Fermi Estimates: from Harry Potter to ET David Wakeham May 30, 2019 Abstract Some notes on Fermi estimates and methods for doing them, presented at the UBC Physics Circle in May, 2019. We’ll look at various applications, including global computer storage, the length of the Harry Potter novels, and the probability of intelligent aliens in the galaxy. Note: This is a rough draft only; I hope to polish these and add pictures and additional exercises at some point. For any comments, suggestions, or corrections, please get in touch at <[email protected]>. Contents 1 Introduction 1 1.1 Why Fermi problems? . 2 1.2 Powers of 10 . 2 2 Fermi problems 4 2.1 Comparison . 4 2.2 Factorisation . 6 2.3 Units . 7 2.4 Piano tuners . 10 3 Aliens 11 3.1 Counting aliens . 12 3.2 Cosmic numbers . 13 1 Introduction I’m going to be talking about Fermi problems: what they are, methods for solving them, and if there’s time, a fun application to the search for extraterrestrial intelligence, i.e. aliens we can talk to. Who knows what a Fermi problem is? Has anyone done one before? A Fermi problem is an order of magnitude estimate. Roughly speaking, it means to guess an answer to the nearest power of 10. 1 1.1 Why Fermi problems? Fermi problems are great because the nearest power of 10 is a very forgiving notion of correct- ness, and because they’re so forgiving, we can accurately estimate many things which seem impossible at first sight. This makes Fermi estimates a great party trick. You can impress your friends and family. But they’re also extremely useful! I’m going to quickly talk about some things you can do with them. • First of all, if you’re doing some sort of fiddly, delicate calculation, you may want a way to double-check your result. Fermi estimates are perfect for this! They’re simple, they’re robust, you can often literally do them on the back of an envelope, so they’re a good thing to compare to a difficult calculation. If the fiddly calculation is very different from the Fermi estimate, maybe something’s gone wrong there. • You can also use them to double-check claims that other people make. If you read a statistic in the newspaper or online which strikes you as fishy, you can do some quick googling and cobble together a Fermi estimate and see if the statistic is plausible. So Fermi problems can help protect you from fake news! • Sometimes, though, you’re not comparing against something more precise, you just want a ballpark figure. For instance, you can use Fermi approximation to settle the question of whether there are more stars in the sky, or grains of sand on the earth’s beaches. I won’t spoil the answer, because you’ll get a chance to try this problem yourself. But the point is that you just need to know roughly how big they are to answer the question. In fact, this idea of ballparking numbers is so useful outside of physics that Fermi problems actually get taught in business degrees! You can Fermi estimate costs, which is really useful in day-to-day life. Of course, impressing friends and family is fun, and I don’t want to discourage that! That’s a legitimate application as well. 1.2 Powers of 10 Before starting on methods for Fermi estimates, let’s talk a bit about orders of magnitude, and how this relates to scientific notation, the usual way we write estimates and measurements. In scientific notation, we write answers as significant figures with a decimal after the first digit, then a power of 10 to shift the decimal point if we need to. For instance, the population of the world is around 7.7 billion, or 7:7 × 109 people: This has two significant figures, 7 and 7. Say we want to approximate by keeping only a single significant figure. We’re going to throw a digit away and round 7.7 up to 8, so the population becomes 8 × 109 people; or 8 billion people. It may seem like that’s as approximate as you can get, but there is another level of approximation: throw away significant figures altogether! We just keep the order of 2 magnitude. Since 8 is closer to 10 than 1, we round up to get 10 × 109 people = 1010 people: To the nearest order of magnitude, there are 10 billion in the world. The two rounding steps (from 7.7 to 8, and from 8 to 10) are subtly different. We all know how to round a decimal to the nearest whole number: 0.5 rounds up to 1, and anything less rounds down to 0. But to round something to the nearest power of 10 means to round it to something of the form 10n, for a whole number n. We really just want to round the index up or down! For instance, 100:5 should round up to 101 = 10, and 100:4 rounds down to 100 = 1. This sounds reasonable, but when you plug in 100:5 on a calculator, you get a bit of surprise: it’s around 3.2! So in powers of 10, 3.2 is closer to 10 than to 1! This will be important for us. But it’s initially counterintuitive. A simple way to perform the rounding with powers of 10 is to first take a logarithm in base 10, which plucks out the index, round this to the nearest whole number, and use this as the index. In a sense, we just want to measure things on a logarithmic ruler, with a line drawn at every whole number index n. Let’s do a quick example. To round the world population to the nearest power of 10, we take a logarithm of 8 and find that log10 8 ≈ 0:9: That clearly rounds up to 1, so the nearest power of 10 is 101 = 10. Hence, 10 billion people. In practice, we don’t always round to a power of 10 when we do Fermi estimates. For instance, we might have 8 billion as an order of magnitude estimate for the world’s population. That seems very confusing at first, because first of all, I’ve told you that we’re guessing things to the nearest power of 10, and seconly, in scientific notation, this guess would have to be correct to the nearest billion. But we want a looser notion of correctness, allowing for half a power of 10 on either side, in the same way that we would allow for an error of half a billion on either side in scientific notation. But since 100:5 ≈ 3, this means that if our Fermi estimate is 8 billion, we only expect it to be right up to a factor of 3 biger or smaller. You can think of this as just shifting marks on the logarithmic ruler, so we have a mark centred at 8 billion, with the next mark up at 80 billion, and the next mark down at 800 million. We’re saying that the real number should be closer to 8 billion than these two other marks. We use the ∼ symbol to indicate order of magnitude estimates. For instance world population ∼ 8 × 109 people means we only expect this to be true up to a factor of 3 bigger or smaller. But if we instead want 8 to be a significant figure, with a much smaller error of half a billion on either side, we use an equals sign: world population = 8 × 109 people: This is a subtle but important difference! 3 2 Fermi problems 2.1 Comparison Now that we know what an order-of-magnitude estimate is, we can start to do some estimates ourselves. Let’s start with a simple question: how many people are in this room, to the nearest order of magnitude? The answer is clearly 10, but there are a couple of different ways to get here. The first is just to do a head count: either counting individuals, or clumps of 2 or 5 or whatever works for you. Counting here is doable, but for an order-of-magnitude estimate it’s unnecessary, and won’t generalise to harder problems where you can’t count things or you don’t want to count things. A different method, which is less precise but does generalise well, is to compare to things we already know. To estimate the number of people in this room, I can compare to the size of the room. It looks around half full. An average classroom can comfortably fit 30 people, which I know because I’ve been in classes and taught classes, so there’s probably 10-15 people here. Exercise 1. Estimate the population of Australia. Since you can’t go out and count, compare to something you know. (You can answer either with a power or 10, or if you’re game, with a digit followed by a power of 10. Either way, you’re correct if you’re within a factor of 3.) Solution. What did you estimate, and how did you do it? One option is to compare to the population of Canada. You probably know that the population of Canada is around 35 million, but Australia is a bit smaller, so a reasonable guess is 30 million. The actual answer is about 25 million. You’re correct if your answer is between 8 million and 80 million.