Do you know how fast the Mississippi River flows? No? Suppose I told you it flowed at eighty miles an hour, would you believe me? Just picture an old steamboat floating gently along with the current. Surely it isn’t outdistancing a speeding automobile. Even though you didn’t know the speed to three decimal places, your prior knowledge let you make an estimate that showed my wild assertion to be invalid.
A very high barrier stands between us and the habit of making rough estimates–the fear of getting the “wrong” answer. Contrary to what most of us learned in school, however, an inexact answer is almost always good enough. All through elementary school, high school, and the first two years of college, I was taught that only the exact answer would do-“July 14, 1789” ; “5,280 feet”; “r-h-o-d-o-d-e-n-d-r-o-n”. It was like a high-wire act. The slightest imbalance would send everything tumbling downward toward an inadequate safety net called “partial credit.” I well remember the moment when the spell of that attitude was broken.
Henry Semat, my college atomic physics instructor, had completed a calculation on the blackboard, and several of us thought we had caught him in a “mistake.” His answer was almost twice as big as the one given in the book-and he had written the book! Nevertheless, he didn’t bat an eye when we pointed out the discrepancy to him. “Same order of magnitude,” he shrugged with an air of total unconcern. “As long as we know that the effect is the right size, we can always fill in better numbers if we need to.”
It was a revelation to find that a real-life practitioner of my intended profession didn’t feel obligated to fill in every decimal place. An important lesson, indeed. Keep your eye on the left-hand digit and put zeros in all the other places. If you don’t know the first digit, make a rough estimate and pick one.
I recently needed to know something about the long-term effect of radioactivity on transoceanic optical fibers. Cosmic rays flash regularly into our atmosphere from high-energy nuclear reactions, the product of massive stellar explosions within our galaxy. These rays constitute the primary source of the naturally created radioactivity that constantly bathes our planet. Some of these rays are so penetrating that they literally plow right through the entire earth and emerge out the other side to continue their travel through space.
To learn what effect, if any, such radiation might have on the useful lifetime of optical glass fibers, I called a friend of mine, an expert on the subject. One input to the rough calculation we wanted to make was the amount of glass contained in a given length of optical fiber. She remembered “27 grams per meter” as the weight per unit length, but that didn’t make sense to me. We had both handled fibers very often and I couldn’t imagine that a one-meter length of hair-thin optical fiber would contain enough material to fill a shot glass.
A shot glass contains one liquid ounce, or about one-sixteenth of a pound of liquid. Two pounds of water make up a quart, or 32 liquid ounces, which is roughly equal to a liter. Since a liter of water weighs a kilogram, or 1,000 grams, dividing 1,000 (the grams) by 32 (the ounces) told me that the contents of a filled shot glass would weigh something like 30 grams. Since glass fiber is denser than water I figured it might take about half as much glass fiber to make the same weight of liquid-but still far more than one meter’s worth. Once I made the connection, my friend quickly corrected her recollection to “27 grams of glass in a kilometer of fiber.” The difference is 1,000 times.
My long chain of connections sounds really complicated, but that’s more or less how I figured out that original number had to be wrong. Since I’m so used to dealing with chains of numerical relationships, I completely overlooked a much easier way-one that only occurred to me as I was writing up this story. I needed simply to start (mentally) coiling up some fiber in a tight enough loop to fit into an envelope to be sent by first-class post to my friend, and stop when I felt I had enough to weigh more than an ounce, or approximately 28 grams, so that I would need an additional postage stamp. Since I had often handled optical fiber, it would have been easy for me to “see” that the coil would contain many dozens of turns-each about a foot long-so a single meter couldn’t possibly weigh an ounce.
While the brute-force approach I took the first time around lacked the simple elegance of a comparison with first-class postage, it led me to the answer I needed. Both methods worked. The trick was to scan through familiar things with similar properties in order to build a path to the object in question. In order to get these, however, I had to make a number of rough estimates. Estimation gives problem solving the benefit of the imprecise knowledge our minds gather through everyday experience.
In most of my rough calculations, I treat the first digit as a 1 or a 3 and all the other digits as zeros. I like using three because it simplifies multiplication: 3 times 3 equals 10 in my system. Multiplying any given number by 10, 100, 1,000, or 1,000,000,000 can be accomplished by shifting the decimal point one, two, three, or nine places to the right, respectively. Similarly, division moves the decimal place to the left. In other words, it’s simply a matter of adding or subtracting zeros.
When my son, David, was looking for his first job, one interviewer asked him, “How many barbers are there in the United States ?” Not every engineering graduate would welcome that kind of question, but it was a good way of finding out whether or not a prospective co-worker could deal with the kinds of things not taught explicitly in engineering school.
David remembered that there were four barbershops on the main street of our town with a total of about ten barbers. Since our town has something over 10,000 people in it, that worked out to one barber per thousand, or about 200,000 barbers in a nation of 200 million people. The interviewer used a different method: one haircut per month for each of the 100 million people who get haircuts, and 400 haircuts per month per barber, which works out to 250,000.
When David presented me with the problem, I used money instead. I figured that each of the 100 million men who patronize barbershops spends $100 a year on haircuts, or $10 billion all together. I guessed that each barber must take in about $30,000 a year, in order to make a living and pay for a share of the shop expenses, which gave me about 300,000 barbers. The actual number is just under 100,000, according to the U.S. Department of Labor.
While some chains of connections miss the actual answer by a wider margin than others, there is only one wrong method for dealing with such problems: trying to memorize answers to all the questions one might be asked. My son’s interviewer was looking for someone who could find ways of probing an unfamiliar idea, rather than someone who relied on memorized facts alone.
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