1.6 - Estimates and Orders of Magnitude

00:00 Order of Magnitude Estimates
Order of magnitude estimates are sometimes called Fermi Estimates, after the famous physicist Enrico Fermi who popularized this technique. These estimates are a way to rapidly come to useful conclusions and results from seemingly no input information. These kinds of problems have come to be known as Fermi Problems.

The key to solving Fermi Estimates or Order of Magnitude Estimates, is to combine three skills: Rounding + Dimensional Analysis + Base Knowledge. We’ve covered Dimensional Analysis in a previous lesson and we will be learning more physics in the future to supplement your base knowledge. Rounding is an important skill to develop.

01:50 Rounding
We are going to learn how to round numbers to the nearest order of magnitude. An order of magnitude is a power of 10. For example, 0.1, 1, 10, 100, 1000 are all powers of ten. Specifically, they are 10^−1, 10^0, 10^1, 10^2, 10^3.

We can round numbers to the nearest power of 10. For example:
8 -- 10
773 -- 1,000
0.002 --- 0.001

Normally when we round numbers the “halfway point” is the number 5. Meaning if a number is greater than 5 then we round up, and if it’s less than 5 then we round down. However, when doing order of magnitude estimates, we want to think more logarithmically and less linearly. This means we are going to be thinking along a logarithmic number line.

On a logarithmic number line we can see that the distance between 0.1 and 1 is the same as the distance between 1 and 10 and also the same distance between 10 and 100. Essentially a unit distance on the number line is equal to a power of 10. You can also see that the halfway point between 1 and 10 is no longer 5. Instead it lies somewhere between 3 and 4. In fact the halfway distance between 10^0 and 10^1 is going to be 10^1/2.

We can calculate 10^1/2=√10=3.16. So 3.16 is actually the halfway point between 1 and 10 on a logarithmic scale. Meaning when we round numbers to the nearest order of magnitude, we will round up if they are greater than 3.16 and round down if they are less than 3.16.

09:16 Fermi Problem
Now that we know how to round numbers, we can combine this with dimensional analysis and some base knowledge to solve lots of problems that may at first seem impossible. I go over one such example in the video, but many more Fermi Problems exist and can be tried. Common Fermi Problem examples include:

How many piano tuners are there in Chicago?
How many hairs do you have on your head?
What is the volume of Mt. Everest?
How many gallons of water are in the Atlantic Ocean?

[martinmakur]

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it says estimate the number of grains of sand needed to make a beach volleyball may u do it for me