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Simply by doubling both sides of the last equation, we find

Subtracting the first equation from the second gives us 2S-S=S = 264-1, which is the exact answer.

How much is it roughly in ordinary base-10 notation? 210 is close to 1,000, or 103 (within 2.4 percent). So 220 = 2<10X2) = (210)2 = roughly (103)2 = 106, which is 10 multiplied by itself 6 times, or a million. Likewise,

260 = (210)6 = roughly (103)6 = 1()18 So 264 = A x 360 =

roughly 16 X 1018, 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 *$ 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 bilh'on years old; and that the Earth itself is 4.6 billion years old. The Cosmos, of course, is billions of years older still. If you understand exponentials, the key to many of the secrets of the Universe is in your hand.

If you know a thing only qualitatively, you know it no more than vaguely. If you know it quantitatively— grasping some numerical measure that distinguishes it from an infinite number of other possibilities—you are beginning to know it deeply. You comprehend some of its beauty and you gain access to its power and the understanding it provides. Being afraid of quantification is tantamount to disenfranchising yourself, giving up on one of the most potent prospects for understanding and changing the world.

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