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! 

Note: additions to blog are marked in blue! 

"Trinomial Method"

From the looks of Mr. Felix’s classroom, you might think he was a narcissist. Through his genealogy, he was graced with the same name as the 1919 silent-film-star-turned-cartoon, Felix the Cat, and he was unashamed to fill the majority of his walls with posters of Felix.

In addition, he was affectionately known by the student body as “that guy with the flat-top hair cut that coaches the basketball team,” but even more than that, he taught the all-so-frightening Algebra II.

I can remember his words clearly: “If you miss tomorrow’s class on factoring, you are going to fail my class.” My fear drove me to class the next day, and I sat through what was, in retrospect, one of the strangest and most unique lectures of my life. I did not realize how strange this lecture was until, well, today when a few of my classmates looked at a paper I had on factoring and asked why I had T-charts all over it. 

You might be thinking to yourself, “T-charts… factoring… what do these even have to do with each other?” Well, sit back, enjoy the show, and let me share with you Mr. Felix’s “Trinomial Method,” because somewhere behind the flat-top and basketballs is a pretty smart guy.

Polynomials may look like big scary monsters, but they are really easy to tackle with the right method.

TRINOMIAL METHOD

Let’s say we’re given the snazzy polynomial

3x² + 12x + 9,

and asked to factor it. First, we’re going to draw a cute little T-chart

that will also double as abstract art. Now, we’re going to take the first and last coefficient of the polynomial and multiply them together, and place this number in the upper left of the T-chart. Then we will take the middle coefficient and place it in the upper right of the T-chart.

Next, we will list all of the ways the leftmost number can be factored. We’ll get something like this:

Don't forget all possible ways to factor! 

So many factors...

Now, we will take each of these numbers and add them together (and also realize it’s easier to type it than to write it by hand), to get something like…

You might notice that 3 + 9 = 12 is in the 12 column… hmmm… I wonder if that means anything.

Let’s use these numbers to split up the middle term in the polynomial. We’ll get something like

3x² + 12x + 9 = 3x² + 3x + 9x + 9.

Now we’ll factor this, and we get

3x² + 3x + 9x + 9

3x (x + 1) + 9 (x + 1)

(x + 1) (3x + 9),

thus factoring our polynomial.

“How does this work, exactly?” you might ask. 

Let's flush this out.

Well, the general thought process behind it is that you need to find two numbers that when multiplied together equal the first and last coefficient multiplied together, but when added together equal the middle coefficient. Obviously, this does not work easily for all cases. You can try a few on your own if you’d like. But, in essence, this is it:

There are a few cool things about the “Trinomial Method.” If your leading coefficient is 1, s and t are how you factor your polynomial. As an example:

Math is trippy sometimes.

Obviously, the more time you spend using this method, the better you will get at it. Eventually you won't need to list out all of the factors and it will come more naturally to you.

Destroy all the polynomials with factoring! 

This can greatly reduce time spent factoring. Although it's important to learn how to factor the traditional way, Mr. Felix's "Trinomial Method" can be very helpful and is definitely worth looking at. 

So, thank you Mr. Felix! You keep doin' you.