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One of the well-known characteristics of professional physicists is the ability to do "back-of-the-envelope" 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 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), when he saw the flash, he dropped some torn-up scraps of paper. As the shock wave went past, he estimated the energy of the explosion from how far they were blown back.

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 extremely powerful.

Learning to develop the skill of quantitative estimation has many great benefits. If you get good at it, you will often be able to save yourself a lot of time helping you to decide what matters and what doesn't. 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. 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.
- Tell what physical principles you are using so we can see how you get from your starting assumptions to your results.
- Be sure you 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 10^{2}(10^{2})^{2} = 10^{6} or one million blades per square meter." That would receive full credit.

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:

- Don't even attempt more than one significant figure of precision.
- Don't guess area, volume, or mass directly. Guess linear dimensions and get area, volume, or mass from them.
- 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.
- Check your final answer to see if it is reasonable. If you estimate that a herd of ten thousand cattle would yield 0.01 m
^{2} of leather, then you have probably made a mistake with conversion factors somewhere.

To these I would add:

- Learn a small number of useful numbers to serve as benchmarks and know them well (like the number of people on the planet).
- 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.
- 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).

*Follow ons:*

For many more estimation problems, check the following websites.

Joe Redish 7/11/11

Wolfgang Losert 8/29/12

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