Pi IS a Rational Number


So this thing, this seemingly unrelated string of digits who has been given a very specific name Pi, why is it such an iconic topic for math?  We even go so far as to name a day after it, the fourteenth of March- Pi day.  I still get quite a few odd looks from my colleagues when I say that, to many Pi is something best forgotten or eaten.  This year is the first year I have struggled with the idea of celebrating Pi day with doing activities, and here’s why.

Pi is no longer Irrational.

What?  You are mad Bryan!  Why would you ever make such a false statement!  I know Pi is irrational, but in our current state of mathematics education Pi is most certainly NOT irrational to our students.  Test this in your own classroom, ask your students to write down Pi, and let me know what your results are.

Here’s mine:


Pi is 3.14

To make things “easier” for students we have created a powerful misconception, that Pi terminates after two decimal places.  This is reinforced almost everywhere for students, how often have you seen (use 3.14 for ∏) in textbooks, on tests, or even given your students those instructions?  We are all guilty of it. Even though we throw out the catch phrase of “but remember this is an approximation”, this is heard and acknowledged just as much as “take out the trash before you go out with your friends”.  Some of my older students remember something about this (or perhaps it is because we talked about it 2 days ago…) and here is how they tried to explain that:


They still think of Pi as 3.14, but add their own catch-all.  I will have to talk with them tomorrow about this thinking, how can you write 3.14 but then say it’s never ending?  I can see some of my students trying to restate however many decimal places of Pi they remember, but not one will use the word irrational.

I think about this as I also reflect on the Pi Day activity I had my students do on Monday (and now I am beginning to regret doing it).  It was an activity similar to Apple Pi from Illuminations, located here.  I had various circular objects in the classroom: soda cans, DVDs, rolls of tape, coasters, plates and cups.  Students measured the circumference and diameter of these and explored the ratio of the circumference to the diameter.

I had thought about posting pictures of the activity, other than the initial hurdle of students completing long division, students caught on quickly and completed their ratios without problems.  We looked at the class average and what that number looked like to them: Pi.  The problem- every student’s calculation was a rational number.  Some students only had to work through the hundredths place while others went beyond the hundred-trillionths.  But they all terminated or repeated.

I asked about Pi then, and I received answers similar to the first picture of this post.  No talk of terminating or repeating, just good ‘ol 3.14.  When I asked about number classifications, and we tried to classify their numbers I finally got some of my older students to talk about rational numbers.  They then had this itch back in their brains about irrational and Pi- but it was after a lengthy discussion before they realized that Pi might be something beyond 3.14.  When we got to irrational, they were lost.  Students still have problems envisioning this, how can a number go on forever without any sort of repetition or end?  All of their numbers did, and fairly quickly considering how “long” an irrational number is.  I asked them for ideas why they didn’t get Pi, and it was only my youngest student who cracked the problem- measurement.  Students are not accurate when doing measurements- in fact there isn’t anyone that I know of that can take a tape measure, make the measurements and not get a rational number.  We have to round somewhere, whether we realize it or not.  That rounding causes our ratio to become a rational number.  For one class I even atempted to be smart and use technology to create numbers for the circumference and diameter, only to find out that just like calculators, graphing technology is restricted by displaying place value and rounding- producing yet another rational number (but better approximation of Pi).

Is it good to take this journey, am I enforcing a misconception or am I creating an atmosphere where students can concretely learn, discuss outcomes, and translate into abstract thinking of a concept rationalized in the brain only?  Typically I thought it was, until here I am 2 days later and I get these responses for what Pi is.  In order to combat this deeply embedded concept I need to revisit this throughout the rest of the year, and hope that it filters down into their long-term memory.

When I get responses like the following:


It further reminds me of how perhaps Pi day is a disservice to students, one that was shared by @delta_dc a few years ago here.  I had many students who were disappointed that they would not eat Pie that day, and disheartened that their expectation of a day that we created to help promote an important mathematical concept had degenerated into a homonym reminding them of a delicious dessert.

This year it seems, Pi day won…



2 thoughts on “Pi IS a Rational Number

  1. Of course, for ALL practical purposes pi is a rational number, and so is root 2. If fact the practical world gets by with a relatively small number of rational numbers. I personally like 3.14159

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