Innumeracy
Mathematical Illiteracy and its Consequences
By John Allen Paulos Hill and Wang
Copyright © 2001 John Allen Paulos
All rights reserved.
ISBN: 978-1-4299-3438-1
CHAPTER 1
Examples and Principles
Two aristocrats are out horseback riding and one challenges the other to see which can come up with the larger number. The second agrees to the contest, concentrates for a few minutes, and proudly announces, "Three." The proposer of the game is quiet for half an hour, then finally shrugs and concedes defeat.
A summer visitor enters a hardware store in Maine and buys a large number of expensive items. The skeptical, reticent owner doesn't say a word as he adds the bill on the cash register. When he's finished, he points to the total and watches as the man counts out $1,528.47. He then methodically recounts the money once, twice, three times. The visitor finally asks if he's given him the right amount of money, to which the Mainer grudgingly responds, "Just barely."
The mathematician G. H. Hardy was visiting his protégé, the Indian mathematician Ramanujan, in the hospital. To make small talk, he remarked that 1729, the number of the taxi which had brought him, was a rather dull number, to which Ramanujan replied immediately, "No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."
Big Numbers, Small Probabilities
People's facility with numbers ranges from the aristocratic to the Ramanujanian, but it's an unfortunate fact that most are on the aristocrats' side of our old Mainer. I'm always amazed and depressed when I encounter students who have no idea what the population of the United States is, or the approximate distance from coast to coast, or roughly what percentage of the world is Chinese. I sometimes ask them as an exercise to estimate how fast human hair grows in miles per hour, or approximately how many people die on earth each day, or how many cigarettes are smoked annually in this country. Despite some initial reluctance (one student maintained that hair just doesn't grow in miles per hour), they often improve their feeling for numbers dramatically.
Without some appreciation of common large numbers, it's impossible to react with the proper skepticism to terrifying reports that more than a million American kids are kidnapped each year, or with the proper sobriety to a warhead carrying a megaton of explosive power — the equivalent of a million tons (or two billion pounds) of TNT.
And if you don't have some feeling for probabilities, automobile accidents might seem a relatively minor problem of local travel, whereas being killed by terrorists might seem to be a major risk when going overseas. As often observed, however, the 45,000 people killed annually on American roads are approximately equal in number to all American dead in the Vietnam War. On the other hand, the seventeen Americans killed by terrorists in 1985 were among the 28 million of us who traveled abroad that year — that's one chance in 1.6 million of becoming a victim. Compare that with these annual rates in the United States: one chance in 68,000 of choking to death; one chance in 75,000 of dying in a bicycle crash; one chance in 20,000 of drowning; and one chance in only 5,300 of dying in a car crash.
Confronted with these large numbers and with the correspondingly small probabilities associated with them, the innumerate will inevitably respond with the non sequitur, "Yes, but what if you're that one," and then nod knowingly, as if they've demolished your argument with their penetrating insight. This tendency to personalize is, as we'll see, a characteristic of many people who suffer from innumeracy. Equally typical is a tendency to equate the risk from some obscure and exotic malady with the chances of suffering from heart and circulatory disease, from which about 12,000 Americans die each week.
There's a joke I like that's marginally relevant. An old married couple in their nineties contact a divorce lawyer, who pleads with them to stay together. "Why get divorced now after seventy years of marriage? Why not last it out? Why now?" The little old lady finally pipes up in a creaky voice: "We wanted to wait until the children were dead."
A feeling for what quantities or time spans are appropriate in various contexts is essential to getting the joke. Slipping between millions and billions or between billions and trillions should in this sense be equally funny, but it isn't, because we too often lack an intuitive feeling for these numbers. Many educated people have little grasp for these numbers and are even unaware that a million is 1,000,000; a billion is 1,000,000,000; and a trillion, 1,000,000,000,000.
A recent study by Drs. Kronlund and Phillips of the University of Washington showed that most doctors' assessments of the risks of various operations, procedures, and medications (even in their own specialties) were way off the mark, often by several orders of magnitude. I once had a conversation with a doctor who, within approximately twenty minutes, stated that a certain procedure he was contemplating (a) had a one-chance-in-a-million risk associated with it; (b) was 99 percent safe; and (c) usually went quite well. Given the fact that so many doctors seem to believe that there must be at least eleven people in the waiting room if they're to avoid being idle, I'm not surprised at this new evidence of their innumeracy.
For very big or very small numbers, so-called scientific notation is often clearer and easier to work with than standard notation and I'll therefore sometimes use it. There's nothing very tricky about it: 10N is 1 with N zeroes following it, so 104 is 10,000 and 109 is a billion. 10-N is 1 divided by 10N, so 10-4 is 1 divided by 10,000 or .0001 and 10-2 is one hundredth. 4 × 106 is 4 × 1,000,000 or 4,000,000; 5.3 × 108 is 5.3 × 100,000,000 or 530,000,000; 2 × 10-3is 2 × 1/1,000 or .002; 3.4 × 10-7 is 3.4 × 1/10,000,000 or .00000034.
Why don't news magazines and newspapers make appropriate use of scientific notation in their stories? The notation is not nearly as arcane as many of the topics discussed in these media, and it's considerably more useful than the abortive switch to the metric system about which so many boring articles were written. The expression 7.39842 × 1010 is more comprehensible and legible than seventy-three billion nine hundred and eighty-four million and two hundred thousand.
Expressed in scientific notation, the answers to the questions posed earlier are: human hair grows at a rate of roughly 10-8 miles per hour; approximately 2.5 × 105 people die each day on earth; and approximately 5 × 1011 cigarettes are smoked each year in the United States. Standard notation for these numbers is: .00000001 miles per hour; about 250,000 people; approximately 500,000,000,000 cigarettes.
Blood, Mountains, and Burgers
In a Scientific American column on innumeracy, the computer scientist Douglas Hofstadter cites the case of the Ideal Toy Company, which stated on the package of the original Rubik cube that there were more than three billion possible states the cube could attain. Calculations show that there are more than 4 × 1019possible states, 4 with 19 zeroes after it. What the package says isn't wrong; there are more than three billion possible states. The understatement, however, is symptomatic of a pervasive innumeracy which ill suits a technologically based society. It's analogous to a sign at the entrance to the Lincoln Tunnel stating: New York, population more than 6; or McDonald's proudly announcing that they've sold more than 120 hamburgers.
The number 4 × 1019 is not exactly commonplace, but numbers like ten thousand, one million, and a trillion are. Examples of collections each having a million elements, a billion elements, and so on, should be at hand for quick comparison. For example, knowing that it takes only about eleven and a half days for a million seconds to tick away, whereas almost thirty-two years are required for a billion seconds to pass, gives one a better grasp of the relative magnitudes of these two common numbers. What about trillions? Modern Homo sapiens is probably less than 10 trillion seconds old; and the subsequent complete disappearance of the Neanderthal version of early Homo sapiens occurred only a trillion or so seconds ago. Agriculture's been here for approximately 300 billion seconds (ten thousand years), writing for about 150 billion seconds, and rock music has been around for only about one billion seconds.
More common sources of such large numbers are the trillion-dollar federal budget and our burgeoning weapons stockpiles. Given a U.S. population of about 250 million people, every billion dollars in the federal budget translates into $4 for every American. Thus, an annual Defense Department budget of almost a third of a trillion dollars amounts to approximately $5,000 per year for a family of four. What have all these expenditures (ours and theirs) bought over the years? The TNT equivalent of all the nuclear weapons in the world amounts to 25,000 megatons, or 50 trillion pounds, or 10,000 pounds for every man, woman, and child on earth. (One pound in a car, incidentally, demolishes the car and kills everyone in it.) The nuclear weapons on board just one of our Trident submarines contain eight times the firepower expended in all of World War II.
To cite some happier illustrations for smaller numbers, the standard I use for the lowly thousand is a section of Veterans Stadium in Philadelphia which I know contains 1,008 seats and which is easy to picture. The north wall of a garage near my house contains almost exactly ten thousand narrow bricks. For one hundred thousand, I generally think of the number of words in a good-sized novel.
To get a handle on big numbers, it's useful to come up with one or two collections such as the above corresponding to each power of ten, up to maybe 13 or 14. The more personal you can make these collections, the better. It's also good practice to estimate whatever quantity piques your curiosity: How many pizzas are consumed each year in the United States? How many words have you spoken in your life? How many different people's names appear in The New York Times each year? How many watermelons would fit inside the U.S. Capitol building?
Compute roughly how many acts of sexual intercourse occur each day in the world. Does the number vary much from day to day? Estimate the number of potential human beings, given all the human ova and sperm there have ever been, and you find that the ones who make it to actuality are ipso facto incredibly, improbably fortunate.
These estimations are generally quite easy and often suggestive. For example, what is the volume of all the human blood in the world? The average adult male has about six quarts of blood, adult women slightly less, children considerably less. Thus, if we estimate that on average each of the approximately 5 billion people in the world has about one gallon of blood, we get about 5 billion (5 × 109) gallons of blood in the world. Since there are about 7.5 gallons per cubic foot, there are approximately 6.7 × 108 cubic feet of blood. The cube root of 6.7 × 108 is 870. Thus, all the blood in the world would fit into a cube 870 feet on a side, less than 1/200th of a cubic mile!
Central Park in New York has an area of 840 acres, or about 1.3 square miles. If walls were built about it, all the blood in the world would cover the park to a depth of something under 20 feet. The Dead Sea on the Israel–Jordan border has an area of 390 square miles. If all the world's blood were put into the Dead Sea, it would add only three-fourths of an inch to its depth. Even without any particular context, these figures are surprising; there isn't that much blood in the world! Compare this with the volume of all the grass, or of all the leaves, or of all the algae in the world, and man's marginal status among life forms, at least volume-wise, is vividly apparent.
Switching dimensions for a moment, consider the ratio of the speed of the supersonic Concorde, which travels about 2,000 miles per hour, to that of a snail, which moves 25 feet per hour, a pace equivalent to about .005 miles per hour. The Concorde's velocity is 400,000 times that of the snail. An even more impressive ratio is that between the speed with which an average computer adds ten-digit numbers and the rate at which human calculators do so. Computers perform this task more than a million times faster than we do with our snail-like scratchings, and for supercomputers the ratio is over a billion to one.
One last earthly calculation that a scientific consultant from M.I.T. uses to weed out prospective employees during job interviews: How long, he asks, would it take dump trucks to cart away an isolated mountain, say Japan's Mount Fuji, to ground level? Assume trucks come every fifteen minutes, twenty-four hours a day, are instantaneously filled with mountain dirt and rock, and leave without getting in each other's way. The answer's a little surprising and will be given later.
Gargantuan Numbers and the Forbes 400
A concern with scale has been a mainstay of world literature from the Bible to Swift's Lilliputians, from Paul Bunyan to Rabelais' Gargantua. Yet it's always struck me how inconsistent these various authors have been in their use of large numbers.
The infant Gargantua (whence "gargantuan") is said to have needed 17,913 cows to supply him with milk. As a young student he traveled to Paris on a mare that was as large as six elephants, and hung the bells of Notre Dame on the mare's neck as jingles. Returning home, he was attacked by cannon fire from a castle, and combed the cannonballs from his hair with a 900-foot-long rake. For a salad he cut lettuces as large as walnut trees, and devoured half a dozen pilgrims who'd hidden among the trees. Can you determine the internal inconsistencies of this story?
The book of Genesis says of the Flood that "... all the high hills that were under the whole heaven were covered ..." Taken literally, this seems to indicate that there were 10,000 to 20,000 feet of water on the surface of the earth, equivalent to more than half a billion cubic miles of liquid! Since, according to biblical accounts, it rained for forty days and forty nights, or for only 960 hours, the rain must have fallen at a rate of at least fifteen feet per hour, certainly enough to sink any aircraft carrier, much less an ark with thousands of animals on board.
Determining internal inconsistencies such as these is one of the minor pleasures of numeracy. The point, however, is not that one should be perpetually analyzing numbers for their consistency and plausibility, but that, when necessary, information can be gleaned from the barest numerical facts, and claims can often be refuted on the basis of these raw numbers alone. If people were more capable of estimation and simple calculation, many obvious inferences would be drawn (or not), and fewer ridiculous notions would be entertained.
Before returning to Rabelais, let's consider two hanging wires of equal cross section. (This latter sentence, I'm sure, has never before appeared in print.) The forces on the wires are proportional to their masses, which are proportional to their lengths. Since the areas of the cross sections of the supporting wires are equal, the stresses in the wire, force divided by cross-sectional area, vary as the lengths of the wires. A wire ten times as long as another will have ten times the stress of the shorter one. Similar arguments show that, of two geometrically similar bridges of the same material, the larger one is necessarily the weaker of the two.
Likewise, a six-foot man cannot be scaled up to thirty feet, Rabelais notwithstanding. Multiplying his height by 5 will increase his weight by a factor of 53, while his ability to support this weight — as measured by the cross-sectional area of his bones — will increase by a factor of only 52. Elephants are big but at the cost of quite thick legs, while whales are relatively immune because they're submerged in water.
Although a reasonable first step in many cases, scaling quantities up or down proportionally is often invalid, as more mundane examples also demonstrate. If the price of bread goes up 6 percent, that's no reason to suspect the price of yachts will go up by 6 percent as well. If a company grows to twenty times its original size, the relative proportions of its departments will not stay the same. If ingesting a thousand grams of some substance causes one rat in one hundred to develop cancer, that's no guarantee that ingesting just one hundred grams will cause one rat in one thousand to develop cancer.
(Continues...)
Excerpted from Innumeracy by John Allen Paulos. Copyright © 2001 John Allen Paulos. Excerpted by permission of Hill and Wang.
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