Almost everyone has heard the story of the man who invented chess. The king of the realm was so pleased with the invention, so the story goes, that he offered the inventor any prize he could name. The prize the inventor wanted was a certain amount of grain, computed as follows: On the first square of the chessboard, the kind would put one grain of wheat; on the second square, two grains of wheat; on the third square, four grains of wheat; on the fourth square, eight grains of wheat... and so on, doubling the number of grains of wheat for each of the chessboard's 64 squares. How much wheat did the inventor receive? Approximately all the grain that has ever been grown in North America.
The powers of doubling are as eerie as they are impressive. Consider this: If you folded a piece of tissue paper so that it doubled its thickness 50 times, it would reach to the moon and back 17 times. If you don't believe it, just pick a number, any number, and start doubling. Notice how the numbers grow: barely perceptibly at first, astronomically later. No wonder the king was fooled. He probably did not even realize what was happening until he was more than half-way through the chessboard.
My favorite example of this was described by Dr. Albert Bartlett in the American Journal of Physics several years ago. Bartlett took a population of bacteria doubling their numbers once a minute inside a Coke bottle. They started at 11 a.m., and by noon the bottle was full. What time would it be, Bartlett asked, when even the most foresighted bacteria realized they were running out of room? 11:58 a.m. At that time, the bottle would still be three-quarters empty -- two doublings away from full. At 11:59, it would still be half empty -- and no doubt the presidents of bacteria bottle companies would be running around bacteria land assuring everyone that there was no reason to limit growth because, after all, there was more room still left than had ever been used in the population's history!
Bartlett wrote that people tended not to believe him when he pointed out that discovering "more oil than has ever been used on this planet" really didn't mean much at current growth rates for the use of oil (even a growth rate of 7 percent doubles consumption in 10 years) any more than people believe it when the prices of houses double -- or the value of the dollar is cut in half -- in just five years at, say, 15 percent inflation. (If you don't believe it, take 15 percent of say, $40,000, five times. Presto! you need $80,000 to buy what $40,000 bought five years before.) The kind of "exponential" growth applies to everything that grows on top of previous growth, that is, where the amount of the next bit of growth depends on the last bit of growth. Each next step is bigger because each previous step is bigger. It explains everything from avalanches and nuclear explosions to compound interest and population explosions.
Curiously, this kind of calculating is barely taught in school and scarcely believed even by those who understand it. One reason may be that our ears and eyes can no more count exponentially than can our minds. This is literally true.For example, the sun is about 100 thousand times as bright as the moon. But our eyes registerregister it as only about nine times as bright (otherwise, we would be blinded). Our eyes, that is, count the number of squares on the chessboard while the actual brightness increases like grains of wheat. The same is true of sound. Like the decibel scale, our ears measure even chessboard-like intervals while loudness increases exponentially.
I remember the first time I realized this innate inability to perceive large numbers.I was an editor on a copy desk of a newspaper and considered a typo that changed a million into a billion a relatively minor error. (After all, both were "a lot.") It seemed to me -- as it seems to most people -- that the difference between a million and a billion was about the same as, say, the difference between a thousand and a million. Then someone pointed out to me that if you started with a millimeter (the thickness of a dime) and multiplied by a thousand, you would have a meter -- about a yard. If you multiplied a millimeter by a million, you would have akilometer -- a little more than a half-mile. Because whenever you add a zero, you don't merely double what you already have; you multiply by 10. The differnce between a thousand and a million -- three zeros -- is 999,000. But the differnce between a million and a billion -- a mere three more zeros -- is 999 million. Add three more zeros and you've added 999 billion -- nearly the size of our national debt.
Ever since I learned about exponentials, I've been paying more attention to the fact that child health budgets, for example, are always counted in millions, while corporate bail-outs and defense boondoggles always add up to billions. Learningabout how things grow tells you a lot about what makes things tick. It's a lot more than a numbers' game.