Fear of Physics (6 page)

At first, I took this to be a sign of grave inadequacies in the social studies curriculum in our educational system. After all, the proximity of New York notwithstanding, this country would be a drastically different place if its population numbered only 1 million. I later came to realize that for most of these students, concepts such as 1 million or 100 million had no objective meaning. They had never learned to associate something containing a million things, like a mid-sized American city, with the number 1 million. Many people, for example, could not tell me the approximate distance across the United States in miles. Even this is too big a number to think about. Yet a short bit of rational deduction, such as combining an estimate of the mileage you can drive comfortably in a day on an interstate (about 500 miles) with an estimate of the number of days it would take to drive across the country (about 5–6 days) tells me this distance is closer to 2,500–3,000 miles than it is to, say, 10,000 miles.

Thinking about numbers in terms of what they represent takes most of the mystery out of the whole business. It is also what physicists specialize in. I don’t want to pretend that mathematical thinking is something that everyone can feel comfortable with, or that there is some magic palliative cure for math anxiety. But it is not so difficult—often even amusing and, indeed, essential for understanding the way physicists think—to appreciate what numbers represent, to play with them in your head a little. At the
very least, one should learn to appreciate the grand utility of numbers, even without necessarily being able to carry out detailed quantitative analyses oneself. In this chapter, I will briefly depart somewhat from Stephen Hawking’s maxim (I hope, of course, that you and all the buying public will prove him wrong!) and show you how physicists approach numerical reasoning, in a way that should make clear why we want to use it and what we gain from the process. The central object lesson can be simply stated: We use numbers to make things never more difficult than they have to be.

In the first place, because physics deals with a wide variety of scales, very large or very small numbers can occur in even the simplest problems. The most difficult thing about dealing with such quantities, as anyone who has ever tried to multiply two 8-digit numbers will attest, is to account properly for all the digits. Unfortunately, however, this most difficult thing is often also the most important thing, because the number of digits determines the overall scale of a number. If you multiply 40 by 40, which is a better answer: 160 or 2,000? Neither is precise, but the latter is much closer to the actual answer of 1,600. If this were the pay you were receiving for 40 hours of work, getting the 16 right would not be much consolation for having lost over $1,400 by getting the magnitude wrong.

To help avoid such mistakes, physicists have invented a way to split numbers up into two pieces, one of which tells you immediately the overall scale or magnitude of the number—is it big or small?—to within a range of a factor of 10, while the other tells you the precise value within this range. Moreover, it is easier to specify the actual magnitude without having to display all the digits explicitly, in other words, without having to write a lot of zeros, as one would if one were writing the size of the visible universe in
centimeters: about 1,000,000,000,000,000,000,000,000,000. Displayed this way, all we know is that the number is big!

Both of these goals are achieved through a way of writing numbers called
scientific notation.
(It should be called
sensible
notation.) Begin by writing 10
ⁿ
to be the number 1 followed by
n
zeros, so that 100 is written as 10
²
, for example, while 10
⁶
represents the number 1 followed by 6 zeros (1 million), and so on. The key to appreciating the size of such numbers is to remember that a number like 10
⁶
has one more zero, and therefore is 10 times bigger, than 10
⁵
. For very small numbers, like the size of an atom in centimeters, about 0.000000001 cm, we can write 10
^–n
to represent the number 1 divided by 10
ⁿ
, which is a number with a 1 in the nth place after the decimal point. Thus, one-tenth would be 10
^–1
, one billionth would be 10
^–9
, and so on.

This not only gets rid of the zeros but it achieves everything we want, because any number can be written simply as a number between 1 and 10 multiplied by a number consisting of 1 followed by
n
zeros. The number 100 is 10
²
, while the number 135 is 1.35 × 10
²
, for example. The beauty of this is that the second piece of a number written this way, called the
exponent,
or
power of ten,
tells us immediately the number of digits it has, or the “order of magnitude” of the number (thus 100 and 135 are the same order of magnitude), while the first piece tells you precisely what the value is within this range (that is, whether it is 100 or 135).

Since the most important thing about a number is probably its magnitude, it gives a better sense of the meaning of a large number, aside from the fact that it is less cumbersome, to write it in a form such as 1.45962 × 10
¹³
rather than as 1,459,620,000,000, or one trillion, four hundred fifty nine billion, six hundred and twenty million. What may be more surprising is the claim I will
shortly make that numbers that represent the physical world make sense
only
when written in scientific notation.

First, however, there is an immediate benefit of using scientific notation. It makes manipulating numbers much easier. For example, if you carry places correctly you find that 100 × 100 = 10,000. Writing this instead as 10
²
× 10
²
= 10
^{(2 + 2)}
= 10
⁴
, multiplication turns into addition. Similarly, writing 1000 ÷ 100 = 10 as 10
³
÷ 10
¹
= 10
^(3–1)
= 10
²
, division becomes as simple as subtraction. Using these rules for the powers of ten, the major headache—keeping track of the overall scale in a calculation—becomes trivial. The only thing you might need a calculator for is multiplying or dividing the first pieces of numbers written in scientific notation, the parts between 1 and 10. But even here things are simpler, since familiarity with multiplication tables up to 10 × 10 allows one to guess closely in advance what the result should be.

The point of this discussion is not to try to turn you into calculational whizzes. Rather, if simplifying the world means approximating it, then scientific notation makes possible one of the most important tools in all of physics: order-of-magnitude estimation. Thinking about numbers in the way scientific notation directs you to allows you quickly to estimate the answers to questions that would otherwise be largely intractable. And since it helps tremendously to know if you are on the right track when making your way through uncharted territory, being able to estimate the correct answer to any physical problem is very useful. It also saves a great deal of embarrassment. Apocryphal stories are told of Ph.D. students who presented theses with complex formulas meant to describe the universe, only to find during their thesis defense that plugging realistic numbers into the formulas shows the estimates to be ridiculously off.

Order-of-magnitude estimation opens up the world at your feet, as Enrico Fermi might have said. Fermi (1901–1954) was one of the last great physicists of this century equally skilled at experimental and theoretical physics. He was chosen to be in charge of the Manhattan Project, the secret U.S. wartime effort to develop a nuclear reactor and thereby demonstrate the feasibility of controlled nuclear
fission
—the splitting of atomic nuclei—in advance of building an atomic bomb. He was also the first physicist to propose a successful theory to describe the interactions that allow such processes to take place, for which he was awarded the Nobel Prize. He died at an untimely age of cancer, probably due to years of radiation exposure in the time before it was known how dangerous this could be. (Those of you who ever land at Logan Airport and get stuck in the massive traffic jams leading into the tunnel that takes you into Boston may amuse yourselves looking for a plaque dedicated to Fermi located at the base of a small overpass just before the tollbooths at the entrance to the tunnel. We name towns after presidents, and stadiums after sports heroes. It is telling that Fermi gets an overpass next to a tollbooth.)

I bring up Fermi because, as leader of a team of physicists working on the Manhattan Project in a basement lab under a football field at the University of Chicago, he used to help keep up morale by regularly offering challenges to the group. These were not really physics problems. Instead, Fermi announced that a good physicist should be able to answer any problem posed of him or her—not necessarily produce the right answer, but develop an algorithm that allows an order-of-magnitude estimate to be obtained based on things one either knows or can reliably estimate. For example, a question often asked on undergraduate
physics quizzes is, How many piano tuners are there in Chicago at any given time?

Let me take you through the kind of reasoning Fermi might have expected. The key point is that if all you are interested in is getting the order of magnitude right, it is not so difficult. First, estimate the population of Chicago. About 5 million? How many people in an average household? About 4? Thus, there are about 1 million (10
⁶
) households in Chicago. How many households have pianos? About 1 in 10? Thus, there are about 100,000 pianos in Chicago. Now, how many pianos does a piano tuner tune each year? If he is to make a living at it, he probably tunes at least two a day, five days a week, or ten a week. Working about fifty weeks a year makes 500 pianos. If each piano is tuned on average once a year, then 100,000 tunings per year are required, and if each tuner does 500, the number of tuners required is 100,000/500 = 200 (since 100,000/500 = 1/5 × 10
⁵
/10
²
= 1/5 × 10
³
= 0.2 × 10
³
= 2 × 10
²
).

The point is not that there may or may not be exactly 200 piano tuners in Chicago. It is that this estimate, obtained quickly, tells us that we would be surprised to find less than about 100 or more than about 1,000 piano tuners. (I believe there are actually about 600.) When you think about the fact that before performing such an estimate, you probably had no idea what the range of the answer might be, the power of this technique becomes manifest.

Order-of-magnitude estimation can give you new insights about things you might never have thought you could expect to estimate or picture. Are there more specks of sand on a beach than there are stars in the sky? How many people on Earth sneeze each second? How long will it take wind and water to wear down Mount Everest? How many people in the world are . . .
(fill in your favorite possibility here
) as you read these words?

Equally important, perhaps, order-of-magnitude estimation gives you new insights about things you
should
understand. Humans can directly picture numbers up to somewhere between 6 and 12. If you see the 6 dots when you roll a die, you don’t have to count them each time to know there are 6. You can picture the “whole” as distinct from the sum of its parts. If I gave you a die with 20 sides, however, it is unlikely that you could look at 20 dots and immediately comprehend them as the number 20. Even if they were arranged in regular patterns, you would still probably have to group the dots in your head, into, say, 4 groups of 5, before you could discern the total. That does not mean we cannot easily intuit what the number 20
represents.
We are familiar with lots of things that are described by that number: the total number of our fingers and toes; perhaps the number of seconds it takes to leave the house and get into your car.

With truly big or small numbers, however, we have no independent way of comprehending what they represent without purposefully making estimates that can be attached to these numbers to give them meaning. A million may be the number of people who live in your city, or the number of seconds in about 10 days. A billion is close to the number of people who live in China. It is also the number of seconds in about 32 years. The more estimates you make of quantities that have these numbers attached to them, the better you can intuitively comprehend them. It can actually be fun. Estimate things you are curious about, or things that seem amusing: How many times will you hear your name called in your life? How much food do you eat, in pounds, in a decade? The pleasure you get from being able to tackle, step by step, what would be an insurmountably difficult problem to answer exactly can be addictive. I think this kind of “rush” provides much of the enjoyment physicists get from doing physics.

The virtue of scientific notation and order-of-magnitude estimation is even more direct for physicists. These allow the conceptual simplifications I discussed in the last chapter to become manifest. If we can understand the correct order of magnitude, we often understand most of what we need to know. That is not to say that getting all the factors of 2 and pi correct is not important. It is, and this provides the acid test that we know what we are talking about, because we can then compare predictions to observations with ever higher precision to test our ideas.

This leads me to the curious statement I made earlier, that numbers that represent the world make sense
only
when written in scientific notation. This is because numbers in physics generally refer to things that can be
measured.
If I measure the distance between the Earth and the sun, I could express that distance as 14,960,000,000,000 or 1.4960 × 10
¹³
centimeters (cm). The choice between the two may seem primarily one of mathematical semantics and, indeed, to a mathematician these are two different but equivalent representations of a single identical number. But to a physicist, the first number not only means something very different from the second but it is highly suspect. You see, the first number suggests that the distance between the Earth and sun is 14,960,000,000,000 cm and not 14,959,790,562,739 cm or even 14,960,000,000,001 cm. It suggests that we know the distance between the Earth and sun to the nearest centimeter!