Chapter 2: Discoveries on the Back of an Envelope.

[1:] It was the bottom of the seventh inning at Shea Stadium and Dwight Gooden was pitching a masterful game for the New York Mets although, given the Mets' woeful offense, it was still a scoreless tie. Then it started to rain.

[2:] I was at the game with my brother-in-law, whose interest in baseball I have never quite fathomed. As a lifelong Red Sox fan, I have a clear understanding of my own motives. Red Sox fandom serves for me the same purpose as Greek tragedy -- one always knows the outcome, yet one experiences catharsis getting there (the end of the season, that is -- don't worry, 2004 was the exception that proves the rule). But Jeffrey is more of a theoretical fan than a passionate one, and his first response to the rain was not that this wonderful pitching performance might go for nought in the record books, but instead:

[3:] "How much do you suppose the ball slows down as it encounters the raindrops? Does it significantly affect his fastball for the hitters?"

[4:] This may well be a question that no one has ever thought to pose before. One is unlikely to find an answer in a textbook. Indeed, an internet search with key words "fastball", "slows" and "raindrops" produced only the following:

[5:] "Tanya: Just go slow, think about what you're doing. The crafty Limbor smoked a couple of fastballs. Oakly (glances at the darkened sky as raindrops... "

[6:] If you want the whole literary gem, click here.

[7:] Not much help from Google for the baseball question, however.

[8:] My mustard-stained napkin was considerably more useful, because it is possible to calculate an answer -- or at least give a reasonable estimate of the size of effect. Right; geeks are like that, I hear you muttering. But the point is that you can do it too. In fact, estimating in this way -- a back-of-the-envelope or napkin calculation -- is one of the easiest of the scientific habits of mind to acquire, because it only involves arithmetic plus something you will begin to gain in the next few pages: confidence.


[9:] Science is often portrayed as a rigidly precise discipline in which the sixth decimal place may reveal a critical cosmic truth. Indeed, sometimes, this is the case: a miniscule discrepancy in the position of Mercury in its orbit around the Sun led to the confirmation of Einstein's revolutionary theory of relativity, while measurements of tiny fluctuations in the cosmic background radiation provide critical clues to the Universe's first moments. But focusing on precision is not always necessary, and doing so often fails to capture the spirit of the scientific process. Scientists do quantify Nature, but often in a more qualitative way.

[10:] Indeed, one of the distinguishing habits of a scientific mind is the ability and the willingness to make rough estimates of unknown (and often, unknowable) quantities. "Back-of-the-envelope" calculations almost always contain few enough steps to fit on the back of the envelope (or, more frequently, on a napkin). The idea is not to calculate something precisely, not to spend hours searching the library or the Web for information, but to get a quick, rough idea of how big, or how many, or how heavy, or how expensive something is.

[11:] Scientists often use this process to explore the feasibility of undertaking some set of observations, to design an experiment, or to evaluate the tractability of a computer simulation. However, it is also often useful in assessing the plausibility of statements one hears or reads, and in contextualizing sensationalized news stories, political claims, or simply one's own life experiences. I also understand from former students that back-of-the-envelope reasoning is a skill highly valued in lucrative "management consulting" jobs (in case you need a crass, practical reason to be interested in finishing this chapter).

[12:] (Here's one such student's story.)

[13:] As noted above, the typical back-of-the-envelope calculation requires no more than arithmetic and a dash of confidence. I will assume here you can do the former; I will foster the latter with a series of examples.


[14:] I frequently use envelope backs to debunk, or place in perspective, sensationalized news stories. For example, every few years the media gets excited about "killer sharks." By the beginning of fall term a few years ago, despite weeks of headline coverage concerning the "shark menace," precisely two people had died in the US from shark bites. What fraction is that, you might ask yourself, of all the people who have died during the year? The answer can be easily determined as follows:

[15:] There are about 300 million (3 x 108) people in the US, and the average life expectancy in this country is about 75 years (averaging men and women -- note that average life expectancy is just what we want here because it tells us how long the average person lives). This means that:

[16:] 3.0 x 108 people / 75 years = 4.0 x 106 people die each year.

[17:] By early September, the year is about 245 days/365 days = 67% over, so roughly 0.67 x 4.0 x 106 = 2.7 x 106 people will have died by the time the semester begins. The shark victims, then, represent less than 1 out of every million deaths. Not exactly a major health threat. In contrast, two US residents die every three minutes from smoking cigarettes, and two die every 25 minutes in car accidents.

[18:] Notice that in this kind of calculation, one does not worry about the fact that there are not exactly the same number of people in each of the 75 years of life, or that the death rate might vary slightly from month to month over the year, or that the US population is not exactly 300 million (in fact, it is currently 295,730,000 and increasing by one every 12 seconds). My interest was in the relative fraction of deaths from sharks -- was it 1 in 100 or 1 in a million? Clearly, it is closer to the latter, and whether it is 1 in 1.35 million or one in 0.98 million doesn't matter at all in getting a feeling for the relative importance of the problem.

[19:] Note also that the input numbers to such calculations are rarely accurate to more than one or two significant figures (300 million people, 75-year life expectancy). In keeping with rules we will enunciate in Chapter 5, one should never quote the answer from such a calculation to more than two significant figures, although it is fine to carry along more decimal places on your calculator until you finish calculating.

[20:] Another favorite news story is forest fires in the West. Some years it is Montana and Idaho, others it is California or Colorado. In each case, however, one is left with the distinct impression that a large fraction of a state is burning: "Colorado on Fire," headlines reported a few years ago. Tourism dropped to zero -- who wanted to camp in the middle of a burned-out forest? -- and the Governor had to call a press conference to say that the entire state was not burning (which, of course, got much less coverage than the fires). The sensationalism lavished on minor stories has several characteristics: 1) it ignores both history and context (there are and, for healthy forests, must be many fires every year), and 2) it quotes numbers in inflammatory (pun intended) ways -- they must be big numbers. Fires are always reported as the number of acres burned. Do you know how big an acre is?

[21:] If not, you are to be forgiven, since there seems to be some confusion on the subject in the English-speaking world (the only portion of the world benighted enough to still abjure metric units). The word comes from the Latin "agrus" which means a plowed or open (but distinctly not forested) field. In the US and Britain, an acre is defined as 160 square rods -- which probably doesn't help much unless you happen to know that a rod (also known as a perch) is 5.5 yards. That makes a US acre 4849 sq. yds. or 43,600 sq.ft. In Scotland, however, an acre is 6150 sq. yds, and in Ireland it is a whopping 7840 sq. yds. Is this absurd or what? In passing, it is worth noting that the metric unit for fields is the hectare, a neat 100 x 100 m = 104 m2 (or roughly 2.5 US acres).

[22:] In any event, during 2002 in Colorado, 500,000 acres burned. That sounds like a huge number. It makes more sense, however, to quote the number in square miles: 500,000 acres is less than 790 sq. miles. The total area of Colorado is 2.7 x 105 sq. miles, so roughly 0.29% -- less than one third of one percent -- of the state actually burned. The equivalent on campus would be the burning of a little under one half of John Jay Dining Commons. The news media might well report this as "Columbia Burns Down", but locals might regard it merely as a small improvement in the quality of campus life.


[23:] Envelopes can also be used to spot errors in apparently authoritative sources. (As you can see, I intend to assault your habitual, passive acceptance of information). I had a Columbia student many years ago who is now an editor at Foreign Affairs magazine, the publication of the Council on Foreign Relations, one of the most authoritative journals in its field. By using back-of-the-envelope calculations almost every week, he frequently catches errors in articles written by UN officials, former Secretaries of State and others. So this skill is not only useful for scientists and management consultants.

[24:] Recently in the New York Times (my favorite place for catching errors given their self-proclaimed status as the "paper of record"), I read an article on the dangers of inadequately inspected food flowing into the US from foreign countries. It stated that "30 billions tons of food are imported annually." So I did a little calculation.

[25:] 30 billions tons = 30 x 109 tons x 2000 pounds/ton = 6 x 1013 pounds of food. There are roughly 300 million people in the US and most of them eat (in fact, most apparently eat too much). But if we all only ate imported food, we would need to consume

[26:] 6 x 1013 pounds/yr /3 x 108 people/ 365 days/ yr = 550 pounds per person per day.

[27:] While we certainly waste a lot of food in this country, and are, as a nation, obese, no one eats 550 pounds of food a day. Clearly, this number must be wrong by a factor of at least 1000. So much for the "paper of record."


[28:] One of the principal expositors of back-of-the-envelope calculations was Enrico Fermi, the great Italian physicist who, fleeing the Facists, ended up at Columbia in 1938, and left three years later when President Butler would not let him drain the old swimming pool to install the world's first nuclear reactor. (Fermi moved to Chicago and built his reactor under their football stadium instead -- have you heard much about the University of Chicago's football team lately?) Indeed, these are often called "Fermi Problems", and without doubt the most famous is, "How many piano tuners are there in New York?"

[29:] While this is clearly not a matter of burning urgency to our nation's geopolitical success or to your own personal financial future (unless you happen to become a tuning wrench salesperson), it is

[30:] 1) a factoid you, or anyone you know, is extremely unlikely to know offhand, and

[31:] 2) a good example of how different people will arrive at similar numbers while making a calculation with several steps and assumptions.

[32:] I have posed this problem to many of my classes and never fail to be impressed that novice calculators all come up with very similar answers.

[33:] To tackle this (or any) problem, begin with what you know (or can easily find):

[34:] Population of New York: 8 million

Time to tune a piano: ~2 hours

Frequency of tuning: ~1 per year

Working days in a year: 365 - 104(weekends) - 15 (holidays) - 20 (vacation) = 226 days

[35:] Note that the accuracy of these numbers varies. The population is right to better than 10%, and the number of days worked per year for the average person is even more precise. The time to tune a piano probably ranges from less than an hour (though we might add a little travel time) to many hours (before a major concert at Carnegie Hall), and the frequency of tuning is even more variable and uncertain (probably more than once a week at major concert venues, to less than once per decade for the piano of an elderly couple in Brooklyn). Nonetheless, these serve as rough averages for the average piano -- I would guess.

[36:] Next, estimate what you don't know:

[37:] Number of pianos per person in New York.

[38:] Since the census bureau does not ask about pianos, there are no official data. There is little doubt, as with almost any question you can dream up, you can find an answer on the Web, but how will you know if the number you find there is valid? I Googled "number of cars in the US" and, on the first page of hits, found answers differing by 26 million (despite the fact that the reported numbers were quoted to between three and nine(!) significant figures). Without being able to do a rough calculation yourself -- without the self-reliance this brings -- you are a dependent creature, doomed to accept what the world of charlatans and hucksters, politicians and professors provides.

[39:] How do I estimate such a number? Let's start with the extremes. Is it zero? Categorically no, since I have a piano and I live in NYC. Is it 8 million? No again, since my wife and I share one. Furthermore, I know most of the people in our building (approximately 45 apartments with an average of 2 or 3 people each, so maybe 100 people), and I only know of one other piano. Maybe there are two or three that I don't know about. But my building, occupied mostly by Columbia affiliates, is clearly not typical of New York City as a whole; my building's mean income is almost certainly well above average, and it is reasonable to assume that piano ownership and income are correlated. I'd guess the true value is around 1% of the population or 80,000 pianos -- maybe we'll round up to 105, just to be generous.

[40:] Wait! You might say. What about all the concert halls? And schools? And universities? Here's an important lesson in relevance, again using the envelope. How many concert venues are there in New York? 100? 200? And suppose they each have several pianos -- so that's an extra 1000 pianos at most -- less than a 1% correction to my estimate. And schools? Each school has one, or a few, pianos, yet each school has hundreds to thousands of kids, and school kids make up only about 12/75ths of the population WHY?; again, a small correction. Columbia has 38 pianos (I asked) and 23,000 students.

[41:] In calculating on the back of the envelope, there is limited space. Don't go off on irrelevant tangents and pursue trivial corrections to your rough estimates. Remember the goal: to obtain an approximate number, good to a factor of two or even ten; all minor effects can be safely ignored.

[42:] So, 105 pianos x 1 tuning/yr x 2 hr/tuning x 1day/8hr x 1yr/225 days = 111 tuners

[43:] Now, of course, it is not exactly 111 -- as I said, quote numbers to one or two significant figures. In this case, more than one would overestimate the accuracy; thus, I'd say "about 100". Many of them may be part-time tuners, so maybe a couple of hundred. But it is highly unlikely that there are thousands (unless they are all very hungry), and it is highly unlikely there are only ten (they would each have to tune thousands of pianos a year).

[44:] In many years of asking this question, my students always

[45:] 1) think I'm weird and say "How can I possibly know that?" and

[46:] 2) go home and come up with a number between 50 and 500.

[47:] That is, they find that the "order of magnitude" of the answer is about 102, which is all we wanted to know. And then at interview time they respond to a management consultant's question about the number of fax machines in Brooklyn the same way, and garner a six-figure starting salary.


[48:] In the third century BCE, Archimedes wrote about the number of grains of sand needed to fill the Universe. Clearly, he had not counted the grains of sand on Earth or measured the size of the Universe -- it was just a poetic way of saying the Universe is a big place. Archimedes' speculation has subsequently been paraphrased as saying the number of stars in the sky is greater than the number of grains of sand on Earth. Telescopes now extend our vision to stars four billion times fainter than the stars Archimedes could see with his naked eye, but we are still very far from being able to count them all. So we estimate:

[49:] stars within 50 light-years of the Sun (these we can count) = 1000

[50:] fractional volume of the Galaxy this 50 ly sphere represents = 10-8,

[51:] which implies about 100 billion (1011) stars in the Milky Way.

[52:] The number of galaxies in the visible Universe comes from taking the number we count in the deepest picture of space ever taken, the Hubble Space Telescope's "Ultra Deep Field", which covers a piece of sky about 10% the size of the full Moon. We then multiply by the number of such patches it takes to cover the sky (about 2.1 million): again, curiously, but purely coincidentally, the number is about 1011, a hundred billion galaxies. So the number of stars in the observable Universe is about 1022. This is an "unknowable" number in that no one can, by dint of stubborn persistence, go out and count them. First of all, 99.99999999999% of them are too faint for even our largest telescopes to record individually and, secondly, if we set all 6.4 billion people on Earth to work on this task twelve hours a day, and they counted 1,2,3,4,5... getting to 20 every five seconds, it would take 25,000 years to finish the count.

[53:] However, this is a number astronomers need to know if, for example, we are to develop an understanding of the distribution of the elements in the Periodic Table. Since stars cook up all the elements in their cores and disgorge them at the time of their deaths, the number of stars making elements is essential in understanding why platinum is rarer than gold, and why both are rare compared to iron. We can compute this useful number on the back of an envelope.

[54:] The number of grains of sand on all the beaches of the world is a less useful number, but for practice, and to gain some perspective on the size of the Universe, it is helpful to explore Archimedes' analogy. What do we need to know?

[55:] How big a grain of sand is, how big a beach is, and how many beaches there are would be a good start. If we then calculate the volume of all the beaches and divide by the volume of a single grain, we'll have the number of grains.

[56:] Sand comes in many varieties, but a typical grain is probably about 0.5 mm across, or 0.25 mm in radius, yielding a volume (assuming a spherical grain) of 1.6 x 10-2 mm3 or 6.5 x 10-11 m3HUH?. Beaches, of course, vary enormously in length, depth, and width. You may know from trying to bury your younger sibling that the sand goes down at least a couple of feet, but it does not go on indefinitely -- eventually you hit rock or dirt; 3 meters is probably a good guess at the depth. The width of beaches varies from miles in the Bay of Fundy (when the tide is out) to just a few meters along many cliff-edged coasts. Probably 200 meters is a good guess for the average. As for the amount of shoreline, we can start by looking up the Earth's circumference: 40,000 km. It is clear from looking at a globe that if you stretched out the world's shorelines, they would wrap around the Earth many times -- probably more than ten but less than a hundred. Since only some fraction of that total coastline is bounded by sandy beaches, let's guess 20 times the Earth's circumference or around 800,000 km (the answer is here -- I guessed first and was damn close!) The result, then, is

[57:] 3m x 200 m x 8 x 105 km x 103 m/km / 6.5 x 10-11 m3/grain = 7 x1021 ~ 1022 grains of sand,

[58:] an inspirational coincidence. Think about it next time you are at the beach; let the sand run through your fingers -- the grains from nearly a million kilometers of beach (which would take you 36 years to cover if you walked 12 hours a day, 365 days a year) just equals the number of stars in the sky (each of which has a mass 1037 times greater).


[59:] I recently attended a conference at which scientists from many disciplines gathered for a weekend of lectures and conversation. After a talk by the discoverer of extrasolar planets -- planetary bodies orbiting parent stars other than the Sun -- an eminent biologist from a very well-known institution to the northeast of Columbia asked

[60:] "So in the next 50 years, how many of these systems will we be able to visit and explore with spacecraft?"

[61:] We astronomers are used to dealing with "astronomical" numbers when it comes to distances, but apparently other scientists are not. The speaker let the questioner off gently, noting that in 30 years no spacecraft has yet come close to the edge of our Solar System, let alone visited others, and this will not happen any time soon.

[62:] Had the questioner scribbled the following on his program before speaking, however, he would not have asked the question:

[63:] Nearest of the stars with planets discussed in the talk: 10 light-years

[64:] Velocity a spacecraft must achieve to leave Earth: 11 km/s (and we can't make them go much faster than that yet)

[65:] Speed of light: 300,000 km/s

Spacecraft speed = 11 km/s / 300,000 km/s = 3.7 x 10-5 light-speed

Time to reach closest planet: 10 ly / 3.7 x 10-5 = 2.7 x 105 years


[66:] It is common to be confronted with a number for which one has absolutely no context; without context, the number is virtually bereft of meaning. In this situation there are two approaches one can adopt: a) ignore the number, and b) clothe it with context. Most people make the former choice; a scientific mind strives for the latter. The US national debt is over $7.2 trillion; no one I know has a visceral feeling for what a "trillion" means. The national debt becomes an abstract concept devoid of real meaning -- so we ignore it. A simple calculation, however, gives it meaning:

[67:] 7.2 x 1012 / 3 x 108 citizens = $24,000 per citizen or $96,000 for a family of four. Is everyone as comfortable with that?

[68:] Part of this debt was accrued building nuclear warheads of which we now have (after dismantling a large number of them, also at great cost) about 10,000 with an average yield of 1 Megaton (roughly fifty times the destructive force of the bombs dropped on Hiroshima and Nagasaki). That works out to about 3500 pounds of dynamite for every person on Earth; 1 pound is enough to blow up a car and kill everyone in it. Do we really need a new manufacturing facility for nuclear warheads?

[69:] Another part of the debt was accumulated buying foreign oil. We currently use about 35% of the world's total energy supply. We have 3 x 108 / 6.4 x 109 = 4.7% of the world's population. How much more energy would be needed if the whole world was using energy as we do?


[70:] I would not be surprised to learn that many of you faced some parental resistance to the idea of coming to school in the big, dangerous City. Here are a couple of calculations to send home this weekend:

[71:] New York City's murder rate is 110th in the country, wedged between Ogden, Utah and Rancho Cucomonga, California; in 2004, 570 people were murdered in the City. Altogether, roughly 8 x 106/75 = 110,000 died, so the chances of dying from a violent act is roughly 0.5%. Since 80% of murder victims know their murderers, the chances of dying in a random act of violence is less than one-tenth of 1%. Of course, you could have gone to school in Ogden, Utah and had your own car on campus. The fraction of deaths among 18-22-year olds from car accidents is 28%.

[72:] Your parents may also have warned you about the dangerous subway system. If so, send them Chapter 4.


[73:] If your credit card bill arrives with a total due that is $10 more than you expected, you wouldn't think twice about paying it. If the total was $10,000 more than you were expecting, you might be a little upset. That's a factor of a thousand more! Well, a billion is a factor of a thousand more than a million, but few people seem to care or notice.

[74:] Scientists, however, think numbers matter. If the Greenland icecap were to melt, would the oceans rise 10 cm or 100 meters? -- it matters. Does smallpox vaccine induce a severe reaction in one in a million soldiers ordered to take it or one in a thousand? -- it matters. Are we destroying the rainforests of the world at 1000 hectares per year or a million hectares per year? -- it matters.

[75:] We are said to be living in the "Information Age." To me, it seems more like the "misinformation age." Technology has now put more misinformation at your fingertips than has existed in all of human history. One tool you have to combat the misinformation glut, to make sense out of nonsense, is the back of an envelope.

[76:] Now, about those raindrops slowing the baseball...