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relatives. But this is far more than the population of the Earth, then or now; it is far more than the total number of human beings who have ever lived. Something is wrong with our calculation. What? Well, we have assumed all those lineal ancestors to be different people. But this, of course, is not the case. The same ancestor is related to us by many different routes. We are repeatedly, multiply connected with each of our relatives—a huge number of times for the more distant relations.
    Something like this is true of the whole human population. If we go far enough back, any two people on Earth have a common ancestor. Whenever a new American President is elected, there’s bound to be someone—generally in England—to discover that the new President is related to the Queen or King of England. This is thought to bind the English-speaking peoples together. When two people derive from the same nation or culture, or from the same small corner of the world, and their genealogies are well-recorded, it is likely that the last common ancestor can be discovered. But whether it can be discovered or not, the relationships are clear. We are all cousins—everyone on Earth.
    THE CALCULATION THE KING SHOULD
HAVE DEMANDED OF HIS VIZIER
    Don’t be scared off. This is really easy. We want to calculate how many grains of wheat were on the entire Persian Chessboard.
    An elegant (and perfectly exact) calculation goes as follows:
    The exponent just tells us how many times we multiply 2 by itself. 2 2 = 4. 2 4 = 16. 2 10 = 1,024, and so on. Call
S
the total number of grains on the chessboard, from 1 in the first square to 2 63 in the 64th square. Then, plainly,
    S = 1 + 2 + 2 2 + 2 3 + … + 2 62 + 2 63
    Simply by doubling both sides of the last equation, we find
    2S
= 2 + 2 2 + 2 3 + 2 4 + … + 2 63 + 2 64
    Subtracting the first equation from the second gives us
    2S –
S = S = 2 64 – l,
    which is the exact answer.
    How much is it roughly in ordinary base-10 notation? 2 10 is close to 1,000, or 10 3 (within 2.4 percent). So 2 20 = 2 (10X2) = (2 10 ) 2 = roughly (10 3 ) 2 = 10 6 , which is 10 multiplied by itself 6 times, or a million. Likewise, 2 60 = (2 10 ) 6 = roughly (10 3 ) 6 = 10 18 . So 2 64 = 2 4 × 2 60 = roughly 16 × 10 18 , or 16 followed by 18 zeros, which is 16 quintillion grains. A more accurate calculation gives the answer 18.6 quintillion grains.
    —
    Another common appearance of exponentials is the idea of half-life. A radioactive “parent” element—plutonium, say, or radium—decays into another, perhaps safer, “daughter” element, but not all at once. It decays statistically. There is a certain time by which half of it has decayed, and this is called its half-life. Half of what is left decays in another half-life, and half of the remainder in still another half-life, and so on. For example, if the half-life were one year, half would decay in a year, half of a half or all but a quarter would be gone in two years, all but an eighth in three years, all but about a thousandth in ten years, etc. Different elements have different half-lives. Half-life is an important idea when trying to decide what to do with radioactive waste from nuclear power plants or in contemplating radioactive fallout in nuclear war. It represents an exponential decay, in the same way that the Persian Chessboard represents an exponential increase.
    Radioactive decay is a principal method for dating the past. If we can measure the amount of radioactive parent material and the amount of daughter decay product in a sample, we can determine how long the sample has been around. In this way we find that the so-called Shroud of Turin is not the burial shroud of Jesus, but a pious hoax from the fourteenth century (when it was denounced by Church authorities); that humans made campfires millions of years ago; that the most ancient fossils of life on Earth are at least 3.5 billion years old; and that the Earth itself is 4.6 billion years old. The