Class Content I  >Modeling with mathematics > Using math in science 

2.1.4

A core idea in this class is the idea of modeling  -- looking at a complex physical situation and deciding what you have to pay attention to and what you can ignore.

This is a key element in all of science. We do this in physics when we decide that for a heavy object falling a fairly small distance we can ignore air resistance. We do it in biology when we draw phylogenetic trees and ignore horizontal gene transfer. We do it in chemistry when we treat chemical bonding in terms of valence orbitals, ignoring the atomic core.

Building a model is like drawing a map. We only put in those elements that are relevant to what we are doing at the accuracy we care about. If we are driving on roads, we want to see the roads but don't care whether houses are surrounded by lawns or trees so we are better off with lines and blank spaces rather than an admittedly more accurate satellite photo. If we are doing a calculation to 1% accuracy, we don't care about factors that only affect the result to one part in a million. 

We have lots of intuition about the physical world just from having lived in it for years. That intuition can help us know what's important. If someone tosses a tennis ball to you, if you have had experience playing catch, you probably know where to put your hands to catch the ball without thinking about it. But as we move to building more complex scientific models, the connection between our physical intuitions and our modeling decisions may not be so obvious. It takes some practice to learn how to connect our physical experiences and knowledge to our modeling decisions -- and to do so quantitatively! Developing your intuition as to what matters and what doesn't will help you throughout your scientific career, whether you go into research or practice.

One of the well-known characteristics of professional physicists is their ability to do estimation problems. This means using their personal knowledge to be able to get semi-quantitative order-of-magnitude estimates for almost anything under the sun (and for many things over it). This is an extremely valuable skill to learn, but it takes some practice. It is not "just guessing" and it is not "I remember from somewhere else that the number is...". These skills are immensely valuable in deciding how important something is.

We say "semi-quantitative order-of-magnitude" since we are trying to decide if something is important or not. The actual numbers are not critical; just developing a sense of the approximate scale of various effects. Then we can calculate only the important ones carefully.

Estimation problems are sometimes called "Fermi problems" after Enrico Fermi, the famous Manhattan Project physicist who was an expert at them. His canonical example was "How many piano tuners are there in the city of Chicago?" Legend has it that when he was watching the first A-bomb test at Alamogordo (from a reasonably safe distance), he dropped some torn-up scraps of paper after he saw the flash. As the shock wave went past, he estimated the energy of the explosion from how far they were blown back. Fermi used to keep a bunch of envelopes in his jacket pocket and do his estimations on them. As a result, these kinds of calculations are often referred to as "back-of-the-envelope" calculations. 

The key idea that Fermi was using about estimation is that you actually have a lot of personal experience that is relevant to almost everything you have to deal with quantitatively. And learning how to quantify and extend that personal experience is surprisingly powerful.

Learning to develop the skill of quantitative estimation is critical in building models, deciding what you need to include and what you can safely ignore. Every physics problem we do in this class has (often hidden) estimations. No scientist ever treats a real world problem exactly -- just "good enough" for the accuracy of the result they need. Learning to see what matters and what you can ignore is an essential part of learning to do problems in physics and in all of science. If you get good at estimation, you will often be able to save yourself a lot of time using it to help you decide what matters and what doesn't. (It also has value in everyday life! If you go into business, the skill to do estimations is a critical one for developing a business plan or detecting a Ponzi scheme.) 

In this class, when we ask you to do an estimation problem, we are asking you to start developing this skill. This means that for estimation problems,

• Do not look up any data in books, on-line, or get it from friends (unless the problem states otherwise). Learn to develop your own numerical estimation skills.
• Do not use your calculator. Do all calculations to one or two significant figure accuracy. You will lose credit if you give sig figs that you have no information about.
• Specify what physical principles you are using so we can see how you get from your starting assumptions to your results.
• Explicitly state your underlying assumptions and be sure they relate to some plausible personal experience.

The last point needs a bit of explaining. If you were asked on an exam: "How many blades of grass are there in a typical lawn in the Maryland suburbs in June?" you might decide you could estimate the size of the lawn, but you would need the density of the grass -- the number of blades per square meter.

If you said, "Let's assume that there are a million blades of grass per square meter" you would receive no credit. If you said, "When I lie down on a grassy lawn, I can see the grass. Knowing that the last joint of my thumb is about 1 inch long, I can easily imagine the grass against it. I can then see about 10 blades of grass against half that thumb joint, or 10 per cm. This makes 100 (= 10x10) per square cm, or 10²(10²)² = 10⁶ or one million blades per square meter." That would receive full credit because it connects to something that your listener can expect you plausibly know.

To see other examples and solutions, check out the examples given in the Follow-ons at the end of this article.

Other good examples are given in the section on "Order-of-magnitude estimates" in the online book Light and Matter by Benjamin Crowell.  There, Crowell gives the following excellent advice:

1. Don't even attempt more than one significant figure of precision.
2. Don't guess area, volume, or mass directly. Guess linear dimensions and get area, volume, or mass from them.
3. When dealing with areas or volumes of objects with complex shapes, idealize them as if they were some simpler shape, a cube or a sphere, for example.
4. Check your final answer to see if it is reasonable. If you estimate that a herd of ten thousand cattle would yield 0.01 m² of leather, then you have probably made a mistake with conversion factors somewhere.

To these we would add:

5. Learn a small number of useful numbers to serve as benchmarks and know them well (like the number of people on the planet).
6. To make powerful estimations relevant to biology, you will also need a small number of benchmark numbers for biological systems.  We will build up and use a set of benchmark numbers during the semester.  A long list of key cell biology benchmark numbers can be found as part of a new effort at Harvard (called BiologyByTheNumbers): [Key cell biology numbers from their website are here].  We will use a few of these among our benchmarks. 
7. Except for the cases in the previous two entries, don't guess. Start from something in your experience that you know and can quantify and scale up (or down).

 Workout: Estimation

Follow ons:

• Useful numbers 
• Example: Sampling 
• Example: Can trees save the planet? 

For many more estimation problems, check the following websites. 

• The Maryland PERG Fermi Problem Site
• More Maryland Estimation Problems
• Links to many other Fermi problem sites.
• Estimation 180 

Joe Redish 7/11/11 and 8/15/16

Wolfgang Losert 8/29/12