Fermi Problem - Counting to 1,000,000!

Enrico Fermi chilling out with some computations.

Enrico Fermi was an Italian physicist and is considered one of the "fathers of the atomic bomb." He accomplished many things in his life, but today we're going to look at a style of problem named after him. He obviously did much more than this, and I highly suggest looking into him if you have the time. He's a pretty cool guy - and interestingly enough, I found out he died on my birthday. Well, years and years before it became my birthday, but I still think it's eerie. 

I've got the heebie-jeebies

Anyway, let's take a look at a Fermi problem. Some of you may be familiar with Fermi problems; for those of you who are not, and also are unmotivated to use Wikipedia, a Fermi question is defined as:  

"an estimation problem designed to teach dimensional analysis, approximation, and the importance of clearly identifying one's assumptions."

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In today's blog post, loyal readers, we will be considering this essential question:

How long would it take to count to one million?

Let's give it a go! 

First, let's assume that it takes exactly one second to say every single number. That means it would take one million seconds to count to one million. So, exactly how long is one million seconds? We'll do some simple arithmetic....

As always, click to enlarge my creations. 

It would be easy to leave this as our answer, but we all know that some numbers take much longer to say than others. The longest number I can think to say between 1 and 1,000,000 is seven hundred seventy seven thousand seven hundred seventy seven and the shortest number I can think to say between 1 and 1,000,000 is one. Let's time both of these: 

It took me about 3.26 seconds to say 777,777 and about 0.6 seconds to say 1. You are welcome to time yourself, but we're going to use my numbers. 

From this, we can conclude each number will take somewhere between 0.6 seconds and 3.26 seconds to say. We'll use the average of these two numbers for our calculations. We find the average is... 

We could take the easy route and round this number to 2 seconds. But, here on "Jennifer Writes About Math," we're about the most accurate way. Time for some more arithmetic! 

So, it would take about 22.338 days to count to one million! Good job finding that out, team!

Now, I would love to check our answer, but I don't have 22.338 days, let alone 1 day, to devote entirely to counting. This is where you lovely readers come in.... <3

Now, you may be asking yourself, "What the heck, why did we do this? This doesn't matter! There isn't a good way to check the answer!" Well, that's the thing. Math helps us explore the world around us. Maybe it doesn't matter in the grand scheme of things how long it takes to count to 1,000,000, but this process of discovery carries over into everything. How long does it take to get to the moon? How many gummy bears can fit into this tub? Fermi problems help us practice estimating and are a fun, valuable way to get people (specifically students!) involved in math. 

Because you're a pretty inquisitive person, you may also be asking yourself whether or not these kind of problems belong in the classroom. In my humble opinion, they do. Math is often approached as serious and intimidating, with one correct answer or procedure. As a result of this, some students understandably become disheartened and disinterested in the material. With Fermi problems, answers are less concrete and mathematical questions seem kind of goofy. Students don't need to have the correct answer, because the questions typically do not have a "correct" answer, nor is there a good way to prove or disprove an answer. Although the computations in the given example are relatively simple, the process of trying to answer the question can spark a new interest in mathematical exploration. Math is exploration, this is math, albeit rudimentary, and I think it belongs in the classroom. 

If you are interested in doing more Fermi problems, there are entire websites devoted to them. Here are a few...

This one lets you submit an answer, if you're into that.  

If you're more interested in how this could apply to a classroom, check out this resource.

Maybe you don't feel like solving any Fermi problems. That's cool, too. If you think of one you'd like to see me try out, leave a comment with your question! 

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