Given how much time I spend writing it's a little ironic that I hated English class. The writing assignments weren't bad, but what they made me read – and then write about – was obnoxious. Literature, at least in my schoolroom experience, was defined by barbers considering murder, people trapped in mental institutions and buffalo going to slaughter. It was simultaneously horrifying and boring, which is kind of a neat trick.
My one escape from the depressing grind of literature was the occasional book report. For a couple of weeks I could indulge in some doomed-buffalo-free reading and then write about it. I took it as a challenge to find books to report on that both satisfied the ground rules of the assignment (it has to be literature after all) and my own desire to read something I cared about which was mostly match and science. Need a book about real life experiences? Fine, how about Michael Crichton's Five Patients a dive into the US health care system. Need a book on the theme of Change versus tradition? How about a little tome the discovery of molecular biology? And so it was when I was doing the Biography of a Famous Person book report that I first came across Enrico Fermi.
And the Winner Is
Although not nearly as well know as the affable Einstein or the uncertain Heisenberg, Fermi was one of the towering figures of 20th century physics. Fermi had his hands in virtually every important during that period, from quantum theory to nuclear and particle physics and even statistical mechanics.
Arguably, Fermi also won the ultimate science naming contest. In recognition of Fermi's work, Fermi's colleagues christened a particular class of particles fermions. Having a entire class of fundamental things named after you is of like winning the physics gold metal, no matter what the thing is. But with fermions Fermi won a diamond encrusted pluatium medal, because diamonds and pluatium – and gold for that matter – is made of fermions. In fact, almost everything else we bump into in daily life is made of femions. Look around you: It's all fermions.
Aside from discovering what the world was made of and setting up the one of the first nuclear reactors, Fermi is also famous for a sort of game that he would play with his students. The rules of the game were simple: Fermi would pose a particular kind of question – questions which rapidly became known as Fermi questions – and the students would try to answer using only their general knowledge and their wits. Here's a sampling of Fermi-style questions: How many taxi cabs are there in New York City? How many M&Ms does it take to equal the weight of the Sun? How many pennies would you have to stack to equal the height of Mount Everest?
Questions of Fermions
It turns out that coming up with answers to Fermi questions is not that difficult and can be fun. Take the pennies to Everest question as an example. I happen to know that Mount Everest is about 30,000 feet high. And, having spent a lifetime getting more or less useless pennies back in change, I would guess that a stack of ten pennies makes an inch. If there are ten pennies in an inch, then there are 120 pennies in a foot and so there would be 120 * 30,0000 pennies in 30,000 feet. Do the multiplication and you come up with a stack of about three and a half million pennies.
How'd I do? Going out to Google I find that a US penny is in fact about 0.06 inches thick, which means there are 16.7 pennies to the inch or 200 to the foot. And Everest is officially 29,029 feet high. Run that math and you get about 5.8 million pennies to the mountain. So my answer was about 40% too low, which isn't bad considering that I started out with absolutely no idea.
And it's not just me: I got this particular Fermi question from The Fermi Estimates Lesson Plan If you read down through the PDF you will find the authors' attempt at working the pennies to Everest problem and if you look a bit further you'll see they came a bit closer than I did: 4.2 million pennies. Note that while neither I nor the lesson plan authors managed to score a bulls eye, we did both manage to hit the dart board. And how that happened is interesting as well: While the lesson plan estimate fell short on Everest (they thought it was only 21,000 feet high) they had a much better ideas of the number of pennies in a foot. None of this is a coincidence: Fermi estimates tend to work because the odds are that the errors in estimates will to some extent cancel each other out.
But maybe the Everest question was just a fluke, so let's give it another go: Here's a Fermi question for today's world: How many people in the world are talking on their cellphones right now?
Let's see, I think there are about 8 billion people in the world and as a guess I'd say that half of them own cell phones. My experience is that most people spend about 20 minutes a day on their phone. Some more, some less, but maybe 20 minutes a day is about average. So at any given time there a four billion people with a cell phone and in a given 24 hour day each of them talks for 20 minutes. Now 20 minutes is about 1/75th of a day so we have 4 billion / 75 or about 53 million people gabbing at any one time. The lesson plan authors came up with a higher number 120 million.
Real Questions, Decent Answers
So what's the right answer? Dunno. Unlike the Everest question, I wasn't able to find hard data on the Internet to answer this question. But what we do have is a couple of independent estimates, ranging from my low of 53 million to the 120 million from my source. We could probably tighten up this range by breaking the Fermi rules and doing a bit of research - How many cell phones are out there? is probably a question we could get some real data on. But I will leave that as an exercise for the reader and point out that with a little math and a bit of ingenuity we have gone from no idea to making a rough estimate. And with that rough estimate in hand, we can see a way to get to a better number. And that is the brilliance of Professor Fermi and his questions.
The lesson in all of this is simple: You worst estimate is probably better than you best guess. Because when it comes to questions involving numbers, math turns out to be really useful.