## Logo ## Tuesday, November 6, 2012

### Scribbling on paper

Quote from a reader: “I've learned that if you ask a physicist a hypothetical question, they will most likely try to give you a real answer. As a result, never ask a physicist a hypothetical unless you're ready for a real explanation and potentially hand-drawn diagrams on napkins. I love you guys.

A back of the envelope calculation is a rough estimate performed on a random scrap of paper (like the back of an envelope). They are synonymous with physicists. http://en.wikipedia.org/wiki/Back-of-the-envelope_calculation

Let's go through an example of one of these calculations:
Estimate the number of pizzas consumed by all the students at the Northwestern University during one quarter. (Adapted from University of Maryland)

Step 1: Estimate known information.
Let’s say that there ~10,000 students at NU.
And that one quarter is about 10 weeks.

Now, obviously, we could look up the exact numbers online, but the point of these types of calculations is to be quick so we estimate.

Step 2: Make an educated guess about the other information in the problem.
For this question, we need to estimate how much pizza each student eats on average. As far as I know, there's no easy way to look up this information so we have to come up with an answer ourselves.

Based on being at college, I'll say that each pizza-eating student eats pizza once or twice a week averaging 1 pizza a week.

But not every student eats pizza. Again, I'll make a guess that approximately a quarter of students eat pizza.

Note: we're trying to make this problem as simple as possible so I'm not worrying about things like some people eating pizza all the time and some every once in awhile. Just try to guess an average value.

$10000$ students $\cdot \frac{1}{4} \cdot 1 \frac{pizza}{week \:student} \cdot 10$ weeks
$= 25,000$ pizzas

Now we've done a back of the envelope calculation!

Figure 1: The calculation done on an actual envelope that used to contain a Halloween card.

If you're worried about facts, we can try to check our answer.

Let’s use some pizza facts from this website.http://www.statisticbrain.com/pizza-statistics/

There are ~3 billion pizzas sold every year in the US.

The population of the US is ~300 million.

So now we can find the proportion of the 300 billion pizzas that should be eaten by NU students during a quarter.

$3$ billion $\frac{pizzas}{year}$ $\cdot$  $\frac{10,000 NU students}{300 million Americans} \cdot \frac{10 weeks (quarter)}{52 weeks (year)}$
$= \sim 20, 000 pizzas$

As you can see, the back of the envelope estimate is the same order of magnitude as this slightly more factual answer. (http://en.wikipedia.org/wiki/Order_of_magnitude)

Our back of the envelope calculation worked! (If this was a more scientific calculation, you'd have more precise answers available after performing the actual experiment.)

Why do physicists perform back of the envelope calculations?

1)      To check if you’re correct. If you have a rough idea of the answer, it’ll be easier to catch mistakes
2)      To find the order of magnitude of an answer (like our pizza example).
3)      A fun party trick.
4)      To help a friend estimate the answer to a question (see quote at the beginning of this post).

I encourage you to do some back of the envelope calculations yourself or find your local physicist and have him/her do it for you (provide your own envelope).

And if you need further convincing of the merits of this type of calcuation, here’s a video of Charles Townes (inventor of the laser) talking about inventing the laser and using a back of the envelope calculation: http://laserfest.org/lasers/video-history.cfm

Some Resources:

Fermi Problem (similar to back of the envelope calcualtions) : http://en.wikipedia.org/wiki/Fermi_problem

* This blog post was inspired by this SMBC comic. “Pi=3, g=10, and anything we don’t like=0.”

http://www.smbc-comics.com/index.php?db=comics&id=2783