name is Mudd, chances are you're a scientist, engineer or mathematician—the school specializes in those fields.
The occasional stabs at humor in Benjamin's book will leave the reader figuratively bloodied but unbowed and buoyed in the brainpan. Whether it's been decades since you last took algebra or you're currently dealing with the aches of solving for x, The Magic of Math is a good read. Even though it includes, gasp, equations.
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For example (and don't bother to try to stop me if you've heard this one), consider a pizza pie to be a very short cylinder. The volume of a cylinder equals pi times the radius squared times the height. That is, V = pi r r h. And so, deep breath, for a pizza of radius z and height a, V = pi z z a. And if you think that exercise was too cheesy, you need a thicker crust.
“I want people not just to learn mathematics, I want people to love mathematics,” Benjamin told me when he visited New York City in September. “That's what Martin Gardner's writing did for me.” Gardner was the longtime writer of the famous Mathematical Games columns for Scientific American, inspiring many young Arthurs to describe their round tables as fulfilling the requirement that x
2 + y2 = r2. Before tests, these individuals might cram a lot. (Sorry, I've been inspired by Arthur's father to embrace the dark side of the forced jokes.)
One of my favorite parts of the book considers a rope tied to the bottoms of the two goalposts on opposite ends of an American football field, 120 yards away from each other. That's 100 yards plus the two 10-yard butt-slapping and dance areas—I mean the end zones. The taut rope is thus 360 feet long as it traverses the grass along the centerline of the field. Now imagine that the rope gains a measly little foot, so that it's now 361 feet long. At the 50-yard line, how high could you lift the rope, while leaving its ends on the ground at the goalposts?
Feel free to stop reading for a moment if you want to do the relatively simple calculation. (Or for any other reason—I am not the boss of you.)
By lifting the rope, you create an imaginary triangle with a base of 360 feet, an as yet unknown height of h feet, and two sides of 180.5 feet, half of the 361-foot-long rope. Now drop an imaginary plumb line from the top of the rope, and the big triangle can be divided into two smaller and equal right triangles, each with a hypotenuse of 180.5 feet and sides of 180 feet and h feet. Perform the primordial Pythagorean prestidigitation (the sum of the squares of the two sides of a right triangle equals the square of the hypotenuse), and you'll find that the one-foot-longer rope can be lifted high enough for even the most gigantic lineman to trundle under, more than 13 feet off the ground.
I'm fond of this example because the result just felt wrong to me. How could a single additional foot of slack have such a large effect? And yet the math is indisputable, as math tends to be. The Magic of Math thus reminds the reader that reality cares not how you feel about it. Which is why I recommend the book to anyone involved in making public policy. No amount of additional analysis, fact-finding commissions, committee hearings or white papers will change the height of that rope. 10Q. 10Q very much.
