Billions & Billions Read Free

pathfinders in mathematics, and understood the enormous numbers that result when you keep on doubling. Had chess been invented with 100 (10 × 10) squares instead of 64 (8 × 8), the resulting debt in grains of wheat would have weighed as much as the Earth. A sequence of numbers like this, where each number is a fixed multiple of the previous one, is called a geometric progression, and the process is called an exponential increase.
    Exponentials show up in all sorts of important areas, unfamiliar and familiar—for example, compound interest. If, say, an ancestor of yours put $10 in the bank for you 200 years ago,or soon after the American Revolution, and it accrued a steady 5 percent annual interest, then by now it would be worth $10 × (1.05) 200 , or $172,925.81. But few ancestors are so solicitous about the fortunes of their remote descendants, and $10 was a lot of money in those days. [(1.05) 200 simply means 1.05 times itself 200 times.] If that ancestor could have gotten a 6 percent rate, you’d now have over a million dollars; at 7 percent, over $7.5 million; and at an extortionate 10 percent, a tidy $1.9 billion.
    Likewise for inflation. If the rate is 5 percent a year, a dollar is worth $0.95 after one year, (0.95) 2 = $0.91 after two years; $0.61 after ten years; $0.37 after twenty; and so on. This is a very practical matter for retirees whose pensions provide a fixed number of dollars per year with no adjustment for inflation.

    The most common circumstance in which repeated doublings, and therefore exponential growth, occurs is in biological reproduction. Consider first the simple case of a bacterium that reproduces by dividing itself in two. After a while, each of the two daughter bacteria divides as well. As long as there’s enough food and no poisons in the environment, the bacterial colony will grow exponentially. Under very favorable circumstances, there can be a doubling every 15 minutes or so. That means 4 doublings an hour and 96 doublings a day. Although a bacterium weighs only about a trillionth of a gram, its descendants, after a day of wild asexual abandon, will collectively weigh as much as a mountain; in a little over a day and a half as much as the Earth; in two days more than the Sun.… And before very long, everything in the Universe will be made of bacteria. This is not a very happy prospect, and fortunately it never happens. Why not? Because exponential growth of this sort always bumps into some natural obstacle. The bugs run out of food, or they poison each other, or are shy about reproducing when they have hardly any privacy. Exponentials can’t go on forever, because they will gobble up everything. Long before then they encounter some impediment. The exponential curve flattens out. (See illustration.)
    This is a distinction very important for the AIDS epidemic. Right now, in many countries the number of people with AIDS symptoms is growing exponentially. The doubling time is around one year. That is, every year there are twice as many AIDS cases as there were in the previous year. AIDS has already taken a disastrous toll among us. If it were to continue exponentially, it would be an unprecedented catastrophe. In 10 years there would be a thousand times more AIDS cases, and in 20years, a million times more. But a million times the number of people who have already contracted AIDS is much more than the number of people on Earth. If there were no natural impediments to the continued doubling of AIDS every year and the disease were invariably fatal (and no cure found), then everyone on Earth would die from AIDS, and soon.
    However, some people seem to be naturally immune to AIDS. Also, according to the Communicable Disease Center of the U.S. Public Health Service, the doubling in America initially was almost entirely restricted to vulnerable groups largely sexually isolated from the rest of the population—especially male homosexuals, hemophiliacs, and intravenous drug users. If no cure for