Billions & Billions (3 page)

Why a King should delight in the invention of a game called “Death to the King” is a mystery. But, so the story goes, he was so pleased that he asked the Grand Vizier to name his own reward for so splendid an invention. The Grand Vizier had his answer ready: He was a modest man, he told the Shah. He wished only for a modest reward. Gesturing to the eight columns and eight rows of squares on the board he had invented, he asked that he be given a single grain of wheat on the first square, twice that on the second square, twice
that
on the third, and so on, until each square had its complement of wheat. No, the King remonstrated, this is too modest a reward for so important an invention. He offered jewels, dancing girls, palaces. But the Grand Vizier, his eyes becomingly lowered, refused them all. It was little piles of wheat that he craved. So, secretly marveling at the humility and restraint of his counselor, the King consented.

When, however, the Master of the Royal Granary began to count out the grains, the King faced an unpleasant surprise. The number of grains starts out small enough: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 … but by the time the 64th square is approached, the number of grains becomes colossal, staggering. In fact, the number is (see box on
this page
) nearly 18.5 quintillion. Perhaps the Grand Vizier was on a high-fiber diet.

How much does 18.5 quintillion grains of wheat weigh? If each grain is a millimeter in size, then all of the grains together would weigh around 75 billion metric tons, which far exceeds what could have been stored in the Shah’s granaries. In fact, this is the equivalent of about 150 years of the world’s
present
wheat production. An account of what happened next has not come down to us. Whether the King, in default, blaming himself for inattentiveness in his study of arithmetic, handed the kingdom over to the Vizier, or whether the latter experienced the tribulations of a new game called
viziermat
, we are not privileged to know.

The story of the Persian Chessboard may be just a fable. But the ancient Persians and Indians were brilliant 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 20
years, 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 AIDS is found, then most of the intravenous drug users who share hypodermic needles will die—not all, because there is a small percentage of people who are naturally resistant, but almost all. The same is true for homosexual men who have many partners and engage in unsafe sex—but not for those who competently use condoms, those in long-term monogamous relations, and again, not for the small fraction who are naturally immune. Strictly heterosexual couples in long-term monogamous relations reaching back into the early 1980s, or who are vigilant in practicing safe sex, and who do not share needles—and there are many of them—are essentially insulated from AIDS. After the curves for the demographic groups most at risk flatten out, other groups will take their place—at present in America it seems to be young heterosexuals of both sexes, who perhaps find prudence overwhelmed by passion and use unsafe sexual practices. Many of them will die, some will be lucky or naturally immune or abstemious, and they will be replaced by another most-at-risk group—perhaps the next generation of homosexual men. Eventually the exponential curve for all of us together is expected to flatten out, having killed many fewer
than everybody on Earth. (Small comfort for its many victims and their loved ones.)

—

Exponentials are also the central idea behind the world population crisis. For most of the time humans have been on Earth the population was stable, with births and deaths almost perfectly in balance. This is called a “steady state.” After the invention of agriculture—including the planting and harvesting of those grains of wheat the Grand Vizier was hankering for—the human population of this planet began increasing, entering an exponential phase, which is very far from a steady state. Right now the doubling time of the world population is about 40 years. Every 40 years there will be twice as many of us. As the English clergyman Thomas Malthus pointed out in 1798, a population increasing exponentially—Malthus described it as a geometrical progression—will outstrip any conceivable increase in food supply. No Green Revolution, no hydroponics, no making the deserts bloom can beat an exponential population growth.

There is also no extraterrestrial solution to this problem. Right now there are something like 240,000 more humans being born than dying every day. We are very far from being able to ship 240,000 people into space every day. No settlements in Earth orbit or on the Moon or on other planets can put a perceptible dent in the population explosion. Even if it were possible to ship everybody on Earth off to planets of distant stars on ships that travel faster than light, almost nothing would be changed—all the habitable planets in the Milky Way galaxy would be full up in a millennium or so. Unless we slow our rate of reproduction. Never underestimate an exponential.

The Earth’s population, as it has grown through time, is shown in the following figure. We are clearly in (or just about to
emerge from) a phase of steep exponential growth. But many countries—the United States, Russia, and China, for example—have reached or will soon reach a situation where their population growth has ceased, where they arrive at something close to a steady state. This is also called zero population growth (ZPG). Still, because exponentials are so powerful, if even a small fraction of the human community continues for some time to reproduce exponentially the situation is essentially the same—the world population increases exponentially, even if many nations are at ZPG.

There is a well-documented worldwide correlation between poverty and high birthrates. In little countries and big countries, capitalist countries and communist countries, Catholic countries and Moslem countries, Western countries and Eastern countries—in almost all these cases, exponential population growth slows down or stops when grinding poverty disappears. This is called the demographic transition. It is in the urgent long-term interest of the human species that every place on Earth achieves this demographic transition. This is why helping other countries to become self-sufficient is not only elementary human decency, but is also in the self-interest of those richer nations able to help. One of the central issues in the world population crisis is poverty.

The exceptions to the demographic transition are interesting. Some nations with high per capita incomes still have high birthrates. But in them, contraceptives are sparsely available, and/or women lack any effective political power. It is not hard to understand the connection.

At present there are around 6 billion humans. In 40 years, if the doubling time stays constant, there will be 12 billion; in 80 years, 24 billion; in 120 years, 48 billion.… But few believe the Earth can support so many people. Because of the power of this exponential increase, dealing with global poverty now will be much cheaper and much more humane, it seems, than whatever solutions will be available to us many decades hence. Our job is to bring about a worldwide demographic transition and flatten out that exponential curve—by eliminating grinding poverty, making safe and effective birth control methods widely available, and extending real political power (executive, legislative, judicial, military, and in institutions influencing public opinion) to women. If we fail, some other process, less under our control, will do it for us.